---
author:
- 'Paul-Emile PARADAN'
title: Jump formulas in Hamiltonian geometry
---

[UMR 5582, Institut Fourier, B.P. 74, 38402, Saint-Martin-d’Hères cedex, France\
e-mail: Paul-Emile.Paradan@ujf-grenoble.fr\
]{}

Introduction
============

This paper is concerned with the Hamiltonian actions of a (compact) torus $T$ on a symplectic manifold $(M,\Omega)$. We assume that the moment map $\Phi: M\to{\ensuremath{\mathfrak{t}}}^*$ is [*proper*]{} and that the generic stabilizer of $T$ on $M$ is [*finite*]{}. We are interested here in two global invariants:

- the Duistermaat-Heckman measure ${\ensuremath{\rm DH}}(M)$ which is the pushforward by $\Phi$ of the Liouville volume form,

- the Riemann-Roch characters $RR(M,L^{\otimes k}),k\geq 1$, which are virtual representations of $T$. Here $M$ is compact and the data $(M,\Omega,\Phi)$ is prequantized by a Kostant-Souriau line bundle $L$. For every couple $(\mu,k)\in\Lambda^*\times{\ensuremath{\mathbb{Z}}}^{>0}$, we denote by ${\ensuremath{\hbox{\rm m}}}(\mu,k)\in{\ensuremath{\mathbb{Z}}}$ the multiplicity of the weight $\mu$ in $RR(M,L^{\otimes k})$.

One can associate to a connected component ${\ensuremath{\mathfrak{c}}}$ of regular values of $\Phi$ the local invariants:

- the Duistermaat-Heckman polynomial ${\ensuremath{\rm DH}}_{\ensuremath{\mathfrak{c}}}:{\ensuremath{\mathfrak{t}}}^*\to{\ensuremath{\mathbb{R}}}$ which coincides with ${\ensuremath{\rm DH}}(M)$ on ${\ensuremath{\mathfrak{c}}}$,

- the periodic polynomial ${\ensuremath{\hbox{\rm m}}}_{\ensuremath{\mathfrak{c}}}:\Lambda^*\times{\ensuremath{\mathbb{Z}}}\to{\ensuremath{\mathbb{Z}}}$ which coincides with the map ${\ensuremath{\hbox{\rm m}}}:\Lambda^*\times{\ensuremath{\mathbb{Z}}}^{>0}\to{\ensuremath{\mathbb{Z}}}$ on the cone of ${\ensuremath{\mathfrak{t}}}^*\times{\ensuremath{\mathbb{R}}}$ generated by ${\ensuremath{\mathfrak{c}}}\times\{1\}$.

The main results of this paper concern the differences ${\ensuremath{\rm DH}}_{{\ensuremath{\mathfrak{c}}}_+}-{\ensuremath{\rm DH}}_{{\ensuremath{\mathfrak{c}}}_-}$ and ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_+}-{\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_-}$ when ${\ensuremath{\mathfrak{c}}}_\pm$ are two [*adjacent*]{} connected components of regular values of $\Phi$. Let us introduce some notations. We denote by ${\ensuremath{\Delta}}\subset{\ensuremath{\mathfrak{t}}}^*$ the hyperplane that separates ${\ensuremath{\mathfrak{c}}}_\pm$, and by $T_{\ensuremath{\Delta}}\subset T$ the subtorus of dimension $1$ that has for Lie algebra the one dimensional subspace ${\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}$ which is orthogonal to the direction of ${\ensuremath{\Delta}}$. We make the choice of a decomposition $T=T_{\ensuremath{\Delta}}\times T/T_{\ensuremath{\Delta}}$, where $T/T_{\ensuremath{\Delta}}$ denotes a subtorus de $T$. At the level of Lie algebras, we have then ${\ensuremath{\mathfrak{t}}}={\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}\oplus({\ensuremath{\mathfrak{t}}}/{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}})$ and ${\ensuremath{\mathfrak{t}}}^*={\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}^*\oplus({\ensuremath{\mathfrak{t}}}/{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}})^*$: hence $\xi + ({\ensuremath{\mathfrak{t}}}/{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}})^*={\ensuremath{\Delta}}$ for any $\xi\in{\ensuremath{\Delta}}$.

Let ${\ensuremath{\mathfrak{c}}}'\subset{\ensuremath{\Delta}}$ be the relative interior of $\overline{{\ensuremath{\mathfrak{c}}}_+}\cap\overline{{\ensuremath{\mathfrak{c}}}_-}$ in ${\ensuremath{\Delta}}$.

[*In order to give a clear idea of our results we suppose in the introduction that $\Phi^{-1}(\xi)\cap M^{T_{\ensuremath{\Delta}}}$ is [*connected*]{} when $\xi\in{\ensuremath{\mathfrak{c}}}'$. It means that there exists only one connected component $Z\subset M^{T_{\ensuremath{\Delta}}}$ such that ${\ensuremath{\mathfrak{c}}}'\subset\Phi(Z)$. We denote by $N_Z$ the normal bundle of $Z$ in $M$.* ]{}

Let $\Omega^{\ensuremath{\Delta}}_\xi$ be the induced symplectic form on the reduced space ${\ensuremath{\mathcal{M}}}^{\ensuremath{\Delta}}_\xi:=$ $(\Phi^{-1}(\xi)\cap M^{T_{\ensuremath{\Delta}}})/(T/T_{\ensuremath{\Delta}})$. Let $\omega^{\ensuremath{\Delta}}_\xi\in{\ensuremath{\mathcal{H}}}^2({\ensuremath{\mathcal{M}}}^{\ensuremath{\Delta}}_\xi)\otimes {\ensuremath{\mathfrak{t}}}/{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}$ be the curvature of the $T/T_{\ensuremath{\Delta}}$-principal bundle $\Phi^{-1}(\xi)\cap
M^{T_{\ensuremath{\Delta}}}\to{\ensuremath{\mathcal{M}}}^{\ensuremath{\Delta}}_\xi$. Let $\beta\in{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}$ be the primitive vector of the lattice $\ker(\exp:{\ensuremath{\mathfrak{t}}}\to T)$ which is orthogonal to ${\ensuremath{\Delta}}$ and is pointing out ${\ensuremath{\mathfrak{c}}}_-$. We prove in Section \[DH-mesures\] the following

[**Theorem A.**]{}

Theorem [**A.**]{} generalizes previous results of Guillemin-Lerman-Sternberg [@GLS96] and Brion-Procesi [@Brion-Procesi]. In Section \[subsec:Jump-dh\] we give the precise definition of the characteristic classes $\alpha_k$.

Suppose now that $M$ is compact and is prequantized by a Kostant-Souriau line bundle. The hyperplane ${\ensuremath{\Delta}}$ is defined by the equation $\frac{\langle\xi,\beta\rangle}{2\pi}=r_{\ensuremath{\Delta}},\ \xi\in{\ensuremath{\mathfrak{t}}}^*$, for some $r_{\ensuremath{\Delta}}\in{\ensuremath{\mathbb{Z}}}$. The bundle $N_Z$ decomposes as the sum of two polarized sub-bundles $N_Z^{\pm,\beta}$. Let $s_Z^\pm$ be the absolute value of the trace of $\frac{1}{2\pi}{\ensuremath{\mathcal{L}}}_\beta$ on $N_Z^{\pm,\beta}$. In (\[eq:bundle-xi-mu\]) we define a familly of orbifold vector bundles $${\ensuremath{\mathbb{S}}}^k_{Z,\mu}\to{\ensuremath{\mathcal{M}}}^{\ensuremath{\Delta}}_\xi,\quad\quad
(\mu,k)\ \in\ \Lambda^*\times{\ensuremath{\mathbb{Z}}},$$ with the fundamental property that ${\ensuremath{\mathbb{S}}}^k_{Z,\mu}=0$ when $$- s_Z^- < \frac{\langle\mu,\beta\rangle}{2\pi}- k r_{\ensuremath{\Delta}}< s_Z^+.$$ We prove in Section \[subsec:jump-formula\] the following

[**Theorem B.**]{}

In Section \[sec:cas-coadjoint\] we apply Theorem [**B**]{} to the particular cases where $M$ is a integral coadjoint orbit of a compact Lie group $G$. In Section \[subsec:su-n\], we study more precisely the case $G=\SU(n)$: here our result precises some of the results of Billey-Guillemin-Rassart [@BGR03].

In Section \[sec:cas-C-d\], we study the case where $M$ is a complex vector space. In this situation, we interpret Theorem [**B**]{} as a combinatorial formula between vector partition functions. We recover also a Theorem of Szenes-Vergne [@Szenes-Vergne02].

[**Acknowledgments.**]{} I am very grateful to Michèle Vergne for bringing me the reference [@BGR03] to my attention, and for explaining me her work with András Szenes [@Szenes-Vergne02].

[**Notations**]{}

*Throughout the paper $T$ will denote a compact, connected abelian Lie group, and ${\ensuremath{\mathfrak{t}}}$ its Lie algebra. The integral lattice $\Lambda\subset{\ensuremath{\mathfrak{t}}}$ is defined as the kernel of $\exp:{\ensuremath{\mathfrak{t}}}\to T$, and the real weight lattice $\Lambda^* \subset{\ensuremath{\mathfrak{t}}}^*$ is defined by : $\Lambda^*:=\hom(\Lambda,2\pi{\ensuremath{\mathbb{Z}}})$. Every $\mu\in\Lambda^*$ defines a 1-dimensional $T$-representation, denoted by ${\ensuremath{\mathbb{C}}}_{\mu}$, where $t=\exp X$ acts by $t^{\mu}:=e^{i\langle\mu,X\rangle}$. We denote by $R(T)$ the ring of characters of finite-dimensional $T$-representations. We denote by $R^{-\infty}(T)$ the set of generalized characters of $T$. An element $\chi\in R^{-\infty}(T)$ is of the form $\chi=\sum_{\mu\in\Lambda^{*}}a_{\mu}\, {\ensuremath{\mathbb{C}}}_{\mu}$, where $\mu\mapsto a_{\mu}, \Lambda^{*}\to{\ensuremath{\mathbb{Z}}}$ has at most polynomial growth.*

The symplectic manifolds are oriented by their Liouville volume forms. If $(Z,o_Z)$ is an oriented submanifold of an oriented manifold $(M,o_M)$, we take on the fibers of the normal bundle $N$ of $Z$ in $M$, the orientation $o_N$ satisfying $o_M=o_Z\cdot o_N$.

Duistermaat-Heckman measures {#DH-mesures}
============================

Let $(M,\Omega)$ be a symplectic manifold of dimension $2n$ equipped with an Hamiltonian action of a torus $T$, with Lie algebra ${\ensuremath{\mathfrak{t}}}$. The moment map $\Phi: M\to{\ensuremath{\mathfrak{t}}}^*$ satisfies the relations $\Omega(X_M,-)+d\langle\Phi,X\rangle=0$, $X\in{\ensuremath{\mathfrak{t}}}$. We assume in this section that $\Phi$ is [*proper*]{}, and that the generic stabiliser $\Gamma_M$ of $T$ on $M$ is [*finite*]{}.

The Duistermaat-Heckman measure ${\ensuremath{\rm DH}}(M)$ is defined as the pushforward by $\Phi$ of the Liouville volume form $\frac{\Omega^n}{n !}$ on $M$. For every $f\in{\ensuremath{\mathcal{C}^{\infty}}}({\ensuremath{\mathfrak{t}}}^*)$ with compact support one has $\int_{{\ensuremath{\mathfrak{t}}}^*}{{\ensuremath{\rm DH}}(M)}(a)f(a)=\int_M f(\Phi)\frac{\Omega^n}{n !}$. In other terms ${{\ensuremath{\rm DH}}(M)}(a)= \int_M \delta(a-\Phi)\frac{\Omega^n}{n !}$. We can define ${\ensuremath{\rm DH}}(M)$ in terms of equivariant forms as follows. Let ${\ensuremath{\mathcal{A}}}(M)$ be the space of differential forms on $M$ with complex coefficients. We denote by ${\ensuremath{\mathcal{A}}}^{-\infty}_{temp}({\ensuremath{\mathfrak{t}}},M)$ the space of tempered generalized functions over ${\ensuremath{\mathfrak{t}}}$ with values in ${\ensuremath{\mathcal{A}}}(M)$, and by ${\ensuremath{\mathcal{M}}}^{-\infty}_{temp}({\ensuremath{\mathfrak{t}}}^*,M)$ the space of tempered distributions over ${\ensuremath{\mathfrak{t}}}^*$ with values in ${\ensuremath{\mathcal{A}}}(M)$. Let ${\ensuremath{\mathcal{F}}}:{\ensuremath{\mathcal{A}}}^{-\infty}_{temp}({\ensuremath{\mathfrak{t}}},M)\to 
{\ensuremath{\mathcal{M}}}^{-\infty}_{temp}({\ensuremath{\mathfrak{t}}}^*,M)$ be the Fourier transform normalized by the condition that ${\ensuremath{\mathcal{F}}}(X\mapsto e^{i\langle \xi,X\rangle})$ is equal to the Dirac distribution $a\mapsto\delta(a-\xi)$. Let $\Omega_{\ensuremath{\mathfrak{t}}}(X)=\Omega -\langle\Phi,X\rangle$ be the equivariant symplectic form. We have then ${\ensuremath{\mathcal{F}}}(e^{-i\Omega_{\ensuremath{\mathfrak{t}}}})=e^{-i\Omega}\delta(a-\Phi)$ and so $$\label{eq:dh-equi}
    {{\ensuremath{\rm DH}}(M)}=(i)^n \int_M {\ensuremath{\mathcal{F}}}(e^{-i\Omega_{\ensuremath{\mathfrak{t}}}}).$$

Equivariant cohomology and localization
---------------------------------------

We first recall the Cartan model of equivariant cohomology with polynomial coefficients and the extension to generalized coefficients defined by Kumar and Vergne [@Kumar-Vergne]. We give after a brief account to the method of localization developped in [@pep1; @pep2],

Let $M$ be a manifold provided with an action of a compact connected Lie group $K$ with Lie algebra ${\ensuremath{\mathfrak{k}}}$. Let $d:{\ensuremath{\mathcal{A}}}(M)\to{\ensuremath{\mathcal{A}}}(M)$ be the exterior differentiation. Let ${\ensuremath{\mathcal{A}}}_{c}(M)$ be the sub-algebra of compactly supported differential forms. If $\xi$ is a vectors field on $M$ we denote by $c(\xi):{\ensuremath{\mathcal{A}}}(M)\to{\ensuremath{\mathcal{A}}}(M)$ the contraction by $\xi$. The action of $K$ on $M$ gives a morphism $X\to X_{M}$ from ${\ensuremath{\mathfrak{k}}}$ to the Lie algebra of vectors fields on $M$.

We consider the space of $K$-equivariant maps ${\ensuremath{\mathfrak{k}}}\to{\ensuremath{\mathcal{A}}}(M),\ X\mapsto\eta(X)$, equipped with the derivation $(D\eta)(X):= (d-c(X_{M}))(\eta(X)),\ X\in{\ensuremath{\mathfrak{k}}}$. Since $D^2=0$, one can define the cohomology space $\ker D/{\rm Im} D$. The Cartan model [@B-G-V; @Guill-Stern00] considers [*polynomial*]{} maps and the associated cohomology is denoted ${\ensuremath{\mathcal{H}}}_{K}^{*}(M)$. Kumar and Vergne [@Kumar-Vergne] studied the cohomology spaces ${\ensuremath{\mathcal{H}}}_{K}^{\pm\infty}(M)$ obtained by taking ${\ensuremath{\mathcal{C}}}^{\pm\infty}$ maps. Recall the construction ${\ensuremath{\mathcal{H}}}_{K}^{-\infty}(M)$.

The space ${\ensuremath{\mathcal{C}}}^{-\infty}({\ensuremath{\mathfrak{k}}},{\ensuremath{\mathcal{A}}}(M))$ of generalized functions on ${\ensuremath{\mathfrak{k}}}$ with values in the space ${\ensuremath{\mathcal{A}}}(M)$ is, by definition, the space ${{\ensuremath{\rm Hom}}}({\hbox{\sl m}}_c({\ensuremath{\mathfrak{k}}}),{\ensuremath{\mathcal{A}}}(M))$ of continuous ${\ensuremath{\mathbb{C}}}$-linear maps from the space ${\hbox{\sl m}}_c({\ensuremath{\mathfrak{k}}})$ of smooth compactly supported densities on ${\ensuremath{\mathfrak{k}}}$ to the space ${\ensuremath{\mathcal{A}}}(M)$, both endowed with the ${\ensuremath{\mathcal{C}^{\infty}}}$-topologies. We define ${\ensuremath{\mathcal{A}}}_K^{-\infty}(M):={\ensuremath{\mathcal{C}}}^{-\infty}({\ensuremath{\mathfrak{k}}},{\ensuremath{\mathcal{A}}}(M))^K$ as the space of $K$-equivariant ${\ensuremath{\mathcal{C}}}^{-\infty}$-maps from ${\ensuremath{\mathfrak{k}}}$ to ${\ensuremath{\mathcal{A}}}(M)$. The differential $D$ defined on ${\ensuremath{\mathcal{C}}}^{\infty}({\ensuremath{\mathfrak{k}}},{\ensuremath{\mathcal{A}}}(M))$ admits a natural extension to ${\ensuremath{\mathcal{C}}}^{-\infty}({\ensuremath{\mathfrak{k}}},{\ensuremath{\mathcal{A}}}(M))$ and $D^2=0$ on ${\ensuremath{\mathcal{A}}}_K^{-\infty}(M)$ [@Kumar-Vergne]. The cohomology associated to $({\ensuremath{\mathcal{A}}}_K^{-\infty}(M),D)$ is called the $K$-equivariant cohomology with generalized coefficients and is denoted by ${\ensuremath{\mathcal{H}}}_K^{-\infty}(M)$. The subspace ${\ensuremath{\mathcal{A}}}_{K,c}^{-\infty}(M):=
{\ensuremath{\mathcal{C}}}^{-\infty}({\ensuremath{\mathfrak{k}}},{\ensuremath{\mathcal{A}}}_{c}(M))^K$ is stable under the differential $D$, and we denote by ${\ensuremath{\mathcal{H}}}_{K,c}^{-\infty}(M)$ the associated cohomology. When $M$ is oriented, the integration over $M$ gives rise to a map $\int_M:
{\ensuremath{\mathcal{H}}}_{K,c}^{-\infty}(M)\to{\ensuremath{\mathcal{C}}}^{-\infty}({\ensuremath{\mathfrak{k}}})^K$.

[*Localization procedure.*]{} Let $\lambda$ be a $K$-invariant $1$-form on $M$ and let $$\label{eq:phi-lambda}
  \Phi_\lambda:M\to{\ensuremath{\mathfrak{k}}}^*$$ be the $K$-equivariant map defined by $\langle\Phi_\lambda(m),X\rangle=
\lambda(X_M)_m$ : then $D\lambda(X)=d\lambda-\langle\Phi_\lambda,X\rangle$. The localization procedure developped in [@pep1; @pep2] is based on the existence of an inverse $[D\lambda]^{-1}$ of the $K$-equivariant form $D\lambda$. It is an equivariantly closed element of ${\ensuremath{\mathcal{A}}}_K^{-\infty}(M- \Phi_\lambda^{-1}(0))$ defined by the integral $$\label{eq:lambda-inverse}
  [D\lambda]^{-1}(X)=i\int_0^{\infty}e^{-i\, t\, D\lambda(X)}dt.$$ An open subset ${\ensuremath{\mathcal{U}}}\subset M$ is called [*adapted to*]{} $\lambda$ if ${\ensuremath{\mathcal{U}}}$ is $K$-invariant and if $(\partial{\ensuremath{\mathcal{U}}})
\cap\Phi_\lambda^{-1}(0)=\emptyset$. In [@pep2], we associate to an open subset ${\ensuremath{\mathcal{U}}}$ adapted to $\lambda$, the following equivariantly closed form with generalized coefficients $$\label{eq:def-P}
{\ensuremath{\hbox{\rm P}}}_{\lambda}^{\ensuremath{\mathcal{U}}}=\chi^{\ensuremath{\mathcal{U}}}+ d\chi^{\ensuremath{\mathcal{U}}}[D\lambda]^{-1}\lambda\ .$$ Here $\chi^{\ensuremath{\mathcal{U}}}\in{\ensuremath{\mathcal{C}^{\infty}}}(M)$ is a $K$-invariant function supported in ${\ensuremath{\mathcal{U}}}$ which is equal to $1$ in a neighborhood of ${\ensuremath{\mathcal{U}}}\cap\Phi_\lambda^{-1}(0)$. The cohomology class defined by ${\ensuremath{\hbox{\rm P}}}_{\lambda}^{\ensuremath{\mathcal{U}}}$ in ${\ensuremath{\mathcal{H}}}_K^{-\infty}(M)$ does not depend of $\chi^{\ensuremath{\mathcal{U}}}$ (in particular ${\ensuremath{\hbox{\rm P}}}_{\lambda}^{\ensuremath{\mathcal{U}}}=0$ in ${\ensuremath{\mathcal{H}}}_K^{-\infty}(M)$ if ${\ensuremath{\mathcal{U}}}\cap\Phi_\lambda^{-1}(0)=\emptyset$). If ${\ensuremath{\mathcal{U}}}\cap
\Phi_\lambda^{-1}(0)$ is compact, we take $\chi^{\ensuremath{\mathcal{U}}}$ with compact support, then ${\ensuremath{\hbox{\rm P}}}_{\lambda}^{\ensuremath{\mathcal{U}}}$ defines a cohomology class in ${\ensuremath{\mathcal{H}}}_{K,c}^{-\infty}(M)$.

Localization of ${\ensuremath{\rm DH}}(M)$ {#sec:hd}
------------------------------------------

We come back to the situation of a Hamiltonian action of a torus $T$ on a symplectic manifold $(M,\omega)$. We keep the same notations and hypothesys of the introduction. We need two auxilliary data : a $T$-invariant Riemannian metric on $M$ denoted $(-,-)_{_M}$, and a scalar product $(-,-)$ on ${\ensuremath{\mathfrak{t}}}^*$ which induces an identification ${\ensuremath{\mathfrak{t}}}^*\simeq{\ensuremath{\mathfrak{t}}}$.

Let ${\ensuremath{\mathcal{H}}}$ be the Hamiltonian vectors field of the function $\frac{-1}{2}\|\Phi\|^2: M\to {\ensuremath{\mathbb{R}}}$ : for every $m\in M$ we have ${\ensuremath{\mathcal{H}}}_m=(\Phi(m))_M\vert_m$. Then for every $\xi\in{\ensuremath{\mathfrak{t}}}^*$, the Hamiltonian vectors field of $\frac{-1}{2}\|\Phi-\xi\|^2$ is ${\ensuremath{\mathcal{H}}}-\xi_M$. For every $\xi\in{\ensuremath{\mathfrak{t}}}^*$, we consider the following $T$-invariant $1$-form $$\label{eq:lambda-xi}
  \lambda_\xi=({\ensuremath{\mathcal{H}}}-\xi_M,-)_{M}$$ and the corresponding map $\Phi_{\lambda_\xi}:M\to{\ensuremath{\mathfrak{t}}}^*$ (see (\[eq:phi-lambda\])). Here $\Phi_{\lambda_\xi}^{-1}(0)$ coincides with the subset ${\ensuremath{\hbox{\rm Cr}}}(\|\Phi-\xi\|^2)\subset M$ of critical points of the function $\|\Phi-\xi\|^2$, and $m\in {\ensuremath{\hbox{\rm Cr}}}(\|\Phi-\xi\|^2)$ if and only if $(\Phi(m)-\xi)_M$ vanishes at $m$ [@pep1; @pep2].

\[def:p-xi\] Let ${\ensuremath{\hbox{\rm P}}}_\xi\ \in\ {\ensuremath{\mathcal{H}}}_{T,c}^{-\infty}(M)$ be the cohomology class defined by ${\ensuremath{\hbox{\rm P}}}_{\lambda_\xi}^{\ensuremath{\mathcal{U}}}$, where ${\ensuremath{\mathcal{U}}}$ is a $T$-invariant relatively compact neighborhood of $\Phi^{-1}(\xi)$ such that $\overline{{\ensuremath{\mathcal{U}}}}\cap {\ensuremath{\hbox{\rm Cr}}}(\|\Phi-\xi\|^2)=\Phi^{-1}(\xi)$.

The cohomology class ${\ensuremath{\hbox{\rm P}}}_\xi$ will be used to localized the Duitermaat-Heckman measure. For every $\xi\in{\ensuremath{\mathfrak{t}}}^*$, we define the distribution ${\ensuremath{\rm DH}}_\xi(M)$ by $$\label{eq:dh-xi}
  {{\ensuremath{\rm DH}}_\xi(M)}=(i)^n {\ensuremath{\mathcal{F}}}\left(\int_M {\ensuremath{\hbox{\rm P}}}_\xi e^{-i\Omega_{\ensuremath{\mathfrak{t}}}}\right).$$ Here we can put the Fourier transform outside the integral because ${\ensuremath{\hbox{\rm P}}}_\xi$ is compactly supported on $M$. For any $\xi\in{\ensuremath{\mathfrak{t}}}^*$ let $r_\xi>0$ be the smallest non-zero critical value of the function $\|\Phi-\xi\|^2$. As a particular case of Proposition 3.8 in [@pep2], we have

\[prop:dh-loc\] Let $\xi$ be any point in ${\ensuremath{\mathfrak{t}}}^*$. The following equality of distributions on ${\ensuremath{\mathfrak{t}}}^*$ $${\ensuremath{\rm DH}}(M)={\ensuremath{\rm DH}}_\xi(M)$$ holds in the open ball $B(\xi,r_\xi)\subset {\ensuremath{\mathfrak{t}}}^*$.

We will now use the last Proposition, first to recover the classical result of Duistermaat and Heckman [@D-H] concerning the polynomial behaviour of ${\ensuremath{\rm DH}}(M)$ on the open subset of regular values of $\Phi$. After we determine the difference taken by ${\ensuremath{\rm DH}}(M)$ between two adjacent regions of regular values.

Polynomial behaviour
--------------------

We recall now the computation of the cohomology class ${\ensuremath{\hbox{\rm P}}}_\xi$ when $\xi$ is a regular value of $\Phi$, that is given in [@pep1]\[Section 6\] for the torus case (and in [@pep2] \[Section 3.1\] for the case of Hamiltonian action of a compact Lie group). First recall the following basic result which shows that $\xi\mapsto{\ensuremath{\rm DH}}_\xi(M)$ is locally constant on the open subset of regular values of $\Phi$.

\[lem:P-xi-0\] If $\xi$ and $\xi'$ belong to the same connected component of regular values of $\Phi$, we have ${\ensuremath{\hbox{\rm P}}}_\xi={\ensuremath{\hbox{\rm P}}}_{\xi'}$ in ${\ensuremath{\mathcal{H}}}_{T,c}^{-\infty}(M)$.

Associated to a regular value of $\xi$, we have the $T$-principal bundle $\Phi^{-1}(\xi)\to {\ensuremath{\mathcal{M}}}_\xi:=\Phi^{-1}(\xi)/T$ with curvature form $\omega_\xi\in{\ensuremath{\mathcal{H}}}^2({\ensuremath{\mathcal{M}}}_\xi)\otimes{\ensuremath{\mathfrak{t}}}$. The orbifold ${\ensuremath{\mathcal{M}}}_\xi$ carries a canonical symplectic $2$-form $\Omega_\xi$. We denote ${\ensuremath{\hbox{\rm \bf Kir}}}_{\xi}:{\ensuremath{\mathcal{H}}}_T^{\infty}(M)\to{\ensuremath{\mathcal{H}}}^*({\ensuremath{\mathcal{M}}}_\xi)$ the Kirwan morphism. For any $\psi\in{\ensuremath{\mathcal{C}^{\infty}}}({\ensuremath{\mathfrak{t}}})$ and $\eta\in{\ensuremath{\mathcal{H}}}_T^{\infty}(M)$ we have ${\ensuremath{\hbox{\rm \bf Kir}}}_{\xi}(\eta\psi)={\ensuremath{\hbox{\rm \bf Kir}}}_{\xi}(\eta)\psi(\omega_\xi)$, where the characteristic class $\psi(\omega_\xi)$ is the value of the differential operator $e^{\omega_\xi(\frac{\partial}{\partial X}|_0)}$ against $\psi$. After [@pep2]\[Prop. 3.11\], the integral $\int_{\ensuremath{\mathfrak{t}}}\int_{M}{\ensuremath{\hbox{\rm P}}}_\xi(X)\eta(X)\psi(X)dX$ is equal to $$\label{eq:loc-xi-psi}
\frac{(-2i\pi)^{dimT}{\rm vol}(T,dX)}{|\Gamma_M|}\int_{{\ensuremath{\mathcal{M}}}_\xi}
{\ensuremath{\hbox{\rm \bf Kir}}}_{\xi}(\eta)\psi(\omega_\xi)$$ for every equivariant class $\eta\in{\ensuremath{\mathcal{H}}}_{T}^{\infty}(M)$. Here ${\ensuremath{\hbox{\rm vol}}}(T,dX)$ is the volume of $T$ for the Haar mesure compatible with $dX$, and $\vert \Gamma_M\vert$ is the cardinal of $\Gamma_M$ (Note that the generic stabilizer of $T$ on $\Phi^{-1}(\xi)$ is $\Gamma_M$). In other words, for every $\eta\in{\ensuremath{\mathcal{H}}}_{T}^{\infty}(M)$ we have the following equality of generalized functions on ${\ensuremath{\mathfrak{t}}}^*$ supported at $0$ $$\label{eq:formule-p}
\int_{M}{\ensuremath{\hbox{\rm P}}}_\xi(X)\eta(X)=\frac{(-2i\pi)^{dimT}}{|\Gamma_M|}
\int_{{\ensuremath{\mathcal{M}}}_\xi}{\ensuremath{\hbox{\rm \bf Kir}}}_{\xi}(\eta)
e^{\omega_\xi(\frac{\partial}{\partial X}|_0)}{\ensuremath{\hbox{\rm vol}}}(T,-).$$

For $\eta=e^{-i\Omega_{\ensuremath{\mathfrak{t}}}}$ we have ${\ensuremath{\hbox{\rm \bf Kir}}}_\xi(\eta)=
e^{-i(\Omega_\xi-\langle\xi,\omega_\xi\rangle)}$, and a small computation shows that $$\label{eq:fourier-omega}
  {\ensuremath{\mathcal{F}}}\left(e^{\omega_\xi(\frac{\partial}{\partial X}|_0)}{\ensuremath{\hbox{\rm vol}}}(T,-)\right)(a)=
e^{-i\langle a ,\omega_\xi\rangle}  \frac{da}{(2\pi)^{dimT}}, \quad a\in{\ensuremath{\mathfrak{t}}}^*.$$ where $da$ is the Lebesgue measure on ${\ensuremath{\mathfrak{t}}}^*$ normalized by the condition: ${\ensuremath{\hbox{\rm vol}}}(T,dX)=1$ for the Lebesgue measure $dX$ on ${\ensuremath{\mathfrak{t}}}$ which is dual to $da$.

Finally (\[eq:dh-xi\]), (\[eq:formule-p\]) and (\[eq:fourier-omega\]) give $$\begin{aligned}
  \label{eq:dh-xi-calcul}
   {{\ensuremath{\rm DH}}_\xi(M)}(a)&=& \frac{(i)^{p}}{|\Gamma_M|}\int_{{\ensuremath{\mathcal{M}}}_\xi}
e^{-i(\Omega_\xi+\langle a-\xi,\omega_\xi\rangle)} \ da \nonumber\\
&=&\frac{1}{|\Gamma_M|}\int_{{\ensuremath{\mathcal{M}}}_\xi}
\frac{(\Omega_\xi+\langle a-\xi,\omega_\xi\rangle)^p}{p!}\ da,\end{aligned}$$ where $2p= {\rm dim}{\ensuremath{\mathcal{M}}}_\xi$.

\[def:DH-c\] For any connected component ${\ensuremath{\mathfrak{c}}}$ of regular values of $\Phi$ we denote ${\ensuremath{\rm DH}}_{{\ensuremath{\mathfrak{c}}}}$ the polynomial function $a\mapsto\frac{1}{|\Gamma_M|}\int_{{\ensuremath{\mathcal{M}}}_\xi}
\frac{(\Omega_\xi+\langle a-\xi,\omega_\xi\rangle)^p}{p!}$, where $\xi$ is any point of ${\ensuremath{\mathfrak{c}}}$. Hence, if $\xi\in{\ensuremath{\mathfrak{c}}}$ we have the equality ${\ensuremath{\rm DH}}_\xi(M)(a)={\ensuremath{\rm DH}}_{{\ensuremath{\mathfrak{c}}}}(a) da,\ a\in{\ensuremath{\mathfrak{t}}}^*$, where $da$ is the Lebesgue measure on ${\ensuremath{\mathfrak{t}}}^*$ normalized by the condition: ${\ensuremath{\hbox{\rm vol}}}(T,dX)=1$ for the Lebesgue measure $dX$ on ${\ensuremath{\mathfrak{t}}}$ which is dual to $da$.

