---
abstract: 'We study connections between the topology of generic character varieties of fundamental groups of punctured Riemann surfaces, Macdonald polynomials, quiver representations, Hilbert schemes on $\C^\times\times\C^\times$, modular forms and multiplicities in tensor products of irreducible characters of finite general linear groups.'
author:
- |
    Tamás Hausel\
    [*University of Oxford*]{}\
    [hausel@maths.ox.ac.uk]{}
- |
    Emmanuel Letellier\
    [ *Université de Caen*]{}\
    [ letellier.emmanuel@math.unicaen.fr]{}
- |
    Fernando Rodriguez-Villegas\
    [*University of Texas at Austin*]{}\
    [ villegas@math.utexas.edu]{}\
    \
    [with an appendix by Gergely Harcos]{}
title: |
    Arithmetic harmonic analysis on\
    character and quiver varieties II
---

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

Character varieties {#char}
-------------------

Given a non-negative integer $g$ and a $k$-tuple $\muhat=(\mu^1,\mu^2,\dots,\mu^k)$ of partitions of $n$, we define the generic character variety $\M_\muhat$ of type $\muhat$ as follows (see [@hausel-letellier-villegas] for more details). Choose a *generic* tuple $(\calC_1,\dots,\calC_k)$ of semisimple conjugacy classes of $\GL_n(\C)$ such that for each $i=1,2,\dots,k$ the multiplicities of the eigenvalues of $\calC_i$ are given by the parts of $\mu^i$.

Define $\cal{Z}_\muhat$ as $$\calZ_\muhat:= \left\{(a_1,b_1,\dots,a_g,b_g,x_1,\dots,x_k)\in
  (\GL_n)^{2g} \times\calC_1\times\cdots\times\calC_k\,\left|\,
    \prod_{j=1}^g(a_i,b_i)\prod_{i=1}^kx_i=1\right.\right\},$$ where $(a,b)=aba^{-1}b^{-1}$. The group $\GL_n$ acts diagonally by conjugation on $\calZ_\muhat$ and we define $\M_\muhat$ as the affine GIT quotient $$\M_\muhat:=\calZ_\muhat/\!/\GL_n:={\rm
  Spec}\,\left(\C[\calZ_\muhat]^{\GL_n}\right).$$ We prove in [@hausel-letellier-villegas] that, if non-empty, $\M_\muhat$ is nonsingular of pure dimension $$d_\muhat:=n^2(2g-2+k)-\sum_{i,j}(\mu^i_j)^2+2.$$ We also defined an [*a priori*]{} rational function $\H_\muhat(z,w)\in\Q(z,w)$ in terms of Macdonald symmetric functions (see § \[Cauchy\] for a precise definition) and we conjecture that the compactly supported mixed Hodge numbers $\{h_c^{i,j;k}(\M_\muhat)\}_{i,j,k}$ satisfies $h_c^{i,j;k}(\M_\muhat)=0$ unless $i=j$ and $$H_c(\M_\muhat;q,t){\stackrel{?}{=}}(t\sqrt{q})^{d_\muhat}\H_\muhat 
\left(-t\sqrt{q},\frac{1}{\sqrt{q}}\right),
\label{mainconj}$$ where $H_c(\M_\muhat;q,t):=\sum_{i,j}h_c^{i,i;j}(\M_\muhat)q^it^j$ is the compactly supported mixed Hodge polynomial.

In particular, $\H_\muhat(-z,w)$ should actually be a polynomial with non-negative integer coefficients of degree $d_\muhat$ in each variable.

In [@hausel-letellier-villegas] we prove that (\[mainconj\]) is true under the specialization $(q,t)\mapsto (q,-1)$, namely, $$E(\M_\muhat;q):=H_c(\M_\muhat;q,-1)= 
q^{\frac{1}{2}d_\muhat}\H_\muhat\left(\sqrt{q},\frac{1}{\sqrt{q}}\right).
\label{mainresult}$$

This formula is obtained by counting points of $\M_\muhat$ over finite fields (after choosing a spreading out of $\M_\muhat$ over a finitely generated subalgebra of $\C$). We compute $\M_\muhat(\F_q)$ using a formula involving the values of the irreducible characters of $\GL_n(\F_q)$ (a formula that goes back to Frobenius [@frobenius]). The calculation shows that $\M_\muhat$ is *polynomial count*; i.e., there exists a polynomial $P\in\C[T]$ such that for any finite field $\F_q$ of sufficiently large characteristic, $\#\M_\muhat(\F_q)=P(q)$. Then by a theorem of Katz [@hausel-letellier-villegas Appendix] $E(\M_\muhat;q)=P(q)$.

Recall also that the $E(\M_\muhat;q)$ satisfies the following identity

$$E(\M_\muhat;q)=q^{d_\muhat}E(\M_\muhat;q^{-1}).
\label{curious}$$

In this paper we use Formula (\[mainresult\]) to prove the following theorem.

If non-empty, the character variety $\M_\muhat$ is connected.

The proof of the theorem reduces to proving that the coefficient of the lowest power of $q$ in $\H_\muhat(\sqrt{q},1/\sqrt{q})$, namely $q^{-d_\muhat/2}$, equals $1$. This turns out to require a rather delicate argument, by far the most technical of the paper, that uses the inequality of § \[appendix\] in a crucial way.

Relations to Hilbert schemes on $\C^\times\times\C^\times$ and modular forms
----------------------------------------------------------------------------

Here we assume that $g=k=1$. Put $X=\C^\times\times\C^\times$ and denote by $X^{[n]}$ the Hilbert scheme of $n$ points in $X$. Define $\H^{[n]}(z,w)\in \Q(z,w)$ by $$\sum_{n\geq 0}\H^{[n]}(z,w)T^n:=\prod_{n\geq
1}\frac{(1-zwT^n)^2}{(1-z^2T^n)(1-w^2T^n)},$$ with the convention that $\H^{[0]}(z,w):=1$. It is known by work of Göttsche and Soergel [@Gottsche-Soergel] that the mixed Hodge polynomial $H_c\left(X^{[n]};q,t\right)$ is given by $$H_c\left(X^{[n]};q,t\right)
=(qt^2)^n\H^{[n]}\left(-t\sqrt{q}, \frac{1}{\sqrt{q}}\right).$$

We have $$\H^{[n]}(z,w)=\H_{(n-1,1)}(z,w).$$ \[conjHS\]

This together with the conjectural formula (\[mainconj\]) implies that the Hilbert scheme $X^{[n]}$ and the character variety $\M_{(n-1,1)}$ should have the same mixed Hodge polynomial. Although this is believed to be true (in the analogous additive case this is well-known; see Theorem \[adpure\]) there is no complete proof in the literature. (The result follows from known facts modulo some missing arguments in the non-Abelian Hodge theory for punctured Riemann surfaces; see the comment after Conjecture \[conjCV=HS\].) We prove the following results which give evidence for Conjecture \[conjHS\].

We have $$\begin{aligned}
&\H^{[n]}(0,w)=\H_{(n-1,1)}\left(0,w\right),\\
&\H^{[n]}(w^{-1},w)=\H_{(n-1,1)}(w^{-1},w).\end{aligned}$$ \[theospe\]

The second identity means that the $E$-polynomials of $X^{[n]}$ and $\M_{(n-1,1)}$ agree. As a consequence of Theorem \[theospe\] we have the following relation between character varieties and quasi-modular forms.

We have $$1+\sum_{n\geq 1}\H_{(n-1,1)}\left(e^{u/2},e^{-u/2}\right)T^n
=\frac{1}{u}\left(e^{u/2}-e^{-u/2}\right)\exp\left(2\sum_{k\geq
    2}G_k(T)\frac{u^k}{k!}\right),$$ where $$G_k(T)=\frac{-B_k}{2k}+\sum_{n\geq 1}\sum_{d\,|\, n}
d^{k-1}T^n$$(with $B_k$ is the $k$-th Bernoulli number) is the classical Eisenstein series for $SL_2(\Z)$.

In particular, the coefficient of any power of $u$ in the left hand side is in the ring of *quasi-modular* forms, generated by the $G_k$, $k\geq 2$, over $\Q$.

Relation between Hilbert schemes and modular forms was first investigated by Göttsche [@Gottsche].

Quiver representations {#quiver-repns}
----------------------

For a partition $\mu=\mu_1\geq\dots \geq \mu_r>0$ of $n$ we denote by $l(\mu)=r$ its length. Given a non-negative integer $g$ and a $k$-tuple $\muhat=(\mu^1,\mu^2,\dots,\mu^k)$ of partitions of $n$ we define a *comet-shaped* quiver $\Gamma_\muhat$ with $k$ legs of length $s_1,s_2,\dots,s_k$ (where $s_i=l(\mu^i)-1$) and with $g$ loops at the central vertex (see picture in §\[comet\]). The multi-partition $\muhat$ defines also a dimension vector $\v_\muhat$ of $\Gamma_\muhat$ whose coordinates on the $i$-th leg are $(n,n-\mu^i_1,n-\mu^i_1-\mu^i_2,\dots,n-\sum_{r=1}^{s_i}\mu^i_r)$.

By a theorem of Kac [@kacconj] there exists a monic polynomial $A_\muhat(T)\in \Z[T]$ of degree $d_\muhat/2$ such that the number of absolutely indecomposable representations over $\F_q$ (up to isomorphism) of $\Gamma_\muhat$ of dimension $\v_\muhat$ equals $A_\muhat(q)$.

Let us state our first main result.

We have $$A_\muhat(q)=\H_\muhat(0,\sqrt{q}).
\label{purity0}$$

If we assume that $\v_\muhat$ is indivisible, i.e., the gcd of all the parts of the partitions $\mu^1,\dots,\mu^k$ equals $1$, then, as mentioned in [@hausel-letellier-villegas Remark 1.4.3], the formula can be proved using the results of Crawley-Boevey and van den Bergh [@crawley-boevey-etal] together with the results in [@hausel-letellier-villegas]. More precisely the results of Crawley-Boevey and van den Bergh say that $A_\muhat(q)$ equals (up to some power of $q$) the compactly supported Poincaré polynomial of some quiver variety $\calQ_\muhat$ (which exists only if $\v_\muhat$ is indivisible). In [@hausel-letellier-villegas] we show that the Poincaré polynomial of $\calQ_\muhat$ agrees with $\H_\muhat(0,\sqrt{q})$ up to the same power of $q$, hence the formula (\[purity0\]).

The proof of Formula (\[purity0\]) we give in this paper is completely combinatorial (and works also in the divisible case). It is based on Hua’s formula [@hua] for the number of absolutely indecomposable representations of quivers over finite fields.

The conjectural formula (\[mainconj\]) together with Formula (\[purity0\]) implies the following conjecture.

We have $$A_\muhat(q)=q^{-\frac{d_\muhat}{2}}PH_c(\M_\muhat;q),$$where $PH_c(\M_\muhat;q):=\sum_ih_c^{i,i;2i}(\M_\muhat)q^i$ is the *pure part* of $H_c(\M_\muhat;q,t)$. \[purconj\]

Conjecture \[purconj\] implies Kac’s conjecture [@kacconj] for comet shaped quivers, namely, $A_\muhat(q)$ is a polynomial in $q$ with non-negative coefficients (see §\[genquiv\] for more details).

Characters of general linear groups over finite fields
------------------------------------------------------

Given two irreducible complex characters $\calX_1,\calX_2$ of $\GL_n(\F_q)$ it is a natural and difficult question to understand the decomposition of the tensor product $\calX_1\otimes\calX_2$ as a sum of irreducible characters. Note that the character table of $\GL_n(\F_q)$ is known (Green, 1955) and so we can compute in theory the multiplicity $\langle\calX_1\otimes\calX_2,\calX\rangle$ of any irreducible character $\calX$ of $\GL_n(\F_q)$ in $\calX_1\otimes\calX_2$ using the scalar product formula $$\langle\calX_1\otimes\calX_2,\calX\rangle=\frac{1}{|\GL_n(\F_q)|}\sum_{g\in\GL_n(\F_q)}\calX_1(g)\calX_2(g)\overline{\calX(g)}.\label{scalprod}$$However it is very difficult to extract any interesting information from this formula. In his thesis Mattig uses this formula to compute (with the help of a computer) the multiplicities $\langle
\calX_1\otimes\calX_2,\calX\rangle$ when $\calX_1,\calX_2,\calX$ are *unipotent characters* and when $n\leq 8$ (see [@Hiss]), and he noticed that $\langle \calX_1\otimes\calX_2,\calX\rangle$ is a polynomial in $q$ with positive integer coefficients.

In [@hausel-letellier-villegas] we define the notion of *generic* tuple $(\calX_1,\dots,\calX_k)$ of irreducible characters of $\GL_n(\F_q)$. We also consider the character $\Lambda:\GL_n(\F_q)\rightarrow \C$, $x\mapsto q^{g\cdot {\rm dim}\,
  C_{\GL_n}(x)}$ where $C_{\GL_n}(x)$ denotes the centralizer of $x$ in $\GL_n(\overline{\F}_q)$ and where $g$ is a non-negative integer. If $g=1$, this is the character of the conjugation action of $\GL_n(\F_q)$ on the group algebra $\C[\gl_n(\F_q)]$.

If $\mu=(\mu_1,\mu_2,\dots,\mu_r)$ is a partition of $n$, an irreducible character of $\GL_n(\F_q)$ is said to be of type $\mu$ if it is of the form $R_{L_\mu}^{GL_n}(\alpha)$ where $L_\mu=\GL_{\mu_1}\times\GL_{\mu_2}\times\cdots\times\GL_{\mu_r}$ and where $\alpha$ is a *regular* linear character of $L_\mu(\F_q)$, see §\[applichar\] for definitions. Characters of this form are called *semisimple split*.

In [@hausel-letellier-villegas] we prove that for a generic tuple $(\calX_1,\dots,\calX_k)$ of semisimple split irreducible characters of $\GL_n(\F_q)$ of type $\muhat$, we have $$\langle
  \Lambda\otimes\calX_1\otimes\cdots\otimes\calX_k,1\rangle
  =\H_\muhat(0,\sqrt{q})\label{multi0}.$$ Note that in particular this implies that the left hand side only depends on the combinatorial type $\muhat$ not on the specific choice of characters.

Together with Formula (\[purity0\]) we deduce the following formula.

We have $$\langle
  \Lambda\otimes\calX_1\otimes\cdots\otimes\calX_k,1\rangle=A_\muhat(q).$$

Using Kac’s results on quiver representations (see §\[genquiv\]) the above theorem has the following consequence.

Let $\Phi(\Gamma_\muhat)$ denote the root system associated with $\Gamma_\muhat$ and let $(\calX_1,\dots,\calX_k)$ be a generic $k$-tuple of irreducible characters of $\GL_n(\F_q)$ of type $\muhat$.

We have $\langle
  \Lambda\otimes\calX_1\otimes\cdots\otimes\calX_k,1\rangle\neq 0$ if and only if $\v_\muhat\in\Phi(\Gamma_\muhat)$. Moreover $\langle
  \Lambda\otimes\calX_1\otimes\cdots\otimes\calX_k,1\rangle=1$ if and only if $\v_\muhat$ is a real root. \[coromulti0\]

In [@letellier] the second author proves that Corollary \[coromulti0\] extends to any type of generic tuples of irreducible characters of $\GL_n(\F_q)$ (not necessarily semisimple split).

Recall that there is a natural parametrization, $\mu\mapsto
\calU_\mu$, of the unipotent characters of $\GL_n(\F_q)$ by partitions of $n$, fixed by requiring that $\calU_{(1^n)}$ is trivial. Using again quiver representations, we also prove in §\[applichar\] the following result on multiplicities for unipotent characters (confirming partly Mattig’s observation for $n\leq 8$).

For a $k$-tuple of partitions $\muhat=(\mu^1,\ldots,\mu^k)$ of $n$ there exists a polynomial $U_\muhat\in \Z[T]$ such that $\langle
  \Lambda\otimes \calU_{\mu^1}\otimes\cdots\otimes
  \calU_{\mu^k},1\rangle=U_\muhat(q)$.

#### Acknowledgements.

We would like to thank the Mathematisches Forschungsinstitut Oberwolfach for a research in pairs stay where much of the work was done. TH was supported by a Royal Society University Research Fellowship. EL was supported by ANR-09-JCJC-0102-01. FRV was supported by NSF grant DMS-0200605, an FRA from the University of Texas at Austin, EPSRC grant EP/G027110/1, Visiting Fellowships at All Souls and Wadham Colleges in Oxford and a Research Scholarship from the Clay Mathematical Institute.

Preliminaries
=============

We denote by $\F$ an algebraic closure of a finite field $\F_q$.

Symmetric functions
-------------------

### Partitions, Macdonald polynomials, Green polynomials

We denote by $\calP$ the set of all partitions including the unique partition $0$ of $0$, by $\calP^*$ the set of non-zero partitions and by $\calP_n$ be the set of partitions of $n$. Partitions $\lambda$ are denoted by $\lambda=(\lambda_1,\lambda_2,\ldots)$, where $\lambda_1\geq \lambda_2\geq\cdots\geq 0$. We will also sometimes write a partition as $(1^{m_1},2^{m_2},\ldots,n^{m_n})$ where $m_i$ denotes the multiplicity of $i$ in $\lambda$. The [*size*]{} of $\lambda$ is $|\lambda|:=\sum_i\lambda_i$; the [*length*]{} $l(\lambda)$ of $\lambda$ is the maximum $i$ with $\lambda_i>0$. For two partitions $\lambda$ and $\mu$, we define $\langle\lambda,\mu\rangle$ as $\sum_i\lambda_i'\mu_i'$ where $\lambda'$ denotes the dual partition of $\lambda$. We put $n(\lambda)=\sum_{i>0}(i-1)\lambda_i$. Then $\langle\lambda,\lambda\rangle=2n(\lambda)+|\lambda|$. For two partitions $\lambda=(1^{n_1},2^{n_2},\dots)$ and $\mu=(1^{m_1},2^{m_2},\dots)$, we denote by $\lambda\cup\mu$ the partition $(1^{n_1+m_1},2^{n_2+m_2},\dots)$. For a non-negative integer $d$ and a partition $\lambda$, we denote by $d\cdot\lambda$ the partition $(d\lambda_1,d\lambda_2,\dots)$. The [*dominance ordering*]{} for partitions is defined as follows: $\mu \unlhd \lambda$ if and only if $\mu_1+\cdots+\mu_j\leq \lambda_1+\cdots+\lambda_j$ for all $j\geq 1$.

Let $\x=\{x_{1},x_{2},\dots\}$ be an infinite set of variables and $\Lambda(\x)$ the corresponding ring of symmetric functions. As usual we will denote by $s_\lambda(\x),h_\lambda(\x),p_\lambda(\x)$, and $m_\lambda(\x)$, the Schur symmetric functions, the complete symmetric functions, the power symmetric functions and the monomial symmetric functions.

We will deal with elements of the ring $\Lambda(\x) \otimes_\Z \Q(z,w)$ and their images under two specializations: their [*pure part*]{}, $z=0,w=\sqrt q$ and their [*Euler specialization*]{}, $z=\sqrt
q,w=1/\sqrt q$.

For a partition $\lambda$, let $\tilde{H}_\lambda(\x;q,t) \in
\Lambda(\x) \otimes_\Z \Q(q,t)$ be the [*Macdonald symmetric function*]{} defined in Garsia and Haiman [@garsia-haiman I.11]. We collect in this section some basic properties of these functions that we will need.

We have the duality $$\tilde{H}_\lambda(\x;q,t)=\tilde{H}_{\lambda^\prime}(\x;t,q)
\label{Hduality}$$ see [@garsia-haiman Corollary 3.2]. We define the (transformed) [*Hall-Littlewood symmetric function*]{} as $$\label{HL-defn}
\tilde{H}_\lambda(\x;q):=\tilde{H}_\lambda(\x;0,q).$$ In the notation just introduced then $\tilde{H}_\lambda(\x;q)$ is the pure part of $\tilde{H}_\lambda(\x;z^2,w^2)$.

Under the Euler specialization of $\tilde{H}_\lambda(\x;z^2,w^2)$ we have [@hausel-letellier-villegas Lemma 2.3.4] $$\tilde{H}_\lambda(\x;q,q^{-1})=q^{-n(\lambda)}H_\lambda(q)s_\lambda(\x\y),\label{eulermac}$$where $y_i=q^{i-1}$ and $H_\lambda(q):=\prod_{s\in\lambda}(1-q^{h(s)})$ is the *hook polynomial* [@macdonald I, 3, example 2].

Define the [*$(q,t)$-Kostka polynomials*]{} $\tilde{K}_{\nu\lambda}(q,t)$ by $$\label{K-defn}
\tilde{H}_{\lambda}(\x;q,t)=
\sum_{\nu}\tilde{K}_{\nu\lambda}(q,t)s_{\nu}(\x).$$ These are $(q,t)$ generalizations of the $\tilde{K}_{\nu\lambda}(q)$ Kostka-Foulkes polynomial in Macdonald [@macdonald III, (7.11)], which are obtained as $q^{n(\lambda)}K_{\nu\lambda}(q^{-1})=
\tilde{K}_{\nu\lambda}(q)=\tilde{K}_{\nu\lambda}(0,q)$, i.e., by taking their pure part. In particular, $$\label{Hall-Littlewood}
\tilde{H}_{\lambda}(\x;q)=\sum_{\nu}\tilde{K}_{\nu\lambda}(q)s_{\nu}(\x).$$

For a partition $\lambda$, we denote by $\chi^\lambda$ the corresponding irreducible character of $S_{|\lambda|}$ as in Macdonald [@macdonald]. Under this parameterization, the character $\chi^{(1^n)}$ is the sign character of $S_{|\lambda|}$ and $\chi^{(n^1)}$ is the trivial character. Recall also that the decomposition into disjoint cycles provides a natural parameterization of the conjugacy classes of $S_n$ by the partitions of $n$. We then denote by $\chi^\lambda_\mu$ the value of $\chi^\lambda$ at the conjugacy class of $S_{|\lambda|}$ corresponding to $\mu$ (we use the convention that $\chi^\lambda_\mu=0$ if $|\lambda|\neq|\mu|$). The *Green polynomials* $\{Q_\lambda^\tau(q)\}_{\lambda,\tau\in\calP}$ are defined as

$$Q_\lambda^\tau(q)=\sum_\nu \chi^\nu_\lambda\tilde{K}_{\nu\tau}(q)$$

if $|\lambda|=|\tau|$ and $Q_\lambda^\tau=0$ otherwise.

### Exp and Log

Let $\Lambda(\x_1,\ldots,\x_k):=
\Lambda(\x_1)\otimes_\Z\cdots\otimes_\Z\Lambda(\x_k)$ be the ring of functions separately symmetric in each set $\x_1,\x_2,\ldots,\x_k$ of infinitely many variables. To ease the notation we will simply write $\Lambda_k$ for the ring $\Lambda(\x_1,\ldots,\x_k)\otimes_\Z \Q(q,t)$.

The power series ring $\Lambda_k[[T]]$ is endowed with a natural $\lambda$-ring structure in which the Adams operations are $$\psi_d(f(\x_1,\x_2,\dots,\x_k,q,t;T)):=f(\x_1^d,\x_2^d,\dots,\x_k^d,q^d,t^d;T^d).$$

Let $\Lambda_k[[T]]^+$ be the ideal $T\Lambda_k[[T]]$ of $\Lambda_k[[T]]$. Define $\Psi: \Lambda_k[[T]]^+\rightarrow\Lambda_k[[T]]^+$ by

$$\Psi(f):=\sum_{n\geq 1}\frac{\psi_n(f)}{n},$$ and $\Exp:\Lambda_k[[T]]^+\rightarrow 1+\Lambda_k[[T]]^+$ by

$$\Exp(f)=\exp(\Psi(f)).$$

The inverse $\Psi^{-1}:\Lambda_k[[T]]^+\rightarrow\Lambda_k[[T]]^+$ of $\Psi$ is given by

$$\Psi^{-1}(f)=\sum_{n\geq 1}\mu(n)\frac{\psi_n(f)}{n}$$where $\mu$ is the ordinary Möbius function.

The inverse $\Log:1+\Lambda_k[[T]]\rightarrow\Lambda_k[[T]]$ of $\Exp$ is given by

$$\Log(f)=\Psi^{-1}(\log(f)).$$

Let $f=1+\sum_{n\geq 1}f_nT^n\in 1+\Lambda_k[[T]]^+$. If we write

$$\log\,(f)=\sum_{n\geq 1}\frac{1}{n}U_nT^n,\hspace{1cm}\Log\,(f)=\sum_{n\geq 1}V_nT^n,$$ then

$$V_r=\frac{1}{r}\sum_{d | r}\mu(d)\psi_d(U_{r/d}).$$

We have the following propositions (details may be found for instance in Mozgovoy [@mozgovoy]).

For $g\in \Lambda_k$ and $n\geq 1$ we put

$$g_n:=\frac{1}{n}\sum_{d | n}\mu(d)\psi_{\frac{n}{d}}(g).$$ This is the Möbius inversion formula of $\psi_n(g)=\sum_{d| n}d\cdot g_d$.