With the help of Proposition \[prop:dh-loc\] we recover the classical result of Duistermaat and Heckman that says that the measure ${\ensuremath{\rm DH}}(M)$ is locally polynomial[^1] on the open subset of regular values of $\Phi$, and it’s value at a regular element $\xi$ is equal to the symplectic volume of the reduce space ${\ensuremath{\mathcal{M}}}_\xi$. More precisely we have shown that for a connected component ${\ensuremath{\mathfrak{c}}}$ of regular values of $\Phi$ we have $$\label{eq:f-C}
{{\ensuremath{\rm DH}}(M)}(a)={{\ensuremath{\rm DH}}_{{\ensuremath{\mathfrak{c}}}}}(a) d a,\qquad a\in{\ensuremath{\mathfrak{c}}}.$$

Jump formulas {#subsec:Jump-dh}
-------------

Consider now two connected regions ${\ensuremath{\mathfrak{c}}}_\pm$ of regular values of $\Phi$ separated by an hyperplane ${\ensuremath{\Delta}}\subset{\ensuremath{\mathfrak{t}}}^*$. In this section we compute the polynomial ${\ensuremath{\rm DH}}_{{\ensuremath{\mathfrak{c}}}_+}-{\ensuremath{\rm DH}}_{{\ensuremath{\mathfrak{c}}}_-}$. It generalizes previous results of Guillemin-Lerman-Sternberg [@GLS96] and Brion-Procesi [@Brion-Procesi].

Let $\xi_+,\xi_-$ be respectively two elements of ${\ensuremath{\mathfrak{c}}}_+$ and ${\ensuremath{\mathfrak{c}}}_-$. We know from (\[prop:dh-loc\]), (\[eq:dh-xi-calcul\]) and Definition (\[def:DH-c\]) that $$\label{eq:f-difference}
 {({\ensuremath{\rm DH}}_{{\ensuremath{\mathfrak{c}}}_+}}-{{\ensuremath{\rm DH}}_{{\ensuremath{\mathfrak{c}}}_-})}(a)da=
(i)^n {\ensuremath{\mathcal{F}}}\left(\int_M ({\ensuremath{\hbox{\rm P}}}_{\xi_+}-{\ensuremath{\hbox{\rm P}}}_{\xi_-}) 
e^{-i\Omega_{\ensuremath{\mathfrak{t}}}}\right)(a),\quad a\in{\ensuremath{\mathfrak{t}}}^*.$$ We recall now the computation of the cohomology class ${\ensuremath{\hbox{\rm P}}}_{\xi_+}-{\ensuremath{\hbox{\rm P}}}_{\xi_-}\in
{\ensuremath{\mathcal{H}}}^{-\infty}_{T,c}(M)$ [@cras04].

Let $T_{{\ensuremath{\Delta}}}\subset T$ be the subtorus of dimension $1$, with Lie algebra ${\ensuremath{\mathfrak{t}}}_{{\ensuremath{\Delta}}}:=
\{X\in{\ensuremath{\mathfrak{t}}}\vert\ \langle \xi-\xi',X\rangle=0,\ \forall \xi,\xi'\in{\ensuremath{\Delta}}\}$. We make the choice of a decomposition $T=T_{\ensuremath{\Delta}}\times T/T_{\ensuremath{\Delta}}$, where $T/T_{\ensuremath{\Delta}}$ denotes a subtorus de $T$. At the level of Lie algebras, we have then ${\ensuremath{\mathfrak{t}}}={\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}\oplus({\ensuremath{\mathfrak{t}}}/{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}})$ and ${\ensuremath{\mathfrak{t}}}^*={\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}^*\oplus({\ensuremath{\mathfrak{t}}}/{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}})^*$: we see that $\xi + ({\ensuremath{\mathfrak{t}}}/{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}})^*={\ensuremath{\Delta}}$ for any $\xi\in{\ensuremath{\Delta}}$.

Let ${\ensuremath{\mathfrak{c}}}'$ be the relative interior of $\overline{{\ensuremath{\mathfrak{c}}}_+}\cap\overline{{\ensuremath{\mathfrak{c}}}_-}$ in ${\ensuremath{\Delta}}$. It is not difficult to see that for any $m\in\Phi^{-1}({\ensuremath{\mathfrak{c}}}')$ the stabilizer ${\ensuremath{\mathfrak{t}}}_m\subset{\ensuremath{\mathfrak{t}}}$ is either equal to ${\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}$ or reduced to $\{0\}$.

\[def:M-delta-quantum\] We denote $M_{{\ensuremath{\mathfrak{c}}}'}$ the union of the connected component $Z$ of the fixed point set $M^{T_{{\ensuremath{\Delta}}}}$ for which we have ${\ensuremath{\mathfrak{c}}}'\subset\Phi(Z)$. Let $M_{{\ensuremath{\mathfrak{c}}}'}^o$ be the $T$-invariant open subset of $M_{{\ensuremath{\mathfrak{c}}}'}$ where $T/T_{\ensuremath{\Delta}}$ acts locally freely.

The symplectic manifold $M_{{\ensuremath{\mathfrak{c}}}'}$ carries a Hamiltonian action of $T/T_{{\ensuremath{\Delta}}}$ with moment map $\Phi_{{\ensuremath{\mathfrak{c}}}'}:M_{{\ensuremath{\mathfrak{c}}}'}\to{\ensuremath{\Delta}}$ equal to the restriction of $\Phi$ on $M_{{\ensuremath{\mathfrak{c}}}'}$. Let $\xi$ be a point in ${\ensuremath{\mathfrak{c}}}'$. We remark before that for all $m\in\Phi^{-1}(\xi)$ the stabilizer ${\ensuremath{\mathfrak{t}}}_m$ is either equal to ${\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}$ or to $\{0\}$: in particular $\xi$ is a regular value of $\Phi_{{\ensuremath{\mathfrak{c}}}'}$, i.e. $\Phi^{-1}(\xi)\in
M_{{\ensuremath{\mathfrak{c}}}'}^o$. Following Definition \[def:p-xi\] we associate to $\xi$ the cohomology class ${\ensuremath{\hbox{\rm P}}}_\xi^{\ensuremath{\Delta}}\in{\ensuremath{\mathcal{H}}}^{-\infty}_{T/T_{\ensuremath{\Delta}},c}(M_{{\ensuremath{\mathfrak{c}}}'}^o)$.

Let ${\ensuremath{\mathcal{H}}}^*(M_{{\ensuremath{\mathfrak{c}}}'}^o)^{bas}$ be the sub-algebra of ${\ensuremath{\mathcal{H}}}^*(M_{{\ensuremath{\mathfrak{c}}}'}^o)$ formed by the $T$-basic elements. Since the $T_{\ensuremath{\Delta}}$-action on $M_{{\ensuremath{\mathfrak{c}}}'}^o$ is trivial we have a canonical product operation $$\label{eq:produit}
  {\ensuremath{\mathcal{H}}}^{-\infty}_{T/T_{\ensuremath{\Delta}},c}(M_{{\ensuremath{\mathfrak{c}}}'}^o)
\times{\ensuremath{\mathcal{C}^{-\infty}}}({\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}, {\ensuremath{\mathcal{H}}}^*(M_{{\ensuremath{\mathfrak{c}}}'}^o)^{bas})
\longrightarrow {\ensuremath{\mathcal{H}}}^{-\infty}_{T,c}(M_{{\ensuremath{\mathfrak{c}}}'}^o).$$

\[prop:p-diff\] There exists a generalized function supported at $0$, $\delta^{\ensuremath{\Delta}}\in{\ensuremath{\mathcal{C}^{-\infty}}}({\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}, {\ensuremath{\mathcal{H}}}^*(M_{{\ensuremath{\mathfrak{c}}}'}^o)^{bas})$, such that $${\ensuremath{\hbox{\rm P}}}_{\xi^+}-{\ensuremath{\hbox{\rm P}}}_{\xi^-}=
(i_{\ensuremath{\Delta}})_*\left({\ensuremath{\hbox{\rm P}}}_\xi^{\ensuremath{\Delta}}\delta^{\ensuremath{\Delta}}\right)\qquad {\rm in} 
\quad {\ensuremath{\mathcal{H}}}^{-\infty}_{T,c}(M).$$ Here $(i_{\ensuremath{\Delta}})_*:{\ensuremath{\mathcal{H}}}^{-\infty}_{T,c}(M_{{\ensuremath{\mathfrak{c}}}'}^o)\to{\ensuremath{\mathcal{H}}}^{-\infty}_{T,c}(M)$ is the direct image map relative to the inclusion $i_{\ensuremath{\Delta}}: M_{{\ensuremath{\mathfrak{c}}}'}^o{\ensuremath{\hookrightarrow}}M$.

We will now give the precise definition of $\delta^{\ensuremath{\Delta}}$. The decomposition $T=T_{\ensuremath{\Delta}}\times T/T_{\ensuremath{\Delta}}$ and the trivial action of $T_{\ensuremath{\Delta}}$ on $M_{{\ensuremath{\mathfrak{c}}}'}^o$ determine a canonical isomorphism $$j_{\ensuremath{\Delta}}:{\ensuremath{\mathcal{H}}}^*_{T}(M_{{\ensuremath{\mathfrak{c}}}'}^o)\stackrel{\sim}{\longrightarrow}
{\ensuremath{\mathcal{S}^{*}}}({\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}})\otimes{\ensuremath{\mathcal{H}}}^*_{T/T_{{\ensuremath{\Delta}}}}(M_{{\ensuremath{\mathfrak{c}}}'}^o),$$ where ${\ensuremath{\mathcal{S}^{*}}}({\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}})$ is the algebra of complex polynomial functions on ${\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}$. Since the $T/T_{\ensuremath{\Delta}}$-action on $M_{{\ensuremath{\mathfrak{c}}}'}^o$ is locally free, we have the Chern-Weil isomorphism $${\ensuremath{\hbox{\rm \bf cv}}}_{\ensuremath{\Delta}}:{\ensuremath{\mathcal{H}}}^*_{T/T_{{\ensuremath{\Delta}}}}(M_{{\ensuremath{\mathfrak{c}}}'}^o)
\stackrel{\sim}{\longrightarrow}{\ensuremath{\mathcal{H}}}^*(M_{{\ensuremath{\mathfrak{c}}}'}^o)^{bas}.$$

Let $N_{{\ensuremath{\Delta}}}$ be the $T$-equivariant normal bundle of $M^{T_{\ensuremath{\Delta}}}$ in $M$, and let ${\ensuremath{\hbox{\rm Eul}}}{}(N_{{\ensuremath{\Delta}}})\in{\ensuremath{\mathcal{H}}}_{T}^*(M^{T_{\ensuremath{\Delta}}})$ be the $T$-equivariant Euler class of $N_{\ensuremath{\Delta}}$. Now we consider the restriction of ${\ensuremath{\hbox{\rm Eul}}}{}(N_{{\ensuremath{\Delta}}})$ on the open subset $M_{{\ensuremath{\mathfrak{c}}}'}^o\subset
M^{T_{\ensuremath{\Delta}}}$, that we look through the isomorphism ${\ensuremath{\hbox{\rm \bf cv}}}_{\ensuremath{\Delta}}\circ j_{\ensuremath{\Delta}}$ as an element of ${\ensuremath{\mathcal{S}^{*}}}({\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}})\otimes{\ensuremath{\mathcal{H}}}^*(M_{{\ensuremath{\mathfrak{c}}}'}^o)^{bas}$ (for simplicity we keep the same notations ${\ensuremath{\hbox{\rm Eul}}}{}(N_{{\ensuremath{\Delta}}})$ for this element). Following [@pep1], we define inverses ${\ensuremath{\hbox{\rm Eul}}}{}^{-1}_{\pm}(N_{{\ensuremath{\Delta}}})\in{\ensuremath{\mathcal{C}^{-\infty}}}({\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}},{\ensuremath{\mathcal{H}}}^*(M_{{\ensuremath{\mathfrak{c}}}'}^o)^{bas})$ by $$\label{eq:eul-1}
  {\ensuremath{\hbox{\rm Eul}}}{}^{-1}_{\pm\beta}(N_{{\ensuremath{\Delta}}})(X) 
=\lim_{s\to +\infty}\frac{1}{{\ensuremath{\hbox{\rm Eul}}}{}(N_{{\ensuremath{\Delta}}})(X\pm is\beta)}.$$ Here $\beta\in{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}-\{0\}$ is choosen such that $\langle\xi^+ - \xi^-,\beta\rangle>0$.

The generalized function $\delta^{\ensuremath{\Delta}}\in{\ensuremath{\mathcal{C}^{-\infty}}}({\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}},{\ensuremath{\mathcal{H}}}^*(M_{{\ensuremath{\mathfrak{c}}}'}^o)^{bas})$ is defined by $$\label{eq:def-delta}
  \delta^{\ensuremath{\Delta}}:={\ensuremath{\hbox{\rm Eul}}}{}^{-1}_{\beta}(N_{{\ensuremath{\Delta}}})-{\ensuremath{\hbox{\rm Eul}}}{}^{-1}_{-\beta}(N_{{\ensuremath{\Delta}}}).$$ Since the polynomial ${\ensuremath{\hbox{\rm Eul}}}{}(N_{{\ensuremath{\Delta}}})$ is invertible in a smooth manner on ${\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}-\{0\}$ the generalized function $\delta^{\ensuremath{\Delta}}$ is supported at $0$. If we restrict $\delta^{\ensuremath{\Delta}}$ to the submanifold $\Phi_{{\ensuremath{\mathfrak{c}}}'}^{-1}(\xi)$ we get the generalized function $\delta^{\ensuremath{\Delta}}_\xi\in{\ensuremath{\mathcal{C}^{-\infty}}}({\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}},{\ensuremath{\mathcal{H}}}^*({\ensuremath{\mathcal{M}}}^{\ensuremath{\Delta}}_\xi))$, where ${\ensuremath{\mathcal{M}}}^{\ensuremath{\Delta}}_\xi$ denotes the reduced space $\Phi_{{\ensuremath{\mathfrak{c}}}'}^{-1}(\xi)/(T/T_{\ensuremath{\Delta}})=(M^{T_{\ensuremath{\Delta}}}\cap\Phi^{-1}(\xi))/(T/T_{\ensuremath{\Delta}})$.

Now we are able to compute the RHS of (\[eq:f-difference\]). Let $\omega^{\ensuremath{\Delta}}_\xi\in{\ensuremath{\mathcal{H}}}^2({\ensuremath{\mathcal{M}}}^{\ensuremath{\Delta}}_\xi)\otimes{\ensuremath{\mathfrak{t}}}/{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}$ be the curvature of the $T/T_{\ensuremath{\Delta}}$-principal bundle $\Phi_{{\ensuremath{\mathfrak{c}}}'}^{-1}(\xi)\to{\ensuremath{\mathcal{M}}}^{\ensuremath{\Delta}}_\xi$. Let $\vert S^{\ensuremath{\Delta}}_\xi\vert$ be locally constant function on $\Phi_{{\ensuremath{\mathfrak{c}}}'}^{-1}(\xi)$ which is equal to the cardinal of the generic stabilizer of $T/T_{\ensuremath{\Delta}}$. From (\[eq:formule-p\]) and Proposition \[prop:p-diff\] we have $$\begin{aligned}
  \label{eq:diff-1}
 \lefteqn{\int_M ({\ensuremath{\hbox{\rm P}}}_{\xi_+}-{\ensuremath{\hbox{\rm P}}}_{\xi_-})(X)e^{-i\Omega_{\ensuremath{\mathfrak{t}}}(X)}}\nonumber\\
& & =\int_{M_{{\ensuremath{\mathfrak{c}}}'}^o}{\ensuremath{\hbox{\rm P}}}_\xi^{\ensuremath{\Delta}}(X')\delta^{\ensuremath{\Delta}}(X'')e^{-i\Omega_{\ensuremath{\mathfrak{t}}}(X'+X'')}\nonumber\\
& & = \frac{(-2i\pi)^{dim T-1}}{|S^{\ensuremath{\Delta}}_\xi|}\int_{{\ensuremath{\mathcal{M}}}^{\ensuremath{\Delta}}_\xi}
e^{\omega^{\ensuremath{\Delta}}_\xi(\frac{\partial}{\partial X'}|_0)}{\ensuremath{\hbox{\rm vol}}}(T/T_{\ensuremath{\Delta}},-)
{\ensuremath{\hbox{\rm \bf Kir}}}^{\ensuremath{\Delta}}_{\xi}(e^{-i\Omega_{\ensuremath{\mathfrak{t}}}})(X'')
\delta^{\ensuremath{\Delta}}_\xi(X'')\end{aligned}$$ In the last equation the notations are the following:

- $X=X'+X''$ with $X'\in{\ensuremath{\mathfrak{t}}}/{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}$ and $X''\in{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}$,

- the Kirwan map ${\ensuremath{\hbox{\rm \bf Kir}}}^{\ensuremath{\Delta}}_{\xi}: {\ensuremath{\mathcal{H}}}^{\infty}_{T}(M)\to 
{\ensuremath{\mathcal{C}^{\infty}}}({\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}},{\ensuremath{\mathcal{H}}}^*({\ensuremath{\mathcal{M}}}^{\ensuremath{\Delta}}_\xi))$ is the composition of the restriction ${\ensuremath{\mathcal{H}}}^{\infty}_{T}(M)\to{\ensuremath{\mathcal{H}}}^{\infty}_{T}(\Phi_{{\ensuremath{\mathfrak{c}}}'}^{-1}(\xi))$ with the Chern-Weil isomorphism ${\ensuremath{\mathcal{H}}}^{\infty}_{T}(\Phi_{{\ensuremath{\mathfrak{c}}}'}^{-1}(\xi))
\stackrel{\sim}{\to}{\ensuremath{\mathcal{C}^{\infty}}}({\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}},{\ensuremath{\mathcal{H}}}^*({\ensuremath{\mathcal{M}}}^{\ensuremath{\Delta}}_\xi))$.

A direct computation gives that ${\ensuremath{\hbox{\rm \bf Kir}}}^{\ensuremath{\Delta}}_{\xi}(\Omega_{\ensuremath{\mathfrak{t}}})(X'')=\Omega_\xi^{\ensuremath{\Delta}}-\langle\xi,\omega^{\ensuremath{\Delta}}_\xi+X''\rangle$ where $\Omega_\xi^{\ensuremath{\Delta}}$ is the induced symplectic form on the reduced space ${\ensuremath{\mathcal{M}}}^{\ensuremath{\Delta}}_\xi$. If we take the fourier transform in (\[eq:diff-1\]) we get $$\begin{aligned}
  \label{eq:diff-2}
\lefteqn{{({\ensuremath{\rm DH}}_{{\ensuremath{\mathfrak{c}}}_+}-{\ensuremath{\rm DH}}_{{\ensuremath{\mathfrak{c}}}_-})}(a)da}\nonumber\\
&= &  \frac{(i)^{n+1-dim T }}{|S^{\ensuremath{\Delta}}_\xi|}\left(\int_{{\ensuremath{\mathcal{M}}}^{\ensuremath{\Delta}}_\xi}
e^{-i(\Omega^{\ensuremath{\Delta}}_\xi+\langle a',\omega^{\ensuremath{\Delta}}_\xi\rangle)}\, da'\, 
{\ensuremath{\mathcal{F}}}_{{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}}(\delta^{\ensuremath{\Delta}}_\xi)(a'')\right)(a-\xi),\nonumber\\
&= & \sum_{Z} \frac{(i)^{n+1-dim T }}{|S^Z_\xi|}\left(\int_{{\ensuremath{\mathcal{Z}}}_\xi}
e^{-i(\Omega^Z_\xi+\langle a',\omega^Z_\xi\rangle)}\, da'\, 
{\ensuremath{\mathcal{F}}}_{{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}}(\delta^Z_\xi)(a'')\right)(a-\xi)\end{aligned}$$ where $a=a'+a''$ with $a'\in({\ensuremath{\mathfrak{t}}}/{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}})^*$ and $a''\in({\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}})^*$. In (\[eq:diff-2\]), we write $\int_{{\ensuremath{\mathcal{M}}}^{\ensuremath{\Delta}}_\xi}=\sum_{Z}\int_{{\ensuremath{\mathcal{Z}}}^{\ensuremath{\Delta}}_\xi}$ where the sum is taken over the connected components $Z$ of $M_{{\ensuremath{\mathfrak{c}}}'}^o$, and ${\ensuremath{\mathcal{Z}}}_\xi=(Z\cap\Phi^{-1}(\xi))/(T/T_{\ensuremath{\Delta}})$. The $2$-forms $\Omega^{\ensuremath{\Delta}}_\xi,\omega^{\ensuremath{\Delta}}_\xi$, the generic stabiliser $S^{\ensuremath{\Delta}}_\xi$, the vector bundle $N^{\ensuremath{\Delta}}$, the generalized function $\delta^{\ensuremath{\Delta}}_\xi$ restrict to each component $Z$: we denote them respectively $\Omega^Z_\xi,\omega^Z_\xi$, $S^Z_\xi$, $N_Z$, $\delta^Z_\xi$.

We recall now the computation of the Fourier tranform of the inverses ${\ensuremath{\hbox{\rm Eul}}}{}^{-1}_{\pm\beta}(N_Z)$ $={\ensuremath{\hbox{\rm Eul}}}{}^{-1}_{\pm\beta}(N_{\ensuremath{\Delta}})_{\vert Z}$ that is given in [@pep1]\[Proposition 4.8.\]. We consider a $T$-invariant scalar product on the fibers of the bundle $N_{\ensuremath{\Delta}}$. Let $R\in{\ensuremath{\mathcal{A}}}^2(M_{{\ensuremath{\mathfrak{c}}}'}^o,{\rm so}(N_{\ensuremath{\Delta}}))^{bas}$ be the curvature of a $T$-invariant and $T/T_{\ensuremath{\Delta}}$-horizontal Euclidean connexion on $N_{\ensuremath{\Delta}}$: we denote by $R^Z\in{\ensuremath{\mathcal{A}}}^2(Z,{\rm so}(N_Z))^{bas}$ the restriction of $R$ to a component $Z$. The curvature commutes with the infinitesimal action ${\ensuremath{\mathcal{L}}}_X$ of $X\in{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}$, and with the complex structure $J_\beta={\ensuremath{\mathcal{L}}}_\beta(-{\ensuremath{\mathcal{L}}}_\beta^2)^{1/2}$ on $N_{\ensuremath{\Delta}}$ defined by $\beta\in{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}$.

We denote by $S^\bullet$ the symmetric algebra of the complex vector bundle $(N_{\ensuremath{\Delta}},J_\beta)$. We keep the same notation for the restriction of $S^\bullet$ on the submanifolds $Z$, $\Phi_{{\ensuremath{\mathfrak{c}}}'}^{-1}(\xi)$, and for the induced orbifold vector bundle on the reduced spaces ${\ensuremath{\mathcal{Z}}}_\xi$ and ${\ensuremath{\mathcal{M}}}^{\ensuremath{\Delta}}_\xi$. For each $k\in{\ensuremath{\mathbb{N}}}$, we denote by ${\ensuremath{\hbox{\bf Tr}}}_{S^k}$ the trace operator defined on the complex endomorphism of $S^k$. For a complex endomorphism $A$ of $N_{\ensuremath{\Delta}}$, we denote by $A^{\otimes k}$ the induced endomorphism on $S^k$. For any $X\in{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}$, the complex endomorphism ${\ensuremath{\mathcal{L}}}_X^{-1}R^Z$ is symmetric. Hence the trace ${\ensuremath{\hbox{\bf Tr}}}_{S^k}(({\ensuremath{\mathcal{L}}}_X^{-1}R^Z)^{\otimes k})$ is a basic [*real*]{} differential form of degree $2k$ on $Z$ which does not depend of the choice of complex structures ($J_\beta$ or $J_{-\beta}$).

For a smooth function $f$ on ${\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}^*$ with compact support we have $\int_{{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}^*}{\ensuremath{\mathcal{F}}}_{{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}}({\ensuremath{\hbox{\rm Eul}}}{}^{-1}_{\beta}(N_Z))(a'')f(a'')=\int_0^\infty
\hbox{\rm P}_Z(t)f(t\beta^*)dt$ where $\hbox{P}_Z$ is the polynomial on ${\ensuremath{\mathbb{R}}}$ defined by: $$\label{eq:p-beta}
  \hbox{\rm P}_Z(t)=
\frac{(2\pi i)^{r_Z}}{\det^{1/2}_Z({\ensuremath{\mathcal{L}}}_\beta)}\left(\frac{t^{r_Z-1}}{(r_Z-1)!}+
\sum_{k=1}^{dim(Z)/2} \,(i)^k \,{\ensuremath{\hbox{\bf Tr}}}_{S^k}(({\ensuremath{\mathcal{L}}}_\beta^{-1}R^Z)^{\otimes k})
\frac{t^{r_Z-1+k}}{(r_Z-1+k)!}
\right).$$ Here $\det^{1/2}_Z({\ensuremath{\mathcal{L}}}_\beta)$ is the Pfaffian of ${\ensuremath{\mathcal{L}}}_\beta$ on $N_Z$, and $r_Z=\hbox{\rm rk}_{\ensuremath{\mathbb{C}}}(N_Z)$. For \* ${\ensuremath{\mathcal{F}}}_{{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}}({\ensuremath{\hbox{\rm Eul}}}{}^{-1}_{-\beta}(N_Z))$ we have $$\begin{aligned}
  \int_{({\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}})^*}{\ensuremath{\mathcal{F}}}_{{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}}({\ensuremath{\hbox{\rm Eul}}}{}^{-1}_{-\beta}(N_Z))(a'')f(a'')
&=& \int_0^\infty -\hbox{\rm P}_Z(-t)f(-t\beta^*)dt\\
&=& -\int_{-\infty}^0 \hbox{\rm P}_Z(t)f(t\beta^*)dt.\end{aligned}$$ Hence the distribution ${\ensuremath{\mathcal{F}}}_{{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}}(\delta^Z)$ is equal to $\hbox{\rm P}_Z(\beta)d\beta$.

Let $R^Z_\xi$ be the restriction of the curvature $R^Z$ to the submanifold $Z\cap\Phi^{-1}(\xi)$. Since $R^Z$ is $T/T_{\ensuremath{\Delta}}$-basic, ${\ensuremath{\hbox{\bf Tr}}}_{S^k}(({\ensuremath{\mathcal{L}}}_\beta^{-1}R^Z_\xi)^{\otimes k})$ can be seen as a real differential form of degree $2k$ on the orbifold ${\ensuremath{\mathcal{Z}}}_\xi=(Z\cap\Phi^{-1}(\xi))/(T/T_{\ensuremath{\Delta}})$.

Each connected component $Z$ of $M_{{\ensuremath{\mathfrak{c}}}'}^o$ is a $T/T_{\ensuremath{\Delta}}$ Hamiltonian manifold: we take for moment map $\Phi_Z: Z\to ({\ensuremath{\mathfrak{t}}}/{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}})^*$ the restriction of $\Phi-\xi$ to $Z$. Hence $0$ is a regular value of $\Phi_Z$. Let ${\ensuremath{\rm DH}}_0(Z)$ be the polynomial function on $({\ensuremath{\mathfrak{t}}}/{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}})^*=\{a\in{\ensuremath{\mathfrak{t}}}^*\,\vert\,\langle\beta,a\rangle=0\}$ such that ${{\ensuremath{\rm DH}}(Z)}(a')={{\ensuremath{\rm DH}}_0(Z)}(a')da'$ near $0$. Finally (\[eq:diff-2\]) together with the last proposition give the following

\[DH-theo-principal\]

[**(a).**]{}We have ${\ensuremath{\rm DH}}_{{\ensuremath{\mathfrak{c}}}_+}-{\ensuremath{\rm DH}}_{{\ensuremath{\mathfrak{c}}}_-}= \sum_{Z\subset
  M_{{\ensuremath{\mathfrak{c}}}'}}\hbox{\bf D}_Z$ where $$\label{eq:hd-plus-moins}
 \hbox{\bf D}_Z(a)=\frac{(-2\pi)^{r_Z}}{\det^{1/2}_Z({\ensuremath{\mathcal{L}}}_\beta)}
\left(\sum_{k=0}^{d_Z} \hbox{\rm Q}_{Z,k}\frac{\beta^{r_Z-1+k}}{(r_Z-1+k)!} 
\right)(a-\xi).$$ The $\hbox{\rm Q}_{Z,k}$ are the polynomials of degree $d_Z-k$ on $({\ensuremath{\mathfrak{t}}}/{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}})^*$ defined by $$\label{eq:pol-q-k}
  \hbox{\rm Q}_{Z,k}(a')=\frac{(-1)^k}{|S^Z_\xi|}\int_{{\ensuremath{\mathcal{Z}}}_\xi}
\frac{(\Omega^Z_\xi+\langle a',\omega^Z_\xi\rangle)^{d_Z-k}}{(d_Z-k)!}
{\ensuremath{\hbox{\bf Tr}}}_{S^k}(({\ensuremath{\mathcal{L}}}_\beta^{-1}R^Z_\xi)^{\otimes k}).$$ Here $2d_Z=\dim{\ensuremath{\mathcal{Z}}}_\xi$ and $2r_Z=\dim M-\dim Z$. The polynomial $\hbox{\rm Q}_{Z,0}$ correspond to the Duistermaat-Heckmann polynomial ${\ensuremath{\rm DH}}_0(Z)$. In particular we see that the polynomial ${\ensuremath{\rm DH}}_{{\ensuremath{\mathfrak{c}}}_+}-{\ensuremath{\rm DH}}_{{\ensuremath{\mathfrak{c}}}_-}$ is divisible by the factor $a\mapsto \langle\beta,\xi-a\rangle^{r-1}$ where $r=\inf_Z r_Z$. If ${\ensuremath{\Delta}}\cap\Phi(M)$ is not a facet of the polytope $\Phi(M)$ we have $r_Z\geq 2$ for all connected component $Z$ of $M_{{\ensuremath{\mathfrak{c}}}`}$, hence $r\geq 2$.

[**(b).**]{} Suppose now that ${\ensuremath{\mathfrak{c}}}_-$ is a connected component of regular values of $\Phi$ bording a facet $\Phi(M)\cap{\ensuremath{\Delta}}$ of the polytope $\Phi(M)$. Here $Z=\Phi^{-1}({\ensuremath{\Delta}})$ is a connected component of the fixed point set $M^{T_{\ensuremath{\Delta}}}$. In this situation we have ${\ensuremath{\rm DH}}_{{\ensuremath{\mathfrak{c}}}_-}= -\hbox{\bf D}_Z$ where the polynomial ${\bf D}_Z$ is defined by (\[eq:hd-plus-moins\]).

In (\[eq:hd-plus-moins\]) and (\[eq:pol-q-k\]) the vector $\beta\in{\ensuremath{\mathfrak{t}}}$ is normalized by the following conditions: $\beta$ is a primitive vector of the lattice $\ker(\exp:{\ensuremath{\mathfrak{t}}}\to T)$, orthogonal to the hyperplane containing $\overline{{\ensuremath{\mathfrak{c}}}_+}\cap\overline{{\ensuremath{\mathfrak{c}}}_-}$, and pointing out ${\ensuremath{\mathfrak{c}}}_-$.