Let $g\in \Lambda_k$ and $f_1,f_2\in 1+\Lambda_k[[T]]^+$ such that

$$\log\,(f_1)=\sum_{d=1}^\infty g_d\cdot\log\,(\psi_d(f_2)).$$ Then $$\Log\,(f_1)=g\cdot \Log\,(f_2).$$

\[moz\]

Assume that $f\in 1+\Lambda_k[[T]]^+$ belongs to $\Lambda(\x_1,\ldots,\x_k)\otimes_\Z\Z[q,t]$, then $\Exp(f)\in
  \Lambda(\x_1,\ldots,\x_k)\otimes_\Z\Z[q,t]$.

\[Exp\]

### Types

We choose once and for all a total ordering $\geq$ on $\calP$ (e.g. the lexicographic ordering) and we continue to denote by $\geq$ the total ordering defined on the set of pairs $\N^*\times\calP^*$ as follows: If $\lambda\neq\mu$ and $\lambda\geq\mu$, then $(d,\lambda)\geq(d',\mu)$, and $(d,\lambda)\geq(d',\lambda)$ if $d\geq
d'$. We denote by $\mathbf{T}$ the set of non-increasing sequences $\omega=(d_1,\omega^1)\geq (d_2,\omega^2)\geq \cdots \geq
(d_r,\omega^r)$, which we will call a [*type*]{}. To alleviate the notation we will then omitt the symbol $\geq$ and write simply $\omega=(d_1,\omega^1)(d_2,\omega^2)\cdots (d_r,\omega^r)$. The [ *size*]{} of a type $\omega$ is $|\omega|:=\sum_id_i|\lambda^i|$. We denote by $\mathbf{T}_n$ the set of types of size $n$. We denote by $m_{d,\lambda}(\omega)$ the multiplicity of $(d,\lambda)$ in $\omega$. As with partitions it is sometimes convenient to consider a type as a collection of integers $m_{d,\lambda}\geq 0 $ indexed by pairs $(d,\lambda)\in \Nstar\times \; \calP^*$. For a type $\omega=(d_1,\omega^1)(d_2,\omega^2)\cdots (d_r,\omega^r)$, we put $n(\omega)=\sum_id_i n(\omega^i)$ and $[\omega]:=\cup_id_i\cdot\omega^i$.

When considering elements $a_\muhat\in \Lambda_k$ indexed by multi-partitions $\muhat=(\mu^1,\ldots,\mu^k)\in \calP^k$, we will always assume that they are homogeneous of degree $(|\mu^1|,\ldots,|\mu^k|)$ in the set of variables $\x_1,\dots,\x_k$.

Let $\{a_\muhat\}_{\muhat\in\calP^k}$ be a family of symmetric functions in $\Lambda_k$ indexed by multi-partitions.

We extend its definition to a *multi-type* $\omhat=(d_1,\omhat^1)\cdots(d_s,\omhat^s)$ with $\omhat^p\in(\calP_{n_p})^k$, by

$$a_\omhat:=\prod_p\psi_{d_p}(A_{\omhat^p}).$$

For a multi-type $\omhat$ as above, we put $$C_\omhat^o:=\begin{cases}\frac{\mu(d)}{d}(-1)^{r-1}\frac{(r-1)!}{\prod_\muhat m_{d,\muhat}(\omhat)!}\,\text{ if } d_1=\cdots=d_r=d.\\0\,\text{ otherwise.}
\end{cases}$$ where $m_{d,\muhat}(\omhat)$ with $\muhat\in\calP^k$ denotes the multiplicity of $(d,\muhat)$ in $\omhat$.

We have the following lemma (see [@hausel-letellier-villegas §2.3.3] for a proof).

Let $\{A_\muhat\}_{\muhat\in\calP^k}$ be a family of symmetric functions in $\Lambda_k$ with $A_0=1$. Then $$\Log\left(\sum_{\muhat\in\calP^k}A_\muhat T^{|\muhat|}\right)=\sum_{\omhat}C_\omhat^oA_\omhat T^{|\omhat|}$$ where $\omhat$ runs over multi-types $(d_1,\omhat^1)\cdots(d_s,\omhat^s)$.\[Log-w\]

The formal power series $\sum_{n\geq 0}a_nT^n$ with $a_n\in \Lambda_k$ that we will consider in what follows will all have $a_n$ homogeneous of degree $n$. Hence we will typically scale the variables of $\Lambda_k$ by $1/T$ and eliminate $T$ altogether.

Given any family $\{a_\mu\}$ of symmetric functions indexed by partitions $\mu\in \P$ and a multi-partition $\muhat \in
\P^k$ as above define $$a_\muhat:=a_{\mu^1}(\x_1)\cdots a_{\mu^k}(\x_k).$$ Let $\langle\cdot ,\cdot \rangle$ be the Hall pairing on $\Lambda(\x),$ extend its definition to $\Lambda(\x_1,\ldots,\x_k)$ by setting \[extendedhall\] a\_1(\_1)a\_k(\_k), b\_1(\_1)b\_k(\_k)= a\_1, b\_1 a\_k, b\_k , for any $a_1,\ldots,a_k;b_1,\ldots,b_k\in \Lambda(\x)$ and to formal series by linearity.

### Cauchy identity {#Cauchy}

Given a partition $\lambda\in\calP_n$ we define the genus $g$ [*hook function*]{} $\calH_{\lambda}(z,w)$ by $$\calH_{\lambda}(z,w):=
\prod_{s\in \lambda}\frac{(z^{2a(s)+1}-w^{2l(s)+1})^{2g}}
{(z^{2a(s)+2}-w^{2l(s)})(z^{2a(s)}-w^{2l(s)+2})},$$ where the product is over all cells $s$ of $\lambda$ with $a(s)$ and $l(s)$ its arm and leg length, respectively. For details on the hook function we refer the reader to [@hausel-villegas].

Recall the specialization (cf. [@hausel-letellier-villegas §2.3.5]) $$\calH_\lambda(0,\sqrt{q})=\frac{q^{g\langle\lambda,\lambda\rangle}}{a_\lambda(q)}\label{alambda}$$where $a_\lambda(q)$ is the cardinality of the centralizer of a unipotent element of $\GL_n(\F_q)$ with Jordan form of type $\lambda$.

It is also not difficult to verify that the Euler specialization of $\calH_\lambda$ is

$$\calH_\lambda(\sqrt{q},1/\sqrt{q})=\left(q^{-\frac{1}{2}\langle\lambda,\lambda\rangle}H_\lambda(q)\right)^{2g-2}.
\label{H-specializ}$$

We have $$\calH_\lambda(z,w)=\calH_{\lambda'}(w,z)\,\,\,{\rm and }\,\,\,\calH_\lambda(-z,-w)=\calH_\lambda(z,w).\label{hook-duality}$$

Let $$\Omega(z,w)=\Omega(\x_1,\dots,\x_k;z,w):=\sum_{\lambda\in \calP} \calH_{\lambda}(z,w)
\prod_{i=1}^k\tilde{H_\lambda}(\x_i;z^2,w^2).$$By (\[Hduality\]) and (\[hook-duality\]) we have $$\Omega(z,w)=\Omega(w,z)\,\,\,{\rm and }\,\,\,\Omega(-z,-w)=\Omega(z,w).\label{Oduality}$$

For $\muhat=(\mu^1,\cdots,\mu^k)\in\P^k$, we let

$$\H_\muhat(z,w):=(z^2-1)(1-w^2)\left\langle\Log\,\Omega(z,w),h_\muhat\right\rangle.
 \label{H}$$

By (\[Oduality\]) we have the symmetries

$$\H_\muhat(z,w)=\H_\muhat(w,z) \,\,\,{\rm and}\,\,\,\H_\muhat(-z,-w)=\H_\muhat(z,w). \label{HHduality}$$

We may recover $\Omega(z,w)$ from the $\H_\muhat(z,w)$’s by the formula:

(z,w)=(\_[\^k]{}m\_). \[exp\]

From Formula (\[eulermac\]) and Formula (\[H-specializ\]) we have:

With the specialization $y_i=q^{i-1}$,

$$\Omega\left(\sqrt{q},\frac{1}{\sqrt{q}}\right)=\sum_{\lambda\in\calP}q^{(1-q)|\lambda|}\left(q^{-n(\lambda)}H_\lambda(q)\right)^{2g+k-2}\prod_{i=1}^ks_\lambda(\x_i\y).$$

\[specializ\]

The rational function $\H_\muhat(z,w)$ is a polynomial with integer coefficients. It has degree

$$d_\muhat:=n^2(2g-2+k)-\sum_{i,j}(\mu^i_j)^2+2$$ in each variable and the coefficients of $\H_\muhat(-z,w)$ are non-negative. \[conjH\]

The function $\H_\muhat(z,w)$ is computed in many cases in [@hausel-letellier-villegas §1.5].

Characters and Fourier transforms {#finite-groups}
---------------------------------

### Characters of finite general linear groups {#charGL}

For a finite group $H$ let us denote by ${\rm Mod}_H$ the category of finite dimensional $\C[H]$ left modules. Let $K$ be an other finite group. By an *$H$-module-$K$* we mean a finite dimensional $\C$-vector space $M$ endowed with a left action of $H$ and with a right action of $K$ which commute together. Such a module $M$ defines a functor $R_K^H:{\rm Mod}_K\rightarrow{\rm Mod}_H$ by $V\mapsto M\otimes_{\C[K]} V$. Let $\C(H)$ denotes the $\C$-vector space of all functions $H\rightarrow\C$ which are constant on conjugacy classes. We continue to denote by $R_K^H$ the $\C$-linear map $\C(K)\rightarrow\C(H)$ induced by the functor $R_K^H$ (we first define it on irreducible characters and then extend it by linearity to the whole $\C(K)$). Then for any $f\in\C(K)$, we have

$$R_K^H(f)(g)=|K|^{-1}\sum_{k\in K}{\rm Trace}\,\left((g,k^{-1})\,|\, M\right)f(k).\label{bimod}$$

Let $G=\GL_n(\F_q)$ with $\F_q$ a finite field. Fix a partition $\lambda=(\lambda_1,\dots,\lambda_r)$ of $n$ and let $\calF_\lambda=\calF_\lambda(\F_q)$ be the variety of partial flags of $\F_q$-vector spaces $$\{0\}=E^r\subset E^{r-1}\subset\cdots\subset E^1\subset E^0=(\F_q)^n$$such that ${\rm dim}(E^{i-1}/E^i)=\lambda_i$.

Let $G$ acts on $\calF_\lambda$ in the natural way. Fix an element $$X_o=\left(\{0\}=E^r\subset E^{r-1}\subset\cdots\subset E^1\subset E^0=(\F_q)^n\right)\in\calF_\lambda$$ and denote by $P_\lambda$ the stabilizer of $X_o$ in $G$ and by $U_\lambda$ the subgroup of elements $g\in P_\lambda$ which induces the identity on $E^i/E^{i+1}$ for all $i=0,1,\dots,r-1$.

Put $L_\lambda:=\GL_{\lambda_r}(\F_q)\times\cdots\times\GL_{\lambda_1}(\F_q)$. Recall that $U_\lambda$ is a normal subgroup of $P_\lambda$ and that $P_\lambda=L_\lambda\ltimes U_\lambda$.

Denote by $\C[G/U_\lambda]$ the $\C$-vector space generated by the finite set $G/U_\lambda=\{gU_\lambda\,|\, g\in G\}$. The group $L_\lambda$ (resp. $G$) acts on $\C[G/U_\lambda]$ as $(gU_\lambda)\cdot l=glU_\lambda$ (resp. as $g\cdot (hU_\lambda)=ghU_\lambda$). These two actions make $\C[G/U_\lambda]$ into a $G$-module-$L_\lambda$. The associated functor $R_{L_\lambda}^G:{\rm Mod}_{L_\lambda}\rightarrow {\rm Mod}_G$ is the so-called *Harish-Chandra functor*.

We have the following well-known lemma.

We denote by $1$ the identity character of $L_\lambda$. Then for all $g\in G$, we have

$$R_{L_\lambda}^G(1)(g)=\#\{X\in\calF_\lambda\,|\, g\cdot X=X\}.$$ \[R=F\]

By Formula (\[bimod\]) we have

$$\begin{aligned}
 R_{L_\lambda}^G(1)(g)&=|L_\lambda|^{-1}\sum_{k\in L_\lambda}\#\{hU_\lambda\,|\, ghU_\lambda=hkU_\lambda\}\\
&=|L_\lambda|^{-1}\sum_{k\in L_\lambda}\#\{hU_\lambda\,|\, gh\in hkU_\lambda\}\\
&=|L_\lambda|^{-1}\#\{hU_\lambda\,|\, gh\in hP_\lambda\}\\
&=\#\{hP_\lambda\,|\, ghP_\lambda=hP_\lambda\}.
 \end{aligned}$$

We deduce the lemma from last equality by noticing that the map $G\rightarrow\calF_\lambda$, $g\mapsto g\cdot X_o$ induces a bijection $G/P_\lambda\rightarrow\calF_\lambda$.