Quantum version of Duistermaat-Heckman measures {#sec:quantum-version}
===============================================

We suppose here that the Hamiltonian $T$-manifold $(M,\omega,\Phi)$ is prequantized by a $T$-equivariant Hermitian line bundle $L$ over $M$, which is equipped with an Hermitian connection $\nabla$ satisfying the Kostant formula $$\label{eq:kostant}
  {\ensuremath{\mathcal{L}}}(X)-\nabla_{X_M}=i\langle\Phi,X\rangle,\quad X\in{\ensuremath{\mathfrak{t}}}.$$ The former equation implies that the first Chern class of $L$ is equal to $\frac{\Omega}{2\pi}$. In this section we suppose that $M$ is [*compact*]{} and we still assume that the generic stabiliser $\Gamma_M$ of $T$ on $M$ is [*finite*]{}. The quantization of $(M,\Omega)$ is defined by the Riemann-Roch character $RR(M,L)\in
R(T)$ which is compute with a $T$-equivariant almost complex stucture on $M$ [*compatible*]{} with $\Omega$. For $k\geq 1$, we consider the tensor product $L^{\otimes k}$. Its Riemann-Roch character $RR(M,L^{\otimes k})$ decomposes as $$\label{eq:mult-k}
  RR(M,L^{\otimes k})=\sum_{\mu\in\Lambda^*}{\ensuremath{\hbox{\rm m}}}(\mu,k)\,{\ensuremath{\mathbb{C}}}_\mu.$$

Let us recall the well-known properties of the map ${\ensuremath{\hbox{\rm m}}}:\Lambda^*\times
{\ensuremath{\mathbb{Z}}}^{\geq 0}\to {\ensuremath{\mathbb{Z}}}$. When $\frac{\mu}{k}$ is a regular value of $\Phi$, the "Quantization commutes with Reduction Theorem” [@Meinrenken98; @Meinrenken-Sjamaar] tell us that $$\label{eq:Q-R}
  {\ensuremath{\hbox{\rm m}}}(\mu,k)=RR({\ensuremath{\mathcal{M}}}_{\frac{\mu}{k}},{\ensuremath{\mathcal{L}}}_\mu^k)$$ where ${\ensuremath{\mathcal{L}}}_\mu^k=(L^{\otimes  k}\vert_{\Phi^{-1}(\frac{\mu}{k})}\otimes{\ensuremath{\mathbb{C}}}_{-\mu})/T$ is an orbifold line bundle over the symplectic orbifold ${\ensuremath{\mathcal{M}}}_{\frac{\mu}{k}}=\Phi^{-1}(\frac{\mu}{k})/T$. In particular if $\frac{\mu}{k}$ does not belong to $\Phi(M)$ we have ${\ensuremath{\hbox{\rm m}}}(\mu,k)=0$. When $\frac{\mu}{k}\in\Phi(M)$ is not necessarilly a regular value of $\Phi$, one procceed by shift desingularization. If $\xi\in\Phi(M)$ is a regular value of $\Phi$ close enough to $\frac{\mu}{k}$ then (\[eq:Q-R\]) becomes $$\label{eq:Q-R-general}
  {\ensuremath{\hbox{\rm m}}}(\mu,k)=RR({\ensuremath{\mathcal{M}}}_{\xi},{\ensuremath{\mathcal{L}}}_{\xi,\mu}^k)$$ where ${\ensuremath{\mathcal{L}}}_{\xi,\mu}^k=(L^{\otimes k}\vert_{\Phi^{-1}(\xi)}\otimes{\ensuremath{\mathbb{C}}}_{-\mu})/T$ (for a proof see [@Meinrenken-Sjamaar; @pep4]).

A function $f:\Xi\to{\ensuremath{\mathbb{Z}}}$ defined over a lattice $\Xi\simeq{\ensuremath{\mathbb{Z}}}^r$ is called [*periodic polynomial*]{} if $$f(x)=\sum_{i=1}^p e^{i\frac{\langle\alpha_j,x\rangle}{N}}P_j(x),\quad x\in\Xi,$$ where $\alpha_1,\cdots,\alpha_p\in\Xi^*$, $N\geq 1$, and the functions $P_1,\cdots,P_p$ are polynomials with complex coefficients.

Let ${\ensuremath{\mathcal{C}}}$ a cone with [*non-empty*]{} interior in the real vector space $\Xi\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}$. Any [*periodic-polynomial*]{} function $f:\Xi\to{\ensuremath{\mathbb{Z}}}$ is completely determined by its restriction on ${\ensuremath{\mathcal{C}}}\cap\Xi$.

Let ${\ensuremath{\mathfrak{c}}}\subset{\ensuremath{\mathfrak{t}}}^*$ be a connected component of regular values of $\Phi$. In [@Meinrenken-Sjamaar] Meinrenken an Sjamaar proved that there exits a periodic polynomial function ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}}:\Lambda^*\times{\ensuremath{\mathbb{Z}}}\to {\ensuremath{\mathbb{Z}}}$ such that ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}}(\mu,k)={\ensuremath{\hbox{\rm m}}}(\mu,k)$ for every $(\mu,k)$ in the cone $$\label{eq:cone-c}
  {{\ensuremath{\rm Cone}}}({\ensuremath{\mathfrak{c}}})=\{(\xi,s)\in{\ensuremath{\mathfrak{t}}}^*\times{\ensuremath{\mathbb{R}}}^{>0}\,\vert\, \xi\in s\cdot{\ensuremath{\mathfrak{c}}}\}.$$

Consider now two adjacent connected regions ${\ensuremath{\mathfrak{c}}}_\pm$ of regular values of $\Phi$ separated by an hyperplane ${\ensuremath{\Delta}}\subset{\ensuremath{\mathfrak{t}}}^*$. When ${\ensuremath{\Delta}}$ does not contain a facet of the polytope $\Phi(M)$, Meinrenken an Sjamaar proved also that $$\label{eq:MS-egalite}
  {\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_+}(\mu,k)={\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_-}(\mu,k)={\ensuremath{\hbox{\rm m}}}(\mu,k)$$ for every $(\mu,k)\in \overline{{\ensuremath{\rm Cone}}({\ensuremath{\mathfrak{c}}}_+)}\cap\overline{{\ensuremath{\rm Cone}}({\ensuremath{\mathfrak{c}}}_-)}
={\ensuremath{\rm Cone}}(\overline{{\ensuremath{\mathfrak{c}}}_+}\cap\overline{{\ensuremath{\mathfrak{c}}}_-})\subset{\ensuremath{\rm Cone}}({\ensuremath{\Delta}})$. Our main objective is to prove that (\[eq:MS-egalite\]) extends to a “strip” containing ${\ensuremath{\rm Cone}}({\ensuremath{\Delta}})$. Let $\beta\in\Lambda$ be the primitive orthogonal vector to the hyperplane ${\ensuremath{\Delta}}\subset{\ensuremath{\mathfrak{t}}}^*$ which is pointing out of ${\ensuremath{\mathfrak{c}}}_-$. Then $$\label{eq:delta-n}
  {\ensuremath{\Delta}}=\{\xi\in{\ensuremath{\mathfrak{t}}}^*\,\vert\, 
\frac{\langle\mu,\beta\rangle}{2\pi} = r_{\ensuremath{\Delta}}\}$$ for some $r_{\ensuremath{\Delta}}\in{\ensuremath{\mathbb{Z}}}$, ${{\ensuremath{\rm Cone}}}({\ensuremath{\Delta}})=
\{(\xi,s)\in{\ensuremath{\mathfrak{t}}}^*\times{\ensuremath{\mathbb{R}}}^{\geq 0}\,\vert\,
\frac{\langle\xi,\beta\rangle}{2\pi} - s r_{\ensuremath{\Delta}}=0 \}$ and ${\ensuremath{\mathfrak{c}}}_- \subset$ $\{\xi\in{\ensuremath{\mathfrak{t}}}^*\,\vert\,
\frac{\langle\mu,\beta\rangle}{2\pi} < r_{\ensuremath{\Delta}}\}$.

Let $T_{{\ensuremath{\Delta}}}$ be the subtorus of $T$ generated by $\beta$. Let $N_{\ensuremath{\Delta}}$ be the normal vector bundle of $M^{T_{\ensuremath{\Delta}}}$ in $M$. The almost complex structure on $M$ induces a complex structure $J$ on the fibers of $N_{\ensuremath{\Delta}}$. We have a decomposition $N_{\ensuremath{\Delta}}=\sum_s N_{\ensuremath{\Delta}}^s$ where $N_{\ensuremath{\Delta}}^s=\{v\in N_{\ensuremath{\Delta}}\,\vert\, {\ensuremath{\mathcal{L}}}_\beta v= s\, Jv\ \}$. We write $N_{\ensuremath{\Delta}}=N_{\ensuremath{\Delta}}^{+,\beta}\oplus N_{\ensuremath{\Delta}}^{-,\beta}$ where $$\label{eq:N-beta-plus}
 N_{\ensuremath{\Delta}}^{\pm,\beta}=\sum_{\pm s >0}N_{\ensuremath{\Delta}}^s.$$

For every connected component $Z\subset M^{T_{\ensuremath{\Delta}}}$ we define $s^\pm_Z\in{\ensuremath{\mathbb{N}}}$ respectively as the absolute value of the trace of $\frac{1}{2\pi}{\ensuremath{\mathcal{L}}}_\beta$ on $N_{\ensuremath{\Delta}}^{\pm,\beta}\vert_Z$. Note that $s^+_Z + s^-_Z$ is larger than half of the codimension of $Z$ in $M$.

We prove in Section \[subsec:jump-formula\] the following

\[theo:m-pm-section\] We have ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_+}(\mu,k)={\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_-}(\mu,k)$ for all $(\mu,k)\in\Lambda^*\times{\ensuremath{\mathbb{Z}}}$ such that $$\label{eq:encadrement}
  - s^- < \frac{\langle\mu,\beta\rangle}{2\pi}- k\, r_{\ensuremath{\Delta}}< s^+ \ .$$ The number $s^-,s^+\in{\ensuremath{\mathbb{N}}}$ are defined as follows. We take $s^\pm=\inf_Z s^\pm_Z$ where the minimum is taken over the connected components $Z$ of $M^{T_{\ensuremath{\Delta}}}$ for which $\overline{{\ensuremath{\mathfrak{c}}}_+}\cap\overline{{\ensuremath{\mathfrak{c}}}_-}\subset\Phi(Z)$.

Similar results were obtained by Billey-Guillemin-Rassart [@BGR03] in the case where $M$ is a coadjoint orbit of $\SU(n)$, and by Szenes-Vergne [@Szenes-Vergne02] in the case where $M$ is a complex vector space. See Sections \[subsec:su-n\] and \[sec:cas-C-d\] where we study these two particular cases in details. In Proposition \[prop:inegalite-optimale\], we give also a criterium which says when the inequalities in (\[eq:encadrement\]) are optimal. This criterium is fullfilled when there is [*only one*]{} component $Z$ of $M^{T_{\ensuremath{\Delta}}}$ such that $\overline{{\ensuremath{\mathfrak{c}}}_+}\cap\overline{{\ensuremath{\mathfrak{c}}}_-}\subset\Phi(Z)$. Then (\[eq:encadrement\]) is [*optimal*]{} and $s^+ + s^-$ is larger than half of the codimension of $Z$ in $M$.

The following easy Lemma (see Lemma 7.3. of [@pep4]) gives some basic informations about the integer $s_Z^\pm$.

\[lem:pep4\] Let $(M,\Omega,\Phi)$ be a compact Hamiltonian $T$-manifold equipped with a $T$-invariant almost complex structure compatible with $\Omega$. Consider a non-zero vector $\gamma\in{\ensuremath{\mathfrak{t}}}$ and let $Z$ be a connected component of the fixed point set $M^\gamma$. Let $N$ be the normal vector of $Z$ in $M$ and let $N^{-,\gamma}$ be the negative polarized normal bundle (see (\[eq:N-beta-plus\])). Then $N^{-,\gamma}=0$ if and only if the function $\langle\Phi,\gamma\rangle: M\to{\ensuremath{\mathbb{R}}}$ takes its maximal value on $Z$.

This Lemma insures that $s^\pm\geq 1$ in Theorem \[theo:m-pm-section\] when ${\ensuremath{\Delta}}\cap\Phi(M)$ is not a facet of the polytope $\Phi(M)$.

Consider the situation where ${\ensuremath{\Delta}}\cap\Phi(M)$ is a facet of the polytope $\Phi(M)$ so that ${\ensuremath{\mathfrak{c}}}_+\cap\Phi(M)=\emptyset$: hence ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_+}=0$. If we apply Lemma \[lem:pep4\] to $\gamma=\beta$, one gets $N^{-,\beta}=0$ and so $s^-=0$. In this situation we get

Let ${\ensuremath{\mathfrak{c}}}_-$ be a connected component of regular values of $\Phi$ bording a facet $\Phi(M)\cap{\ensuremath{\Delta}}$ of the polytope $\Phi(M)$. Let $\beta\in\Lambda$ be the unimodular orthogonal vector to the hyperplane ${\ensuremath{\Delta}}\subset{\ensuremath{\mathfrak{t}}}^*$ which is pointing out of ${\ensuremath{\mathfrak{c}}}_-$. Here $Z=\Phi^{-1}({\ensuremath{\Delta}})$ is a connected component of the fixed point set $M^{T_{\ensuremath{\Delta}}}$. We have ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_-}(\mu,k)=0$ for all $(\mu,k)\in\Lambda^*\times{\ensuremath{\mathbb{Z}}}$ such that $$\label{eq:encadrement-bord}
 0< \frac{\langle\mu,\beta\rangle}{2\pi}- k r_{\ensuremath{\Delta}}< s^+_Z .$$ Here $s^+_Z\in {\ensuremath{\mathbb{N}}}$ is the value of the trace of $\frac{1}{2\pi}{\ensuremath{\mathcal{L}}}_\beta$ on the normal bundle of $Z$ in $M$, and then is larger than half of the codimension of $Z$ in $M$. Moreover the inequalities (\[eq:encadrement-bord\]) are optimal.

We first review some of the results of [@pep4].

Elliptic and transversally elliptic symbols
-------------------------------------------

We work in the setting of a compact manifold $M$ equipped with a smooth action of a torus $T$.

Let $p:{\ensuremath{\hbox{\bf T}}}M\to M$ be the projection, and let $(-,-)_M$ be a $T$-invariant Riemannian metric. If $E^{0},E^{1}$ are $T$-equivariant vector bundles over $M$, a $T$-equivariant morphism $\sigma \in \Gamma({\ensuremath{\hbox{\bf T}}}M,\hom(p^{*}E^{0},p^{*}E^{1}))$ is called a [*symbol*]{}. The subset of all $(m,v)\in {\ensuremath{\hbox{\bf T}}}M$ where $\sigma(m,v): E^{0}_{m}\to E^{1}_{m}$ is not invertible is called the [*characteristic set*]{} of $\sigma$, and is denoted by ${\ensuremath{\hbox{\rm Char}}}(\sigma)$.

Let ${\ensuremath{\hbox{\bf T}}}_{T}M$ be the following subset of ${\ensuremath{\hbox{\bf T}}}M$ : $${\ensuremath{\hbox{\bf T}}}_{T}M\ = \left\{(m,v)\in {\ensuremath{\hbox{\bf T}}}M,\ (v,X_{M}(m))_{_{M}}=0 \quad {\rm for\ all}\ 
   X\in{\ensuremath{\mathfrak{k}}}\right\} .$$

A symbol $\sigma$ is [*elliptic*]{} if $\sigma$ is invertible outside a compact subset of ${\ensuremath{\hbox{\bf T}}}M$ (${\ensuremath{\hbox{\rm Char}}}(\sigma)$ is compact), and is [*transversally elliptic*]{} if the restriction of $\sigma$ to ${\ensuremath{\hbox{\bf T}}}_{T}M$ is invertible outside a compact subset of ${\ensuremath{\hbox{\bf T}}}_{T}M$ (${\ensuremath{\hbox{\rm Char}}}(\sigma)\cap {\ensuremath{\hbox{\bf T}}}_{T}M$ is compact). An elliptic symbol $\sigma$ defines an element in the equivariant $K$-theory of ${\ensuremath{\hbox{\bf T}}}M$ with compact support, which is denoted by ${\ensuremath{\hbox{\bf K}}}_{T}({\ensuremath{\hbox{\bf T}}}M)$, and the index of $\sigma$ is a virtual finite dimensional representation of $T$ [@Atiyah-Segal68; @Atiyah-Singer-1; @Atiyah-Singer-2; @Atiyah-Singer-3].

A [*transversally elliptic*]{} symbol $\sigma$ defines an element of ${\ensuremath{\hbox{\bf K}}}_{T}({\ensuremath{\hbox{\bf T}}}_{T}M)$, and the index of $\sigma$ is defined as a trace class virtual representation of $T$ (see [@Atiyah74] for the analytic index and [@B-V.inventiones.96.1; @B-V.inventiones.96.2] for the cohomological one). Remark that any elliptic symbol of ${\ensuremath{\hbox{\bf T}}}M$ is transversally elliptic, hence we have a restriction map ${\ensuremath{\hbox{\bf K}}}_{T}({\ensuremath{\hbox{\bf T}}}M)\to {\ensuremath{\hbox{\bf K}}}_{T}({\ensuremath{\hbox{\bf T}}}_{T}M)$, and a commutative diagram $$\label{indice.generalise}
\xymatrix{
{\ensuremath{\hbox{\bf K}}}_{T}({\ensuremath{\hbox{\bf T}}}M)\ar[r]\ar[d]_{{\ensuremath{\hbox{\rm Index}}}_{M}^T} & 
{\ensuremath{\hbox{\bf K}}}_{T}({\ensuremath{\hbox{\bf T}}}_{T}M)\ar[d]^{{\ensuremath{\hbox{\rm Index}}}_{M}^T}\\
R(T)\ar[r] &  R^{-\infty}(T)\ .
   }$$

Using the [*excision property*]{}, one can easily show that the index map ${\ensuremath{\hbox{\rm Index}}}_{{\ensuremath{\mathcal{U}}}}^{T}:{\ensuremath{\hbox{\bf K}}}_{T}({\ensuremath{\hbox{\bf T}}}_{T}{\ensuremath{\mathcal{U}}})\to R^{-\infty}(T)$ is still defined when ${\ensuremath{\mathcal{U}}}$ is a $T$-invariant relatively compact open subset of a $T$-manifold (see [@pep4]\[section 3.1\]).

Localization of the Riemann-Roch character {#ssection.thom}
------------------------------------------

We suppose now that the compact $T$-manifold $M$ is equipped with a $T$-invariant almost complex structure $J$. Let us recall the definitions of the Thom symbol ${\ensuremath{\hbox{\rm Thom}}}(M,J)$ and of the Riemann-Roch character [@pep4].

Consider a $T$-invariant Riemannian metric $q$ on $M$ such that $J$ is orthogonal relatively to $q$, and let $h$ be the Hermitian structure on ${\ensuremath{\hbox{\bf T}}}M$ defined by : $h(v,w)=q(v,w) -\imath q(Jv,w)$ for $v,w\in {\ensuremath{\hbox{\bf T}}}M$. The symbol $${\ensuremath{\hbox{\rm Thom}}}_{_{T}}(M,J)\in 
\Gamma\left(M,\hom(p^{*}(\wedge_{{\ensuremath{\mathbb{C}}}}^{even} {\ensuremath{\hbox{\bf T}}}M),\,p^{*}
(\wedge_{{\ensuremath{\mathbb{C}}}}^{odd} {\ensuremath{\hbox{\bf T}}}M))\right)$$ at $(m,v)\in {\ensuremath{\hbox{\bf T}}}M$ is equal to the Clifford map $$\label{eq.thom.complex}
 \Clif_{m}(v)\ :\ \wedge_{{\ensuremath{\mathbb{C}}}}^{even} {\ensuremath{\hbox{\bf T}}}_m M
\longrightarrow \wedge_{{\ensuremath{\mathbb{C}}}}^{odd} {\ensuremath{\hbox{\bf T}}}_m M,$$ where $\Clif_{m}(v).w= v\wedge w - c_{h}(v).w$ for $w\in 
\wedge_{{\ensuremath{\mathbb{C}}}}^{\bullet} {\ensuremath{\hbox{\bf T}}}_{x}M$. Here $c_{h}(v):\wedge_{{\ensuremath{\mathbb{C}}}}^{\bullet} 
{\ensuremath{\hbox{\bf T}}}_{m}M\to\wedge^{\bullet -1} {\ensuremath{\hbox{\bf T}}}_{m}M$ denotes the contraction map relative to $h$. Since the map $\Clif_{m}(v)$ is invertible for all $v\neq 0$, the symbol ${\ensuremath{\hbox{\rm Thom}}}_{_{T}}(M,J)$ is elliptic.

The Riemann-Roch character $RR(M,-):{\ensuremath{\hbox{\bf K}}}_{T}(M)\to R(T)$ is defined by the following relation $$\label{def.RR}
RR(M,E)={\ensuremath{\hbox{\rm Index}}}^{T}_{M}\left({\ensuremath{\hbox{\rm Thom}}}_{_{T}}(M,J)\otimes p^{*}E\right)\ .$$ The important point is that for any $T$-vector bundle $E$, ${\ensuremath{\hbox{\rm Thom}}}_{_{T}}(M,J)\otimes p^{*}E$ corresponds to the [*principal symbol*]{} of the twisted $\spinc$ Dirac operator ${\ensuremath{\mathcal{D}}}_{E}$ [@Duistermaat96], hence $RR(M,E)\in R(T)$ is also defined as the (analytical) index of the elliptic operator ${\ensuremath{\mathcal{D}}}_E$.

Consider now the case of a [*compact*]{} Hamiltonian $T$-manifold $(M,\omega,\Phi)$. Here $J$ is a $T$-invariant almost comlex structure compatible with $\Omega$: $(v,w)\mapsto\Omega(v,Jw)$ defines a Riemannian metric on $M$. Like in Section \[sec:hd\], we make the choice of a scalar product $(-,-)$ on ${\ensuremath{\mathfrak{t}}}^*$ (which induces an identification ${\ensuremath{\mathfrak{t}}}^*\simeq{\ensuremath{\mathfrak{t}}}$) and we consider for any $\xi\in{\ensuremath{\mathfrak{t}}}^*$ the function $\frac{-1}{2}\parallel\Phi-\xi\parallel^2: M\to {\ensuremath{\mathbb{R}}}$ and its Hamiltonian vectors field ${\ensuremath{\mathcal{H}}}-\xi_M$.

\[def.thom.loc\] For any $\xi\in{\ensuremath{\mathfrak{t}}}^*$ and any $T$-invariant open subset ${\ensuremath{\mathcal{U}}}\subset M$ we define the symbol ${\ensuremath{\hbox{\rm Thom}}}_{\xi}({\ensuremath{\mathcal{U}}})$ by the relation $${\ensuremath{\hbox{\rm Thom}}}_{\xi}({\ensuremath{\mathcal{U}}})(m,v):={\ensuremath{\hbox{\rm Thom}}}(M,J)(m,v-({\ensuremath{\mathcal{H}}}-\xi_M)(m))\quad (m,v)\in{\ensuremath{\hbox{\bf T}}}{\ensuremath{\mathcal{U}}}$$

The characteristic set of ${\ensuremath{\hbox{\rm Thom}}}_{\xi}({\ensuremath{\mathcal{U}}})$ corresponds to $\{(m,v)\in {\ensuremath{\hbox{\bf T}}}{\ensuremath{\mathcal{U}}},\ v=({\ensuremath{\mathcal{H}}}-\xi_M)(m)\}$, the graph of the vector field ${\ensuremath{\mathcal{H}}}-\xi_M$ over ${\ensuremath{\mathcal{U}}}$. Since ${\ensuremath{\mathcal{H}}}-\xi_M$ belongs to the set of tangent vectors to the $T$-orbits, we have $$\begin{aligned}
{\ensuremath{\hbox{\rm Char}}}\left({\ensuremath{\hbox{\rm Thom}}}_{\xi}({\ensuremath{\mathcal{U}}})\right)\cap {\ensuremath{\hbox{\bf T}}}_{T}{\ensuremath{\mathcal{U}}}&=&\{(m,0)\in {\ensuremath{\hbox{\bf T}}}{\ensuremath{\mathcal{U}}}\,\vert\, ({\ensuremath{\mathcal{H}}}-\xi_M)(m)=0 \}\\
&\cong& \{m\in {\ensuremath{\mathcal{U}}},\ d\parallel\Phi-\xi\parallel^2_m=0 \} \ .\end{aligned}$$ Therefore the symbol ${\ensuremath{\hbox{\rm Thom}}}_{\xi}({\ensuremath{\mathcal{U}}})$ is transversally elliptic if and only if $$\label{eq:U-xi}
  {\ensuremath{\hbox{\rm Cr}}}(\parallel\Phi-\xi\parallel^2)\cup\partial{\ensuremath{\mathcal{U}}}=\emptyset.$$ Here ${\ensuremath{\hbox{\rm Cr}}}(\parallel\Phi-\xi\parallel^2)$ denotes the set of critical points of the function $\parallel\Phi-\xi\parallel^2$. When (\[eq:U-xi\]) holds we say that the couple $({\ensuremath{\mathcal{U}}},\xi)$ is [*good*]{}.

\[def:RR-U-xi\] Let $({\ensuremath{\mathcal{U}}},\xi)$ be a good couple. For any $T$-vector bundle $E\to M$, the tensor product ${\ensuremath{\hbox{\rm Thom}}}_{\xi}({\ensuremath{\mathcal{U}}})\otimes p^{*}E$ belongs to ${\ensuremath{\hbox{\bf K}}}_{T}({\ensuremath{\hbox{\bf T}}}_{T}{\ensuremath{\mathcal{U}}})$ and we denote by $$RR_{{\ensuremath{\mathcal{U}}}}^{\xi}(M,E)\in R^{-\infty}(T)$$ its index.

\[prop:U-xi\] Let $({\ensuremath{\mathcal{U}}},\xi)$ be a good couple.

$a)$ If ${\ensuremath{\mathcal{U}}}$ possess two $T$-invariant open subsets ${\ensuremath{\mathcal{U}}}^1,{\ensuremath{\mathcal{U}}}^2$ such that $\overline{{\ensuremath{\mathcal{U}}}^1}\cap \overline{{\ensuremath{\mathcal{U}}}^2}\cap$ ${\ensuremath{\hbox{\rm Cr}}}(\parallel\Phi-\xi\parallel^2)=\emptyset$ and $({\ensuremath{\mathcal{U}}}^1\cup{\ensuremath{\mathcal{U}}}^2)\cap{\ensuremath{\hbox{\rm Cr}}}(\parallel\Phi-\xi\parallel^2)=
{\ensuremath{\mathcal{U}}}\cap{\ensuremath{\hbox{\rm Cr}}}(\parallel\Phi-\xi\parallel^2)$, then the couples $({\ensuremath{\mathcal{U}}}^1,\xi)$ and $({\ensuremath{\mathcal{U}}}^2,\xi)$ are good and $$RR_{{\ensuremath{\mathcal{U}}}}^{\xi}(M,-)=RR_{{\ensuremath{\mathcal{U}}}^1}^{\xi}(M,-)+RR_{{\ensuremath{\mathcal{U}}}^2}^{\xi}(M,-).$$ In particular $RR_{{\ensuremath{\mathcal{U}}}}^{\xi}(M,-)=RR_{{\ensuremath{\mathcal{U}}}^1}^{\xi}(M,-)$ if ${\ensuremath{\mathcal{U}}}^1$ is an open subset of ${\ensuremath{\mathcal{U}}}$ such that ${\ensuremath{\mathcal{U}}}^1\cap{\ensuremath{\hbox{\rm Cr}}}(\parallel\Phi-\xi\parallel^2)=
{\ensuremath{\mathcal{U}}}\cap{\ensuremath{\hbox{\rm Cr}}}(\parallel\Phi-\xi\parallel^2)$.

$b)$ If $\xi'\in{\ensuremath{\mathfrak{t}}}^*$ is close enough to $\xi$, then $({\ensuremath{\mathcal{U}}},\xi')$ is good and $$RR_{{\ensuremath{\mathcal{U}}}}^{\xi}(M,-)=RR_{{\ensuremath{\mathcal{U}}}}^{\xi'}(M,-).$$

[*Proof.*]{} The part $a)$ is a direct consequence of the excision property (see Proposition 4.1. in [@pep4]). Consider now the scalar product $({\ensuremath{\mathcal{H}}}-\xi^s_M,{\ensuremath{\mathcal{H}}}-\xi_M)$ where $\xi^s=s\xi'+(1-s)\xi,\ s\in[0,1]$ and $(-,-)$ is a $T$-invariant Riemannian metric on $M$. We have $({\ensuremath{\mathcal{H}}}-\xi^s_M,{\ensuremath{\mathcal{H}}}-\xi_M)=\|{\ensuremath{\mathcal{H}}}-\xi_M\|^2+ s((\xi-\xi')_M,{\ensuremath{\mathcal{H}}}-\xi_M)$ and then the following inequality holds on $M$ $$\label{eq:maj-prime}
  ({\ensuremath{\mathcal{H}}}-\xi^s_M,{\ensuremath{\mathcal{H}}}-\xi_M)\geq \|{\ensuremath{\mathcal{H}}}-\xi_M\|^2 
\Big( \|{\ensuremath{\mathcal{H}}}-\xi_M\|-s\|(\xi-\xi')_M\|\Big).$$ Since $\partial{\ensuremath{\mathcal{U}}}$ is compact we have the following inequalities on it: $\|{\ensuremath{\mathcal{H}}}-\xi_M\|\geq c_1>0$ and $\|a_M\|\leq c_2\|a\|$ for any $a\in{\ensuremath{\mathfrak{t}}}$. So (\[eq:maj-prime\]) implies the following inequality on $\partial{\ensuremath{\mathcal{U}}}$: $$({\ensuremath{\mathcal{H}}}-\xi^s_M,{\ensuremath{\mathcal{H}}}-\xi_M)\geq c_1( c_1-s\|\xi-\xi'\|)\quad {\rm for}\quad s\in[0,1].$$ So if $\xi'$ is close enough to $\xi$, we have $\|{\ensuremath{\mathcal{H}}}-\xi^s_M\|\geq
c_3>0$ on $\partial{\ensuremath{\mathcal{U}}}$ for any $s\in[0,1]$. We have first prove that the couple $({\ensuremath{\mathcal{U}}},\xi^s)$ is good for any $s\in[0,1]$. We see then that the familly of transversally elliptic symbols ${\ensuremath{\hbox{\rm Thom}}}_{\xi^s}({\ensuremath{\mathcal{U}}}),\, s\in[0,1]$ defines an homotopy between ${\ensuremath{\hbox{\rm Thom}}}_{\xi}({\ensuremath{\mathcal{U}}})$ and ${\ensuremath{\hbox{\rm Thom}}}_{\xi'}({\ensuremath{\mathcal{U}}})$. Hence ${\ensuremath{\hbox{\rm Thom}}}_{\xi}({\ensuremath{\mathcal{U}}})={\ensuremath{\hbox{\rm Thom}}}_{\xi'}({\ensuremath{\mathcal{U}}})$ in ${\ensuremath{\hbox{\bf K}}}_{T}({\ensuremath{\hbox{\bf T}}}_{T}{\ensuremath{\mathcal{U}}})$. $\Box$

Part $a)$ of Proposition \[prop:U-xi\] tells us that $RR^\xi_{\ensuremath{\mathcal{U}}}(M,-)$ depends closely of the intersection ${\ensuremath{\mathcal{U}}}\cap{\ensuremath{\hbox{\rm Cr}}}(\parallel\Phi-\xi\parallel^2)$. In particular $RR^\xi_{\ensuremath{\mathcal{U}}}(M,-)=0$ when ${\ensuremath{\mathcal{U}}}\cap{\ensuremath{\hbox{\rm Cr}}}(\parallel\Phi-\xi\parallel^2)=\emptyset$. Recall that $$\label{eq:decom-kirwan}
 {\ensuremath{\hbox{\rm Cr}}}(\parallel\Phi-\xi\parallel^2)=
 \bigcup_{\gamma\in{\ensuremath{\mathcal{B}}}_\xi}M^\gamma\cap\Phi^{-1}(\gamma+\xi)$$ where ${\ensuremath{\mathcal{B}}}_\xi\subset{\ensuremath{\mathfrak{t}}}^*$ is a finite set [@Kirwan84].

\[def:RR-xi\] For any $\xi\in{\ensuremath{\mathfrak{t}}}^*$ and $\gamma\in{\ensuremath{\mathcal{B}}}_\xi$, we denote simply by $$RR^\xi_\gamma(M,-):{\ensuremath{\hbox{\bf K}}}_{T}(M)\to R^{-\infty}(T)$$ the map $RR_{{\ensuremath{\mathcal{U}}}}^{\xi}(M,-)$, where ${\ensuremath{\mathcal{U}}}$ is a $T$-invariant open neighborhood of $M^\gamma\cap\Phi^{-1}(\gamma+\xi)$ such that ${\ensuremath{\hbox{\rm Cr}}}(\parallel\Phi-\xi\parallel^2)\cap\overline{{\ensuremath{\mathcal{U}}}}=
M^\gamma\cap\Phi^{-1}(\gamma+\xi)$.