We now recall the definition of the type of a conjugacy class $C$ of $G$ (cf. [@hausel-letellier-villegas 4.1]). The Frobenius $f:\F\rightarrow\F$, $x\mapsto x^q$ acts on the set of eigenvalues of $C$. Let us write the set of eigenvalues of $C$ as a disjoint union

$$\{\gamma_1,\gamma_1^q,\dots\}\coprod\{\gamma_2,\gamma_2^q,\dots\}\coprod\cdots\coprod\{\gamma_r,\gamma_r^q,\dots\}$$of $\langle f\rangle$-orbits, and let $m_i$ be the multiplicity of $\gamma_i$. The unipotent part of an element of $C$ defines a unique partition $\omega^i$ of $m_i$. Re-ordering if necessary we may assume that $(d_1,\omega^1)\geq (d_2,\omega^2)\geq\cdots\geq (d_r,\omega^r)$. We then call $\omega=(d_1,\omega^1)\cdots(d_r,\omega^r)\in{\bf T}_n$ the *type* of $C$.

Put $T:=L_{(1,1,\dots,1)}$. It is the subgroup of diagonal matrices of $G$. The decomposition of $R_T^{L_\lambda}(1)$ as a sum of irreducible characters reads $$R_T^{L_\lambda}(1)=\sum_{\chi\in {\rm
    Irr}(W_{L_\lambda})}\chi(1)\cdot \calU_\chi,$$where $W_{L_\lambda}:=N_{L_\lambda}(T)/T$ is the Weyl group of $L_\lambda$. We call the irreducible characters $\{\calU_\chi\}_\chi$ the *unipotent* characters of $L_\lambda$. The character $\calU_1$ is the trivial character of $L_\lambda$. Since $W_{L_\lambda}\simeq
S_{\lambda_1}\times\cdots\times S_{\lambda_r}$, the irreducible characters of $W_{L_\lambda}$ are $\chi^\tau:=\chi^{\tau^1}\cdots\chi^{\tau^r}$ where $\tau$ runs over the set of types $\tau=\{(1,\tau^i)\}_{i=1,\dots,r}$ with $\tau^i$ a partition of $\lambda_i$. We denote by $\calU_\tau$ the unipotent character of $L_\lambda$ corresponding to such a type $\tau$.

Let $\calU_\tau$ be a unipotent character of $L_\lambda$ and let $C$ be a conjugacy class of type $\omega$. Then

$$R_{L_\lambda}^G(\calU_\tau)(C)=\left\langle \tilde{H}_\omega(\x,q),s_\tau(\x)\right\rangle.$$

\[Rtau\]

The proof is contained in [@hausel-letellier-villegas] although the formula is not explicitely written there. For the convenience of the reader we now explain how to extract the proof from [@hausel-letellier-villegas]. For $w\in W_\lambda$, we denote by $R_{T_w}^G(1)$ the corresponding *Deligne-Lusztig character* of $G$. Its construction is outlined in [@hausel-letellier-villegas 2.6.4]. The character $\calU_\tau$ of $L_\lambda$ decomposes as,

$$\calU_\tau=|W_\lambda|^{-1}\sum_{w\in W_\lambda}\chi^\tau_w\cdot R_{T_w}^{L_\lambda}(1)$$where $\chi^\tau_w$ denotes the value of $\chi^\tau$ at $w$. Applying the Harish-Chandra induction $R_{L_\lambda}^G$ to both side and using the transitivity of induction we find that

$$R_{L_\lambda}^G(\calU_\tau)=|W_\lambda|^{-1}\sum_{w\in W_\lambda}\chi^\tau_w\cdot R_{T_w}^G(1).$$We are now in position to use the calculation in [@hausel-letellier-villegas]. Notice that the right handside of the above formula is the right hand side of the first formula displayed in the proof of [@hausel-letellier-villegas Theorem 4.3.1] with $(M,\theta^{T_w},\tilde{\varphi})=(L_\lambda,1,\chi^\tau)$ and so the same calculation to get [@hausel-letellier-villegas (4.3.2)] together with [@hausel-letellier-villegas (4.3.3)] gives in our case

$$R_{L_\lambda}^G(\calU_\tau)(C)=\sum_\alpha z_\alpha^{-1}\chi^\tau_\alpha\sum_{\{\beta\,|\,[\beta]=[\alpha]\}}Q_\beta^\omega(q)z_{[\alpha]}z_\beta^{-1}$$where the notation are those of [@hausel-letellier-villegas § 4.3]. We now apply [@hausel-letellier-villegas Lemma 2.3.5] to get

$$R_{L_\lambda}^G(\calU_\tau)(C)=\left\langle \tilde{H}_\omega(\x;q),s_\tau(\x)\right\rangle.$$

If $\alpha$ is the type $(1,(\lambda_1))\cdots(1,(\lambda_r))$, then $s_\alpha(\x)=h_\lambda(\x)$. Hence we have:

If $C$ is a conjugacy class of $G$ type $\omega$, then

$$R_{L_\lambda}^G(1)(C)=\left\langle \tilde{H}_\omega(\x,q),h_\lambda(\x)\right\rangle.$$ \[R\]

Put $\calF^\#_{\lambda,\omega}(q):=\#\{X\in\calF_\lambda\,|\, g\cdot X=X\}$ where $g\in G$ is an element in a conjugacy class of type $\omega$. Then

$$\tilde{H}_\omega(\x,q)=\sum_\lambda \calF^\#_{\lambda,\omega}(q)m_\lambda(\x).$$

It follows from Lemma \[R=F\] and Corollary \[R\].

We now recall how to construct from a partition $\lambda=(\lambda_1,\dots,\lambda_r)$ of $n$ a certain family of irreducible characters of $G$. Choose $r$ distinct linear character $\alpha_1,\dots,\alpha_r$ of $\F_q^{\times}$. This defines for each $i$ a linear characters $\tilde{\alpha}_i:\GL_{\lambda_i}(\F_q)\rightarrow\C^{\times}$, $g\mapsto \alpha_i\left({\rm det}(g)\right)$, and hence a linear character $\tilde{\alphahat}: L_\lambda\rightarrow\C^{\times}$, $(g_i)\mapsto \tilde{\alpha}_r(g_r)\cdots\tilde{\alpha}_1(g_1)$. This linear character has the following property: for an element $g\in N_G(L_\lambda)$, we have $\tilde{\alpha}(g^{-1}lg)=\tilde{\alpha}(l)$ for all $l\in L_\lambda$ if and only if $g\in L_\lambda$. A linear character of $L_\lambda$ which satifies this property is called a *regular* character of $L_\lambda$.

It is a well-known fact that $R_{L_\lambda}^G(\tilde{\alphahat})$ is an irreducible character of $G$. Note that the irreducible characters of $G$ are not all obtained in this way (see [@LSr] for the complete description of the irreducible characters of $G$ in terms of Deligne-Luzstig induction).

We now recall the definition of generic tuples of irreducible characters (cf. [@hausel-letellier-villegas Definition 4.2.2]). Since in this paper we are only considering irreducible characters of the form $R_{L_\lambda}^G(\tilde{\alphahat})$, the definition given in [@hausel-letellier-villegas Definition 4.2.2] simplifies.

Consider irreducible characters $R_{L_{\lambda^1}}^G(\tilde{\alphahat}_1),\dots,R_{L_{\lambda^k}}^G(\tilde{\alphahat}_k)$ of $G$ as above for a multi-partition $\lambdahat=(\lambda^1,\dots,\lambda^k)\in(\calP_n)^k$. Let $T$ be the subgroup of $G$ of diagonal matrices. Note that $T\subset L_\lambda$ for all partition $\lambda$, and so $T$ contains the center $Z_\lambda$ of any $L_\lambda$. Consider the linear character $\alphahat=\left(\tilde{\alphahat}_1|_T\right)\cdots\left(\tilde{\alphahat}_k|_T\right)$ of $T$. Then we say that the tuple $\left(R_{L_{\lambda^1}}^G(\tilde{\alphahat}_1),\dots,R_{L_{\lambda^k}}^G(\tilde{\alphahat}_k)\right)$ is *generic* if the restriction $\alphahat|_{Z_\lambda}$ of $\alphahat$ to any subtori $Z_\lambda$, with $\lambda\in\calP_n-\{(n)\}$, is non-trivial and if $\alphahat|_{Z_{(n)}}$ is trivial (the center $Z_{(n)}\simeq \F_q^{\times}$ consists of scalar matrices $a.I_n$).

We can show as for conjugacy classes [@hausel-letellier-villegas Lemma 2.1.2] that if the characteristic $p$ of $\F_q$ and $q$ are sufficiently large, generic tuples of irreducible characters of a given type $\lambdahat$ always exist.

Put $\mathfrak{g}:=\gl_n(\F_q)$. For $X\in\mathfrak{g}$, put $$\Lambda^1(X):=\#\{Y\in\mathfrak{g}\,|\, [X,Y]=0\}.$$

The restriction $\Lambda^1:G\rightarrow\C$ of $\Lambda^1$ to $G\subset\mathfrak{g}$ is the character of the representation $G\rightarrow\GL\left(\C[\mathfrak{g}]\right)$ induced by the conjugation action of $G$ on $\mathfrak{g}$. Fix a non-negative integer $g$ and put $\Lambda:=(\Lambda^1)^{\otimes g}$.

For a multi-partition $\muhat=(\mu^1,\dots,\mu^k)\in(\calP_n)^k$ and a generic tuple $\left(R_{L_{\mu^1}}^G(\tilde{\alphahat}_1),\dots,R_{L_{\mu^k}}^G(\tilde{\alphahat}_k)\right)$ of irreducible characters we put $$R_\muhat:=R_{L_{\mu^1}}^G(\tilde{\alphahat}_1)\otimes\cdots\otimes R_{L_{\mu^k}}^G(\tilde{\alphahat}_k).$$

For two class functions $f,g\in\C(G)$, we define $$\langle f,g\rangle:=|G|^{-1}\sum_{h\in G}f(h)\overline{g(h)}.$$

We have the following theorem [@hausel-letellier-villegas Theorem 1.4.1].

We have $$\left\langle \Lambda\otimes R_\muhat,1\right\rangle=\H_\muhat\left(0,\sqrt{q}\right)$$where $\H_\muhat(z,w)$ is the function defined in §\[Cauchy\].

\[multi\]

The multiplicity $\left\langle \Lambda\otimes
    R_\muhat,1\right\rangle$ depends only on $\muhat$ and not on the choice of linear characters $(\tilde{\alphahat}_1,\dots,\tilde{\alphahat}_k)$.

### Fourier transforms {#Fourier}

Let ${\rm Fun}(\mathfrak{g})$ be the $\C$-vector space of all functions $\mathfrak{g}\rightarrow\C$ and by $\C(\mathfrak{g})$ the subspace of functions $\mathfrak{g}\rightarrow\C$ which are contant on $G$-orbits of $\mathfrak{g}$ for the conjugation action of $G$ on $\mathfrak{g}$.

Let $\Psi:\F_q\rightarrow\C^{\times}$ be a non-trivial additive character and consider the trace pairing ${\rm Tr}:\mathfrak{g}\times\mathfrak{g}\rightarrow\C^{\times}$. Define the Fourier transform $\calF^\mathfrak{g}:{\rm Fun}(\mathfrak{g})\rightarrow{\rm Fun}(\mathfrak{g})$ by the formula

$$\calF^\mathfrak{g}(f)(x)=\sum_{y\in\mathfrak{g}}\Psi\left({\rm Tr}\,(xy)\right)f(y)$$for all $f\in{\rm Fun}(\mathfrak{g})$ and $x\in\mathfrak{g}$.

The Fourier transform satisfies the following easy property.

For any $f\in{\rm Fun}(\mathfrak{g})$ we have:

$$|\mathfrak{g}|\cdot f(0)=\sum_{x\in\mathfrak{g}}\calF^\mathfrak{g}(f)(x).$$ \[fourprop1\]

Let $*$ be the convolution product on ${\rm Fun}(\mathfrak{g})$ defined by $$(f*g)(a)=\sum_{x+y=a}f(x)g(y)$$for any two functions $f,g\in{\rm Fun}(\mathfrak{g})$.