Part $a)$ of Proposition \[prop:U-xi\] insures that the maps $RR_\gamma^\xi(M,-)$ are well defined, and for any good couple $({\ensuremath{\mathcal{U}}},\xi)$ we have $$\label{eq:RR-U-gamma}
   RR^\xi_{\ensuremath{\mathcal{U}}}(M,-)=\sum_{\gamma\in{\ensuremath{\mathcal{B}}}_\xi\cap\Phi({\ensuremath{\mathcal{U}}})} RR_{\gamma}^{\xi}(M,-).$$ If one takes ${\ensuremath{\mathcal{U}}}=M$, we have $RR^\xi_{\ensuremath{\mathcal{U}}}(M,-)=RR(M,-)=\sum_{\gamma\in{\ensuremath{\mathcal{B}}}_\xi} RR_{\gamma}^{\xi}(M,-)$ (see [@pep4]\[Section 4\]).

Periodic polynomial behaviour of the multiplicities
---------------------------------------------------

We suppose here that the Hamiltonian $T$-manifold $(M,\Omega,\Phi)$ is prequantized by a $T$-complex line bundle $L$ satisfying (\[eq:kostant\]) for a suitable invariant connection. In this section we will characterize the periodic polynomial behaviour of the multiplicities ${\ensuremath{\hbox{\rm m}}}(\mu,k)$ with the help of the localized Riemann-Roch character $RR^\xi_0(M,-)$.

Let us introduce some vocabulary. We say that two generalized characters $\chi^\pm=\sum_{\mu\in\Lambda^*}a^\pm_\mu\,{\ensuremath{\mathbb{C}}}_\mu$ [*coincide*]{} on a region $D\subset{\ensuremath{\mathfrak{t}}}^*$, if $a^+_\mu=a^{-}_\mu$ for every $\mu \in D\cap\lambda^*$. A generalized character $\chi=\sum_{\mu}a_\mu\, {\ensuremath{\mathbb{C}}}_\mu$ is [*supported*]{} on a region $D\subset{\ensuremath{\mathfrak{t}}}^*$ if $a_\mu=0$ for $\mu\notin D$. A weight $\mu\in\Lambda^*$ [*occurs*]{} in $\chi=\sum_{\mu}a_\mu\, {\ensuremath{\mathbb{C}}}_\mu$ if $a_\mu\neq 0$.

For $\xi\in{\ensuremath{\mathfrak{t}}}^*$, we define $r_\xi>0$ as the smallest non-zero critical value of the function $\parallel\Phi-\xi\parallel$, and we denote by $B(\xi,r_\xi)$ the open ball of center $\xi$ and radius $r_\xi$.

\[th-RR-xi\] For any $\xi\in{\ensuremath{\mathfrak{t}}}^*$, the generalized character $RR_0^\xi(M,L^{\otimes k})$ coincides with $RR(M,L^{\otimes k})$ on the open ball $k\cdot B(\xi,r_\xi)$.

The arguments of [@pep4] for the proof of this Theorem will be needed another time, so we recall them. Let $\xi\in{\ensuremath{\mathfrak{t}}}^*$. We start with the decomposition $$\label{eq:loc-gene}
 RR(M,L^{\otimes k})=\sum_{\gamma\in{\ensuremath{\mathcal{B}}}_\xi} RR_{\gamma}^{\xi}(M,L^{\otimes k}).$$ We recall now, for a non-zero $\gamma\in{\ensuremath{\mathcal{B}}}_\xi$, the localization of the map $RR_{\gamma}^{\xi}$ on the fixed point set $M^\gamma$ [@pep4]

Let $N$ be the normal bundle of $M^\gamma$ in $M$. The almost complex structure on $M$ induces an almost complex struture on $M^\gamma$ and a complex structure on the bundles $N$ and $N_{\ensuremath{\mathbb{C}}}:=N\otimes {\ensuremath{\mathbb{C}}}$. Following (\[eq:N-beta-plus\]) we define the $\gamma$-polarized complex vector bundles $N^{+,\gamma}$ and $(N_{\ensuremath{\mathbb{C}}})^{+,\gamma}$.

The manifold $M^\gamma$ is a symplectic submanifold of $M$ equipped with an induced Hamiltonian action of $T$: its moment map is the restriction of $\Phi$ on $M^\gamma$. Following Definition \[def:RR-xi\], we have on $M^\gamma$ a localized Riemann-Roch character $RR_\gamma^\xi(M^\gamma,-)$. On $M^\gamma$, the Hamiltonian vectors fields of the functions $\parallel\Phi-\xi\parallel^2$ and $\parallel\Phi-(\xi+\gamma)\parallel^2$ coincide, hence $$\label{eq:xi-gamma}
  RR_\gamma^\xi(M^\gamma,-)=RR_0^{\xi+\gamma}(M^\gamma,-).$$

We prove in [@pep4]\[Theorem 5.8.\] that $$\label{eq:loc-RR-beta}
  RR_\gamma^\xi(M,E)=\sum_{k\in{\ensuremath{\mathbb{N}}}}(-1)^l
RR_\gamma^\xi(M^\gamma,E\vert_{M^\gamma}\otimes \det(N^{+,\gamma})\otimes
S^k(N_{\ensuremath{\mathbb{C}}}^{+,\gamma}))$$ for every $T$-vector bundle $E$. Here $l$ is the locally constant fonction on $M^\gamma$ equal to the complex rank of $N^{+,\gamma}$.

[@pep4]\[Section 5\]\[def:lambda-N\] Let $\overline{N}$ be the $T$-vector bundle $N$ with the opposite complex structure on the fibers. The sum $(-1)^l\sum_{k\in{\ensuremath{\mathbb{N}}}} \det(N^{+,\gamma})\otimes S^k(N_{\ensuremath{\mathbb{C}}}^{+,\gamma})$ is an inverse of $\wedge_{{\ensuremath{\mathbb{C}}}}^{\bullet}\overline{N}$ that we denote $\left[\wedge_{{\ensuremath{\mathbb{C}}}}^{\bullet}\overline{N}\right]^{-1}_{\gamma}$.

If we use the notations of Proposition \[def:lambda-N\] and (\[eq:xi-gamma\]), the localization (\[eq:loc-RR-beta\]) can be rewritten as $$\label{eq:loc-RR-beta-bis}
  RR_\gamma^\xi(M,E)=
RR_0^{\xi+\gamma}\left(M^\gamma,E\vert_{M^\gamma}\otimes
\left[\wedge_{{\ensuremath{\mathbb{C}}}}^{\bullet}\overline{N}\right]^{-1}_{\gamma}\right).$$

Let $i:T_\gamma{\ensuremath{\hookrightarrow}}T$ be the inclusion of the subtorus generated by $\gamma$. For a $T$-vector bundle $F\to M^\gamma$, it is easy to show that a weight $\mu\in\Lambda^*$ occurs in $RR_\gamma^\xi(M^\gamma,F)$ only if $i^*(\mu)$ occurs as a weight for the $T_\gamma$-action on the fibers of $F$ (see Lemma 9.4. in [@pep4]). Since the $T_\gamma$ weights on the bundles $N_{\ensuremath{\mathbb{C}}}^{+,\gamma}$ and $N^{+,\gamma}$ are polarized by $\gamma$, the localization (\[eq:loc-RR-beta\]) gives the following

For a non-zero $\gamma\in{\ensuremath{\mathcal{B}}}_\xi$, the generalized character $RR_\gamma^\xi(M,L^{\otimes k})$ is supported on the half space $\{a\in{\ensuremath{\mathfrak{t}}}^*\,\vert\, (\gamma, a-k(\xi+\gamma))\geq 0\}$.

Since the condition $(\gamma, a-k(\xi+\gamma))\geq 0$ implies that $\parallel a-k\xi\parallel\geq k \parallel\gamma\parallel\geq k r_\xi$, the last proposition shows that every weights of the open ball $k\cdot B(\xi,r_\xi)$ does not occurs in $RR_\gamma^\xi(M,L^{\otimes k})$. This last remark together with (\[eq:loc-gene\]) prove Theorem \[th-RR-xi\].

For the localized Riemann-Roch character $RR_0^\xi(M,-)$ we have the following Lemma which is very similar to Lemma \[lem:P-xi-0\].

\[lem:RR-xi-prime\] Let ${\ensuremath{\mathfrak{c}}}\subset{\ensuremath{\mathfrak{t}}}^*$ be a connected component of regular values of $\Phi$. For every $\xi,\xi'\in{\ensuremath{\mathfrak{c}}}$, we have $RR_0^\xi(M,-)=RR_0^{\xi'}(M,-)$.

[*Proof.*]{} We have to show that the map $\xi\mapsto RR_0^\xi(M,-)$ is locally constant on ${\ensuremath{\mathfrak{c}}}$. Let $\xi\in{\ensuremath{\mathfrak{c}}}$ and take an open neigborhood ${\ensuremath{\mathcal{U}}}$ of $\Phi^{-1}(\xi)$ small enough such that the stabilizer $T_m=\{t\in T\,\vert\, t\cdot m=m\}$ is finite for every $m\in\overline{{\ensuremath{\mathcal{U}}}}$. We see then that $\overline{{\ensuremath{\mathcal{U}}}}\cap{\ensuremath{\hbox{\rm Cr}}}(\parallel \Phi-\xi'\parallel^2)=
\Phi^{-1}(\xi')$ and $\partial{\ensuremath{\mathcal{U}}}\cap{\ensuremath{\hbox{\rm Cr}}}(\parallel \Phi-\xi'\parallel^2)=
\emptyset$ if $\xi'$ is close enough to $\xi$: hence $RR_0^{\xi'}(M,-)=RR_{\ensuremath{\mathcal{U}}}^{\xi'}(M,-)$ for $\xi'$ close enough to $\xi$. Part $b)$ of Proposition \[prop:U-xi\] finishes the proof. $\Box$

When $\xi$ is a regular value of $\Phi$, the localized Riemann-Roch character $RR_0^\xi(M,-)$ as been computed in [@pep4] as follows. Let $RR({\ensuremath{\mathcal{M}}}_\xi,-)$ be the Riemann-Roch map defined on the orbifold ${\ensuremath{\mathcal{M}}}_\xi=\Phi^{-1}(\xi)/T$ by means of an almost complex structure compatible with the induced symplectic structure. For every $T$-vector bundle $E\to M$ we define the following familly of orbifold vector bundles over ${\ensuremath{\mathcal{M}}}_\xi$: $$\label{eq:ecal-mu}
  {\ensuremath{\mathcal{E}}}_{\xi,\mu}:=\Big(E\vert_{\Phi^{-1}(\xi)}\otimes{\ensuremath{\mathbb{C}}}_{-\mu}\Big)/T,
  \quad \mu\in\Lambda^*.$$ For every $T$-vector bundle $E$ on $M$, we proved in [@pep4]\[Section 6.2.\] the following equality in $R^{-\infty}(T)$ $$\label{eq:RR-xi-O}
 RR_0^\xi(M,E)=\sum_{\mu\in \Lambda^*}RR({\ensuremath{\mathcal{M}}}_\xi,{\ensuremath{\mathcal{E}}}_{\xi,\mu})\,{\ensuremath{\mathbb{C}}}_\mu.$$ This decomposition was first obtained by Vergne [@Vergne96] when $T$ is the circle group and when $M$ is Spin. The number $RR({\ensuremath{\mathcal{M}}}_\xi, {\ensuremath{\mathcal{E}}}_{\xi,\mu})\in {\ensuremath{\mathbb{Z}}}$ is then equal to the $T$-invariant part of the index $RR_0^\xi(M,E)\otimes{\ensuremath{\mathbb{C}}}_{-\mu}$.

\[lem:discrete\] Let $t\to t^\lambda$ be a character of $T$. Suppose that a subgroup $H\subset T$ acts trivially on $M$ and with the character $t\in H\to t^\lambda$ on the the fibers of the $T$-vector bundle $E$. Then $H$ acts with the character $t\in H\to t^{\lambda-\mu}$ on $RR_0^\xi(M,E)\otimes{\ensuremath{\mathbb{C}}}_{-\mu}$, and then $RR({\ensuremath{\mathcal{M}}}_\xi, {\ensuremath{\mathcal{E}}}_{\xi,\mu})\neq 0$ only if $t^{\lambda-\mu}=1$ for every $t\in H$. So the sum in (\[eq:RR-xi-O\]) can be restricted to $\lambda + \Lambda^*_H$, where $\Lambda^*_H$ is the sub-lattice of $\Lambda^*$ formed by the element $\alpha\in\Lambda^*$ satisfying $t^\alpha=1,\ \forall\ t\in H$.

This remark applies also on the usual character $RR(M,E)=\sum_{\mu\in\Lambda^*}m_\mu{\ensuremath{\mathbb{C}}}_\mu$. The mutiplicity $m_\mu\in{\ensuremath{\mathbb{Z}}}$ is equal to the (virtual) dimension of the $T$-invariant part of $RR(M,E)\otimes{\ensuremath{\mathbb{C}}}_{-\mu}$. With the same hypothesis than above we see that $m_\mu\neq 0$ only if $\mu\in \lambda + \Lambda^*_H$.

Let $\Gamma_M$ be the generic stabilizer for the action of $T$ on $M$. Consider a weight $\alpha_{o}$ such that $\Gamma_M$ acts on the fibers of $L$ with the character $t\mapsto t^{\alpha_{o}}$. We define the sublattice $\Xi(M,L)\subset \Lambda^*\times {\ensuremath{\mathbb{Z}}}$ by $$\label{eq:xi-lattice}
 \Xi(M,L):=\{(\mu,k)\in \Lambda^*\times {\ensuremath{\mathbb{Z}}}\ \vert\ k\alpha_{o}-\mu\in
\Lambda^*_{\Gamma_M}\}.$$

We know then that ${\ensuremath{\hbox{\rm m}}}(\mu,k)=0$ if $(\mu,k)\notin \Xi(M,L)$.

\[prop:m-poly\] Let ${\ensuremath{\mathfrak{c}}}$ be a connected component of regular values of $\Phi$ and let ${\ensuremath{\rm Cone}}({\ensuremath{\mathfrak{c}}})$ be the corresponding cone in ${\ensuremath{\mathfrak{t}}}^*\times{\ensuremath{\mathbb{R}}}^{> 0}$ (see (\[eq:cone-c\])). Let $\xi\in{\ensuremath{\mathfrak{c}}}$. For any $(\mu,k)\in{\ensuremath{\rm Cone}}({\ensuremath{\mathfrak{c}}})
\cap\Xi(M,L)$ we have ${\ensuremath{\hbox{\rm m}}}(\mu,k)=RR({\ensuremath{\mathcal{M}}}_\xi, {\ensuremath{\mathcal{L}}}^k_{\xi,\mu})$ where $$\label{eq:line-k-xi-mu}
 {\ensuremath{\mathcal{L}}}^k_{\xi,\mu}=
(L^{\otimes k}\vert_{\Phi^{-1}(\xi)}\otimes{\ensuremath{\mathbb{C}}}_{-\mu})/T.$$

[*Proof.*]{} Let $(\mu,k)\in{\ensuremath{\rm Cone}}({\ensuremath{\mathfrak{c}}})$ and let $\xi'=\frac{\mu}{k}\in
{\ensuremath{\mathfrak{c}}}$. We known from Theorem \[th-RR-xi\] that the generalized character $RR_0^{\xi'}(M,L^{\otimes k})$ coincides with $RR(M,L^{\otimes k})$ on the open ball $k\cdot B(\xi',r_{\xi'})=
B(\mu,k r_{\xi'})$. So ${\ensuremath{\hbox{\rm m}}}(\mu,k)$ is equal to the $\mu$-multiplicity in $RR_0^{\xi'}(M,L^{\otimes k})$. Take now any $\xi\in{\ensuremath{\mathfrak{c}}}$. We know after Lemma \[lem:RR-xi-prime\] that $RR_0^\xi(M,-)=RR_0^{\xi'}(M,-)$ and (\[eq:RR-xi-O\]) shows that the $\mu$-multiplicity in $RR_0^{\xi}(M,L^{\otimes k})$ is equal to $RR({\ensuremath{\mathcal{M}}}_\xi, {\ensuremath{\mathcal{L}}}^k_{\xi,\mu})$. $\Box$

Take $\xi\in{\ensuremath{\mathfrak{c}}}$. The map ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}}:\Lambda^*\times{\ensuremath{\mathbb{Z}}}\to {\ensuremath{\mathbb{Z}}}$ is defined by the equation $$\label{eq:def:m-c}
    {\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}}(\mu,k)=RR({\ensuremath{\mathcal{M}}}_\xi, {\ensuremath{\mathcal{L}}}^k_{\xi,\mu}),$$ where ${\ensuremath{\mathcal{L}}}^k_{\xi,\mu}$ is the orbifold line bundle defined by (\[eq:line-k-xi-mu\]). In other words, the map ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}}$ is defined by the following equality in $R^{-\infty}(T)$ $$\sum_{\mu\in\Lambda^*}{\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}}(\mu,k)\ {\ensuremath{\mathbb{C}}}_\mu=RR_0^\xi(M,L^{\otimes k}).$$ for all $k\in{\ensuremath{\mathbb{Z}}}$. After remark \[lem:discrete\], we know that ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}}$ is supported on the sublattice $\Xi(M,L)$ defined in (\[eq:xi-lattice\]).

We will now exploit the Riemann-Roch for orbifold due to Atiyah-Kawasaki [@Atiyah74; @Kawasaki79] to show that the map ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}}$ is a periodic polynomial.

Riemann-Roch theorem on orbifolds {#subsec:RR-theo}
---------------------------------

First we recall how is defined the Riemann-Roch character $RR({\ensuremath{\mathcal{M}}}_\xi,{\ensuremath{\mathcal{E}}}_\xi)$ when $\xi$ is a regular value of $\Phi$, and ${\ensuremath{\mathcal{E}}}_\xi= E_{\vert\Phi^{-1}(\xi)}/T$ is the reduction of a complex $T$-vector bundle $E$ over $M$. The number $RR({\ensuremath{\mathcal{M}}}_\xi,{\ensuremath{\mathcal{E}}}_\xi)\in{\ensuremath{\mathbb{Z}}}$ is defined has the $T$-invariant part of the index of a transversally elliptic operator $D_E$ on $\Phi^{-1}(\xi)$. Since the index of $D_E$ depend only of the class of its symbol $\sigma(D_E)$ in $K_T({\ensuremath{\hbox{\bf T}}}_T\Phi^{-1}(\xi))$, it is enough to define the transversally elliptic symbol $\sigma(D_E)$. Since the action of $T$ on $\Phi^{-1}(\xi)$ is locally free, $V:={\ensuremath{\hbox{\bf T}}}_T\Phi^{-1}(\xi)$ is a vector bundle. It carries a canonical symplectic structure on the fibers and we choose any compatible complex structure making $V$ into a Hermitian vector bundle. At $(m,v)\in{\ensuremath{\hbox{\bf T}}}\Phi^{-1}(\xi)$, the map $\sigma(D_E)(m,v)$ is the Clifford action $$\Clif_{m}(v_1)\otimes Id_{E_m}\ :\ (\wedge_{{\ensuremath{\mathbb{C}}}}^{even} V_m)\otimes E_m
\longrightarrow (\wedge_{{\ensuremath{\mathbb{C}}}}^{odd} V_m)\otimes E_m.$$ where $v_1\in V_m$ is the $V$-component of the vector $v\in{\ensuremath{\hbox{\bf T}}}_m\Phi^{-1}(\xi)$. We explain now the formula of Atiyah-Kawasaki for $RR({\ensuremath{\mathcal{M}}}_\xi,{\ensuremath{\mathcal{E}}}_\xi)$ when $\xi\in\Phi(M)$ is a regular value of $\Phi$ [@Atiyah74; @Kawasaki79].

Let ${\ensuremath{\mathbb{F}}}$ be the collection of the finite subgroup of $T$ which are stabilizer of points in $M$. Consider the orbit type stratification of $\Phi^{-1}(\xi)$ and denote by ${\ensuremath{\mathbb{S}}}_\xi$ the set of its orbit type strata. Each statum $S$ is a connected component of the smooth submanifold $$\label{eq:stab-H}
  \Phi^{-1}(\xi)_{H_S}:= \{m\in\Phi^{-1}(\xi)\,\vert\, {\ensuremath{\hbox{\rm Stab}}}_T(m)=H_S\}.$$ for a unique $H_S\in{\ensuremath{\mathbb{F}}}$. The orbifold ${\ensuremath{\mathcal{M}}}_\xi$ decomposes as a disjoint union $\cup_{S\in{\ensuremath{\mathbb{S}}}_\xi}S/T$ of smooth components, and each quotient $\overline{S}/T$ is a suborbifold of ${\ensuremath{\mathcal{M}}}_\xi$. The generic stabilizer $\Gamma_M$ of $T$ on $M$ is also the generic stabilizer of $T$ on the fiber[^2] $\Phi^{-1}(\xi)$, and is associated to an open and dense stratum $S_{max}$.

Suppose that $E\to M$ is an Hermitian $T$-vector bundle. On each suborbifold $\overline{S}/T$, we get the orbifold complex vector bundle $$\label{eq:bundle-E-H}
  {\ensuremath{\mathcal{E}}}_S:=E_{\vert\overline{S}}/T.$$ We define twisted characteristic classes ${\ensuremath{\hbox{\rm Ch}}}^-({\ensuremath{\mathcal{E}}}_S)$ and $D^-({\ensuremath{\mathcal{E}}}_S)$ by $$\label{eq:chern}
  {\ensuremath{\hbox{\rm Ch}}}^\gamma({\ensuremath{\mathcal{E}}}_S):={\ensuremath{\hbox{\bf Tr}}}\left(\gamma^{{\ensuremath{\mathcal{E}}}_S}\cdot e^{\frac{i}{2\pi}R({\ensuremath{\mathcal{E}}}_S)}\right),\quad \gamma\in H,$$ and $$\label{eq:D-class}
  D^\gamma({\ensuremath{\mathcal{E}}}_S):=\det 
\left(1-(\gamma^{{\ensuremath{\mathcal{E}}}_S})^{-1}\cdot e^{-\frac{i}{2\pi}R({\ensuremath{\mathcal{E}}}_S)}\right),
\quad \gamma\in H.$$ Here $R({\ensuremath{\mathcal{E}}}_S)\in{\ensuremath{\mathcal{A}}}^2(\overline{S}/T,{\ensuremath{\hbox{\rm End}}}({\ensuremath{\mathcal{E}}}_S))$ is the curvature of an horizontal Hermitian connection on $E_{\vert\overline{S}}$, and $\gamma\mapsto
\gamma^{{\ensuremath{\mathcal{E}}}_S}$ is the linear action of $H_S$ on the fibers of $E_{\vert\overline{S}}$.

Let $N_S$ be the normal bundle of $\overline{S}$ in $\Phi^{-1}(\xi)$. The symplectic struture on $M$ induces a symplectic form $\Omega_S$ on each suborbifold $\overline{S}/T$, and a symplectic structure on the fibers of the bundle $N_S$. Choose a compatible almost complex structure on $\overline{S}/T$, and a compatible complex structure on the fibers of $N_S$ making the tangent bundle of $\overline{S}/T$ and ${\ensuremath{\mathcal{N}}}_S:=N_S/T$ into Hermitian vector bundle. Consider a Hermitian connexion on ${\ensuremath{\hbox{\bf T}}}(\overline{S}/T)$, with curvature $R(\overline{S}/T)$, and let $$\label{eq:todd-form}
  {\ensuremath{\hbox{\rm Td}}}(\overline{S}/T)=\det\left(\frac{(i/2\pi)R(\overline{S}/T)}
{1-e^{-(i/2\pi)R(\overline{S}/T)}}\right)$$ be the corresponding Todd forms. Like in (\[eq:D-class\]), we associate to the complex orbifold vector bundle ${\ensuremath{\mathcal{N}}}_S$, the twisted form $D^-({\ensuremath{\mathcal{N}}}_S)$ which is a map form $H_S$ to ${\ensuremath{\mathcal{A}}}^{even}(\overline{S}/T)$. The $0$-degree part of $D^\gamma({\ensuremath{\mathcal{N}}}_S)$ is equal to $\det(1-(\gamma^{{\ensuremath{\mathcal{N}}}_S})^{-1})$, hence $D^\gamma({\ensuremath{\mathcal{N}}}_S)$ is invertible in ${\ensuremath{\mathcal{A}}}^{even}(\overline{S}/T)$ when $\gamma$ belongs to $$\label{eq:H-S-o}
  H_S^o=\{\gamma\in H_S\ \vert\ \det(1-(\gamma^{{\ensuremath{\mathcal{N}}}_S})^{-1})\neq 0\}.$$ Note that $H_S^o$ corresponds to the set of $\gamma\in H_S$ for which $\overline{S}$ is a connected component of $(\Phi^{-1}(\xi))^\gamma$.

\[th:RR-AK\] The number $RR({\ensuremath{\mathcal{M}}}_\xi,{\ensuremath{\mathcal{E}}}_\xi)\in{\ensuremath{\mathbb{Z}}}$ is given by the formula $$\label{eq:RR-AK}
 RR({\ensuremath{\mathcal{M}}}_\xi,{\ensuremath{\mathcal{E}}}_\xi) = \sum_{S\in{\ensuremath{\mathbb{S}}}_\xi}\frac{1}
{\vert H_S\vert}\sum_{\gamma\in H_S^o}\int_{\overline{S}/T}
\frac{{\ensuremath{\hbox{\rm Td}}}(\overline{S}/T){\ensuremath{\hbox{\rm Ch}}}^\gamma({\ensuremath{\mathcal{E}}}_S)}
{D^\gamma({\ensuremath{\mathcal{N}}}_S)}.$$

We exploit now Theorem \[th:RR-AK\] to show that the map ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}}:\Lambda^*\times{\ensuremath{\mathbb{Z}}}\to {\ensuremath{\mathbb{Z}}}$ which is defined by (\[eq:def:m-c\]) is periodic polynomial. We need the classical computation of the first Chern class of the line bundle $$\label{eq:L-S-mu}
  {\ensuremath{\mathcal{L}}}^k_{S,\mu}=(L^{\otimes k}\otimes{\ensuremath{\mathbb{C}}}_{-\mu})_{\vert\overline{S}}/T.$$ The curvature form $\omega_\xi\in H^2({\ensuremath{\mathcal{M}}}_\xi)\otimes{\ensuremath{\mathfrak{t}}}$ of the principal $T$-bundle $\Phi^{-1}(\xi)\to{\ensuremath{\mathcal{M}}}_\xi$ restricts to a curvature form $\omega_S\in H^2(\overline{S}/T)\otimes{\ensuremath{\mathfrak{t}}}$ on each strata.

\[lem:c-1-L\] The first Chern class of the line bundle ${\ensuremath{\mathcal{L}}}^k_{S,\mu}$ is given by $$c_1({\ensuremath{\mathcal{L}}}^k_{S,\mu})=\frac{1}{2\pi}
\Big(k\Omega_S -\langle k\xi-\mu,\omega_S\rangle\Big).$$

For a strata $S$, we consider $\alpha_S\in\Lambda^*$ such that $\gamma\in H_S\mapsto \gamma^{\alpha_S}$ corresponds to the action of $H_S$ on the fibers of $L_{\vert\overline{S}}$. Finally we have the decomposition $$\label{eq:m-c-decompose}
  {\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}}(\mu,k)=\sum_{S\in{\ensuremath{\mathbb{S}}}_\xi} P_S(\mu,k),$$ where $$\label{eq:Pol-S}
P_S(\mu,k)=\frac{1}{\vert H_S\vert}
\sum_{\gamma\in H_S^o}\gamma^{k\alpha_S-\mu}\int_{\overline{S}/T}
\frac{{\ensuremath{\hbox{\rm Td}}}(\overline{S}/T)}{D^\gamma({\ensuremath{\mathcal{N}}}_S)}\ 
e^{\frac{1}{2\pi}\Big(k\Omega_S -\langle k\xi-\mu,\omega_S\rangle\Big)}.$$ When $S$ is the principal open dense stratum $S_{max}$ the map $P_S$ is $$\label{eq:Pol-S-max}
P_{max}(\mu,k)=\frac{\sum_{\gamma\in \Gamma_M}
\gamma^{k\alpha_{o}-\mu}}{\vert \Gamma_M\vert}
\int_{{\ensuremath{\mathcal{M}}}_\xi}{\ensuremath{\hbox{\rm Td}}}({\ensuremath{\mathcal{M}}}_\xi)
e^{\frac{1}{2\pi}(k\Omega_\xi -\langle k\xi-\mu,\omega_\xi\rangle)}.$$ The term $\frac{\sum_{\gamma\in \Gamma_M}
\gamma^{k\alpha_{o}-\mu}}{\vert \Gamma_M\vert}$ is equal to $1$ when $(\mu,k)$ belongs to the lattice $\Xi(M,L)$ (see (\[eq:xi-lattice\])), and is equal to $0$ in the other cases. From (\[eq:Pol-S\]) we see that $P_S$ is a periodic polynomial of degree less than $\frac{dim(\overline{S}/T)}{2}$, and for $S=S_{max}$ we have on $\Xi(M,L)$ $$\label{eq:Pol-S-bis}
P_{max}(\mu,k)=\frac{1}{(2\pi)^l}\int_{{\ensuremath{\mathcal{M}}}_\xi}
\frac{(k\Omega_\xi -\langle k\xi-\mu,\omega_\xi\rangle)^l}{l !} +O(l-1)$$ where $l=\frac{dim{\ensuremath{\mathcal{M}}}_\xi}{2}$ and $O(l-1)$ denotes a polynomial of degree less than $l-1$. If we use the polynomial ${\ensuremath{\rm DH}}_{{\ensuremath{\mathfrak{c}}}}$ defined in Section \[DH-mesures\] we can conclude our computations with the following

\[prop:m-c-poly\] The map ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}}$ is a periodic polynomial of degree $l=\frac{dim{\ensuremath{\mathcal{M}}}_\xi}{2}$ supported on $\Xi(M,L)$. For $(\mu,k)\in\Xi(M,L)$ we have $${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}}(\mu,k)=\vert \Gamma_M\vert\frac{k^l}{(2\pi)^l}{{\ensuremath{\rm DH}}_{{\ensuremath{\mathfrak{c}}}}}
(\frac{\mu}{k}) \ +\ O(l-1),$$ where $O(l-1)$ means a periodic polynomial of degree less than $l-1$.

If ${\ensuremath{\mathfrak{c}}}_\pm$ are two adjacent connected components of regular values of $\Phi$, we know after Theorem \[DH-theo-principal\] that ${\ensuremath{\rm DH}}_{{\ensuremath{\mathfrak{c}}}_+}\neq{\ensuremath{\rm DH}}_{{\ensuremath{\mathfrak{c}}}_-}$ so we can conclude from Proposition \[prop:m-c-poly\] that $$\label{eq:mm-plus-moins}
 {\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_+}\neq{\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_-}.$$

Jump formulas for the ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}}$ {#subsec:jump-formula}
---------------------------------------------------------------------------------

Let ${\ensuremath{\mathfrak{c}}}_+$ and ${\ensuremath{\mathfrak{c}}}_-$ be two adjacent connected component of regular values of $\Phi$ separated by an hyperplane ${\ensuremath{\Delta}}$. The aim of this section is to compute the periodic polynomial ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_+}-{\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_-}$.