Recall that

$$\calF^\mathfrak{g}(f*g)=\calF^\mathfrak{g}(f)\cdot\calF^\mathfrak{g}(g).\label{conv}$$

For a partition $\lambda$ of $n$, let $\mathfrak{p}_\lambda$, $\mathfrak{l}_\lambda$, $\mathfrak{u}_\lambda$ be the Lie sub-algebras of $\mathfrak{g}$ corresponding respectively to the subgroups $P_\lambda$, $L_\lambda$, $U_\lambda$ defined in §\[finite-groups\], namely $\mathfrak{l}_\lambda=\bigoplus_i\gl_{\lambda_i}(\F_q)$, $\mathfrak{p}_\lambda$ is the parabolic sub-algebra of $\mathfrak{g}$ having $\mathfrak{l}_\lambda$ as a Levi sub-algebra and containing the upper triangular matrices. We have $\mathfrak{p}_\lambda=\mathfrak{l}_\lambda\oplus\mathfrak{u}_\lambda$.

Define the two functions $R_{\mathfrak{l}_\lambda}^{\mathfrak{g}}(1), Q_{\mathfrak{l}_\lambda}^\mathfrak{g} \in\C(\mathfrak{g})$ by

$$\begin{aligned}
&R_{\mathfrak{l}_\lambda}^{\mathfrak{g}}(1)(x)=|P_\lambda|^{-1}\#\{g\in G\,|\, g^{-1}xg\in \mathfrak{p}_\lambda\},\\&Q_{\mathfrak{l}_\lambda}^{\mathfrak{g}}(x)=|P_\lambda|^{-1}\#\{g\in G\,|\, g^{-1}xg\in \mathfrak{u}_\lambda\}.\end{aligned}$$

We define the type of a $G$-orbit of $\mathfrak{g}$ similarly as in the group setting (see above Corollary \[R\]). The types of the $G$-orbits of $\mathfrak{g}$ are then also parameterized by $\mathbf{T}_n$.

From Lemma \[R=F\], we see that $R_{L_\lambda}^G(1)(x)=|P_\lambda|^{-1}\#\{g\in G\,|\, g^{-1}xg\in P_\lambda\}$, hence $R_{\mathfrak{l}_\lambda}^{\mathfrak{g}}(1)$ is the Lie algebra analogue of $R_{L_\lambda}^G(1)$ and the two functions take the same values on elements of same type.\[Rl=RL\]

We have $$\calF^\mathfrak{g}\left(Q_{\mathfrak{l}_\lambda}^\mathfrak{g}\right)=q^{\frac{1}{2}(n^2-\sum_i\lambda_i^2)}R_{\mathfrak{l}_\lambda}^\mathfrak{g}(1).$$ \[fourprop2\]

Consider the $\C$-linear map $R_{\mathfrak{l}_\lambda}^\mathfrak{g}:\C(\mathfrak{l}_\lambda)\rightarrow\C(\mathfrak{g})$ defined by

$$R_{\mathfrak{l}_\lambda}^\mathfrak{g}(f)(x)=|P_\lambda|^{-1}\sum_{\{g\in G\,|\, g^{-1}xg\in \mathfrak{p}_\lambda\}}f(\pi(g^{-1}xg))$$where $\pi:\mathfrak{p}_\lambda\rightarrow\mathfrak{l}_\lambda$ is the canonical projection. Then it is easy to see that $Q_{\mathfrak{l}_\lambda}^\mathfrak{g}=R_{\mathfrak{l}_\lambda}^\mathfrak{g}(1_0)$ where $1_0\in\C(\mathfrak{l}_\lambda)$ is the characteristic function of $0\in\mathfrak{l}_\lambda$, i.e., $1_0(x)=1$ if $x=0$ and $1_0(x)=0$ otherwise. The result follows from the easy fact that $\calF^{\mathfrak{l}_\lambda}(1_0)$ is the identity function $1$ on $\mathfrak{l}_\lambda$ and the fact (see Lehrer [@lehrer]) that $$\calF^\mathfrak{g}\circ R_{\mathfrak{l}_\lambda}^\mathfrak{g}=q^{\frac{1}{2}(n^2-\sum_i\lambda_i^2)}R_{\mathfrak{l}_\lambda}^\mathfrak{g}\circ\calF^{\mathfrak{l}_\lambda}.$$

For $x\in\mathfrak{g}$, denote by $1_x\in{\rm Fun}(\mathfrak{g})$ the characteristic function of $x$ that takes the value $1$ at $x$ and the value $0$ elsewhere. Note that $\calF^\mathfrak{g}(1_x)$ is the linear character $\mathfrak{g}\rightarrow\C$, $t\mapsto \Psi({\rm Tr}\,(xt))$ of the abelian group $(\mathfrak{g},+)$. Hence if $f:\mathfrak{g}\rightarrow\C$ is a function which takes integer values, then $\calF^\mathfrak{g}(f)$ is a character (not necessarily irreducible) of $(\mathfrak{g},+)$. Since the Green functions $Q_{\mathfrak{l}_\lambda}^\mathfrak{g}$ take integer values, by Proposition \[fourprop2\] the function $q^{\frac{1}{2}(n^2-\sum_i\lambda_i^2)}R_{\mathfrak{l}_\lambda}^\mathfrak{g}(1)$ is a character of $(\mathfrak{g},+)$.

\[charab\]

Absolutely indecomposable representations
=========================================

Generalities on quiver representations {#genquiv}
--------------------------------------

Let $\Gamma$ be a finite quiver, $I$ be the set of its vertices and let $\Omega$ be the set of its arrows. For $\gamma\in\Omega$, we denote by $h(\gamma),t(\gamma)\in I$ the head and the tail of $\gamma$. A *dimension vector* of $\Gamma$ is a collection of non-negative integers $\v=\{v_i\}_{i\in I}$ and a *representation* $\varphi$ of $\Gamma$ of dimension $\v$ over a field $\K$ is a collection of $\K$-linear maps $\varphihat=\{\varphi_\gamma:V_{t(\gamma)}\rightarrow
V_{h(\gamma)}\}_{\gamma\in\Omega}$ with ${\rm dim}\, V_i=v_i$. Let ${\rm Rep}_{\Gamma,\v}(\K)$ be the $\K$-vector space of all representations of $\Gamma$ of dimension $\v$ over $\K$. If $\varphihat\in {\rm Rep}_{\Gamma,\v}(\K)$, $\varphihat'\in {\rm
  Rep}_{\Gamma,\v'}(\K)$, then a morphism $f:\varphihat\rightarrow\varphihat'$ is a collection of $\K$-linear maps $f_i: V_i\rightarrow V_i'$, $i\in I$ such that for all $\gamma\in\Omega$, we have $f_{h(\gamma)}\circ
\varphi_\gamma=\varphi'_\gamma\circ f_{t(\gamma)}$.

We define in the obvious way direct sums $\varphihat\,\oplus\,\varphihat'\in {\rm Rep}_\K(\Gamma,\v+\v')$ of representations. A representation of $\Gamma$ is said to be *indecomposable* over $\K$ if it is not isomorphic to a direct sum of two non-zero representations of $\Gamma$. If an indecomposable representation of $\Gamma$ remains indecomposable over any finite extension of $\K$, we say that it is *absolutely indecomposable*. Denote by ${\rm M}_{\Gamma,\v}(\K)$ be the set of isomorphism classes of ${\rm Rep}_{\Gamma,\v}(\K)$ and by ${\rm A}_{\Gamma,\v}(\K)$ the subset of absolutely indecomposable representations of ${\rm
  Rep}_{\Gamma,\v}(\K)$.

By a theorem of Kac there exists a polynomial $A_{\Gamma,\v}(T)\in \Z[T]$ such that for any finite field with $q$ elements $A_{\Gamma,\v}(q)=\#{\rm A}_{\Gamma,\v}(\F_q)$. We call $A_{\Gamma,\v}$ the *$A$-polynomial* of $(\Gamma,\v)$.

Let $\Phi(\Gamma)\subset \Z^I$ be the root system associated with the quiver $\Gamma$ following Kac [@kacconj] and let $\Phi(\Gamma)^+\subset \left(\N\right)^I$ be the subset of positive roots. Let ${\bf C}=(c_{ij})_{i,j}$ be the Cartan matrix of $\Gamma$, namely $$c_{ij}=\begin{cases} 2-2(\text{the number of edges joining $i$ to
    itself})\hspace{.2cm}\text{if }i=j\\ 
  - (\text{the number of edges joining $i$ to
    $j$})\hspace{1.2cm}\text{ otherwise}.
         \end{cases}$$

Then we have the following well-known theorem (see Kac [@kacconj]).

$A_{\Gamma,\v}(q)\neq 0$ if and only if $\v\in\Phi(\Gamma)^+$; $A_{\Gamma,\v}(q)=1$ if and only if $\v$ is a real root. The polynomial $A_{\Gamma,\v}$, if non-zero, is monic of degree $2-{^t}\v{\bf C}\v$. \[kactheo\]

We have the following conjecture due to Kac [@kacconj].

The polynomial $A_{\Gamma,\v}(T)$ has non-negative coefficients. \[kac-conj\]

We will say that a dimension vector $\v$ is *indivisible* if ${\rm gcd}\,\{v_i\}_{i\in I}=1$. Conjecture \[kac-conj\] was proved by Crawley-Boevey and van den Bergh [@crawley-boevey-etal] when the dimension vector $\v$ is indivisible. This was achieved by giving a cohomological interpretation of $A_{\Gamma,\v}(q)$. A more recent work by Mozgovoy [@mozgovoy2] proves Conjecture \[kac-conj\] for any dimension vector for quivers with at least one loop at each vertex. His proof is accomplished via work of Kontsevich-Soibelman [@kontsevich-soibelman] and Efimov [@efimov] on motivic Donaldson-Thomas invariants associated to quivers.

By Kac[@kacconj], there exists a polynomial $M_{\Gamma,\v}(q)\in\Q[T]$ such that $M_{\Gamma,\v}(q):=\#{\rm
  M}_{\Gamma,\v}(\F_q)$ for any finite field $\F_q$. The following formula is a reformation of Hua’s formula [@hua].

We have $$\Log\left(\sum_{\v\in(\N)^I}M_{\Gamma,\v}(q)X^{\v}\right)
=\sum_{\v\in(\N)^I-\{0\}}A_{\Gamma,\v}(q)X^{\v},$$ where $X^{\v}$ is the monomial $\prod_{i\in I}X_i^{v_i}$ for some independent commuting variables $\{X_i\}_{i\in I}$. \[hua\].

Since $A_{\Gamma,\v}(q)\in\Z[q]$, we see by Theorem \[hua\] and Lemma \[Exp\], that $M_{\Gamma,\v}(q)$ also has integer coefficients.

Comet-shaped quivers {#comet}
--------------------

Fix strictly positive integers $g,k,s_1,\dots,s_k$ and consider the following (comet-shaped) quiver $\Gamma$ with $g$ loops on the central vertex and with set of vertices $I=\{0\}\cup\left\{[i,j]\,|\, i=1,\dots,k\,;\, j=1,\dots,s_i\right\}$.

0.1in

( 52.1000, 15.4500)( 4.0000,-17.0000)

(19.7000,-2.4500)[(0,0)[$[1,1]$]{}]{}(29.7000,-2.4000)[(0,0)[$[1,2]$]{}]{}(55.7000,-2.5000)[(0,0)[$[1,s_1]$]{}]{}(19.7000,-6.5500)[(0,0)[$[2,1]$]{}]{}(29.7000,-6.4500)[(0,0)[$[2,2]$]{}]{}(55.7000,-6.5500)[(0,0)[$[2,s_2]$]{}]{}(19.7000,-17.8500)[(0,0)[$[k,1]$]{}]{}(29.7000,-17.8500)[(0,0)[$[k,2]$]{}]{}(55.7000,-17.8500)[(0,0)[$[k,s_k]$]{}]{}(14.3000,-7.6000)[(0,0)[$0$]{}]{}

Let $\Omega^0$ denote the set of arrows $\gamma\in\Omega$ such that $h(\gamma)\neq t(\gamma)$.

Let $\K$ be any field. Let $\varphihat\in{\rm
    Rep}_{\Gamma,\v}(\K)$ and assume that $v_0> 0$. If $\varphihat$ is indecomposable, then the linear maps $\varphi_\gamma$, with $\gamma\in\Omega^0$, are all injective. \[injective\]

If $\gamma$ is the arrow $[i,j]\rightarrow [i,j-1]$, with $j=1,\dots,s_i$ and with the convention that $[i,0]=0$, we use the notation $\varphi_{ij}:V_{[i,j]}\rightarrow V_{[i,j-1]}$ rather than $\varphi_\gamma:V_{t(\gamma)}\rightarrow V_{h(\gamma)}$. Assume that $\varphi_{ij}$ is not injective. We define a graded vector subspace $\Vhat'=\bigoplus_{i\in I}V_i'$ of $\Vhat=\bigoplus_{i\in I}V_i$ as follows.