Let $\beta\in\Lambda$ be the unimodular orthogonal vector to the hyperplane ${\ensuremath{\Delta}}\subset{\ensuremath{\mathfrak{t}}}^*$ which is pointing out of ${\ensuremath{\mathfrak{c}}}_-$. Let $T_{{\ensuremath{\Delta}}}\subset T$ be the subtorus of dimension $1$, with Lie algebra ${\ensuremath{\mathfrak{t}}}_{{\ensuremath{\Delta}}}:=\{X\in{\ensuremath{\mathfrak{t}}}\,\vert\, \langle \xi-\xi',X\rangle=0,\ 
\forall \xi,\xi'\in{\ensuremath{\Delta}}\}$. We make the choice of a decomposition $T=T_{\ensuremath{\Delta}}\times T/T_{\ensuremath{\Delta}}$, where $T/T_{\ensuremath{\Delta}}$ denotes a subtorus de $T$.

Let ${\ensuremath{\mathfrak{c}}}'$ be the relative interior of $\overline{{\ensuremath{\mathfrak{c}}}_+}\cap\overline{{\ensuremath{\mathfrak{c}}}_-}$ in ${\ensuremath{\Delta}}$. Following Definition \[def:M-delta-quantum\] we denote by $M_{{\ensuremath{\mathfrak{c}}}'}$ the union of the connected component $Z$ of the fixed point set $M^{T_{{\ensuremath{\Delta}}}}$ for which $\Phi(Z)$ contains ${\ensuremath{\mathfrak{c}}}'$. The symplectic submanifold $M_{{\ensuremath{\mathfrak{c}}}'}$ carries an Hamiltonian action of $T/T_{{\ensuremath{\Delta}}}$ with moment map $\Phi_{{\ensuremath{\mathfrak{c}}}'}$ equal to the restriction of $\Phi$ on $M_{{\ensuremath{\mathfrak{c}}}'}$.

We consider two points $\xi_\pm\in {\ensuremath{\mathfrak{c}}}_\pm$ such that $\xi=
\frac{1}{2}(\xi^+ + \xi^-)\in{\ensuremath{\mathfrak{c}}}'$. We suppose furthermore that $\xi^+ -\xi^-$ is orthogonal to ${\ensuremath{\Delta}}$. Using the identification ${\ensuremath{\mathfrak{t}}}^*\simeq{\ensuremath{\mathfrak{t}}}$ given by the scalar product the vector $\gamma=\frac{1}{2}(\xi_+-\xi_-)$, seen as a vector of ${\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}$, belongs to ${\ensuremath{\mathbb{R}}}^{>0}\beta$. We noticed in Section \[subsec:Jump-dh\] that for all $m\in\Phi^{-1}(\xi)$ the stabilizer ${\ensuremath{\mathfrak{t}}}_m$ is either equal to ${\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}$ or to $\{0\}$. Then $\xi$ is a regular value of $\Phi_{{\ensuremath{\mathfrak{c}}}'}$ and there exists an open $T$-invariant neighborhood ${\ensuremath{\mathcal{U}}}$ of $\Phi^{-1}(\xi)$ in $M$ such that for all $m\in\overline{{\ensuremath{\mathcal{U}}}}$ either ${\ensuremath{\mathfrak{t}}}_m:=\{0\}$, or ${\ensuremath{\mathfrak{t}}}_m={\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}$ and $\Phi(m)\in{\ensuremath{\Delta}}$.

One see easily that the couple $({\ensuremath{\mathcal{U}}},\xi)$ is good and part $b)$ of Proposition \[prop:U-xi\] tells us that $$\label{eq:RR-U-pm}
 RR^\xi_{\ensuremath{\mathcal{U}}}(M,-)=RR^{\xi_-}_{\ensuremath{\mathcal{U}}}(M,-)=RR^{\xi_+}_{\ensuremath{\mathcal{U}}}(M,-)$$ when $\xi_\pm$ are close enough to $\xi$. Since ${\ensuremath{\mathcal{U}}}\cap{\ensuremath{\hbox{\rm Cr}}}(\parallel\Phi-\xi\parallel^2)=\Phi^{-1}(\xi)$ we have $RR^\xi_{\ensuremath{\mathcal{U}}}(M,-)=RR^\xi_0(M,-)$. If $\xi_\pm$ are close enough to $\xi$ we have $$\label{eq:U-cap-xi-pm}
 {\ensuremath{\mathcal{U}}}\cap{\ensuremath{\hbox{\rm Cr}}}(\parallel\Phi-\xi_\pm\parallel^2)=
\Phi^{-1}(\xi_\pm)\bigcup M^{\gamma}\cap\Phi^{-1}(\xi).$$ The former decomposition is due to (\[eq:decom-kirwan\]) and to the fact that the stabiliser of ${\ensuremath{\mathfrak{t}}}$ on ${\ensuremath{\mathcal{U}}}$ are either equal to ${\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}$ or to $\{0\}$. Notice that $\xi_-
+\gamma=\xi_+ +\gamma=\xi$. The decomposition (\[eq:U-cap-xi-pm\]) gives $$\label{eq:RR-U-mp}
RR_{{\ensuremath{\mathcal{U}}}}^{\xi_\pm}(M,-)=RR_{0}^{\xi_\pm}(M,-)+ RR_{\mp\gamma}^{\xi_\pm}(M,-),$$ where $RR_{\gamma}^{\xi_-}(M,-)$ (resp. $RR_{-\gamma}^{\xi_+}(M,-)$) is the Riemann-Roch character localized on $M^\gamma\cap\Phi^{-1}(\xi)$ by the vectors field ${\ensuremath{\mathcal{H}}}- (\xi_-)_M$ (resp. ${\ensuremath{\mathcal{H}}}- (\xi_+)_M$). Now (\[eq:RR-U-pm\]) and (\[eq:RR-U-mp\]) prove the following

\[RR-0-diff\] If $\xi_\pm$ are close enough to ${\ensuremath{\Delta}}$, we have $$RR_0^{\xi_+}(M,-)-RR_0^{\xi_-}(M,-)=
RR_{\gamma}^{\xi_-}(M,-)-RR_{-\gamma}^{\xi_+}(M,-).$$

We know from Proposition \[prop:m-poly\] that ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_\pm}(\mu,k)$ is equal to the $\mu$-mutiplicity of $RR_0^{\xi_\pm}(M,L^{\otimes k})$. Hence ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_+}(\mu,k)-{\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_-}(\mu,k)$ is equal to the $\mu$-mutiplicity of $RR_{\gamma}^{\xi_-}(M,L^{\otimes k})-RR_{-\gamma}^{\xi_+}(M,L^{\otimes k})$.

Let $N_{{\ensuremath{\Delta}}}$ be the normal bundle of $M^{T_{\ensuremath{\Delta}}}$ in $M$, and let $\left[\wedge_{{\ensuremath{\mathbb{C}}}}^{\bullet}\overline{N_{{\ensuremath{\Delta}}}}\right]^{-1}_{\pm\beta}$ be the polarized inverses of $\wedge_{{\ensuremath{\mathbb{C}}}}^{\bullet}\overline{N_{{\ensuremath{\Delta}}}}$ (see Proposition \[def:lambda-N\]). Since $\xi=\xi_+ -\gamma=
\xi_- +\gamma$ and $\gamma\in {\ensuremath{\mathbb{R}}}^{>0}\beta$, the localization (\[eq:loc-RR-beta-bis\]) gives $$\label{eq:loc-xi-1}
RR_{\gamma}^{\xi_-}(M,E)= \sum_{Z}
RR_0^{\xi}\left(Z,E\vert_{Z}\otimes
\left[\wedge_{{\ensuremath{\mathbb{C}}}}^{\bullet}\overline{N_Z}\right]^{-1}_{\beta}\right)$$ and $$\label{eq:loc-xi-2}
  RR_{-\gamma}^{\xi_+}(M,E)= \sum_{Z}
RR_0^{\xi}\left(Z,E\vert_{Z}\otimes
\left[\wedge_{{\ensuremath{\mathbb{C}}}}^{\bullet}\overline{N_Z}\right]^{-1}_{-\beta}\right).$$

In (\[eq:loc-xi-1\]) and (\[eq:loc-xi-2\]) the sum are taken over the connected components $Z$ of $M_{{\ensuremath{\mathfrak{c}}}'}\subset M^{T_{\ensuremath{\Delta}}}$ and we denote $N_Z$ the restriction of the bundle $N_{{\ensuremath{\Delta}}}$ to $Z$. Let us make few remarks concerning the maps $RR_0^{\xi}(Z,-):{\ensuremath{\hbox{\bf K}}}_T(Z)\to R^{-\infty}(T)$. Since $T_{\ensuremath{\Delta}}$ acts trivially on $Z$, the decomposition $T=T/T_{\ensuremath{\Delta}}\times T_{\ensuremath{\Delta}}$ induces a canonical isomorphism ${\ensuremath{\hbox{\bf K}}}_T(Z)\simeq{\ensuremath{\hbox{\bf K}}}_{T/T_{\ensuremath{\Delta}}}(Z)\otimes R(T_{\ensuremath{\Delta}})$: i.e. every $T$-equivariant vector bundle $E\to Z$ decomposes as $$\label{eq:t-p-decompose}
 E= \sum_{\alpha\in\Lambda^*_{{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}}}E^\alpha\otimes{\ensuremath{\mathbb{C}}}_\alpha.$$ Here $\Lambda^*_{{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}}=\Lambda^*\cap{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}^*$ is the set of weights for the subtorus $T_{\ensuremath{\Delta}}$, each $E^\alpha$ is a $T/T_{\ensuremath{\Delta}}$-equivariant vector bundle on $Z$ and ${\ensuremath{\mathbb{C}}}_\alpha$ denotes the one dimensional $T_{\ensuremath{\Delta}}$-representation associated to $\alpha\in\Lambda^*_{{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}}$.

For every $T$-equivariant vector bundle $E\to Z$, the character $RR_0^{\xi}(Z,E)$ is equal to the $T$-equivariant index of the $T$-transversally elliptic symbol ${\ensuremath{\hbox{\rm Thom}}}_{\xi}({\ensuremath{\mathcal{V}}})\otimes
p^*(E)$, where ${\ensuremath{\mathcal{V}}}$ is a small neighborhood of $\Phi^{-1}(\xi)\cap
Z$ in $Z$ (see Definition \[def:RR-U-xi\]). Since the $T_{\ensuremath{\Delta}}$ action is trivial on $Z$ the symbol ${\ensuremath{\hbox{\rm Thom}}}_{\xi}({\ensuremath{\mathcal{V}}})$ is also $T/T_{\ensuremath{\Delta}}$ transversally elliptic and the action of $T_{\ensuremath{\Delta}}$ is trivial on it. We have then $$\label{eq:rr-xi-o}
RR_0^{\xi}(Z,E)=\sum_{\alpha\in\Lambda^*_{{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}}}
RR_0^{\xi}(Z,E^\alpha)\otimes{\ensuremath{\mathbb{C}}}_\alpha$$ where $RR_0^{\xi}(Z,E^\alpha)$ belongs to $R^{-\infty}(T/T_{\ensuremath{\Delta}})$. Since $\xi$ is a regular value of $\Phi_{{\ensuremath{\mathfrak{c}}}'}$, the character $RR_0^{\xi}(Z,E^\alpha)$ is computed by Theorem \[eq:RR-xi-O\] applied to the Hamiltonian $T/T_{\ensuremath{\Delta}}$-manifold $Z$. For every $T$-vector bundle $E\to M$ we define the familly ${\ensuremath{\mathcal{E}}}^{\mu_1}_{\xi,\mu_2}$ of orbifold vector bundles over the reduced space ${\ensuremath{\mathcal{Z}}}_\xi=
Z\cap \Phi^{-1}(\xi)/(T/T_{\ensuremath{\Delta}})$ by $$\label{eq:E-xi-mu}
  {\ensuremath{\mathcal{E}}}^{\mu_1}_{\xi,\mu_2}:=
\left(E^{\mu_1}\otimes{\ensuremath{\mathbb{C}}}_{-\mu_2}\right)\vert_{\Phi^{-1}(\xi)\cap Z}/(T/T_{\ensuremath{\Delta}}).$$ Here $\mu_1\in\Lambda^*_{{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}}$ and $\mu_2\in\Lambda^*_{{\ensuremath{\mathfrak{t}}}/{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}}=
\Lambda^*\cap({\ensuremath{\mathfrak{t}}}/{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}})^*$. Finally Theorem \[eq:RR-xi-O\] and (\[eq:rr-xi-o\]) give the following $$\begin{aligned}
  \label{eq:indice-t-p}
  RR_0^\xi(Z,E)&=&
\sum_{\mu_1\in\Lambda^*_{{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}}}\sum_{\mu_2\in\Lambda^*_{{\ensuremath{\mathfrak{t}}}/{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}}}
RR({\ensuremath{\mathcal{Z}}}_\xi, {\ensuremath{\mathcal{E}}}^{\mu_1}_{\xi,\mu_2})\ \otimes 
\underbrace{{\ensuremath{\mathbb{C}}}_{\mu_1}}_{\in \, R(T_{\ensuremath{\Delta}})} \otimes
\underbrace{{\ensuremath{\mathbb{C}}}_{\mu_2}}_{\in \, R(T/T_{\ensuremath{\Delta}})} 
\nonumber \\
               &=&\sum_{\mu\in\Lambda^*}
RR({\ensuremath{\mathcal{Z}}}_\xi, {\ensuremath{\mathcal{E}}}^{\mu_1}_{\xi,\mu_2})\ {\ensuremath{\mathbb{C}}}_{\mu} \quad .\end{aligned}$$

In (\[eq:indice-t-p\]) we write $\mu\in\Lambda^*$ as a sum of $\mu_1\in\Lambda^*_{{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}}$ with $\mu_2\in\Lambda^*_{{\ensuremath{\mathfrak{t}}}/{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}}$ so that ${\ensuremath{\mathbb{C}}}_{\mu}\in R(T)$ is equal to the tensor product ${\ensuremath{\mathbb{C}}}_{\mu_1}\otimes{\ensuremath{\mathbb{C}}}_{\mu_1}$.

Now we finish the computation of the periodic polynomial ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_+}(\mu,k)-{\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_-}(\mu,k)$. The hyperplane ${\ensuremath{\Delta}}$ is defined by the equation $\frac{\langle\xi,\beta\rangle}{2\pi} = 
r_{\ensuremath{\Delta}},\, \xi\in{\ensuremath{\mathfrak{t}}}^*$, for some $r_{\ensuremath{\Delta}}\in{\ensuremath{\mathbb{Z}}}$. We have the decomposition $$\label{eq:diff-decompose}
 \left[\wedge_{{\ensuremath{\mathbb{C}}}}^{\bullet}\overline{N_Z}\right]^{-1}_{\beta}-
\left[\wedge_{{\ensuremath{\mathbb{C}}}}^{\bullet}\overline{N_Z}\right]^{-1}_{-\beta}=
\sum_{\alpha\in\Lambda^*_{{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}}} (-1)^{n_Z(\alpha)}\ 
S_Z^\alpha\otimes{\ensuremath{\mathbb{C}}}_\alpha$$ where $S_Z^\alpha\to Z$ is a complex $T/T_{\ensuremath{\Delta}}$-vector bundle, and the integers $n_Z(\alpha)$ are defined by the the following relations $$\begin{aligned}
  \label{eq:n-Z-alpha}
 n_Z(\alpha)&=& {\rm rk}_{\ensuremath{\mathbb{C}}}( N_Z^{+,\beta})\quad  if\quad  
\langle\alpha,\beta\rangle\geq 0\nonumber\\
 n_Z(\alpha)&=& {\rm rk}_{\ensuremath{\mathbb{C}}}( N_Z^{+,-\beta}) +1 \quad if \quad 
\langle\alpha,\beta\rangle< 0.\end{aligned}$$

Let $\beta^*$ be the vector of ${\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}^*$ which is defined by the relation $\langle\beta^*,\beta\rangle=2\pi$, so that $\Lambda^*_{{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}}={\ensuremath{\mathbb{Z}}}\beta^*$. The $T_{\ensuremath{\Delta}}$-weight on $\det(N_Z^{+,\pm\gamma})$ is $\pm s_Z^\pm\beta^*$ where $s_Z^\pm\in{\ensuremath{\mathbb{N}}}$ is the absolute value of the trace of $\frac{1}{2\pi}{\ensuremath{\mathcal{L}}}_\beta$ on $N_Z^{+,\pm\gamma}$. Since the $T_{\ensuremath{\Delta}}$-weights on $S^k(N_Z^{+,\gamma})$ (resp. $S^k(N_Z^{+,-\gamma})$) are of the form $p\beta^*$ with $p\geq 0$ (resp. $p\leq 0$), we see that $$\label{eq:S-nul}
  S_Z^{\alpha}=0 \quad {\rm if }\quad -s_Z^- 
< \frac{\langle\alpha,\beta\rangle}{2\pi}< s_Z^+.$$

The line bundles $L^{\otimes k}\vert_{M_{{\ensuremath{\mathfrak{c}}}'}}, k\in {\ensuremath{\mathbb{Z}}}$ can be considered either as $T$-vector bundles or as $T/T_{\ensuremath{\Delta}}$-vector vector bundles: we denote them respectively by $L^{\otimes k}\vert_T$ and by $L^{\otimes k}\vert_{T/T_{\ensuremath{\Delta}}}$. The $T_{\ensuremath{\Delta}}$-weight on $L^{\otimes k}\vert_{M_{{\ensuremath{\mathfrak{c}}}'}}$ is equal to $kr_{\ensuremath{\Delta}}\beta^*$, hence we have $$\label{eq:L-k}
  L^{\otimes k}\vert_T=L^{\otimes k}\vert_{T/T_{\ensuremath{\Delta}}}\otimes {\ensuremath{\mathbb{C}}}_{kr_{\ensuremath{\Delta}}\beta^*} \ .$$

For every $\mu\in\Lambda^*$ and $Z\subset M_{{\ensuremath{\mathfrak{c}}}'}$ we define the orbifold vector bundle[^3] ${\ensuremath{\mathbb{S}}}^k_{Z,\mu}$ on the reduced space ${\ensuremath{\mathcal{Z}}}_\xi=(Z\cap\Phi^{-1}(\xi))/(T/T_{\ensuremath{\Delta}})$ by $$\label{eq:bundle-xi-mu}
  {\ensuremath{\mathbb{S}}}^k_{Z,\mu}:=(-1)^{n_Z(\mu_1-kr_{\ensuremath{\Delta}}\beta^*)}
\left(L^{\otimes k}\vert_{T/T_{\ensuremath{\Delta}}}\otimes 
S_Z^{\mu_1-kr_{\ensuremath{\Delta}}\beta^*}\otimes{\ensuremath{\mathbb{C}}}_{-\mu_2}\right)
\vert_{Z\cap\Phi^{-1}(\xi)}/(T/T_{\ensuremath{\Delta}}),$$ where $\mu=\mu_1+\mu_2$ with $\mu_1\in\Lambda^*_{{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}}$ and $\mu_2\in\Lambda^*_{{\ensuremath{\mathfrak{t}}}/{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}}$.

\[th:th-principal-mult\] Let ${\ensuremath{\mathbb{S}}}^k_{\xi,\mu}$ be the orbifold vector bundle on ${\ensuremath{\mathcal{M}}}^{\ensuremath{\Delta}}_\xi=\cup_{Z\subset M_{{\ensuremath{\mathfrak{c}}}'}}{\ensuremath{\mathcal{Z}}}_\xi$ which is equal to ${\ensuremath{\mathbb{S}}}^k_{Z,\mu}$ over each connected component ${\ensuremath{\mathcal{Z}}}_\xi$. For every $(\mu,k)\in\Lambda^*\times{\ensuremath{\mathbb{Z}}}$, we have $$\label{eq:mm-diff}
 {\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_+}(\mu,k)-{\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_-}(\mu,k)=RR({\ensuremath{\mathcal{M}}}^{\ensuremath{\Delta}}_\xi,{\ensuremath{\mathbb{S}}}^k_{\xi,\mu}).$$ In particular ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_+}(\mu,k)={\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_-}(\mu,k)$ if $$\label{eq:m-egale}
 - s^- < \frac{\langle\mu,\beta\rangle}{2\pi}- k r_{\ensuremath{\Delta}}< s^+,$$ with $s^\pm=\inf_{Z\subset M_{{\ensuremath{\mathfrak{c}}}'}} s^\pm_Z$.

[*Proof.*]{} After (\[eq:loc-xi-1\]), (\[eq:loc-xi-2\]), and Proposition \[RR-0-diff\] we know that ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_+}(\mu,k)-{\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_-}(\mu,k)$ is equal to the $\mu$-mutiplicity in $$\begin{aligned}
\label{eq:RR-T-delta}
 \lefteqn{ \sum_{Z\subset M_{{\ensuremath{\mathfrak{c}}}'}} RR_{0}^{\xi}\left(Z,L^{\otimes k}\vert_Z\otimes
(\left[\wedge_{{\ensuremath{\mathbb{C}}}}^{\bullet}\overline{N_Z}\right]^{-1}_{\beta}-
\left[\wedge_{{\ensuremath{\mathbb{C}}}}^{\bullet}\overline{N_Z}\right]^{-1}_{-\beta})\right)}\nonumber\\
&=&\sum_{\alpha\in\Lambda^*_{{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}}}\sum_{Z\subset M_{{\ensuremath{\mathfrak{c}}}'}}
(-1)^{n_Z(\alpha)}\underbrace{RR_{0}^{\xi}\left(Z,L^{\otimes k}\vert_{T/T_{\ensuremath{\Delta}}}
\otimes S_Z^\alpha\right)}_{\in\ R^{-\infty}(T/T_{\ensuremath{\Delta}})}\otimes
\underbrace{{\ensuremath{\mathbb{C}}}_{\alpha+kr_{\ensuremath{\Delta}}\beta^*}}_{\in \ R(T_{\ensuremath{\Delta}})}\nonumber \\
&=&\sum_{\mu\in\Lambda^*} RR({\ensuremath{\mathcal{M}}}^{\ensuremath{\Delta}}_\xi,{\ensuremath{\mathbb{S}}}^k_{\xi,\mu}) \ {\ensuremath{\mathbb{C}}}_\mu \end{aligned}$$

Here (\[eq:RR-T-delta\]) is a direct consequence of (\[eq:indice-t-p\]) modulo the shift $\mu_1\mapsto\mu_1 - kn_{\ensuremath{\Delta}}\beta^*$ in $\Lambda^*({\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}})$. We get (\[eq:m-egale\]) using the vanishing conditions (\[eq:S-nul\]). $\Box$

When condition (\[eq:m-egale\]) is the optimal one? In other word, do we have ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_+}(\mu,k)\neq {\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_-}(\mu,k)$ for some $\mu\in\Lambda^*$ such that $\frac{\langle\mu,\beta\rangle}{2\pi}- k r_{\ensuremath{\Delta}}= s^+$ , and do we have ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_+}(\mu,k)\neq {\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_-}(\mu,k)$ for some $\mu\in\Lambda^*$ such that $\frac{\langle\mu,\beta\rangle}{2\pi}- k r_{\ensuremath{\Delta}}= s^-$ ?

Until the end of this section we consider couples $(\mu,k)$ such that $\mu=$ $(k r_{\ensuremath{\Delta}}+s^+)\beta^* +\mu_2$ with $\mu_2\in\Lambda^*_{{\ensuremath{\mathfrak{t}}}/{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}}$. For the orbifold vector bundle ${\ensuremath{\mathbb{S}}}^k_{Z,\mu}$ we have ${\ensuremath{\mathbb{S}}}^k_{Z,\mu}= (-1)^{{\rm rk}_{\ensuremath{\mathbb{C}}}( N_Z^{+,\beta})} 
\widetilde{{\ensuremath{\mathbb{S}}}}^k_{Z,\mu_2}$ where $$\label{eq:vector-S-maximal}
 \widetilde{{\ensuremath{\mathbb{S}}}}^k_{Z,\mu_2}= 
\left(L^{\otimes k}\vert_{T/T_{\ensuremath{\Delta}}}\otimes 
\det( N_Z^{+,\beta})\otimes{\ensuremath{\mathbb{C}}}_{-\mu_2}\right)
\vert_{Z\cap\Phi^{-1}(\xi)}/(T/T_{\ensuremath{\Delta}})$$ when $s^+=s^+_Z$ and ${\ensuremath{\mathbb{S}}}^k_{Z,\mu}=0$ when $s^+<s^+_Z$. Hence ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_+}(\mu,k)-{\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_-}(\mu,k)$ is equal to $$\label{eq:mm-pm-maximal}
  \sum_{Z,s^+_Z=s^+} (-1)^{{\rm rk}_{\ensuremath{\mathbb{C}}}( N_Z^{+,\beta})}
RR({\ensuremath{\mathcal{Z}}}_\xi,\widetilde{{\ensuremath{\mathbb{S}}}}^k_{Z,\mu_2})$$

Now we use the results of Section \[subsec:RR-theo\] to compare the behaviour of the maps $$\begin{aligned}
  \label{eq:RR-Z-mu-2}
  \Lambda^*_{{\ensuremath{\mathfrak{t}}}/{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}}\times{\ensuremath{\mathbb{Z}}}&\longrightarrow &{\ensuremath{\mathbb{Z}}},\\ 
(\mu_2,k)&\longmapsto& RR({\ensuremath{\mathcal{Z}}}_\xi,\widetilde{{\ensuremath{\mathbb{S}}}}^k_{Z,\mu_2})\nonumber\end{aligned}$$ for the different components $Z$ satisfying $s^+_Z=s^+$. Let $\Gamma_Z\subset T/T_{\ensuremath{\Delta}}$ be the generic stabiliser of $T/T_{\ensuremath{\Delta}}$ on a component $Z$. Let $\alpha_Z,\delta_Z\in 
\Lambda^*_{{\ensuremath{\mathfrak{t}}}/{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}}$ such that the action of $\Gamma_Z$ on the fibers of $L_{\vert Z}$ and $\det( N_Z^{+,\beta})$ are respectively $t\to t^{\alpha_Z}$ and $t\to t^{\delta_Z}$. After Remark \[lem:discrete\] we know that the map (\[eq:RR-Z-mu-2\]) is supported on the subset $$\label{eq:xi-Z}
 \Xi_Z:=\{(\mu_2,k)\in\Lambda^*_{{\ensuremath{\mathfrak{t}}}/{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}}\times{\ensuremath{\mathbb{Z}}}\ \vert\ 
t^{k\alpha_Z+\delta_Z+\mu_2}=1,\  \forall\ t\in\Gamma_Z\}.$$ The only difference with the computations done in Section \[subsec:RR-theo\] is the presence of the line bundle $\det( N_Z^{+,\beta})$. But this do not change the global behaviour of the map (\[eq:RR-Z-mu-2\]) on $\Xi_Z$: it is a periodic polynomial map of degree $l_Z=\dim({\ensuremath{\mathcal{Z}}}_\xi)/2$ and we have $$\label{eq:pol-behaviour-RR-mu2}
RR({\ensuremath{\mathcal{Z}}}_\xi,\widetilde{{\ensuremath{\mathbb{S}}}}^k_{Z,\mu_2})=\frac{1}{(2\pi)^{l_Z}}\int_{{\ensuremath{\mathcal{Z}}}_\xi}
(k\Omega_{{\ensuremath{\mathcal{Z}}}_\xi} -\langle k\xi-\mu_2,\omega_{{\ensuremath{\mathcal{Z}}}_\xi}\rangle)^{l_Z} +O(l_Z-1)$$ for all $(\mu_2,k)\in\Xi_Z$. We see then that the sum (\[eq:mm-pm-maximal\]) does not vanish for large values of $(\mu_2,k)$ when the number $(-1)^{{\rm rk}_{\ensuremath{\mathbb{C}}}( N_Z^{+,\beta})}$ are equal for all the components $Z$ of maximal dimension. Finally we can conclude with

\[prop:inegalite-optimale\] Consider the set of components $Z\subset M_{{\ensuremath{\mathfrak{c}}}'}$ for which $s^+_Z$ is minimal. Among them consider the subset ${\ensuremath{\mathcal{F}}}$ where $\dim(Z)$ is maximal. If the integers ${\rm rk}_{\ensuremath{\mathbb{C}}}( N_Z^{+,\beta})$, $Z\in{\ensuremath{\mathcal{F}}}$ have the same parity, then the condition “$\frac{\langle\mu,\beta\rangle}{2\pi}- k r_{\ensuremath{\Delta}}< s^+$” is optimal in (\[eq:m-egale\]).

In the same way, consider the set of components $Z\subset M_{{\ensuremath{\mathfrak{c}}}'}$ for which $s^-_Z$ is minimal. Among them consider the subset ${\ensuremath{\mathcal{F}}}'$ where $\dim(Z)$ is maximal. If the integers ${\rm rk}_{\ensuremath{\mathbb{C}}}( N_Z^{+,\beta})$, $Z\in{\ensuremath{\mathcal{F}}}'$ have the same parity, then the condition “$-s^-<\frac{\langle\mu,\beta\rangle}{2\pi}- k r_{\ensuremath{\Delta}}$” is optimal in (\[eq:m-egale\]).

Multiplicities of group representations {#sec:cas-coadjoint}
=======================================

Let $K$ be a semi-simple compact Lie group with Lie algebra ${\ensuremath{\mathfrak{k}}}$, and let $T$ be a maximal torus in $K$ with Lie algebra ${\ensuremath{\mathfrak{t}}}$. In this section we denote $(-,-)$ the scalar product on ${\ensuremath{\mathfrak{k}}}$ induced by the Killing form, and we keep the same notation for the induced scalar products on ${\ensuremath{\mathfrak{t}}}^*$ and on ${\ensuremath{\mathfrak{t}}}$.

An element $\lambda\in {\ensuremath{\mathfrak{t}}}^*$ belong to the real weight lattice $\Lambda^*\subset {\ensuremath{\mathfrak{t}}}^*$ if $\imath\, \lambda$ is the differential of a character of $T$ that we denote $t\mapsto t^\lambda$: if $t=\exp X$ then $t^{\lambda}:=e^{\imath\langle\lambda,X\rangle}$. Let ${\ensuremath{\mathfrak{R}}}\subset\Lambda^*$ be the set of roots for the action of $T$ on ${\ensuremath{\mathfrak{k}}}\otimes{\ensuremath{\mathbb{C}}}$, and let $\Lambda^*_R$ be the sub-lattice of $\Lambda^*$ generated by ${\ensuremath{\mathfrak{R}}}$. We choose a system of positive roots ${\ensuremath{\mathfrak{R}}}^+\subset{\ensuremath{\mathfrak{R}}}$, and we denote ${\ensuremath{\mathfrak{t}}}^*_+$ the corresponding Weyl chamber.

The irreducible representations of $K$ are parametrized by the set $\Lambda^*_+=
\Lambda^*\cap{\ensuremath{\mathfrak{t}}}^*_+$. For $\lambda\in\Lambda^*_+$ we denote by $V_\lambda$ the irreducible representation of $K$ with heighest weight $\lambda$. Here we are interested in the $T$-multiplicities in $V_\lambda\vert_T$. Let ${\ensuremath{\hbox{\rm m}}}: \Lambda^*\times \Lambda^*_+\to{\ensuremath{\mathbb{N}}}$ be the map defined by $$\label{eq:T-mult}
 V_\lambda\vert_T=\sum_{\mu\in\Lambda^*}{\ensuremath{\hbox{\rm m}}}(\mu,\lambda)\ {\ensuremath{\mathbb{C}}}_\mu$$ for every $\lambda\in\Lambda^*_+$.

\[def:mm-lambda\] For every $\lambda\in\Lambda^*_+$, we denote ${\ensuremath{\hbox{\rm m}}}^\lambda: 
\Lambda^*\times {\ensuremath{\mathbb{Z}}}^{>0}\to{\ensuremath{\mathbb{N}}}$ the map defined by ${\ensuremath{\hbox{\rm m}}}^\lambda(\mu,k)={\ensuremath{\hbox{\rm m}}}(\mu,k\lambda)$. So ${\ensuremath{\hbox{\rm m}}}^\lambda(\mu,k)$ is equal to the multiplicity of ${\ensuremath{\mathbb{C}}}_\mu$ in $V_{k\lambda}\vert_T$.