If the vertex $i$ is not one of the vertices $[i,j],[i,j+1],\dots,[i,s_i]$, we put $V'_i:=\{0\}$. We put $V'_{[i,j]}:={\rm Ker}\, \varphi_{ij}$, $V'_{[i,j+1]}:=\varphi_{i(j+1)}^{-1}(V'_{[i,j]}),\dots,V'_{[i,s_i]}:=\varphi_{is_i}^{-1}(V'_{i(s_i-1)})$. Let $\v'$ be the dimension of the graded space $\Vhat'=\bigoplus_{i\in
    I}V_i'$ which we consider as a dimension vector of $\Gamma$. Define $\varphihat'\in{\rm Rep}_{\Gamma,\v'}(\K)$ as the restriction of $\varphihat$ to $\Vhat'$. It is a non-zero subrepresentation of $\varphihat$. It is now possible to define a graded vector subspace $\Vhat''=\bigoplus_{i\in I}V_i''$ of $\Vhat$ such that the restriction $\varphihat''$ of $\varphihat$ to $\Vhat''$ satifies $\varphihat=\varphihat''\oplus\varphihat'$: we start by taking any subspace $V_{[i,j]}''$ such that $V_{[i,j]}=V_{[i,j]}'\oplus V_{[i,j]}''$, then define $V_{[i,j+r]}''$ from $V_{[i,j]}''$ as $V_{[i,j+r]}'$ was defined from $V_{[i,j]}$, and finally put $V''_i:=V_i$ if the vertex $i$ is not one of the vertices $[i,j],[i,j+1],\dots,[i,s_i]$. As $v_0> 0$, the subrepresentation $\varphihat''$ is non-zero, and so $\varphihat$ is not indecomposable.

We denote by ${\rm Rep}_{\Gamma,\v}^*(\F_q)$ be the subspace of representation $\varphihat\in{\rm Rep}_{\Gamma,\v}(\F_q)$ such that $\varphi_\gamma$ is injective for all $\gamma\in \Omega^0$, and by ${\rm M}_{\Gamma,\v}^*(\F_q)$ the set of isomorphism classes of ${\rm
  Rep}^*_{\Gamma,\v}(\F_q)$. Put $M_{\Gamma,\v}^*(q)=\#\left\{{\rm
    M}_{\Gamma,\v}^*(\F_q)\right\}$. Following [@crawley-par] we say that a dimension vector $\v$ of $\Gamma$ is *strict* if for each $i=1,\dots,k$ we have $n_0\geq v_{[i,1]}\geq
v_{[i,2]}\geq\cdots\geq v_{[i,s_i]}$. Let us denote by $\mathcal{S}$ the set of strict dimension vector of $\Gamma$.

$$\Log\left(\sum_{\v\in\mathcal{S}}M_{\Gamma,\v}^*(q)X^{\v}\right)
=\sum_{\v\in\mathcal{S}-\{0\}}A_{\Gamma,\v}(q)X^{\v}.$$ \[hua-inj\]

Let us denote by $I_{\Gamma,\v}(q)$ the number of isomorphism classes of indecomposable representations in ${\rm
    Rep}_{\Gamma,\v}(\F_q)$. By the Krull-Schmidt theorem, a representation of $\Gamma$ decomposes as a direct sum of indecomposable representation in a unique way up to permutation of the summands. Notice that, for $\v\in\mathcal{S}$, each summand of an element of ${\rm Rep}_{\Gamma,\v}^*(\F_q)$ lives in some ${\rm
    Rep}_{\Gamma,\w}^*(\F_q)$ for some $\w\in\mathcal{S}$. On the other hand, by Lemma \[injective\], ${\rm
    Rep}_{\Gamma,\v}^*(\F_q)$ contains all the indecomposable representations in ${\rm Rep}_{\Gamma,\v}(\F_q)$. This implies the following identity $$\sum_{\v\in\mathcal{S}}M_{\Gamma,\v}^*(q)X^{\v}
=\prod_{\v\in\mathcal{S}-\{0\}}(1-X^{\v})^{-I_{\Gamma,\v}(q)},$$ where $X^{\v}$ denotes the monomial $\prod_{i\in I}X_i^{v_i}$ for some fixed independent commuting variables $\{X_i\}_{ i\in I}$. Exactly as Hua [@hua Proof of Lemma 4.5] does we show from this formal identity that $$\Log\left(\sum_{\v\in\mathcal{S}}M_{\Gamma,\v}^*(q)X^{\v}\right)
=\sum_{\v\in\mathcal{S}-\{0\}}A_{\Gamma,\v}(q)\,X^{\v}.$$

It follows from Proposition \[hua-inj\] that since $A_{\Gamma,\v}(T)\in\Z[T]$ the quantity $M_{\Gamma,\v}^*(q)$ is also the evaluation of a polynomial with integer coefficients at $T=q$.

Given a non-increasing sequence $u=(n_0\geq n_1\geq \cdots)$ of non-negative integers we let $\Delta u$ be the sequence of successive differences $n_0-n_1, n_1-n_2\ldots$. We extend the notation of §\[charGL\] and denote by $\calF_{\Delta u}$ the set of partial flags of $\F_q$-vector spaces $$\{0\}\subseteq E^r\subseteq\cdots\subseteq E^1\subseteq
E^0=(\F_q)^{n_0}$$ such that ${\rm dim}(E^i)=n_i$.

Assume that $\v\in\mathcal S$ and let $\muhat=(\mu^1,\dots,\mu^k)$, where $\mu^i$ is the partition obtained from $\Delta \v_i$ by reordering, where $\v_i:=(v_0\geq v_{[i,1]}\geq\cdots\geq
v_{[i,s_i]})$. Consider the set of orbits $${\mathfrak G}_\muhat(\F_q) :=\left.\left({\rm
      Mat}_{n_0}(\F_q)^g\times\prod_{i=1}^k\calF_{\mu^i}(\F_q)\right)
\right/\GL_{v_0}(\F_q),$$ where $\GL_{v_0}(\F_q)$ acts by conjugation on the first $g$ coordinates and in the obvious way on each $\calF_{\mu^i}(\F_q)$.

Let $\varphihat\in {\rm Rep}_{\Gamma,\v}^*(\F_q)$ with underlying graded vector space $\Vhat=V_0\oplus\bigoplus_{i,j} V_{[i,j]}$. We choose a basis of $V_0$ and we identify $V_0$ with $(\F_q)^{v_0}$. In the chosen basis, the $g$ maps $\varphi_\gamma$, with $\gamma\in\Omega-\Omega^0$, give an element in ${\rm
  Mat}_{v_0}(\F_q)^g$. For each $i=1,\dots,k$, we obtain a partial flag by taking the images in $(\F_q)^{v_0}$ of the $V_{[i,j]}$’s via the compositions of the $\varphi_\gamma$’s where $\gamma$ runs over the arrows of the $i$-th leg of $\Gamma$. We thus have defined a map $${\rm Rep}_{\Gamma,\v}^*(\F_q)\longrightarrow\left.\left({\rm
      Mat}_{v_0}(\F_q)^g\times
    \prod_{i=1}^k\calF_{\Delta \v_i}(\F_q)\right)\right/\GL_{v_0}(\F_q).$$ The target set is clearly in bijection with ${\mathfrak G}_\muhat(\F_q)$ and two elements of ${\rm Rep}_{\Gamma,\v}^*(\F_q)$ have the same image if and only if they are isomorphic. Hence this map induces an isomorphism ${\rm M}_{\Gamma,\v}^*(\F_q)\simeq {\mathfrak
  G}_\muhat(\F_q)$.

We define a new comet-shaped quiver $\Gamma_\muhat$ consisting of $g$ loops on a central vertex and $k$ legs of length $l(\mu^i)-1$ and let $\v_\muhat$ be the dimension vector as in §\[quiver-repns\]. Applying the above construction to the pair $\Gamma_\muhat,\v_\muhat$ we obtain a bijection ${\rm
  M}_{\Gamma_\muhat,\v_\muhat}^*(\F_q)\simeq {\mathfrak
  G}_\muhat(\F_q)$. Put $G_\muhat(q):=\#{\mathfrak G}_\muhat(\F_q)$ and let $A_\muhat(q)$ be the $A$-polynomial of the quiver $\Gamma_\muhat$ for the dimension vector $\v_\muhat$.

We have $$\Log\left(\sum_{\muhat\in\calP^k}G_\muhat(q)\,m_\muhat\right)
=\sum_{\muhat\in\calP^k-\{0\}}A_\muhat(q)\,m_\muhat.$$ \[MA\]

In Proposition \[hua-inj\] make the change of variables $$X_0:=x_{1,1}\cdots x_{k,1}, \qquad
X_{[i,j]}:=x_{i,j}^{-1}x_{i,j+1}, \qquad i=1,2,\ldots,k, \quad
j=1,2,\ldots.$$ Since the terms on both sides are invariant under permutation of the entries $v_{[i,1]},v_{[i,2]},\ldots$ of $\v$ we can collect all terms that yield the same multipartition $\muhat$. The resulting sum of $X^\v$ gives the monomial symmetric function $m_\muhat(x)$.

Since $A_\muhat(q)\in\Z[q]$, it follows from Theorem \[MA\] that $G(q)\in \Z[q]$. \[G\]

Recall that $\F$ denotes an algebraic closure of $\F_q$ and $f:\F\rightarrow\F, x\mapsto x^q$ is the Frobenius endomorphism.

We have $$\log\left(\sum_\muhat G_\muhat(q)m_\muhat\right)=\sum_{d=1}^\infty \phi_d(q)\cdot\log\left(\Omega\left(\x_1^d,\dots,\x_k^d;0,q^{d/2}\right)\right)$$where $\phi_n(q)=\frac{1}{n}\sum_{d|n}\mu(d)(q^{n/d}-1)$ is the number of $\langle f\rangle$-orbits of $\F^{\times}:=\F-\{0\}$ of size $n$. \[sumM\]

If $X$ is a finite set on which a finite group $H$ acts, recall Burnside’s formula which says that

$$\#X/H=\frac{1}{|H|}\sum_{h\in H}\#\{x\in X\,|\, h\cdot x=x\}.$$

Denote by ${\bm C}_n$ the set of conjugacy classes of $\GL_n(\F_q)$. Applying Burnside’s formula to ${\mathfrak G}_\muhat(\F_q)$, with $\muhat\in(\calP_n)^k$, we find that

$$\begin{aligned}
G_\muhat(q)&=|\GL_n(\F_q)|^{-1}\sum_{g\in \GL_n(\F_q)}\Lambda(g)\prod_{i=1}^k\#\{X\in\calF_{\mu^i}\,|\, g\cdot X=X\}\\
 &=|\GL_n(\F_q)|^{-1}\sum_{g\in \GL_n(\F_q)}\Lambda(g)\prod_{i=1}^kR_{L_{\mu^i}}^G(1)(g)\\
&=\sum_{\calO\in {\bm C}_n}\frac{\Lambda(\calO)}{|Z_\calO|}\prod_{i=1}^kR_{L_{\mu^i}}^G(1)(\calO)\end{aligned}$$

For a conjugacy class $\calO$ of $\GL_n(\F_q)$, let $\omega(\calO)$ denotes its type. By Formula (\[alambda\]), we have

$$\frac{\Lambda(\calO)}{|Z_\calO|}=\calH_{\omega(\calO)}(0,\sqrt{q}).$$By Corollary \[R\], we deduce that

$$\sum_\muhat G_\muhat(q)m_\muhat=\sum_{\calO\in {\bm C}}\calH_{\omega(\calO)}(0,\sqrt{q})\prod_{i=1}^k\tilde{H}_{\omega(\calO)}(\x_i,q)$$where ${\bm C}:=\bigcup_{n\geq 1}{\bm C}_n$.

We denote by ${\bf F}^{\times}$ the set of $\langle f\rangle$-orbits of $\F^{\times}$. There is a natural bijection from the set ${\bm C}$ to the set of all maps ${\bf F}^{\times}\rightarrow\calP$ with finite support [@macdonald IV, 2]. If $C\in{\bm C}$ corresponds to $\alpha:{\bf F}^{\times}\rightarrow\calP$, then we may enumerate the elements of $\{s\in{\bf F}^{\times}\,|\, \alpha(s)\neq 0\}$ as $c_1,\dots,c_r$ such that $\omega(\alpha):=(d(c_1),\alpha(c_1))\cdots(d(c_r),\alpha(c_r))$, where $d(c)$ denotes the size of $c$, is the type $\omega(C)$.

We have

$$\begin{aligned}
\sum_\muhat G_\muhat(q)m_\muhat&=\sum_{\alpha\in \calP^{{\bf F}^{\times}}}\calH_{\omega(\alpha)}(0,\sqrt{q})\prod_{i=1}^k\tilde{H}_{\omega(\alpha)}(\x_i,q)\\
&=\prod_{c\in{\bf F}^{\times}}\Omega\left(\x_1^{d(c)},\dots,\x_k^{d(c)};0,q^{d(c)/2}\right)\\
&=\prod_{d=1}^\infty\Omega\left(\x_1^d,\dots,\x_k^d;0,q^{d/2}\right)^{\phi_d(q)}\end{aligned}$$

The second formula displayed in the proof of Proposition \[sumM\] shows that

$$G_\muhat(q)=\left\langle\Lambda\otimes
  R_\muhat(1),1\right\rangle$$where $R_\muhat(1):=R_{L_{\mu^1}}^G(1)\otimes\cdots\otimes
R_{L_{\mu^k}}^G(1)$. \[rem326\]

We have $$A_\muhat(q)=\H_\muhat(0,\sqrt{q}).$$ \[purity\]

From Formula (\[exp\]) we have

$$\sum_\muhat\H_\muhat(0,\sqrt{q})\,m_\muhat=(q-1)\, \Log\,\left(\Omega(0,\sqrt{q})\right).$$ We thus need to see that \_A\_(q)m\_=(q-1) ((0,)). \[ExpA\]

From Theorem \[MA\] we are reduced to prove that

$$\Log\left(\sum_\muhat G_\muhat(q)m_\muhat\right)=(q-1)\Log\left(\Omega(0,\sqrt{q})\right).$$

But this follows from Lemma \[moz\] and Proposition \[sumM\].

Another formula for the $A$-polynomial
--------------------------------------

When the dimension vector $\v_\muhat$ is indivisible, it is known by Crawley-Boevey and van den Bergh [@crawley-boevey-etal] that the polynomial $A_\muhat(q)$ equals (up to some power of $q$) to the polynomial which counts the number of points of some quiver variety over $\F_q$.

Here we prove some relation between $A_\muhat(q)$ and some variety which is closely related to quiver varieties. This relation holds for any $\muhat$ (in particular $\v_\muhat$ can be divisible).

We continue to use the notation $G$, $P_\lambda$, $L_\lambda$, $U_\lambda$, $\calF_\lambda$ of §\[finite-groups\] and the notation $\mathfrak{g}$, $\mathfrak{p}_\lambda$, $\mathfrak{l}_\lambda$, $\mathfrak{u}_\lambda$ of §\[Fourier\].

For a partition $\lambda$ of $n$, define $$\X_\lambda:=\left\{(X,gP_\lambda)\in \mathfrak{g}\times(G/P_\lambda)\,\left|\, g^{-1}Xg\in\mathfrak{u}_\lambda\right\}\right.$$It is well-known that the image of the projection $p:\X_\lambda(\F)\rightarrow \mathfrak{g}(\F)$, $(X,gP_\lambda)\mapsto X$ is the Zariski closure $\overline{\calO}_{\lambda'}$ of the nilpotent adjoint orbit $\calO_{\lambda'}$ of $\gl_n(\F)$ whose Jordan form is given by $\lambda'$, and that $p$ is a desingularization.

Put

$$\mathbb{V}_\muhat:=\left\{\left(a_1,b_1,\dots,a_g,b_g,(X_1,g_1P_{\mu^1}),\dots, (X_k,g_kP_{\mu^k})\right)\in\mathfrak{g}^{2g}\times\X_{\mu^1}\times\cdots\times\X_{\mu^k}\,\left|\, \sum_i[a_i,b_i]+\sum_jX_j=0\right\}\right.$$where $[a,b]=ab-ba$.

Define $\Lambda^\sim:\mathfrak{g}\rightarrow\C$, $z\mapsto q^{gn^2}\Lambda(z)$. By [@hausel-letellier-villegas Proposition 3.2.2] we know that

$$\Lambda^\sim=\calF^\mathfrak{g}(F)$$where for $z\in\mathfrak{g}$, $$F(z):=\#\left\{(a_1,b_1,\dots,a_g,b_g)\in\mathfrak{g}^{2g}\,\left|\, \sum_i[a_i,b_i]=z\right.\right\}.$$

By Remark \[charab\], the functions $\Lambda^\sim$ and $\mathfrak{R}_{\mathfrak{l}_\lambda}^\mathfrak{g}:=q^{\frac{1}{2}(n^2-\sum_i\lambda_i^2)}R_{\mathfrak{l}_\lambda}^\mathfrak{g}$ are characters of $\mathfrak{g}$. Put $$\mathfrak{R}_\muhat(1):=\mathfrak{R}_{\mathfrak{l}_{\mu^1}}^\mathfrak{g}(1)\otimes\cdots\otimes \mathfrak{R}_{\mathfrak{l}_{\mu^k}}^\mathfrak{g}(1).$$

For two functions $f,g:\mathfrak{g}\rightarrow\C$, define their inner product as

$$\langle f,g\rangle=|\mathfrak{g}|^{-1}\sum_{X\in\mathfrak{g}}f(X)\overline{g(X)}.$$

We have $$|\mathbb{V}_\muhat|=\left\langle \Lambda^\sim\otimes \mathfrak{R}_\muhat(1),1\right\rangle.$$ \[inner\]

Notice that

$$|\mathbb{V}_\muhat|=\left(F*Q_{\mathfrak{l}_{\mu^1}}^\mathfrak{g}*\cdots*Q_{\mathfrak{l}_{\mu^k}}^\mathfrak{g}\right)(0).$$ Hence the result follows from Proposition \[fourprop1\] and Proposition \[fourprop2\].

The proposition shows that $|\mathbb{V}_\muhat|$ is a rational function in $q$ which is an integer for infinitely many values of $q$. Hence $|\mathbb{V}_\muhat|$ is a polynomial in $q$ with integer coefficients.

Consider

$$V_\muhat(q):=\frac{|\mathbb{V}_\muhat|}{|G|}.$$

Recall that $d_\muhat=n^2(2g-2+k)-\sum_{i,j}(\mu^i_j)^2+2$.

We have

$$\Log\left(\sum_\muhat q^{-\frac{1}{2}(d_\muhat-2)}V_\muhat(q)m_\muhat\right)=\frac{q}{q-1}\sum_\muhat A_\muhat(q)m_\muhat.$$ \[theohua\]

By Lemma \[moz\] and Formula (\[ExpA\]) we are reduced to prove the following.

We have $$\log\,\left(\sum_\muhat  q^{-\frac{1}{2}(d_\muhat-2)}V_\muhat(q)m_\muhat\right)=\sum_{d=1}^\infty\varphi_d(q)\cdot\log\,\left(\Omega\left(\x_1^d,\dots,\x_k^d;0,q^{d/2}\right)\right)$$where $\varphi_n(q)=\frac{1}{n}\sum_{d|n}\mu(d)q^{n/d}$ is the number of $\langle f\rangle$-orbits of $\F$ of size $n$.

By Proposition \[inner\], we have

$$V_\muhat(q)=\frac{q^{-n^2+\frac{1}{2}(kn^2-\sum_{i,j}(\mu^i_j)^2)}}{|G|}\sum_{x\in\mathfrak{g}}\Lambda^\sim(x)R_{\mathfrak{l}_{\mu^1}}^\mathfrak{g}(1)(x)\cdots R_{\mathfrak{l}_{\mu^1}}^\mathfrak{g}(1)(x).$$By Remark \[Rl=RL\] and Corollary \[R\], we see that $R_{\mathfrak{l}_\lambda}^\mathfrak{g}(1)(x)=\left\langle\tilde{H}_\omega(\x;q),h_\lambda(\x)\right\rangle$ when the $G$-orbit of $x$ is of type $\omega$.

We now proceed exactly as in the proof of Proposition \[sumM\] to prove our formula.

Applications to the character theory of finite general linear groups {#applichar}
--------------------------------------------------------------------

The following theorem (which is a consequence of Theorem \[purity\] and Theorem \[multi\]) expresses certain fusion rules in the character ring of $\GL_n(\F_q)$ in terms of absolutely indecomposable representations of comet shaped quivers.

\[mult-thm\] We have $$\langle \Lambda\otimes R_\muhat,1\rangle=A_\muhat(q).$$ \[multi=A\]

From Theorem \[multi=A\] and Theorem \[kactheo\] we have the following result.

$\langle \Lambda\otimes R_\muhat,1\rangle\neq 0$ if and only if $\v_\muhat\in\Phi(\Gamma_\muhat)^+$. Moreover $\langle
  \Lambda\otimes R_\muhat,1\rangle=1$ if and only if $\v_\muhat$ is a real root.

We will see in §\[delta-non-neg\] that $\v_\muhat$ is always an imaginary root when $g\geq 1$, hence the second assertion concerns only the case $g=0$ (i.e. $\Lambda=1$).

A proof of Theorem \[mult-thm\] for $\v_\muhat$ is indivisible is given in [@hausel-letellier-villegas] by expressing $\langle
\Lambda\otimes R_\muhat,1\rangle$ as the Poincaré polynomial of a comet-shaped quiver variety. This quiver variety exists only when $\v_\muhat$ is indivisible.

In [@Hiss] the authors discusses some results of Mattig who showed that the multiplicities $\langle\calX_1\otimes\calX_2,\calX_3\rangle$, with $\calX_1,\calX_2,\calX_3$ unipotent characters of $\GL_n$, are polynomials in $q$ with rational coefficients. Using calcultion with CHEVIE, he also observed when $n\leq 8$ that the coefficients of $\langle\calX_1\otimes\calX_2,\calX_3\rangle$ are non-negative integers. The following proposition confirms part of this prediction.

For a $k$-tuple of partitions $\muhat=(\mu^1,\ldots,\mu^k)$ of $n$ there exists a polynomial $U_\muhat\in \Z[T]$ such that $\langle
  \Lambda\otimes \calU_{\mu^1}\otimes\cdots\otimes
  \calU_{\mu^k},1\rangle=U_\muhat(q)$.

Since the complete symmetric functions $\{h_\lambda(\x)\}_\lambda$ forms a $\Z$-basis of the ring $\Lambda(\x)$, for any partitions $\mu,\lambda$, there exist integers $a_{\mu\lambda}$ such that $$s_\mu(\x)=\sum_\lambda a_{\mu\lambda}h_\lambda(\x).$$The matrix $(a_{\mu\lambda})_{\mu,\lambda}$ is the transpose inverse of the matrix $K=(K_{\mu\lambda})_{\mu,\lambda}$ of Kostka numbers.

If $\calU_\mu$ is the unipotent characters of $\GL_n$ corresponding to $\mu$, by Theorem \[Rtau\] we have $$\calU_\mu(C)=\left\langle
    \tilde{H}_\omega(\x;q),s_{\mu}(\x)\right\rangle$$where $C$ is a conjugacy class of type $\omega$. Hence by Corollary \[R\], we deduce that $$\calU_\mu=\sum_\lambda
  a_{\mu\lambda}R_{L_\lambda}^G(1).$$Hence $\left\langle\Lambda\otimes\calU_1\otimes\cdots\otimes\calU_k,1\right\rangle$ is a $\Z$-linear combination of multiplicities of the form $$\left\langle\Lambda\otimes
    R_{L_{\lambda^1}}^G(1)\otimes\cdots\otimes
    R_{L_{\lambda^k}}^G(1),1\right\rangle$$and so by Remark \[rem326\], it is a $\Z$-linear combination of polynomials of the form $G_\muhat(q)$ which have integer coefficients (see Remark \[G\]).

Example: Hilbert Scheme of $n$ points on $\C^\times\times\C^\times$ {#Hilbert}
===================================================================

Throughout this section we will have $g=k=1$ and $\muhat$ will be either the partition $(n)$ or $(n-1,1)$.

In this section we illustrate our conjectures and formulas in these cases.

Hilbert schemes: Review
-----------------------

For a nonsingular complex surface $S$ we denote by $S^{[n]}$ the Hilbert scheme of $n$ points in $S$. Recall that $S^{[n]}$ is nonsingular and has dimension $2n$.

We denote by $Y^{[n]}$ the Hilbert scheme of $n$ points in $\C^2$.