Borel-Weil Theorem
------------------

First we recall the realization of the $K$-representation $V_\lambda$ given by the Borel-Weil Theorem. The coadjoint orbit $K\cdot\lambda$ is equipped with the Kirillov-Kostant-Souriau symplectic form $\Omega$ which is defined by: $$\label{eq:KKS}
  \Omega(X_M,Y_M)_m=\langle m,[X,Y]\rangle,\quad {\rm for}\quad  
m\in K\cdot\lambda\quad {\rm and}
\quad X,Y\in{\ensuremath{\mathfrak{k}}}.$$ The action of $K$ on $K\cdot\lambda$ is Hamiltonian with moment map $K\cdot\lambda{\ensuremath{\hookrightarrow}}{\ensuremath{\mathfrak{k}}}^*$ equal to the inclusion. The action of $T$ on $K\cdot\lambda$ is also Hamiltonian with moment map $\Phi:K\cdot\lambda \to{\ensuremath{\mathfrak{t}}}^*$ equal to the composition of the inclusion $K\cdot\lambda{\ensuremath{\hookrightarrow}}{\ensuremath{\mathfrak{k}}}^*$ with the projection map ${\ensuremath{\mathfrak{k}}}^*\to{\ensuremath{\mathfrak{t}}}^*$.

There exists a unique $K$-invariant complex structure on $K\cdot\lambda$ compatible with the symplectic form. In this situation the Kostant-Souriau prequantum line bundle over $K\cdot\lambda$ is $${\ensuremath{\mathbb{C}}}_{[\lambda]}=K\times_{K_\lambda}{\ensuremath{\mathbb{C}}}_\lambda.$$ Here we use the canonical identification $K/K_\lambda\simeq K\cdot\lambda,\  
[k]\mapsto k\cdot\lambda$, where $K_\lambda$ is the stabilizer of $\lambda$ in $K$. The line bundle ${\ensuremath{\mathbb{C}}}_{[\lambda]}$ over the complex manifold $K\cdot\lambda$ carries a canonical holomorphic structure. If one work with the symplectic form $k\Omega$, for an integer $k\geq 1$, the corresponding Kostant-Souriau prequantum line bundle is ${\ensuremath{\mathbb{C}}}_{[\lambda]}^{\otimes k}=
K\times_{K_\lambda}{\ensuremath{\mathbb{C}}}_{k\lambda}={\ensuremath{\mathbb{C}}}_{[k\lambda]}$.

Let ${\ensuremath{\mathcal{H}}}^q(K\cdot\lambda,{\ensuremath{\mathbb{C}}}_{[\lambda]}^{\otimes k})$ be $q$th cohomology group of the sheaf of holomorphic section of ${\ensuremath{\mathbb{C}}}_{[\lambda]}^{\otimes k}$ over $K\cdot\lambda$. The Borel-Weil Theorem tells us that $$\label{eq:BW-1}
 {\ensuremath{\mathcal{H}}}^0(K\cdot\lambda,{\ensuremath{\mathbb{C}}}_{[\lambda]}^{\otimes k})= V_{k\lambda}$$ and $$\label{eq:BW-2}
 {\ensuremath{\mathcal{H}}}^q(K\cdot\lambda,{\ensuremath{\mathbb{C}}}_{[\lambda]}^{\otimes k})= 0\quad {\rm for}\quad q\geq 1.$$

If $RR^K(K\cdot\lambda,-):{\ensuremath{\hbox{\bf K}}}_K(K\cdot\lambda)\to R(K)$ is the $K$-Riemann-Roch character defined by the compatible complex structure, (\[eq:BW-1\]) and (\[eq:BW-2\]) give $$\label{eq:RR-V-lambda}
  RR^K(K\cdot\lambda,{\ensuremath{\mathbb{C}}}_{[\lambda]}^{\otimes k})=
V_{k\lambda} \quad {\rm in}\quad  R(K)$$

Now if we denote by $RR(K\cdot\lambda,-):{\ensuremath{\hbox{\bf K}}}_T(K\cdot\lambda)\to R(T)$ the $T$-equivariant Riemann-Roch character, we have $V_{k\lambda}\vert_T=
RR(K\cdot\lambda,{\ensuremath{\mathbb{C}}}_{[\lambda]}^{\otimes k})$. The multiplicity fonction ${\ensuremath{\hbox{\rm m}}}^\lambda:
\Lambda^*_+\times {\ensuremath{\mathbb{N}}}^*\to{\ensuremath{\mathbb{N}}}$ is characterized by the relation $$\label{eq:T-mult-bis}
 RR(K\cdot\lambda,{\ensuremath{\mathbb{C}}}_{[\lambda]}^{\otimes k})= 
\sum_{\mu\in\Lambda^*}{\ensuremath{\hbox{\rm m}}}^\lambda(\mu,k)\ {\ensuremath{\mathbb{C}}}_\mu,\quad {\rm in}\ R(T) ,$$ for $k\geq 1$.

The sub-lattice $\Lambda^*_R$ of $\Lambda^*$ generated by the roots is characterized by the (finite) center $Z(K)$ of $K$ as follows. For $\alpha\in\Lambda^*$ we have $$\label{eq:lambda-R}
\lambda\in\Lambda^*_R\ \Longleftrightarrow\  t^\lambda= 1,\  
\forall t \in Z(K),$$ and for $t\in T$ we have $t\in Z(K)\ \Longleftrightarrow\  t^\lambda= 1,\  \forall 
\lambda\in\Lambda^*_R$. The finite abelian group $\Lambda^*/\Lambda^*_R$ is then naturally identified with the dual of $Z(K)$. We have the following well-known fact.

\[prop:support-R\] We have ${\ensuremath{\hbox{\rm m}}}^\lambda(\mu,k)\neq 0$ only if $\mu-k\lambda\in\Lambda^*_R$.

[*Proof.*]{} The center $Z(K)$ of $K$ acts trivially on $K\cdot\lambda$ and with the character $t\in Z(K)\mapsto t^{k\lambda}$ on the fibers of the line bundle ${\ensuremath{\mathbb{C}}}_{[\lambda]}^{\otimes k}$. Since ${\ensuremath{\hbox{\rm m}}}^\lambda(\mu,k)$ is equal to the dimension of the $T$-invariant subspace of $RR(K\cdot\lambda,{\ensuremath{\mathbb{C}}}_{[\lambda]}^{\otimes k})\otimes{\ensuremath{\mathbb{C}}}_{-\mu}$, we have following Lemma \[lem:discrete\] that ${\ensuremath{\hbox{\rm m}}}^\lambda(\mu,k)\neq 0$ only if $t^{\mu-k\lambda}=1$, $\forall t\in Z(K)$. We conclude then with (\[eq:lambda-R\]). $\Box$

In this section we are interested in the periodic polynomials $$\label{eq:multiplicite-c}
{\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}}^\lambda : \Lambda^* \times {\ensuremath{\mathbb{Z}}}\longrightarrow  {\ensuremath{\mathbb{Z}}}.$$ defined for every connected component ${\ensuremath{\mathfrak{c}}}\subset{\ensuremath{\mathfrak{t}}}^*$ of regular values of the moment map $\Phi:K\cdot\lambda \to{\ensuremath{\mathfrak{t}}}^*$: the map ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}}^\lambda$ coincide with ${\ensuremath{\hbox{\rm m}}}^\lambda$ on the set $\{(\mu,k)\in\Lambda^* \times {\ensuremath{\mathbb{Z}}}^{>0}\ \vert\ \mu\in k{\ensuremath{\mathfrak{c}}}\}$. Like in Proposition \[prop:support-R\], the formula given in Proposition \[prop:m-poly\] for ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}}^\lambda$ tells us that ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}}^\lambda(\mu,k)\neq 0$ only if $\mu-k\lambda\in\Lambda^*_R$.

To apply Theorem \[th:th-principal-mult\] to the periodic polynomials ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}}^\lambda$, we have to compute the critical values of the moment map $\Phi:K\cdot\lambda \to{\ensuremath{\mathfrak{t}}}^*$.

Critical points of $\Phi:K\cdot\lambda \to{\ensuremath{\mathfrak{t}}}^*$
------------------------------------------------------------------------

Let $\{\alpha_1,\cdots,\alpha_{dim T}\}$ be the simple roots of the set ${\ensuremath{\mathfrak{R}}}_+$ of positive weights. The fundamental weights $\varpi_k, 1\leq k\leq \dim T$ are defined by the conditions $$\label{eq:fundamental-w}
 2\frac{(\varpi_i,\alpha_j)}{\vert\alpha_j\vert^2}=\delta_{i,j}\quad {\rm for\ all} 
\quad 1\leq i,j\leq \dim T.$$ Recall that the fundamental weights generate the lattice $\Lambda^*_{alg}$ of [*algebraic*]{} integral element of ${\ensuremath{\mathfrak{t}}}^*$. We have $\Lambda^*\subset \Lambda^*_{alg}$ and equality holds only if $K$ is simply-connected.

Let $W$ be the Weyl group of $(K,T)$. We will see $$\label{eq:Fcal}
 {\ensuremath{\mathcal{F}}}= \{\sigma\cdot\varpi_i\ \vert\ \sigma\in W,\  1\leq i\leq \dim T\}.$$ as a subset of ${\ensuremath{\mathfrak{t}}}$ modulo the identification ${\ensuremath{\mathfrak{t}}}\simeq{\ensuremath{\mathfrak{t}}}^*$ given by the scalar product. The singular points of $\Phi$ have the following nice description. This result first appeared in Heckman’s Thesis [@Heckman-PhD].

\[prop:pt-critiques\] The critical points of $\Phi:K\cdot\lambda \to{\ensuremath{\mathfrak{t}}}^*$ is the union of the fixed points set $(K\cdot\lambda)^\beta,\ \beta\in{\ensuremath{\mathcal{F}}}$. For each $\beta\in{\ensuremath{\mathcal{F}}}$ we have $$(K\cdot\lambda)^\beta=\bigcup_{\sigma\in W} K^\beta\cdot \sigma\lambda.$$ Here $K^\beta$ is the stabilizer of $\beta$ in $K$.

The fixed points of the action of $T$ on $K\cdot\lambda$ characterize the image of $\Phi$ completely: $\Phi(K\cdot\lambda)$ is the convex polytope $$\label{eq:convex}
  {{\ensuremath{\rm conv}}}(W\cdot\lambda):=\ {\rm convex\ hull\ of\ } W\cdot\lambda.$$ This result was first proved by Kostant [@Kostant73]. This is particular case of the convexity theorem of Atiyah, Guillemin and Sternberg [@Atiyah.82; @Guillemin-Sternberg82.bis]. From Proposition \[prop:pt-critiques\], we know that the singular values of $\Phi:K\cdot\lambda \to{\ensuremath{\mathfrak{t}}}^*$ are the convex polytopes $$\label{eq:convex-singular}
  {{\ensuremath{\rm conv}}}(W^\beta\cdot\sigma\lambda),\quad \beta\in{\ensuremath{\mathcal{F}}},\ \sigma\in W/W^\beta.$$ where $W^\beta$ is the stabilizer of $\beta$ in $W$, i.e. $W^\beta$ is the Weyl group of $(K^\beta,T)$. Each convex polytope ${{\ensuremath{\rm conv}}}(W^\beta\cdot\sigma\lambda)$ lies in the hyperplane $$\label{eq:delta-beta-sigma}
  \Delta_{\beta,\sigma}=\{\xi\in{\ensuremath{\mathfrak{t}}}^*\ \vert\ 
(\xi-\sigma\lambda,\beta)=0\}.$$

Note that $K^\beta\cdot \sigma\lambda$ coincide with $K^\beta\cdot \sigma'\lambda$ if and only if $\sigma\lambda\in W^\beta\sigma'\lambda$. Two polytopes ${{\ensuremath{\rm conv}}}(W^\beta\cdot\sigma\lambda)$ and ${{\ensuremath{\rm conv}}}(W^\beta\cdot\sigma'\lambda)$ intersect if and only if $\langle\sigma\lambda,\beta\rangle=\langle\sigma'\lambda,\beta\rangle=0$.

\[def:lambda-gene\] An element $\lambda\in\Lambda^*_+$ is [*generic*]{} if for every fundamental root $\varpi_i$ we have $$\label{eq:lambda-gene}
 (\sigma\lambda,\varpi_i)\neq(\sigma'\lambda,\varpi_i)$$ each times that $\sigma\lambda\notin W^i\sigma'\lambda$ (here $W^i=\{\sigma\in W\,\vert\,\sigma\varpi_i=\varpi_i\}$).

This condition of genericity imposes that the hyperplanes $\Delta_{\beta,\sigma}$ and $\Delta_{\beta,\sigma'}$ are distinct each times that the submanifolds $K^\beta\cdot \sigma\lambda$ and $K^\beta\cdot \sigma'\lambda$ are not equal.

\[ex:su-4\] Consider the case of $\hbox{\rm SU}(4)$. Take the coadjoint orbit trough $\lambda=
(2,1,-1,-2)$, and $\sigma,\sigma'$ such that $\sigma\lambda=(2,-2,1,-1)$ and $\sigma'\lambda=(1,-1,2,-2)$. Take the fundamental weight $\varpi_2=\frac{1}{2}(1,1,-1,-1)$. In this case $\lambda$ is not “generic” since $\sigma\lambda\notin W^i\sigma'\lambda$ but $(\sigma\lambda,\varpi_2)=(\sigma'\lambda,\varpi_2)=0$.

Main theorems
-------------

Let ${\ensuremath{\mathfrak{c}}}_+$ and ${\ensuremath{\mathfrak{c}}}_-$ be two adjacent connected components of regular values of $\Phi:K\cdot\lambda\to{\ensuremath{\mathfrak{t}}}^*$. The intersection $\overline{{\ensuremath{\mathfrak{c}}}_+}\cap\overline{{\ensuremath{\mathfrak{c}}}_+}$ is contained in an hyperplane orthogonal to $\beta\in{\ensuremath{\mathcal{F}}}$.

Let ${\ensuremath{\mathcal{A}}}({\ensuremath{\mathfrak{c}}}_+,{\ensuremath{\mathfrak{c}}}_-)$ be the set of all $\sigma\in  W/W^\beta$ such that the convex polytope ${{\ensuremath{\rm conv}}}(W^\beta\cdot\sigma\lambda)$ contains $\overline{{\ensuremath{\mathfrak{c}}}_+}\cap\overline{{\ensuremath{\mathfrak{c}}}_+}$.

The set $$\bigcup_{\sigma\in{\ensuremath{\mathcal{A}}}({\ensuremath{\mathfrak{c}}}_+,{\ensuremath{\mathfrak{c}}}_-)}K^\beta\cdot \sigma\lambda$$ corresponds to the subset of the critical points of $\Phi$ that intersect $\Phi^{-1}(\xi)$ when $\xi\in\overline{{\ensuremath{\mathfrak{c}}}_+}\cap\overline{{\ensuremath{\mathfrak{c}}}_+}$.

When $\lambda$ is a regular element of ${\ensuremath{\mathfrak{t}}}^*$, all polytopes ${{\ensuremath{\rm conv}}}(W^\beta\cdot\sigma\lambda)$ are of codimension $1$. When $\lambda$ is “generic” (see Def. \[def:lambda-gene\]), the set ${\ensuremath{\mathcal{A}}}({\ensuremath{\mathfrak{c}}}_+,{\ensuremath{\mathfrak{c}}}_-)$ is reduced to one element.

The multiplicity function ${\ensuremath{\hbox{\rm m}}}^\lambda:\Lambda^*\times {\ensuremath{\mathbb{N}}}^*\to{\ensuremath{\mathbb{N}}}$ is invariant under the action of the Weyl group: ${\ensuremath{\hbox{\rm m}}}^\lambda(\sigma\mu,k)={\ensuremath{\hbox{\rm m}}}^\lambda(\mu,k)$ for every $\sigma\in W$. The set of connected component of regular values of $\Phi$ is also invariant under the action of $W$.

So for the rest of this section we restrict our attention to case where ${\ensuremath{\mathfrak{c}}}_+$ and ${\ensuremath{\mathfrak{c}}}_-$ are separated by an hyperplane orthogonal to a fundamental weight $\beta=\varpi_i$: the vector $\beta=\varpi_i$ is pointing out of ${\ensuremath{\mathfrak{c}}}_-$. Consider $\sigma\in {\ensuremath{\mathcal{A}}}({\ensuremath{\mathfrak{c}}}_+,{\ensuremath{\mathfrak{c}}}_-)$ and let $K^i\cdot \sigma\lambda$ be the corresponding connected component of $(K\cdot\lambda)^\beta$ (here $K^i$ denote the stabilizer of $\varpi_i$ in $K$). The tangent space of $K\cdot\lambda$ at $\sigma\lambda$ is the following $K^{\sigma\lambda}$-module $$\label{eq:tangent-s-lambda}
  {\ensuremath{\hbox{\bf T}}}_{\sigma\lambda}(K\cdot\lambda)=\sum_{(\alpha,\sigma\lambda)>0}{\ensuremath{\mathfrak{k}}}_\alpha$$ where ${\ensuremath{\mathfrak{k}}}_\alpha\subset{\ensuremath{\mathfrak{k}}}\otimes{\ensuremath{\mathbb{C}}}$ is the one-dimensional complex subspace associated to the weight $\alpha\in{\ensuremath{\mathfrak{R}}}$. In the same way, the tangent space of $K^i\cdot\sigma\lambda$ at $\sigma\lambda$ is the $K^i\cap K^{\sigma\lambda}$-module defined by $$\label{eq:tangent-s-lambda-i}
  {\ensuremath{\hbox{\bf T}}}_{\sigma\lambda}(K^i\cdot\sigma\lambda)=
\sum_{\stackrel{(\alpha,\sigma\lambda)>0}{(\alpha,\varpi_i)= 0}}
{\ensuremath{\mathfrak{k}}}_\alpha$$ Finally the normal bundle of $K^i\cdot\sigma\lambda\simeq 
K^i/(K^i\cap K^{\sigma\lambda})$ in $K\cdot\lambda$ is ${\ensuremath{\mathcal{N}}}_{\sigma,i}=$ $K^i\times_{K^i\cap K^{\sigma\lambda}}N_{\sigma,i}$ where $$\label{eq:normal-sigma}
  N_{\sigma,i}=\sum_{\stackrel{(\alpha,\sigma\lambda)>0}{(\alpha,\varpi_i)\neq 0}}
{\ensuremath{\mathfrak{k}}}_\alpha$$

For an element $\mu\in{\ensuremath{\mathfrak{t}}}^*$, we have $\mu=\sum_{i=1}^{\dim T}[\mu]_k\, \alpha_k$ where the $[\mu]_k\in{\ensuremath{\mathbb{R}}}$ are defined by the relation $[\mu]_k= 
2\frac{(\varpi_k,\mu)}{\vert\alpha_k\vert^2}$. When $\mu\in\Lambda^*_R$ the coefficients $[\mu]_k$ are all integers.

\[def:s-sigma-pm\] For $\sigma\in {\ensuremath{\mathcal{A}}}({\ensuremath{\mathfrak{c}}}_+,{\ensuremath{\mathfrak{c}}}_-)$ we define the positive integers $$s^\pm_{\sigma,i}=\pm\sum_{\stackrel{(\alpha,\sigma\lambda)>0}{\pm(\alpha,\varpi_i)> 0}}
[\alpha]_i .$$ Note that $s^+_{\sigma,i}+s^-_{\sigma,i}$ is larger than half of the codimension of $K^i\cdot\sigma\lambda$ in $K\cdot\lambda$.

\[theo:mult-coadjoint\] Let ${\ensuremath{\mathfrak{c}}}_+$ and ${\ensuremath{\mathfrak{c}}}_-$ be two adjacent connected component of regular values of $\Phi:K\cdot\lambda\to{\ensuremath{\mathfrak{t}}}^*$ separated by an hyperplane orthogonal to a fundamental weight $\varpi_i$: we denote $r_i$ the commum value $[\xi]_i$ for all $\xi$ in this hyperplane. Let ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_\pm}^\lambda : \Lambda^* \times {\ensuremath{\mathbb{Z}}}\longrightarrow  {\ensuremath{\mathbb{Z}}}$ be the corresponding periodic polynomials: they are supported on the sub-lattice $\Xi_\lambda:=\{(\mu,k)\, \vert\, \mu\in k\lambda +\Lambda^*_R\}$.

For all $(\mu,k)\in\Xi_\lambda$, we have ${\ensuremath{\hbox{\rm m}}}^\lambda_{{\ensuremath{\mathfrak{c}}}_+}(\mu,k)={\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_-}^\lambda(\mu,k)$ when $$\label{eq:m-egale-lambda}
 - s^-_i < [\mu]_i- k r_i< s^+_i.$$ Here the positive integer $s^\pm_i$ are defined by $$\label{eq:n-pm-lambda}
  s^\pm_i=\inf_{\sigma\in {\ensuremath{\mathcal{A}}}({\ensuremath{\mathfrak{c}}}_+,{\ensuremath{\mathfrak{c}}}_-)} s^\pm_{\sigma,i}.$$

When ${\ensuremath{\mathcal{A}}}({\ensuremath{\mathfrak{c}}}_+,{\ensuremath{\mathfrak{c}}}_-)$ is reduced to one element $\sigma$, for example if $\lambda$ is “generic”, the integer $s^+ +s^-$ is larger than half of the codimension of $K^i\cdot\sigma\lambda$ in $K\cdot\lambda$.

Another way to express the result of Theorem \[theo:mult-coadjoint\] is to introduce like in [@Szenes-Vergne02] the convex polytope $$\label{eq:carre-R}
  \Box({\ensuremath{\mathfrak{c}}}_+,{\ensuremath{\mathfrak{c}}}_-)=\bigcap_{\sigma\in{\ensuremath{\mathcal{A}}}({\ensuremath{\mathfrak{c}}}_+,{\ensuremath{\mathfrak{c}}}_-)}
\left(\sum_{(\alpha,\sigma\lambda)>0}[0,1[\,\alpha\right).$$

Let ${\ensuremath{\Delta}}$ be the hyperplane which separates ${\ensuremath{\mathfrak{c}}}_+$ and ${\ensuremath{\mathfrak{c}}}_-$. Equation (\[eq:m-egale-lambda\]) is equivalent to saying that $$\label{eq:m-egale-carre}
 {\ensuremath{\hbox{\rm m}}}^\lambda_{{\ensuremath{\mathfrak{c}}}_+}(\mu,k)={\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_-}^\lambda(\mu,k)\quad {\rm if}\quad 
\mu\in k{\ensuremath{\Delta}}+\Box({\ensuremath{\mathfrak{c}}}_+,{\ensuremath{\mathfrak{c}}}_-).$$

Let ${\ensuremath{\mathfrak{c}}}$ be a connected component of regular values of $\Phi$ which is bording a facet of the polytope $\Phi(K\cdot\lambda)$ orthogonal to the fundamental weight $\varpi_i$: the facet is ${{\ensuremath{\rm conv}}}(W^i\cdot\sigma\lambda)$ for a unique $\sigma\in W/W^i$. We suppose that $\varpi_i$ is pointing out of ${\ensuremath{\mathfrak{c}}}$. We denote $r_i$ the commum value $[\xi]_i$ for all $\xi$ in the facet. For all $(\mu,k)\in\Xi_\lambda$, we have ${\ensuremath{\hbox{\rm m}}}^\lambda_{{\ensuremath{\mathfrak{c}}}}(\mu,k)=0$ when $$\label{eq:m-egale-lambda-bord}
 - s^-_{\sigma,i} < [\mu]_i- k r_i< s^+_{\sigma,i}.$$

[*Proof.* ]{} Theorem \[theo:mult-coadjoint\] is a direct consequence of Theorem \[th:th-principal-mult\]. The main difference between them is the decomposition of the lattice supporting the periodic polynomials. In the former we use the decomposition $\Lambda^*=\Lambda^*_{{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}}\oplus\Lambda^*_{{\ensuremath{\mathfrak{t}}}/{\ensuremath{\mathfrak{t}}}_{\ensuremath{\Delta}}}$ associated to the choice of a subtorus $T/T_{\ensuremath{\Delta}}$. Here, since ${\ensuremath{\hbox{\rm m}}}^\lambda_{{\ensuremath{\mathfrak{c}}}_\pm}$ is supported on $\lambda + \Lambda^*_R$, we use the decomposition $\Lambda^*_R={\ensuremath{\mathbb{Z}}}\alpha_i\oplus\sum_{k\neq i}{\ensuremath{\mathbb{Z}}}\alpha_k$.

We start like after Proposition \[RR-0-diff\]: ${\ensuremath{\hbox{\rm m}}}^\lambda_{{\ensuremath{\mathfrak{c}}}_+}(\mu,k)-
{\ensuremath{\hbox{\rm m}}}^\lambda_{{\ensuremath{\mathfrak{c}}}_-}(\mu,k)$ is equal to the $\mu$-mutiplicity in $\sum_{\sigma\in{\ensuremath{\mathcal{A}}}({\ensuremath{\mathfrak{c}}}_+,{\ensuremath{\mathfrak{c}}}_-)}A^-_\sigma -A^+_\sigma$ where $$\label{eq:mu-lambda-pm}
 A^\pm_\sigma= 
RR_0^{\xi}\left(K^i\cdot\sigma\lambda,{\ensuremath{\mathbb{C}}}_{[\lambda]}^{\otimes k}\otimes
\left[\wedge_{{\ensuremath{\mathbb{C}}}}^{\bullet}\overline{{\ensuremath{\mathcal{N}}}_{\sigma,i}}\right]^{-1}_{\mp\varpi_i}\right)$$

Here $\xi$ belongs to the relative interior of $\overline{{\ensuremath{\mathfrak{c}}}_+}\cap\overline{{\ensuremath{\mathfrak{c}}}_+}$, the line bundle ${\ensuremath{\mathbb{C}}}_{[\lambda]}^{\otimes k}$ is equal to $K^i\times_{K^i\cap K^{\sigma\lambda}}{\ensuremath{\mathbb{C}}}_{k\sigma\lambda}$ and $\left[\wedge_{{\ensuremath{\mathbb{C}}}}^{\bullet}
\overline{{\ensuremath{\mathcal{N}}}_{\sigma,i}}\right]^{-1}_{\pm\varpi_i}$ corresponds to $(-1)^{{\rm rk}_{\ensuremath{\mathbb{C}}}(N_{\sigma,i}^\pm)}$ times $$K^i\times_{K^i\cap K^{\sigma\lambda}}\left(\det(N_{\sigma,i}^\pm)\otimes 
S^\bullet((N_{\sigma,i}\otimes{\ensuremath{\mathbb{C}}})^\pm)\right),$$ with $$N_{\sigma,i}^\pm=\sum_{\stackrel{(\alpha,\sigma\lambda)>0}{\pm(\alpha,\varpi_i)>0}}
{\ensuremath{\mathfrak{k}}}_\alpha\ ,$$ and $$(N_{\sigma,i}\otimes{\ensuremath{\mathbb{C}}})^\pm=
\sum_{\stackrel{(\alpha,\sigma\lambda)\neq 0}{\pm(\alpha,\varpi_i)>0}}
{\ensuremath{\mathfrak{k}}}_\alpha.$$

Now we can apply Remark \[lem:discrete\] with the subgroup $H\subset T$ equal to the center $Z(K^i)$ of $K^i$: an element $\gamma\in\Lambda^*$ belong to $\sum_{k\neq i}{\ensuremath{\mathbb{Z}}}\alpha_k$ if and only if $t^\gamma=1$ for all $t\in Z(K^i)$.

The group $Z(K^i)$ acts trivially on the manifolds $K^i\cdot\sigma\lambda$, and with the characters associated to the weights $$k\sigma\lambda+\sum_{\stackrel{(\alpha,\sigma\lambda)>0}
{(\alpha,\varpi_i)>0}}\alpha+\delta
\quad {\rm with}\quad (\delta,\varpi_i)\geq 0$$ on the bundle ${\ensuremath{\mathbb{C}}}_{[\lambda]}^{\otimes k}\otimes
\left[\wedge_{{\ensuremath{\mathbb{C}}}}^{\bullet}\overline{{\ensuremath{\mathcal{N}}}_{\sigma,i}}\right]^{-1}_{\varpi_i}$, and with the characters associated to the weights $$k\sigma\lambda+\sum_{\stackrel{(\alpha,\sigma\lambda)>0}
{(\alpha,\varpi_i)<0}}\alpha+\delta
\quad {\rm with}\quad (\delta,\varpi_i)\leq 0$$ on the bundle ${\ensuremath{\mathbb{C}}}_{[\lambda]}^{\otimes k}\otimes
\left[\wedge_{{\ensuremath{\mathbb{C}}}}^{\bullet}\overline{{\ensuremath{\mathcal{N}}}_{\sigma,i}}\right]^{-1}_{-\varpi_i}$. Now the $\mu$-multiplicity in $A^\pm_\sigma$ is not equal to $0$ only if $$\label{eq:condition-rem-F}
 k\sigma\lambda+\sum_{\stackrel{(\alpha,\sigma\lambda)>0}{\pm(\alpha,\varpi_i)>0}}
\alpha+\delta
-\mu \in \sum_{k\neq i}{\ensuremath{\mathbb{Z}}}\alpha_k \quad {\rm with}\quad \pm(\delta,\varpi_i)\geq 0.$$ Condition (\[eq:condition-rem-F\]) implies that $[\mu]_i\geq k[\sigma\lambda]_i+s^+_{\sigma,i}$ or $[\mu]_i\leq k[\sigma\lambda]_i- 
s^-_{\sigma,i}$. Finally we have prove that ${\ensuremath{\hbox{\rm m}}}^\lambda_{{\ensuremath{\mathfrak{c}}}_+}(\mu,k)=
{\ensuremath{\hbox{\rm m}}}^\lambda_{{\ensuremath{\mathfrak{c}}}_-}(\mu,k)$ if $$-s^-_{\sigma,i}<[\mu]_i - k[\sigma\lambda]_i<s^+_{\sigma,i}$$ for all $\sigma\in{\ensuremath{\mathcal{A}}}({\ensuremath{\mathfrak{c}}}_+,{\ensuremath{\mathfrak{c}}}_-)$. $\Box$

The case of $\hbox{\rm SU}(n)$ {#subsec:su-n}
------------------------------

Let $T$ be the maximal torus of $\hbox{\rm SU}(n)$ consisting of the diagonal matrices. The dual ${\ensuremath{\mathfrak{t}}}^*$ can be identified with the subspace $x_1 +\cdots +x_n=0$ of ${\ensuremath{\mathbb{R}}}^n$. The roots are ${\ensuremath{\mathfrak{R}}}=\{e_i-e_j\, \vert 1\leq i\neq j\leq n\}$ and we will choose the positives ones to be ${\ensuremath{\mathfrak{R}}}^+=\{e_i-e_j\, \vert \, 1\leq i<j\leq n\}$. The simple roots are then $\alpha_i=e_i -e_{i+1}$, for $1\leq i\leq n-1$, and for these simple roots, the fundamental weights are $$\label{eq:omega-su-n}
  \omega_k=\frac{1}{n}(\underbrace{n-k,n-k,\cdots,n-k}_{k\ {\rm times}}, 
\underbrace{-k,-k,\cdots,-k}_{n-k\ {\rm times}}),\quad 1\leq k\leq n-1.$$

Consider now the coadjoint orbit $O_\lambda$ for $\lambda\in{\ensuremath{\mathfrak{t}}}^*$. Let $\Phi:O_\lambda\to{\ensuremath{\mathfrak{t}}}^*$ the moment map associated to the Hamiltonian action of $T$ on $O_\lambda$. The center of $\hbox{\rm SU}(n)$, that we denote ${\ensuremath{\mathbb{Z}}}_n$ corresponds to the set of matrices $zI$ with $z^n=1$. Recall the following well-known fact.

Let $\xi$ be a regular value of $\Phi:O_\lambda\to{\ensuremath{\mathfrak{t}}}^*$. Then for every $m\in \Phi^{-1}(\xi)$ the stabilizer subgroup $T_m:=\{t\in T\,\vert\, t\cdot m=m\}$ is equal to ${\ensuremath{\mathbb{Z}}}_n$.

[*Proof.*]{} Since $\xi$ is a regular value, we know that $T_m$ is finite for every $m\in \Phi^{-1}(\xi)$. The dual of the Lie algebra ${\ensuremath{\mathfrak{su}}}(n)$ decomposes as ${\ensuremath{\mathfrak{su}}}(n)^*={\ensuremath{\mathfrak{t}}}^*\oplus\sum_{\alpha\in{\ensuremath{\mathfrak{R}}}^+}{\ensuremath{\mathfrak{su}}}(n)^*_\alpha$ where ${\ensuremath{\mathfrak{su}}}(n)^*_\alpha\simeq{\ensuremath{\mathbb{C}}}_{-\alpha}$ as $T$-module. For $m\in\Phi^{-1}(\xi)$, we have $m=m_0 + \sum_{\alpha\in{\ensuremath{\mathfrak{R}}}^+}m_\alpha$ with $m_\alpha\in {\ensuremath{\mathfrak{su}}}(n)^*_\alpha$, and then $T_m=\cap_{m_\alpha\neq 0}\ker(t\mapsto t^\alpha)$. So the lattice $\Lambda^*_m$ generated by the set $\{\alpha\in{\ensuremath{\mathfrak{R}}}^+\,\vert\, m_\alpha\neq 0\}$ is a subgroup of $\Lambda^*_R$ with $\Lambda^*_R/\Lambda^*_m$ finite. We have to show that $\Lambda^*_m=\Lambda^*_R$. For this purpose we introduce the following equivalence relation on $\{1,\ldots, n\}$: $$i\sim j\Longleftrightarrow  e_i-e_j\in \Lambda^*_m.$$ Suppose that $\{1,\ldots, n\}/\sim$ is not reduced to a point: let $C_1$ and $C_2$ be two distinct equivalent classes and let $\beta=(\beta_1,\ldots,\beta_n)$ be the element of ${\ensuremath{\mathfrak{t}}}^*$ defined by: $\beta_i=\frac{1}{\vert C_1\vert}$ if $i\in C_1$, $\beta_i=\frac{-1}{\vert C_2\vert}$ if $i\in C_2$, and $\beta_i=0$ in the other cases. We see then that $(\beta,\alpha)=0$ for all $\alpha\in \Lambda^*_m$: it is in contradiction with the fact that $\Lambda^*_R/\Lambda^*_m$ is finite. We have proved that $e_i-e_j\in \Lambda^*_m$ for all $i,j\in\{1,\ldots, n\}$. $\Box$

Suppose now that $\lambda$ is a positive weight, and let ${\ensuremath{\mathfrak{c}}}$ a connected component of regular values of $\Phi:O_\lambda\to{\ensuremath{\mathfrak{t}}}^*$. We know that the corresponding periodic polynomial ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}}^\lambda : \Lambda^* \times {\ensuremath{\mathbb{Z}}}\longrightarrow  {\ensuremath{\mathbb{Z}}}$ is supported on the sub-lattice $\Xi_\lambda:=\{(\mu,k)\, \vert\, 
\mu\in k\lambda +\Lambda^*_R\}$.

The map ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}}^\lambda : \Xi_\lambda 
\longrightarrow  {\ensuremath{\mathbb{Z}}}$ is a polynomial of degree $\frac{(n-1)(n-2)}{2}-d_\lambda$, where $d_\lambda$ is the number of positive roots orthogonal to $\lambda$. .

[*Proof.*]{} Take $\xi\in{\ensuremath{\mathfrak{c}}}$. Following Proposition \[prop:m-poly\], the periodic-polynomial ${\ensuremath{\hbox{\rm m}}}^\lambda_{{\ensuremath{\mathfrak{c}}}}$ is defined by ${\ensuremath{\hbox{\rm m}}}^\lambda_{{\ensuremath{\mathfrak{c}}}}(\mu,k)=RR((O_\lambda)_\xi, {\ensuremath{\mathcal{L}}}^k_{\xi,\mu})$ for all $(\mu,k)\in\Xi_\lambda$. Here $(O_\lambda)_\xi=\Phi^{-1}(\xi)/T$ is a [*smooth*]{} manifold, and the line bundle ${\ensuremath{\mathcal{L}}}^k_{\xi,\mu}=(L^{\otimes k}\vert_{\Phi^{-1}(\xi)}\otimes{\ensuremath{\mathbb{C}}}_{-\mu})/T$ is also [*smooth*]{} since the center ${\ensuremath{\mathbb{Z}}}_n$ acts trivially on $L^{\otimes k}\vert_{\Phi^{-1}(\xi)}\otimes{\ensuremath{\mathbb{C}}}_{-\mu}$. Now the Atiyah-Singer integral formula for the Riemann-Roch number $RR((O_\lambda)_\xi, {\ensuremath{\mathcal{L}}}^k_{\xi,\mu})$ shows that ${\ensuremath{\hbox{\rm m}}}^\lambda_{{\ensuremath{\mathfrak{c}}}}$ is a polynomial of degree $\frac{\dim (O_\lambda)_\xi}{2} =\frac{\dim O_\lambda}{2} -(n-1)=
\frac{(n-1)(n-2)}{2}-d_\lambda$. $\Box$

Now we rewrite Theorem \[theo:mult-coadjoint\] for the group $\SU(n)$. Let $\lambda=(\lambda_1\geq\cdots\geq\lambda_n)$ be a positive weight and let ${\ensuremath{\mathfrak{c}}}_+$ and ${\ensuremath{\mathfrak{c}}}_-$ be two adjacent connected components of regular values of $\Phi:O_\lambda\to{\ensuremath{\mathfrak{t}}}^*$ separated by an hyperplane orthogonal to a fundamental weight $\varpi_i$: the vector $\varpi_i$ is pointing out of ${\ensuremath{\mathfrak{c}}}_-$. Let $q(\xi)=(\varpi_i,\xi) -r_i$ be the defining equation of this hyperplane.

The conditions $(e_k-e_l,\sigma\lambda)>0$ and $(e_k-e_l,\varpi_i)>0$ are respectively equivalent to $\lambda_{\sigma(k)}>\lambda_{\sigma(l)}$ and $k\leq i<
l$. For $\SU(n)$, the number $[\alpha]_i$ is equal to $0,1$ or $-1$ for any roots $\alpha$ and any $i=1,\cdots,n-1$. Hence for every $\sigma\in{\ensuremath{\mathcal{A}}}({\ensuremath{\mathfrak{c}}}_+,{\ensuremath{\mathfrak{c}}}_-)$, the integers $s_{\sigma,i}^-,s_{\sigma,i}^+\geq 0$ defined in Def. (\[def:s-sigma-pm\]) are equal to $$\label{eq:s-plus-su-n}
  s_{\sigma,i}^+={\rm rk}_{\ensuremath{\mathbb{C}}}(N_{\sigma,i}^{+})=
\sharp\{k\leq i< l\quad {\rm such\ that}\quad 
\lambda_{\sigma(k)}>\lambda_{\sigma(l)} \},$$ $$\label{eq:s-moins-su-n}
  s_{\sigma,i}^-={\rm rk}_{\ensuremath{\mathbb{C}}}( N_{\sigma,i}^{-})=
\sharp\{k\leq i< l\quad {\rm such\ that}\quad 
\lambda_{\sigma(k)}<\lambda_{\sigma(l)} \},$$ and the sum $s_{\sigma,i}^+ + s_{\sigma,i}^-$ is equal to half of the codimension of $K^i\cdot\sigma\lambda$ in $K\cdot\lambda$, that is $s_{\sigma,i}^+ + s_{\sigma,i}^-=i(n-i)-
\dim(K^{\sigma\lambda}/K^i\cap K^{\sigma\lambda})/2$.

Now we precise the results of [@BGR03].

\[th:SU-N\] The polynomial ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_-}^\lambda-{\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_+}^\lambda: 
\Xi_\lambda\to{\ensuremath{\mathbb{Z}}}$ is divible by the linear factors $$(q-s_i^-+1),(q-s_i^-+2),\ldots,q,\ldots,(q+s_i^+ -2),(q+s_i^+ -1),$$ where $q$ is the defining equation of the hyperplane separating ${\ensuremath{\mathfrak{c}}}_\pm$ and $s_i^\pm=$ $\inf_{\sigma\in{\ensuremath{\mathcal{A}}}({\ensuremath{\mathfrak{c}}}_+,{\ensuremath{\mathfrak{c}}}_-)}s_{\sigma,i}^\pm$. Moreover the linear factors $(q-s_i^-)$ and $(q-s_i^-)$ [*do not divide*]{} ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_-}^\lambda-{\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_+}^\lambda$.

[*Proof.*]{} The first part is the translation of Theorem \[theo:mult-coadjoint\]. We have just to prove that the linear factors $(q-s_i^-)$ and $(q-s_i^-)$ do not divide ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_-}^\lambda-{\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_+}^\lambda$. This point is a direct application of Proposition \[prop:inegalite-optimale\]. The only fact we use here is that ${\rm rk}_{\ensuremath{\mathbb{C}}}( N_{\sigma,i}^{\pm})=s_{\sigma,i}^\pm$. So the number ${\rm rk}_{\ensuremath{\mathbb{C}}}( N_{\sigma,i}^{\pm})$ is constant for all $\sigma\in{\ensuremath{\mathcal{A}}}({\ensuremath{\mathfrak{c}}}_+,{\ensuremath{\mathfrak{c}}}_-)$ for which $s_{\sigma,i}^\pm=s_{i}^\pm$. $\Box$

We rewrite now Theorem \[th:SU-N\] in the particular case where ${\ensuremath{\mathcal{A}}}({\ensuremath{\mathfrak{c}}}_+,{\ensuremath{\mathfrak{c}}}_-)$ contains just one element: it happens when $\lambda$ is a “generic” positive weight (see Definition \[def:lambda-gene\]), or when ${\ensuremath{\mathfrak{c}}}_+$ does not intersect $\Phi(O_\lambda)$. Here a positive weight $\lambda=(\lambda_1\geq\cdots\geq\lambda_n)$ is “generic” if for every couple of permutations $\sigma,\sigma'$ and any $k=1,\cdots,n-1$, we have $$\sum_{i=1}^k\lambda_{\sigma(i)}\neq\sum_{i=1}^k\lambda_{\sigma'(i)}$$ when $(\lambda_{\sigma(1)},\cdots,\lambda_{\sigma(n)})\notin {\ensuremath{\mathfrak{S}}}_k\times{\ensuremath{\mathfrak{S}}}_{n-k}
(\lambda_{\sigma'(1)},\cdots,\lambda_{\sigma'(n)})$.

Let $\lambda$ be a [*regular*]{} weight. Let ${\ensuremath{\mathfrak{c}}}_+$ and ${\ensuremath{\mathfrak{c}}}_-$ be two adjacent connected components of regular values of $\Phi:O_\lambda\to{\ensuremath{\mathfrak{t}}}^*$ and suppose that ${\ensuremath{\mathcal{A}}}({\ensuremath{\mathfrak{c}}}_+,{\ensuremath{\mathfrak{c}}}_-)$ contains just one element $\sigma$. Then the polynomial ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_-}^\lambda-{\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_+}^\lambda: 
\Xi_\lambda\to{\ensuremath{\mathbb{Z}}}$ is divible by the $i(n-i)$ linear factors $$(q-s_i^-+1),(q-s_i^-+2),\ldots,q,\ldots,(q+s_i^+ -2),(q+s_i^+ -1),$$ where $s_i^\pm=s_{\sigma,i}^\pm$ are defined by (\[eq:s-plus-su-n\]) and (\[eq:s-moins-su-n\]). Moreover the linear factors $(q-s_i^-)$ and $(q-s_i^+)$ [*do not divide*]{} ${\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_-}^\lambda-{\ensuremath{\hbox{\rm m}}}_{{\ensuremath{\mathfrak{c}}}_+}^\lambda$.

Vector partition functions {#sec:cas-C-d}
==========================

Let $T$ be a torus with Lie algebra ${\ensuremath{\mathfrak{t}}}$ and let $\Lambda^*\subset{\ensuremath{\mathfrak{t}}}^*$ be the weight lattice. Let $R=\{\alpha_1,\ldots,\alpha_d\}$ be a subset of not necessarily distint elements of $\Lambda^*$ which lie enterely in an open halfspace of ${\ensuremath{\mathfrak{t}}}^*$. We associate with the collection $R$ a function $$N_R: \Lambda^*\longrightarrow {\ensuremath{\mathbb{N}}}$$ called the [*vector partition function*]{} associated to $R$. By definition, for a weight $\mu$, the value $N_R(\mu)$ is the number of solutions of the equation $$\label{eq:partition-function}
  \sum_{j=1}^d k_j\alpha_j=\mu,\quad k_j\in{\ensuremath{\mathbb{Z}}}^{\geq 0},\quad j=1,\ldots,d.$$

Let $C(R)\subset{\ensuremath{\mathfrak{t}}}^*$ be the closed convex cone generated by the elements of $R$, and denote by $\Lambda^*_R\subset\Lambda^*$ the sublattice generated by $R$. Obviously, $N_R(\mu)$ vanishes if $\mu$ does not belong to $C(R)\cap\Lambda^*_R$.

Suppose now that $R$ generates the vector space ${\ensuremath{\mathfrak{t}}}^*$. Following [@Szenes-Vergne02], we will call a vector [*singular*]{} with respect to $R$ if it is in a cone $C(\nu)$ generated by a subset $\nu\in R$ of cardinality strictly less than $dim T$. The connected components of ${\ensuremath{\mathfrak{t}}}^*-\{singular\  vectors\}$ are called [*conic chambers*]{}. The periodic polynomial behavior of $N_R$ on closures of conic chambers of the cone $C(R)$ is proved in [@Sturmfels95]. We have the following refinement due to Szenes and Vergne [@Szenes-Vergne02]. Let us introduce the convex polytope $$\label{eq:box-phi}
  \Box(\Phi)=\sum_{j=1}^d[0,1]\alpha_j.$$ We remark that ${\ensuremath{\mathfrak{c}}}-\Box(\Phi)$ is a neighborhood of $\overline{{\ensuremath{\mathfrak{c}}}}$ for any conic chamber ${\ensuremath{\mathfrak{c}}}$ of the cone $C(R)$. We have the following qualitative result.

\[prop:N-conic-chamber-SV\] Let ${\ensuremath{\mathfrak{c}}}$ be a conic chamber of the cone $C(R)$. There exists a periodic polynomial $P_{\ensuremath{\mathfrak{c}}}$ on $\Lambda^*$ such that for each $\mu\in {\ensuremath{\mathfrak{c}}}-\Box(\Phi)$, we have $$N_R(\mu)=P_{\ensuremath{\mathfrak{c}}}(\mu).$$

In Section \[proof:N-conic-chamber-SV\] we will give another proof of Theorem \[prop:N-conic-chamber-SV\]. Let ${\ensuremath{\mathfrak{c}}}_\pm\subset{\ensuremath{\mathfrak{t}}}^*$ be two adjacent conic chambers. The aim of this Section is to give a formula for the periodic polynomial $P_{{\ensuremath{\mathfrak{c}}}_+}-P_{{\ensuremath{\mathfrak{c}}}_-}$.

Let ${\ensuremath{\Delta}}\subset{\ensuremath{\mathfrak{t}}}^*$ be the hyperplane that separates ${\ensuremath{\mathfrak{c}}}_+$ and ${\ensuremath{\mathfrak{c}}}_-$. Let $\beta\in{\ensuremath{\mathfrak{t}}}$ be such that ${\ensuremath{\Delta}}=\{\xi\in{\ensuremath{\mathfrak{t}}}^* \ \vert\ \langle\xi,\beta\rangle=0\}$ and ${\ensuremath{\mathfrak{c}}}_\pm\subset \{\xi\in{\ensuremath{\mathfrak{t}}}^* \ \vert\ \pm\langle\xi,\beta\rangle>0\}$. Note that the vector space ${\ensuremath{\Delta}}$ is generated by $R\cap{\ensuremath{\Delta}}$. We will now polarize the elements of $R$ that are outside ${\ensuremath{\Delta}}$. We define $$\label{eq:R-delta-prime}
  R'=\{\epsilon_j\alpha_j\ \vert\ \langle\alpha_j,\beta\rangle\neq 0
\ {\rm and}\ \epsilon_j={\rm sign}\ 
\langle\alpha_j,\beta\rangle\},$$ $$\label{eq:delta-pm}
  \delta^\pm=\sum_{\pm\langle\alpha_j,\beta\rangle>0}\alpha_j\ ,$$ and $$\label{eq:r-pm}
  r^\pm= \sharp\{j\ \vert \ \pm\langle\alpha_j,\beta\rangle>0\}.$$

We now look at the vector space ${\ensuremath{\Delta}}$ equipped with the subset $R\cap{\ensuremath{\Delta}}\subset\Lambda^*\cap{\ensuremath{\Delta}}$ which lie enterely in an open halfspace: let $N_{R\cap{\ensuremath{\Delta}}}:\Lambda^*\cap{\ensuremath{\Delta}}\to{\ensuremath{\mathbb{N}}}$ be the corresponding vector partition function. Let ${\ensuremath{\mathfrak{c}}}'$ be the cone in ${\ensuremath{\Delta}}$ which is equal to the relative interior of $\overline{{\ensuremath{\mathfrak{c}}}_+}\cap\overline{{\ensuremath{\mathfrak{c}}}_-}$. It is easy to see that ${\ensuremath{\mathfrak{c}}}'$ is a conic chamber in ${\ensuremath{\Delta}}$ with respect to $R\cap{\ensuremath{\Delta}}$. Following Proposition \[prop:N-conic-chamber-SV\] there exists a periodic polynomial $P_{{\ensuremath{\mathfrak{c}}}'}$ on $\Lambda^*\cap{\ensuremath{\Delta}}$ such that for each $\mu\in\overline{{\ensuremath{\mathfrak{c}}}'}\cap\Lambda^*$, we have $$N_{R\cap{\ensuremath{\Delta}}}(\gamma)=P_{{\ensuremath{\mathfrak{c}}}'}(\gamma).$$

Let $N_{R'}:\Lambda^*\to{\ensuremath{\mathbb{N}}}$ be the vector partition function associated to the polarized set of weight $R'$ (see (\[eq:R-delta-prime\])). The main result of this Section is the following

\[theo:cas-C-d\] The periodic polynomial $P_{{\ensuremath{\mathfrak{c}}}_+}-P_{{\ensuremath{\mathfrak{c}}}_-}:\Lambda^*\to {\ensuremath{\mathbb{Z}}}$ satisfies $$\label{eq:P-pm-C-d}
P_{{\ensuremath{\mathfrak{c}}}_+}(\mu)-P_{{\ensuremath{\mathfrak{c}}}_-}(\mu)=\sum_{\gamma\in\Lambda^*\cap{\ensuremath{\Delta}}}
D(\mu-\gamma)P_{{\ensuremath{\mathfrak{c}}}'}(\gamma),\quad \mu\in\Lambda^*,$$ where $D:\Lambda^*\to {\ensuremath{\mathbb{Z}}}$ is defined by $$D(\mu)=(-1)^{r^-}N_{R'}(\mu+\delta^-)-(-1)^{r^+}N_{R'}(-\mu-\delta^+).$$

The proof of Theorem \[theo:cas-C-d\] will be given in Section \[proof-theo-cas-C-d\].

$P_{{\ensuremath{\mathfrak{c}}}_+}(\mu)=P_{{\ensuremath{\mathfrak{c}}}_-}(\mu)$ for all the weights $\mu\in\Lambda^*$ satisfying the condition $$- \langle\delta^+,\beta\rangle<\langle\mu,\beta\rangle
< -\langle\delta^-,\beta\rangle.$$ The former ineqalities are optimal since $$\left(P_{{\ensuremath{\mathfrak{c}}}_+}-P_{{\ensuremath{\mathfrak{c}}}_-}\right)(-\delta^- +\gamma)=
(-1)^{r^-}P_{{\ensuremath{\mathfrak{c}}}'}(\gamma)$$ and $$\left(P_{{\ensuremath{\mathfrak{c}}}_+}-P_{{\ensuremath{\mathfrak{c}}}_-}\right)(-\delta^+ +\gamma)=
(-1)^{1+r^+}P_{{\ensuremath{\mathfrak{c}}}'}(\gamma)$$ for all $\gamma\in\Lambda^*\cap{\ensuremath{\Delta}}$.

[*Proof.*]{} In (\[eq:P-pm-C-d\]), the term $D(\mu-\gamma)P_{{\ensuremath{\mathfrak{c}}}'}(\gamma)$ does not vanish only if $\mu-\gamma\in -\delta^- + C(R')$ or $-(\mu-\gamma)\in \delta^+ + C(R')$ for some $\gamma\in C(R\cap{\ensuremath{\Delta}})$. These two conditions impose respectively that $\langle\mu,\beta\rangle\geq -\langle\delta^-,\beta\rangle$ and $\langle\mu,\beta\rangle\leq -\langle\delta^+,\beta\rangle$. If one take $\mu=-\delta^- +\gamma$ with $\gamma\in\Lambda^*\cap{\ensuremath{\Delta}}$, (\[eq:P-pm-C-d\]) becomes $(P_{{\ensuremath{\mathfrak{c}}}_+}-P_{{\ensuremath{\mathfrak{c}}}_-})(-\delta^- +\gamma)=
\sum_{\gamma'\in\Lambda^*\cap{\ensuremath{\Delta}}}D(-\delta^- +\gamma-\gamma')
P_{{\ensuremath{\mathfrak{c}}}'}(\gamma')$ with $$D(-\delta^- +\gamma-\gamma')=
(-1)^{r^-}N_{R'}(\gamma  -\gamma')-(-1)^{r^+}N_{R'}(\delta^- -\delta^+ 
-\gamma+\gamma').$$ Since the cone $C(R')$ intersects ${\ensuremath{\Delta}}$ only at $\{0\}$, $N_{R'}(\gamma-\gamma')=0$ if $\gamma\neq\gamma'$. Since $\langle\delta^- -\delta^+,\beta\rangle< 0$ we always have $N_{R'}(\delta^- -\delta^+ -\gamma+\gamma')=0$. We get finally that $(P_{{\ensuremath{\mathfrak{c}}}_+}-P_{{\ensuremath{\mathfrak{c}}}_-})(-\delta^- +\gamma)=
(-1)^{r^-}P_{{\ensuremath{\mathfrak{c}}}'}(\gamma)$. One can show in the same way that $(P_{{\ensuremath{\mathfrak{c}}}_+} -P_{{\ensuremath{\mathfrak{c}}}_-})(-\delta^+ +\gamma)=
-(-1)^{r^+}P_{{\ensuremath{\mathfrak{c}}}'}(\gamma)$. $\Box$

Quantization of ${\ensuremath{\mathbb{C}}}^d$
---------------------------------------------

We consider the complex vector space ${\ensuremath{\mathbb{C}}}^d$ equipped with the canonical symplectic form $\Omega=\frac{i}{2}\sum_{i=1}^d dz_j\wedge d\overline{z}_j$. The standard complex struture $J$ on ${\ensuremath{\mathbb{C}}}^d$ is compatible with $\Omega$. Let $T$ be a torus, let $\alpha_j\in{\ensuremath{\mathfrak{t}}}^*$, $j=1,\ldots,d$ be weights of $T$, and let $T$ acts on ${\ensuremath{\mathbb{C}}}^d$ as $$\label{eq:action-T-C}
  t\cdot(z_1,\ldots,z_d)=(t^{-\alpha_1}z_1,\ldots,t^{-\alpha_d}z_d).$$ The action of $T$ preserve the symplectic form $\Omega$ and the moment map associated with this action is $$\label{eq:moment-C}
  \Phi(z)= \frac{1}{2}\sum_{i=1}^d \vert z_j\vert^2 \alpha_j.$$

The pre-quantization data $(L,\langle,\rangle,\nabla)$ on the Hamiltonian $T$-manifold $({\ensuremath{\mathbb{C}}}^d,\Omega,\Phi)$ is a trivial line bundle $L$ with a trivial action of $T$ equipped with the Hermitian structure $\langle s,s'\rangle_z= 
e^{\frac{-\vert z\vert^2}{2}}s\overline{s'}$ and the Hermitian connexion $\nabla=d-\beta$ where $\beta=
\frac{1}{2}\sum_{i=1}^d \overline{z}_j dz_j$.

The quantization of the Hamiltonian $T$-manifold $({\ensuremath{\mathbb{C}}}^d,\Omega)$, that we denote ${\ensuremath{\mathcal{Q}}}^T({\ensuremath{\mathbb{C}}}^d)$, is the [*Bargman space*]{} of entire holomorphic functions on ${\ensuremath{\mathbb{C}}}^d$ which are ${\ensuremath{\mathcal{L}}}^2$ integrable with respect to the Gaussian measure $e^{\frac{-\vert z\vert^2}{2}}\Omega^d$.

We suppose now that the set of weights $R=\{\alpha_1,\ldots,\alpha_d\}$ is [*polarized*]{} by $\eta\in{\ensuremath{\mathfrak{t}}}$, which means that $\langle\alpha_j,\eta\rangle >0$ for all $j$. The $T$-representation ${\ensuremath{\mathcal{Q}}}^T({\ensuremath{\mathbb{C}}}^d)$ is then admissible and we have the following equality in $R^{-\infty}(T)$: $$\label{eq:Q-T-C}
 {\ensuremath{\mathcal{Q}}}^T({\ensuremath{\mathbb{C}}}^d)=\sum_{\mu\in\Lambda^*} N_R(\mu)\, {\ensuremath{\mathbb{C}}}_\mu,$$ where $N_R: \Lambda^*\to {\ensuremath{\mathbb{N}}}$ is the vector partition function associated to $R$. In other words, the generalized character of ${\ensuremath{\mathcal{Q}}}^T({\ensuremath{\mathbb{C}}}^d)$ coincides with the generalized character of the symmetric algebra $S^\bullet(\overline{{\ensuremath{\mathbb{C}}}^d})$, where $\overline{{\ensuremath{\mathbb{C}}}^d}$ means ${\ensuremath{\mathbb{C}}}^d$ with the opposite complex structure.

For the remaining part of Section \[sec:cas-C-d\], we assume that the set of weights $R=\{\alpha_1,\ldots,\alpha_d\}$ is [ *polarized*]{}, and [*generate*]{} the vector space ${\ensuremath{\mathfrak{t}}}^*$. The first assumption is equivalent to the fact that the moment map $\Phi:{\ensuremath{\mathbb{C}}}^d\to{\ensuremath{\mathfrak{t}}}^*$ is [*proper*]{}, and the second assumption is equivalent to the fact that the generic stabiliser of $T$ on ${\ensuremath{\mathbb{C}}}^d$ is [*finite*]{}. Notice that the vectors of ${\ensuremath{\mathfrak{t}}}^*$ which are [*singular*]{} with respect to $R$ correspond to the [*singular values*]{} of $\Phi$.

In the next section we will show that ${\ensuremath{\mathcal{Q}}}^T({\ensuremath{\mathbb{C}}}^d)$, viewed as an element of $R^{-\infty}(T)$, can be realized as the index of transversally elliptic symbols on ${\ensuremath{\mathbb{C}}}^d$. After we will apply the techniques developped in Section \[sec:quantum-version\]. The main difference here is that we work with a [*non-compact*]{} manifold.

Transversally elliptic symbols on ${\ensuremath{\mathbb{C}}}^d$ {#subsec:C-d}
---------------------------------------------------------------

Let $p:{\ensuremath{\hbox{\bf T}}}{\ensuremath{\mathbb{C}}}^d\to{\ensuremath{\mathbb{C}}}^d$ be the canonical projection. The Thom symbol $${\ensuremath{\hbox{\rm Thom}}}({\ensuremath{\mathbb{C}}}^d)\in\Gamma\left({\ensuremath{\hbox{\bf T}}}{\ensuremath{\mathbb{C}}}^d,\hom(p^*(\wedge^{\rm even}_{\ensuremath{\mathbb{C}}}{\ensuremath{\hbox{\bf T}}}{\ensuremath{\mathbb{C}}}^d), 
p^*(\wedge^{\rm odd}_{\ensuremath{\mathbb{C}}}{\ensuremath{\hbox{\bf T}}}{\ensuremath{\mathbb{C}}}^d))\right)$$ is defined as follows. At $(z,v)\in{\ensuremath{\hbox{\bf T}}}{\ensuremath{\mathbb{C}}}^d$, the Thom symbol ${\ensuremath{\hbox{\rm Thom}}}({\ensuremath{\mathbb{C}}}^d)$ is equal to the Clifford map $$Cl(v):\wedge^{\rm even}_{\ensuremath{\mathbb{C}}}{\ensuremath{\mathbb{C}}}^d\longrightarrow \wedge^{\rm odd}_{\ensuremath{\mathbb{C}}}{\ensuremath{\mathbb{C}}}^d$$ which is defined by $Cl(v)w=v\wedge w -c(v)w$. Here $c(v):\wedge^{\bullet}_{\ensuremath{\mathbb{C}}}{\ensuremath{\mathbb{C}}}^d\mapsto
\wedge^{\bullet-1}_{\ensuremath{\mathbb{C}}}{\ensuremath{\mathbb{C}}}^d$ denotes the contraction map relatively to the standard Hermitian structure on ${\ensuremath{\mathbb{C}}}^d$. Obviously the symbol ${\ensuremath{\hbox{\rm Thom}}}({\ensuremath{\mathbb{C}}}^d)$ is not elliptic since its characteristic set is equal to the zero section in ${\ensuremath{\hbox{\bf T}}}{\ensuremath{\mathbb{C}}}^d$ (hence is not compact).

Now we deform the symbol ${\ensuremath{\hbox{\rm Thom}}}({\ensuremath{\mathbb{C}}}^d)$ in order to obtain transversally elliptic symbols. Since ${\ensuremath{\mathbb{C}}}^d$ can be realized as an open subset of a compact $T$-manifold we have a well defined index map $${\ensuremath{\hbox{\rm Index}}}^T_{{\ensuremath{\mathbb{C}}}^d}:{\ensuremath{\hbox{\bf K}}}_T({\ensuremath{\hbox{\bf T}}}_T{\ensuremath{\mathbb{C}}}^d)\longrightarrow R^{-\infty}(T).$$

\[def:Thom-deforme\] For any $\eta\in{\ensuremath{\mathfrak{t}}}$, we define the symbol ${\ensuremath{\hbox{\rm Thom}}}^\eta({\ensuremath{\mathbb{C}}}^d)$ by $${\ensuremath{\hbox{\rm Thom}}}^\eta({\ensuremath{\mathbb{C}}}^d)(z,v)={\ensuremath{\hbox{\rm Thom}}}({\ensuremath{\mathbb{C}}}^d)(z,v-\eta_{{\ensuremath{\mathbb{C}}}^d}(z)), 
\quad (z,v)\in{\ensuremath{\hbox{\bf T}}}{\ensuremath{\mathbb{C}}}^d,$$ where $\eta_{{\ensuremath{\mathbb{C}}}^d}$ is the vectors field on ${\ensuremath{\mathbb{C}}}^d$ generated by $\eta$.

The symbols ${\ensuremath{\hbox{\rm Thom}}}^\eta({\ensuremath{\mathbb{C}}}^d)$ were studied in [@pep4]. It is easy to see that ${\ensuremath{\hbox{\rm Thom}}}^\eta({\ensuremath{\mathbb{C}}}^d)$ is tranversally elliptic if and only if the vector subspace $({\ensuremath{\mathbb{C}}}^d)^\eta$ is reduced to $\{0\}$, i.e. if $\langle\alpha_j,\eta\rangle\neq 0$ for all $j=1,\ldots,d$. We prove in Proposition 5.4. of [@pep4] that $$\label{eq:indice-thom-eta}
 {\ensuremath{\hbox{\rm Index}}}^T_{{\ensuremath{\mathbb{C}}}^d}\left({\ensuremath{\hbox{\rm Thom}}}^\eta({\ensuremath{\mathbb{C}}}^d)\right)=
S^\bullet(\overline{{\ensuremath{\mathbb{C}}}^d})\quad {\rm in}
\quad R^{-\infty}(T),$$ when $\langle\alpha_j,\eta\rangle> 0$ for all $j=1,\ldots,d$.

In order to compute the multiplicities $N_R(\mu)$ of ${\ensuremath{\mathcal{Q}}}^T({\ensuremath{\mathbb{C}}}^d)$ we introduce the following tranversally elliptic symbols. Take a scalar product $b(\cdot,\cdot)$ on ${\ensuremath{\mathfrak{t}}}^*$, and denote by $\xi\mapsto\xi^b,{\ensuremath{\mathfrak{t}}}^*\simeq{\ensuremath{\mathfrak{t}}}$ the induced isomorphism. For each $\xi\in{\ensuremath{\mathfrak{t}}}^*$, the Hamiltonian vectors field of the function $\frac{-1}{2}\|\Phi-\xi\|^2_b$ is the vectors field $$z\mapsto \Big( (\Phi(z)-\xi)^b\Big)_{{\ensuremath{\mathbb{C}}}^d}(z).$$ that we denote ${\ensuremath{\mathcal{H}}}^b-\xi^b_{{\ensuremath{\mathbb{C}}}^d}$.

\[def:Thom-deforme-phi\] For any $\xi\in{\ensuremath{\mathfrak{t}}}^*$, and any scalar product $b(\cdot,\cdot)$ on ${\ensuremath{\mathfrak{t}}}^*$, we define the symbol ${\ensuremath{\hbox{\rm Thom}}}_{\xi,b}({\ensuremath{\mathbb{C}}}^d)$ by $${\ensuremath{\hbox{\rm Thom}}}_{\xi,b}({\ensuremath{\mathbb{C}}}^d)(z,v)={\ensuremath{\hbox{\rm Thom}}}({\ensuremath{\mathbb{C}}}^d)(z,v-({\ensuremath{\mathcal{H}}}^b-\xi^b_{{\ensuremath{\mathbb{C}}}^d})(z)), 
\quad (z,v)\in{\ensuremath{\hbox{\bf T}}}{\ensuremath{\mathbb{C}}}^d.$$

Let ${\ensuremath{\hbox{\rm Char}}}({\ensuremath{\hbox{\rm Thom}}}_{\xi,b}({\ensuremath{\mathbb{C}}}^d))\subset{\ensuremath{\hbox{\bf T}}}{\ensuremath{\mathbb{C}}}^d$ be the characterictic set of ${\ensuremath{\hbox{\rm Thom}}}_{\xi,b}({\ensuremath{\mathbb{C}}}^d)$. We know that ${\ensuremath{\hbox{\rm Char}}}({\ensuremath{\hbox{\rm Thom}}}_{\xi,b}({\ensuremath{\mathbb{C}}}^d))\cap{\ensuremath{\hbox{\bf T}}}_T{\ensuremath{\mathbb{C}}}^d$ is equal to the critical set ${\ensuremath{\hbox{\rm Cr}}}(\|\Phi-\xi\|^2_b)$ of the function $\|\Phi-\xi\|^2_b:{\ensuremath{\mathbb{C}}}^d\to{\ensuremath{\mathbb{R}}}$ (see Section \[ssection.thom\]). A straightforward computation gives that $z\in{\ensuremath{\hbox{\rm Cr}}}(\|\Phi-\xi\|^2_b)$ if and only if $$\label{eq:critique-C-d}
  b(\Phi(z)-\xi,\alpha_j)\, z_j=0 \quad {\rm for\ all}
\quad j=1,\ldots,d.$$ The former relations implies in particular that $b(\Phi(z)-\xi,\Phi(z))=\frac{1}{2}\sum_j b(\Phi(z)-\xi,\alpha_j)\, 
\vert z_j\vert^2=0$. Hence $\|\Phi(z)\|^2_b=b(\Phi(z),\xi)$ which implies $$\label{eq:phi-less-xi}
  \|\Phi(z)\|_b\leq \|\xi\|_b.$$ Take now $\eta\in{\ensuremath{\mathfrak{t}}}$ such that $\langle\alpha_j,\eta\rangle> 0$ for all $j$, and let $\eta_b\in{\ensuremath{\mathfrak{t}}}^*$ such that $(\eta_b)^b=\eta$. We have then $$\label{eq:C-eta}
C_{\eta} \|z\|^2\leq \langle\Phi(z),\eta\rangle=b(\Phi(z),\eta_b)\leq 
\|\Phi(z)\|_b\|\eta_b\|_b$$ where $C_\eta=\frac{1}{2}\inf_j \langle\alpha_j,\eta\rangle$, and $z\mapsto\|z\|^2$ is the usual hermitian form on ${\ensuremath{\mathbb{C}}}^d$. With (\[eq:critique-C-d\]) and (\[eq:C-eta\]) we get the following

\[lem:crit-C-d\] The critical set ${\ensuremath{\hbox{\rm Cr}}}(\|\Phi-\xi\|^2_b)\subset{\ensuremath{\mathbb{C}}}^d$ is contained in the ball of radius $$\frac{\|\xi\|_b \,\|\eta_b\|_b}{C_\eta},$$ where $\eta\in{\ensuremath{\mathfrak{t}}}$ is such that $C_\eta=\frac{1}{2}\inf_j \langle\alpha_j,\eta\rangle >0$.

We have then proved that the symbols ${\ensuremath{\hbox{\rm Thom}}}_{\xi,b}({\ensuremath{\mathbb{C}}}^d)$ are transversally elliptic.

\[prop:indice-thom-xi\] The class of the transversally elliptic symbol ${\ensuremath{\hbox{\rm Thom}}}_{\xi,b}({\ensuremath{\mathbb{C}}}^d)$ in ${\ensuremath{\hbox{\bf K}}}_T({\ensuremath{\hbox{\bf T}}}_T{\ensuremath{\mathbb{C}}}^d)$ does not depend of the data $\xi,b$, and is equal to the class defined by ${\ensuremath{\hbox{\rm Thom}}}^\eta({\ensuremath{\mathbb{C}}}^d)$ where $\eta\in{\ensuremath{\mathfrak{t}}}$ is choosen so that $\langle\alpha_j,\eta\rangle >0$ for all $j$.

[*Proof*]{}. After Lemma \[lem:crit-C-d\], we know that for any scalar product $b(\cdot,\cdot)$ on ${\ensuremath{\mathfrak{t}}}^*$, the characteristic set of ${\ensuremath{\hbox{\rm Thom}}}_{0,b}({\ensuremath{\mathbb{C}}}^d)$ intersects ${\ensuremath{\hbox{\bf T}}}_T{\ensuremath{\mathbb{C}}}^d$ at $\{0\}$. If $b_0$ and $b_1$ are two scalar products on ${\ensuremath{\mathfrak{t}}}^*$ we consider the family $b_t=t b_1 +(1-t)b_0,\ 0\leq t\leq 1$, of scalar products on ${\ensuremath{\mathfrak{t}}}^*$. Hence ${\ensuremath{\hbox{\rm Thom}}}_{0,b_t}({\ensuremath{\mathbb{C}}}^d)$, $t\in[0,1]$, defines an homotopy of transversally elliptic symbols. We have proved that ${\ensuremath{\hbox{\rm Thom}}}_{0,b_0}({\ensuremath{\mathbb{C}}}^d)={\ensuremath{\hbox{\rm Thom}}}_{0,b_1}({\ensuremath{\mathbb{C}}}^d)$ in ${\ensuremath{\hbox{\bf K}}}_T({\ensuremath{\hbox{\bf T}}}_T{\ensuremath{\mathbb{C}}}^d)$ for any $\xi\in{\ensuremath{\mathfrak{t}}}^*$.

Fix now the scalar product $b$ and an element $\xi\in{\ensuremath{\mathfrak{t}}}^*$. For any $t\in[0,1]$ the characteristic set of ${\ensuremath{\hbox{\rm Thom}}}_{t\xi,b}({\ensuremath{\mathbb{C}}}^d)$ intersects ${\ensuremath{\hbox{\bf T}}}_T{\ensuremath{\mathbb{C}}}^d$ in the ball of radius $$\frac{\|\xi\|_{b} \,\|\eta_b\|_{b}}{C_\eta}.$$ Hence ${\ensuremath{\hbox{\rm Thom}}}_{t\xi,b}({\ensuremath{\mathbb{C}}}^d)$, $t\in[0,1]$, defines an homotopy of transversally elliptic symbols: ${\ensuremath{\hbox{\rm Thom}}}_{\xi,b}({\ensuremath{\mathbb{C}}}^d)={\ensuremath{\hbox{\rm Thom}}}_{0,b}({\ensuremath{\mathbb{C}}}^d)$ in ${\ensuremath{\hbox{\bf K}}}_T({\ensuremath{\hbox{\bf T}}}_T{\ensuremath{\mathbb{C}}}^d)$. We have proved that the class of the transversally elliptic symbol ${\ensuremath{\hbox{\rm Thom}}}_{\xi,b}({\ensuremath{\mathbb{C}}}^d)$ in ${\ensuremath{\hbox{\bf K}}}_T({\ensuremath{\hbox{\bf T}}}_T{\ensuremath{\mathbb{C}}}^d)$ does not depend of the data $\xi,b$.

Since the weights $\alpha_j$ lie enterely in an open halfspace of ${\ensuremath{\mathfrak{t}}}^*$, there exists a scalar product $b_+(\cdot,\cdot)$ on ${\ensuremath{\mathfrak{t}}}^*$ for which we have $$b_+(\alpha_i,\alpha_j)>0$$ for all $i,j=1,\ldots,d$. Let ${\ensuremath{\mathcal{H}}}^{b_+}$ be the Hamiltonian vectors field of the function $\frac{-1}{2}\|\Phi\|^2_{b_+}$, and let $\eta_{{\ensuremath{\mathbb{C}}}^d}$ be the vectors field on ${\ensuremath{\mathbb{C}}}^d$ generated by $\eta\in{\ensuremath{\mathfrak{t}}}$ such that $\langle\alpha_j,\eta\rangle >0$ for all $j$. A straightforward computation gives that $$\label{eq:H-eta-plus}
  ({\ensuremath{\mathcal{H}}}^{b_+}(z),\eta_{{\ensuremath{\mathbb{C}}}^d}(z))>0$$ for all non zero $z\in{\ensuremath{\mathbb{C}}}^d$. Consider now the following familly of symbols on ${\ensuremath{\mathbb{C}}}^d$ $$\sigma_t(z,v)={\ensuremath{\hbox{\rm Thom}}}({\ensuremath{\mathbb{C}}}^d)(z,v-(t{\ensuremath{\mathcal{H}}}^{b_+}+(1-t)\eta_{{\ensuremath{\mathbb{C}}}^d})(z)), 
\quad (z,v)\in{\ensuremath{\hbox{\bf T}}}{\ensuremath{\mathbb{C}}}^d.$$ so that $\sigma_0={\ensuremath{\hbox{\rm Thom}}}^\eta({\ensuremath{\mathbb{C}}}^d)$ and $\sigma_1={\ensuremath{\hbox{\rm Thom}}}_{0,b_+}({\ensuremath{\mathbb{C}}}^d)$. The inequality (\[eq:H-eta-plus\]) shows that ${\ensuremath{\hbox{\rm Char}}}(\sigma_t)\cap{\ensuremath{\hbox{\bf T}}}_T{\ensuremath{\mathbb{C}}}^d=\{0\}$ for all $t\in[0,1]$. Hence $\sigma_t$, $t\in[0,1]$, defines an homotopy of transversally elliptic symbols: ${\ensuremath{\hbox{\rm Thom}}}^\eta({\ensuremath{\mathbb{C}}}^d)=
{\ensuremath{\hbox{\rm Thom}}}_{0,b_+}({\ensuremath{\mathbb{C}}}^d)$ in ${\ensuremath{\hbox{\bf K}}}_T({\ensuremath{\hbox{\bf T}}}_T{\ensuremath{\mathbb{C}}}^d)$. $\Box$

For the remaining part of this paper we fix a scalar product on ${\ensuremath{\mathfrak{t}}}^*$, and we consider the family of transversally elliptic symbols ${\ensuremath{\hbox{\rm Thom}}}_{\xi}({\ensuremath{\mathbb{C}}}^d)$, $\xi\in{\ensuremath{\mathfrak{t}}}^*$ (to simplify, we do not mention the scalar product in the notation). Proposition \[prop:indice-thom-xi\] and (\[eq:indice-thom-eta\]) imply the following

\[prop:N-R-indice\] For every $\xi\in{\ensuremath{\mathfrak{t}}}^*$, ${\ensuremath{\mathcal{Q}}}^T({\ensuremath{\mathbb{C}}}^d)$ is equal to the generalized character ${\ensuremath{\hbox{\rm Index}}}^T_{{\ensuremath{\mathbb{C}}}^d}\left({\ensuremath{\hbox{\rm Thom}}}_{\xi}({\ensuremath{\mathbb{C}}}^d)\right)$.

Now we apply the techniques developped in Section \[sec:quantum-version\] in order to compute the multiplicities of ${\ensuremath{\hbox{\rm Index}}}^T_{{\ensuremath{\mathbb{C}}}^d}\left({\ensuremath{\hbox{\rm Thom}}}_{\xi}({\ensuremath{\mathbb{C}}}^d)\right)$.

Localization in a non-compact setting {#subsec:loc-non-compact}
-------------------------------------

Like in Section \[ssection.thom\] we start with the

\[def.thom.loc-C-d\] For any $\xi\in{\ensuremath{\mathfrak{t}}}^*$ and any $T$-invariant [*relatively compact*]{} open subset ${\ensuremath{\mathcal{U}}}\subset {\ensuremath{\mathbb{C}}}^d$ we define the symbol ${\ensuremath{\hbox{\rm Thom}}}_{\xi}({\ensuremath{\mathcal{U}}})$ by the relation $${\ensuremath{\hbox{\rm Thom}}}_{\xi}({\ensuremath{\mathcal{U}}})(z,v):={\ensuremath{\hbox{\rm Thom}}}({\ensuremath{\mathbb{C}}}^d)(z,v-({\ensuremath{\mathcal{H}}}-\xi_{{\ensuremath{\mathbb{C}}}^d})(z))
\quad (z,v)\in{\ensuremath{\hbox{\bf T}}}{\ensuremath{\mathcal{U}}}.$$

The symbol ${\ensuremath{\hbox{\rm Thom}}}_{\xi}({\ensuremath{\mathcal{U}}})$ is transversally elliptic when ${\ensuremath{\hbox{\rm Cr}}}(\parallel\Phi-\xi\parallel^2)\cap\partial{\ensuremath{\mathcal{U}}}=\emptyset$ (the couple $({\ensuremath{\mathcal{U}}},\xi)$ is called [*good*]{}) and we denote by $$RR_{{\ensuremath{\mathcal{U}}}}^{\xi}({\ensuremath{\mathbb{C}}}^d)\in R^{-\infty}(T)$$ its index. Proposition \[prop:U-xi\] is still valid here. In particular, for a good couple $({\ensuremath{\mathcal{U}}},\xi)$, we have $RR_{{\ensuremath{\mathcal{U}}}}^{\xi'}({\ensuremath{\mathbb{C}}}^d)=RR_{{\ensuremath{\mathcal{U}}}}^{\xi}({\ensuremath{\mathbb{C}}}^d)$ if $\xi'$ is close enough to $\xi$. Consider now the decomposition $${\ensuremath{\hbox{\rm Cr}}}(\parallel\Phi-\xi\parallel^2)=
\bigcup_{\gamma\in{\ensuremath{\mathcal{B}}}_\xi}({\ensuremath{\mathbb{C}}}^d)^\gamma\cap\Phi^{-1}(\gamma+\xi).$$ Here ${\ensuremath{\mathcal{B}}}_\xi\subset{\ensuremath{\mathfrak{t}}}^*$ is finite set since ${\ensuremath{\mathbb{C}}}^d$ has a finite number of stabilizer. Since $0\in({\ensuremath{\mathbb{C}}}^d)^\gamma$ and $z\mapsto \langle\Phi(z),\gamma\rangle$ is constant on $({\ensuremath{\mathbb{C}}}^d)^\gamma$, we have $$\label{eq:ps-gamma-xi}
  (\gamma +\xi,\gamma)=0$$ for all $\gamma\in{\ensuremath{\mathcal{B}}}_\xi$.

\[def:RR-xi-C\] For any $\xi\in{\ensuremath{\mathfrak{t}}}^*$ and $\gamma\in{\ensuremath{\mathcal{B}}}_\xi$, we denote simply by $$RR^\xi_\gamma({\ensuremath{\mathbb{C}}}^d)\in R^{-\infty}(T)$$ the generalized character $RR_{{\ensuremath{\mathcal{U}}}}^{\xi}({\ensuremath{\mathbb{C}}}^d)$, where ${\ensuremath{\mathcal{U}}}$ is a $T$-invariant relatively compact open neighborhood of $({\ensuremath{\mathbb{C}}}^d)^\gamma\cap\Phi^{-1}(\gamma+\xi)$ such that ${\ensuremath{\hbox{\rm Cr}}}(\parallel\Phi-\xi\parallel^2)\cap\overline{{\ensuremath{\mathcal{U}}}}=
({\ensuremath{\mathbb{C}}}^d)^\gamma\cap\Phi^{-1}(\gamma+\xi)$.

Since $RR_{{\ensuremath{\mathbb{C}}}^d}^{\xi}({\ensuremath{\mathbb{C}}}^d)$ is equal to ${\ensuremath{\mathcal{Q}}}^T({\ensuremath{\mathbb{C}}}^d)$ (see Proposition \[prop:N-R-indice\]), part $a)$ of Proposition \[prop:U-xi\] insures that we have the decomposition $${\ensuremath{\mathcal{Q}}}^T({\ensuremath{\mathbb{C}}}^d)=\sum_{\gamma\in{\ensuremath{\mathcal{B}}}_\xi} RR^\xi_\gamma({\ensuremath{\mathbb{C}}}^d).$$

Let ${\ensuremath{\mathfrak{c}}}\subset{\ensuremath{\mathfrak{t}}}^*$ be a conic chamber of the cone $C(R)$, and take $\xi$ in ${\ensuremath{\mathfrak{c}}}$. Then $\xi$ is a regular value of the moment map $\Phi:{\ensuremath{\mathbb{C}}}^d\to{\ensuremath{\mathfrak{t}}}^*$ defined in (\[eq:moment-C\]). Let $\Omega_\xi$ be the symplectic structure on the orbifold $({\ensuremath{\mathbb{C}}}^d)_\xi=\Phi^{-1}(\xi)/T$ that is induced from $\Omega$. The orbifold $({\ensuremath{\mathbb{C}}}^d)_\xi$ is also equipped with a [*complex structure*]{} $J_\xi$ that is induced from the standard complex structure on ${\ensuremath{\mathbb{C}}}^d$, in such a way that the orbifold $(({\ensuremath{\mathbb{C}}}^d)_\xi,\Omega_\xi,J_\xi))$ is a Kähler orbifold. If $\xi$ belongs to the lattice $\Lambda^*$, the reduced space $({\ensuremath{\mathbb{C}}}^d)_\xi$ is the Kähler toric variety corresponding to the polytope $\{s\in({\ensuremath{\mathbb{R}}}^{\geq 0})^d\ \vert\ \sum s_j\alpha_j=\xi\}$ of ${\ensuremath{\mathbb{R}}}^d$. For every $\mu\in\Lambda$ we consider the holomorphic orbifold line bundle $${\ensuremath{\mathcal{L}}}_{\xi,\mu}=(\Phi^{-1}(\xi)\times{\ensuremath{\mathbb{C}}}_{-\mu})/T$$ on $({\ensuremath{\mathbb{C}}}^d)_\xi$.

\[def:Pol-C-d\] The periodic polynomial $P_{\ensuremath{\mathfrak{c}}}:\Lambda^*\to{\ensuremath{\mathbb{Z}}}$ associated to the conic chamber ${\ensuremath{\mathfrak{c}}}$ is given by $$\label{eq:P-c-vector}
  P_{\ensuremath{\mathfrak{c}}}(\mu)=RR(({\ensuremath{\mathbb{C}}}^d)_\xi,{\ensuremath{\mathcal{L}}}_{\xi,\mu}),$$ where the right hand side is the Riemann-Roch number associated to the holomorphic orbifold line bundle ${\ensuremath{\mathcal{L}}}_{\xi,\mu}$.

Another way to define the periodic polynomial $P_{\ensuremath{\mathfrak{c}}}$ is to consider the generalized character $RR^\xi_0({\ensuremath{\mathbb{C}}}^d)$ for $\xi\in{\ensuremath{\mathfrak{c}}}$: here $\gamma=0$ parametrizes the component $\Phi^{-1}(\xi)$ of ${\ensuremath{\hbox{\rm Cr}}}(\parallel\Phi-\xi\parallel^2)$. Following (\[eq:RR-xi-O\]) we have $$\label{eq:pol-P-c}
 RR^\xi_0({\ensuremath{\mathbb{C}}}^d)=\sum_{\mu\in \Lambda^*}P_{\ensuremath{\mathfrak{c}}}(\mu)\,{\ensuremath{\mathbb{C}}}_\mu\quad 
{\rm in}\quad R^{-\infty}(T).$$ After Lemma \[lem:RR-xi-prime\], we know that $RR^{\xi'}_0({\ensuremath{\mathbb{C}}}^d)=RR^\xi_0({\ensuremath{\mathbb{C}}}^d)$ when $\xi,\xi'$ are two elements of ${\ensuremath{\mathfrak{c}}}$: hence the polynomial $P_{\ensuremath{\mathfrak{c}}}$ does not depend of the choice of $\xi$ in ${\ensuremath{\mathfrak{c}}}$.

Proof of Theorem \[prop:N-conic-chamber-SV\] {#proof:N-conic-chamber-SV}
--------------------------------------------

Consider a weight $\mu\in({\ensuremath{\mathfrak{c}}}-\Box(\Phi))\cap\Lambda^*$ of the form $\mu=\xi'-\sum_j
t_j\alpha_j$ with ${\xi'}\in{\ensuremath{\mathfrak{c}}}$ and $t_j\in[0,1]$. We start with the decomposition $${\ensuremath{\mathcal{Q}}}^T({\ensuremath{\mathbb{C}}}^d)=
\sum_{\gamma\in{\ensuremath{\mathcal{B}}}_{\xi'}} RR^{\xi'}_\gamma({\ensuremath{\mathbb{C}}}^d).$$ Since $N_R(\mu)$ and $P_{\ensuremath{\mathfrak{c}}}(\mu)$ are respectively the multiplicity of ${\ensuremath{\mathbb{C}}}_\mu$ in ${\ensuremath{\mathcal{Q}}}^T({\ensuremath{\mathbb{C}}}^d)$ and in $RR^{\xi'}_0({\ensuremath{\mathbb{C}}}^d)$, the proof will be complete if we show that the multiplicity of ${\ensuremath{\mathbb{C}}}_\mu$ in $RR^{\xi'}_\gamma({\ensuremath{\mathbb{C}}}^d)$ is equal to zero when $\gamma\neq 0$.

Consider a [*non-zero*]{} element $\gamma$ in ${\ensuremath{\mathcal{B}}}_{\xi'}$. For the character $RR^{\xi'}_\gamma({\ensuremath{\mathbb{C}}}^d)$ the localization (\[eq:loc-RR-beta-bis\]) gives $$\label{eq:loc-RR-gamma}
  RR^{\xi'}_\gamma({\ensuremath{\mathbb{C}}}^d)=
RR_0^{{\xi'}+\gamma}(({\ensuremath{\mathbb{C}}}^d)^\gamma)\otimes
\left[\wedge_{{\ensuremath{\mathbb{C}}}}^{\bullet}\overline{N}\right]^{-1}_{\gamma},$$ where $N=\sum_{(\alpha_j,\gamma)\neq
  0}{\ensuremath{\mathbb{C}}}_{-\alpha_j}$ corresponds to the normal bundle of $({\ensuremath{\mathbb{C}}}^d)^\gamma$ in ${\ensuremath{\mathbb{C}}}^d$. The inverse $\left[\wedge_{{\ensuremath{\mathbb{C}}}}^{\bullet}\overline{N}\right]^{-1}_{\gamma}$ is equal to $(-1)^l{\ensuremath{\mathbb{C}}}_{\delta(\gamma)}\otimes S^\bullet(N_{\ensuremath{\mathbb{C}}}^{+,\gamma})$ where $$\delta(\gamma)=-\sum_{(\alpha_j,\gamma)<0}\alpha_j.$$ Since $\gamma$ acts trivially on $({\ensuremath{\mathbb{C}}}^d)^\gamma$ all the weights $\mu'\in\Lambda^*$ that appear in $RR_0^{{\xi'}+\gamma}(({\ensuremath{\mathbb{C}}}^d)^\gamma)$ satisfy $(\mu',\gamma)=0$. Since the weights of $N_{\ensuremath{\mathbb{C}}}^{+,\gamma}$ are polarized by $\gamma$, we see from (\[eq:loc-RR-gamma\]) that all the weights $\mu'\in\Lambda^*$ that appear in $RR^{\xi'}_\gamma({\ensuremath{\mathbb{C}}}^d)$ must satisfy $$\label{eq:inegalite-mu-delta}
  (\mu',\gamma)\geq (\delta(\gamma),\gamma).$$ Consider now the weight $\mu={\xi'}-\sum_j t_j\alpha_j$. Since $\xi'\in{\ensuremath{\mathfrak{c}}}$, the equality (\[eq:ps-gamma-xi\]) implies $({\xi'},\gamma)<0$ and then $$(\mu,\gamma)=\underbrace{({\xi'},\gamma)}_{<0}+
\underbrace{\sum_{(\alpha_j,\gamma)>0}-
t_j(\alpha_j,\gamma)}_{\leq
0}-\sum_{(\alpha_j,\gamma)<0}t_j(\alpha_j,\gamma)<
-\sum_{(\alpha_j,\gamma)<0}(\alpha_j,\gamma).$$ So we have proved that $(\mu,\gamma) < (\delta(\gamma),\gamma)$, hence the multiplicity of ${\ensuremath{\mathbb{C}}}_\mu$ in $RR^{\xi'}_\gamma({\ensuremath{\mathbb{C}}}^d)$ is equal to zero. $\Box$

Proof of Theorem \[theo:cas-C-d\] {#proof-theo-cas-C-d}
---------------------------------

Let ${\ensuremath{\mathfrak{c}}}_\pm$ be two adjacent conic chambers and let ${\ensuremath{\Delta}}\subset{\ensuremath{\mathfrak{t}}}^*$ be the hyperplane that separates ${\ensuremath{\mathfrak{c}}}_+$ and ${\ensuremath{\mathfrak{c}}}_-$. Let $\beta\in{\ensuremath{\mathfrak{t}}}$ be such that ${\ensuremath{\Delta}}=\{\xi\in{\ensuremath{\mathfrak{t}}}^* \ \vert\ \langle\xi,\beta\rangle=0\}$ and ${\ensuremath{\mathfrak{c}}}_\pm\subset \{\xi\in{\ensuremath{\mathfrak{t}}}^* \ \vert\ \pm\langle\xi,\beta\rangle>0\}$.

We consider two points $\xi_\pm\in {\ensuremath{\mathfrak{c}}}_\pm$ such that $\xi=
\frac{1}{2}(\xi^+ + \xi^-)\in{\ensuremath{\Delta}}$ belongs to the relative interior ${\ensuremath{\mathfrak{c}}}'$ of $\overline{{\ensuremath{\mathfrak{c}}}_+}\cap\overline{{\ensuremath{\mathfrak{c}}}_-}$. We suppose also that the orthogonal projection of $\xi_\pm$ on ${\ensuremath{\Delta}}$ are equal to $\xi$. We know that $P_{{\ensuremath{\mathfrak{c}}}_+}(\mu)-P_{{\ensuremath{\mathfrak{c}}}_-}(\mu)$ is equal to the $\mu$-mutiplicity of $RR_0^{\xi_+}({\ensuremath{\mathbb{C}}}^d)-
RR_0^{\xi_-}({\ensuremath{\mathbb{C}}}^d)$. Proposition \[RR-0-diff\] tells us that $$RR_0^{\xi_+}({\ensuremath{\mathbb{C}}}^d)- RR_0^{\xi_-}({\ensuremath{\mathbb{C}}}^d)=
RR_{\gamma}^{\xi_-}({\ensuremath{\mathbb{C}}}^d)-RR_{-\gamma}^{\xi_+}({\ensuremath{\mathbb{C}}}^d),$$ where $\gamma\in{\ensuremath{\mathbb{R}}}^{>0}\beta$ is such that $\xi_-+\gamma=\xi_+ -
\gamma=\xi$. The localization (\[eq:loc-RR-beta-bis\]) gives then $$\label{eq:P-pm-rhs}
  RR_{\gamma}^{\xi_-}({\ensuremath{\mathbb{C}}}^d)-RR_{-\gamma}^{\xi_+}({\ensuremath{\mathbb{C}}}^d)=
RR_{0}^{\xi}(({\ensuremath{\mathbb{C}}}^d)^\beta)\otimes
\left(\left[\wedge_{{\ensuremath{\mathbb{C}}}}^{\bullet}\overline{N}\right]^{-1}_{\beta}-
\left[\wedge_{{\ensuremath{\mathbb{C}}}}^{\bullet}\overline{N}\right]^{-1}_{-\beta}\right).$$

The element $\xi$ belongs to the relative interior ${\ensuremath{\mathfrak{c}}}'$ of $\overline{{\ensuremath{\mathfrak{c}}}_+}\cap\overline{{\ensuremath{\mathfrak{c}}}_-}$ which is a conic chamber ${\ensuremath{\mathfrak{c}}}'$ in ${\ensuremath{\Delta}}$ with respect to $R\cap{\ensuremath{\Delta}}$. Let $P_{{\ensuremath{\mathfrak{c}}}'}:
\Lambda^*\cap{\ensuremath{\Delta}}\to{\ensuremath{\mathbb{Z}}}$ be the periodic polynomial map which coincides with the vector partition function $N_{R\cap{\ensuremath{\Delta}}}$ on $\overline{{\ensuremath{\mathfrak{c}}}'}\cap\Lambda^*$. If we work with the vector space $({\ensuremath{\mathbb{C}}}^d)^\beta$ equipped with the hamiltonian action of $T/T_{\ensuremath{\Delta}}$, (\[eq:RR-xi-O\]) gives the following equality in $R^{-\infty}(T/T_{\ensuremath{\Delta}})$ $$\label{eq:RR-xi-p-0}
  RR_{0}^{\xi}(({\ensuremath{\mathbb{C}}}^d)^\beta)
=\sum_{\gamma\in\Lambda^*\cap{\ensuremath{\Delta}}}P_{{\ensuremath{\mathfrak{c}}}'}(\gamma){\ensuremath{\mathbb{C}}}_\gamma.$$ A straightforward computation gives $$\label{eq:lambda-N-plus}
  \left[\wedge_{{\ensuremath{\mathbb{C}}}}^{\bullet}\overline{N}\right]^{-1}_{\beta}=
(-1)^{r^-}\sum_{\mu\in\Lambda^*} N_{R'}(\mu+\delta^-){\ensuremath{\mathbb{C}}}_{\mu}$$ and $$\label{eq:lambda-N-moins}
 \left[\wedge_{{\ensuremath{\mathbb{C}}}}^{\bullet}\overline{N}\right]^{-1}_{-\beta}=
(-1)^{r^+}\sum_{\mu\in\Lambda^*} N_{-R'}(\mu+\delta^+){\ensuremath{\mathbb{C}}}_{\mu},$$ where $r^\pm,\delta^\pm,R'$ are defined in (\[eq:R-delta-prime\]), (\[eq:delta-pm\]) and (\[eq:r-pm\]). Since $N_{-R'}(\mu)=N_{R'}(-\mu)$, the equations (\[eq:RR-xi-p-0\]), (\[eq:lambda-N-plus\]) and (\[eq:lambda-N-moins\]) show that the RHS of (\[eq:P-pm-rhs\]) is equal to $$\sum_{\mu\in\Lambda^*}\sum_{\gamma\in\Lambda^*\cap{\ensuremath{\Delta}}}
D(\mu)P_{{\ensuremath{\mathfrak{c}}}'}(\gamma){\ensuremath{\mathbb{C}}}_{\mu+\gamma}$$ with $D(\mu)=(-1)^{r^-}N_{R'}(\mu+\delta^-)-(-1)^{r^+}N_{R'}(-\mu-\delta^+)$. Finally we have proved that $P_{{\ensuremath{\mathfrak{c}}}_+}(\mu)-P_{{\ensuremath{\mathfrak{c}}}_-}(\mu)=
\sum_{\gamma\in\Lambda^*\cap{\ensuremath{\Delta}}}D(\mu-\gamma)P_{{\ensuremath{\mathfrak{c}}}'}(\gamma)$. $\Box$

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[^1]: It is a polynomial times a Lebesgue measure on ${\ensuremath{\mathfrak{t}}}^*$.

[^2]: Since a neighborhood of $\Phi^{-1}(\xi)$ in $M$ is $T$-equivariantly diffeomorphic to $\Phi^{-1}(\xi)\times{\ensuremath{\mathfrak{t}}}^*$.

[^3]: More precisely ${\ensuremath{\mathbb{S}}}^k_{Z,\mu}$ is $\pm 1$ times a orbifold vector bundle over the reduced space ${\ensuremath{\mathcal{Z}}}_\xi$.