Recall (see for instance [@NakHilbert §5.2]) that $h_c^i(Y^{[n]})=0$ unless $i$ is even and that the compactly supported Poincaré polynomial $P_c(Y^{[n]};q):=\sum_ih_c^{2i}(Y^{[n]})q^i$ is given by the following explicit formula

$$\sum_{n\geq 0}P_c(Y^{[n]};q)T^n=\prod_{m\geq 1}\frac{1}{1-q^{m+1}T^m}.\label{Yn}$$

which is equivalent to

$$\Log\left(\sum_{n\geq 0}q^{-n}\cdot P_c(Y^{[n]};q)T^n\right)=\sum_{n\geq 1}q T^n.\label{Ynbis}$$

For $n\geq 2$, consider the partition $\mu=(n-1,1)$ of $n$ and let $C$ be a semisimple adjoint orbit of $\gl_n(\C)$ with characteristic polynomial of the form $(-1)^n(x-\alpha)^{n-1}(x-\beta)$ with $\beta=-(n-1)\alpha$ and $\alpha\neq 0$. Consider the variety

$$\calV_{(n-1,1)}=\{(a,b,X)\in (\gl_n)^2\times C\,|\,
[a,b]+X=0\}.$$

The group $\GL_n$ acts on $\calV_{(n-1,1)}$ diagonally by conjugating the coordinates. This action induces a free action of $\PGL_n$ on $\calV_{(n-1,1)}$ and we put $$\calQ_{(n-1,1)}:=\V_{(n-1,1)}/\!/\PGL_n={\rm
  Spec}\left(\C[\V_{(n-1,1)}]^{\PGL_n}\right).$$ The variety $\calQ_{(n-1,1)}$ is known to be nonsingular of dimension $2n$ (see for instance [@hausel-letellier-villegas §2.2] and the references therein).

We have the following well-known theorem.

The two varieties $\calQ_{(n-1,1)}$ and $Y^{[n]}$ have isomorphic cohomology supporting pure mixed Hodge structures. \[adpure\]

By [@hausel-letellier-villegas Appendix B] it is enough to prove that there is a smooth morphism $f:\mathfrak{M}\rightarrow\C$ which satisfies the two following properties:

\(1) There exists an action of $\C^\times$ on $\mathfrak{M}$ such that the fixed point set $\mathfrak{M}^{\C^\times}$ is complete and for all $x\in X$ the limit ${\rm lim}_{\lambda\mapsto 0}\lambda x$ exists.

\(2) $\calQ_{(n-1,1)}=f^{-1}(\lambda)$ and $Y^{[n]}=f^{-1}(0)$.

Denote by $\v$ the dimension vector of $\Gamma_{(n-1,1)}$ which has coordinate $n$ on the central vertex (i.e., the vertex supporting the loop) and $1$ on the other vertex. It is well-known (see Nakajima [@NakHilbert]) that $Y^{[n]}$ can be identified with the quiver variety $\mathfrak{M}_{0,\theta}(\v)$ where $\theta$ is the stability parameter with coordinate $-1$ on the central vertex and $n$ on the other vertex. If we let $\xi$ be the parameter with coordinate $-\alpha$ at the central vertex and $\alpha-\beta$ at the other vertex, then the variety $\calQ_{(n-1,1)}$ is isomorphic to the quiver variety $\mathfrak{M}_{\xi,\theta}(\v)$ (see for instance [@hausel-letellier-villegas] and the references therein). Now we can define as in [@hausel-letellier-villegas §2.2] a map $f:\mathfrak{M}\rightarrow\C$ such that $f^{-1}(0)=\mathfrak{M}_{0,\theta}(\v)$ and $f^{-1}(\lambda)=\mathfrak{M}_{\xi,\theta}(\v)$ and which satisfies the required properties.

We have $$P_c(Y^{[n]};q)=q^n\cdot A_{(n-1,1)}(q).$$ \[Ypure\]

We have $P_c(\calQ_{(n-1,1)};q)=q^n\cdot \H_{(n-1,1)}(0,\sqrt{q})$ by [@hausel-letellier-villegas Theorem 1.3.1] and so by Theorem \[purity\] we see that $P_c(\calQ_{(n-1,1)};q)=q^n\cdot A_{(n-1,1)}(q)$. Hence the result follows from Theorem \[adpure\].

Now put $X:=\C^*\times\C^*$. Unlike $Y^{[n]}$, the mixed Hodge structure on $X^{[n]}$ is not pure. By Göttsche and Soergel [@Gottsche-Soergel] we have the following result.

We have $h_c^{i,j;k}(X^{[n]})=0$ unless $i=j$ and

$$1+\sum_{n\geq 1}H_c\big(X^{[n]};q,t)T^n=\prod_{n\geq
  1}\frac{(1+t^{2n+1}q^nT^n)^2}{(1-q^{n-1}t^{2n}T^n)(1-t^{2n+2}q^{n+1}
  T^n)}
\label{GS}$$

with $H_c\left(X^{[n]};q,t\right):=\sum_{i,k}h_c^{i,i;k}(X^{[n]})q^it^k$.

Define $\H^{[n]}(z,w)$ such that $$H_c\left(X^{[n]};q,t\right)
=(t\sqrt{q})^{2n}\H^{[n]}\left(-t\sqrt{q},\frac{1}{\sqrt{q}}\right).$$ Then Formula (\[GS\]) reads $$\sum_{n\geq 0}\H^{[n]}(z,w)T^n=\prod_{n\geq
    1}\frac{(1-zwT^n)^2}{(1-z^2T^n)(1-w^2T^n)},
\label{GS1}$$ with the convention that $\H^{[0]}(z,w)=1$. Hence we may re-write Formula (\[GS\]) as $$\Log\left(\sum_{n\geq
    0}\H^{[n]}(z,w)T^n\right)=\sum_{n\geq 1}(z-w)^2T^n. 
\label{GS2}$$

Specializing Formula (\[GS2\]) with $(z,w)\mapsto (0,\sqrt{q})$ we see from Formula (\[Ynbis\]) that $$P_c(Y^{[n]};q)=q^n\cdot
  \H^{[n]}(0,\sqrt{q}).\label{PH=P}$$We thus have the following result.

We have

$$PH_c(X^{[n]};T)=P_c(Y^{[n]};T).$$ where $PH_c(X^{[n]};T):=\sum_ih_c^{i,i;2i}(X^{[n]})T^i$ is the Poincaré polynomial of the pure part of the cohomology of $X^{[n]}$.

A conjecture
------------

The aim of this section is to discuss the following conjecture.

We have $$\H_{(n-1,1)}(z,w)=\H^{[n]}(z,w).
\label{CV=HS}$$ \[conjCV=HS\]

Modulo the conjectural formula (\[mainconj\]), Formula (\[CV=HS\]) says that the two mixed Hodge polynomials $H_c(X^{[n]};q,t)$ and $H_c(\M_{(n-1,1)};q,t)$ agree. This would be a multiplicative analogue of Theorem \[adpure\]. Unfortunately the proof of Theorem \[adpure\] does not work in the multiplicative case. This is because the natural family $g:\mathfrak{X}\to \C$ with $X^{[n]}=g^{-1}(0)$ and $\M_{(n-1,1)}=g^{-1}(\lambda)$ for $0\neq \lambda \in \C$ does not support a $\C^\times$-action with a projective fixed point set and so [@hausel-letellier-villegas Appendix B] does not apply.

One can still attempt to prove that the restriction map $H^*(\mathfrak{X};\Q)\to H^*(g^{-1}(\lambda);\Q)$ is an isomorphism for every fibre over $\lambda\in \C$ by using a family version of the non-Abelian Hodge theory as developed in the tamely ramified case in [@simpson]. In other words one would construct a family $g_\Dol:\mathfrak{X}_{\Dol} \to \C$ such that $g^{-1}_\Dol(0)$ would be isomorphic with the moduli space of parabolic Higgs bundles on an elliptic curve $C$ with one puncture and flag type $(n-1,1)$ and meromorphic Higgs field with a nilpotent residue at the puncture, and $g^{-1}_\Dol(\lambda)$ for $\lambda\neq 0$ would be isomorphic with parabolic Higgs bundles on $C$ with one puncture and semisimple residue at the puncture of type $(n-1,1)$. In this family one should have a $\C^\times$ action satisfying the assumptions of [@hausel-letellier-villegas Appendix B] and so could conclude that $H^*(\mathfrak{X}_\Dol;\Q)\to H^*(g_\Dol^{-1}(\lambda);\Q)$ is an isomorphism for every fibre over $\lambda\in \C$. Then a family version of non-Abelian Hodge theory in the tamely ramified case would yield that the two families $\mathfrak{X}_\Dol$ and $\mathfrak{X}$ are diffeomorphic, and so one could conclude the desired isomorphism $H^*(X^{[n]};\Q)\cong H^*(\M_{(n-1,1)})$ preserving mixed Hodge structures. However a family version of the non-Abelian Hodge theory in the tamely ramified case (which was initiated in [@simpson]) is not available in the literature.

Conjecture \[conjCV=HS\] is true under the specialization $z=0,w=\sqrt q$.

The left hand side specializes to $A_{(n-1,1)}(q)$ by Theorem \[purity\], which by  and Proposition \[Ypure\] agrees with the right hand side.

The [*Young diagram*]{} of a partition $\lambda=(\lambda_1,\lambda_2,\dots)$ is defined as the set of points $(i,j)\in\Z^2$ such that $1\leq j\leq\lambda_i$. We adopt the convention that the coordinate $i$ of $(i,j)$ increases as one goes down and the second coordinate $j$ increases as one goes to the right.

For $\lambda\neq 0$, we define $\phi_\lambda(z,w):=\sum_{(i,j)\in\lambda}z^{j-1}w^{i-1}$, and for $\lambda=0$, we put $\phi_\lambda(z,w)=0$. Define $$\begin{aligned}
&A_1(z,w;T):=\sum_\lambda\calH_\lambda(z,w)\phi_\lambda(z^2,w^2)T^{|\lambda|},\\ 
        &A_0(z,w;T):=\sum_\lambda\calH_\lambda(z,w)T^{|\lambda|}.\end{aligned}$$

\[Log-fmla\] We have $$\sum_{n\geq
  1}\H_{(n-1,1)}(z,w)T^n=(z^2-1)(1-w^2)\frac{A_1(z,w;T)}{A_0(z,w;T)}.$$

The coefficient of the monomial symmetric function $m_{(n-1,1)}(\x)$ in a symmetric function in $\Lambda(\x)$ of homogeneous degree $n$ is the coefficient of $u$ when specializing the variables $\x=\{x_1,x_2,\dots\}$ to $\{1,u,0,0\dots\}$. Hence, the generating series $\sum_{n\geq 1}\H_{(n-1,1)}(z,w)T^n$ is the coefficient of $u$ in $$(z^2-1)(1-w^2)\Log\left(\sum_\lambda\calH_\lambda(z,w)
  \tilde{H}_\lambda(1,u,0,0,\dots;z^2,w^2)\,T^{|\lambda|}\right).$$ We know that $$\tilde{H}_\lambda(\x;z,w)
=\sum_\rho\tilde{K}_{\rho\lambda}(z,w)s_\rho(\x),$$ and $s_\rho(\x)=\sum_{\mu\unlhd\rho} K_{\rho\mu}m_\mu(\x)$ where $K_{\rho\mu}$ are the Kostka numbers. We have $$\begin{aligned}
&s_{(n)}(1,u,0,0,\dots)=1+u+O(u^2)\\
&s_{(n-1,1)}(1,u,0,0,\dots)=u+O(u^2)\end{aligned}$$ and $$s_\rho(1,u,0,0,\dots)=O(u^2)$$ for any other partition $\rho$. Hence, $$\tilde{H}_\lambda(1,u,0,0,\dots;z,w)
=\tilde{K}_{(n)\lambda}(z,w)(1+u)
+\tilde{K}_{(n-1,1)\lambda}(z,w)u+O(u^2).$$ From Macdonald [@macdonald p. 362] we obtain $\tilde{K}_{(n)\lambda}(a,b)=1$ and $\tilde{K}_{(n-1,1)\lambda}(a,b)=\phi_\lambda(a,b)-1$. Hence, finally, $$\tilde{H}_\lambda(1,u,0,0,\dots;z,w)=
1+\phi_\lambda(z,w)u+O(u^2).$$

It follows that $(z^2-1)^{-1}(1-w^2)^{-1}\sum_{n\geq
  1}\H_{(n-1,1)}(z,w)T^n$ equals the coefficient of $u$ in $$\Log\left(\sum_\lambda\calH_\lambda(z,w)\left(1
    +\phi_\lambda(z^2,w^2)u+O(u^2)\right) 
  T^{|\lambda|}\right)=\Log\left(A_0(T)+A_1(T)u+O(u^2)\right).$$ The claim follows from the general fact $$\Log\left(A_0(T)+A_1(T)u+O(u^2)\right)=\Log \,
A_0(T)+\frac{A_1(T)}{A_0(T)}u+O(u^2).$$

Combining Proposition \[Log-fmla\] with  we obtain the following.

Conjecture \[conjCV=HS\] is equivalent to the following combinatorial identity $$1+(z^2-1)(1-w^2)\frac{A_1(z,w;T)}{A_0(z,w;T)}
 =\prod_{n\geq 1}\frac{(1-zwT^n)^2}{(1-z^2T^n)(1-w^2T^n)}.
\label{comb}$$

The main result of this section is the following theorem.

\[euler-spec\] Formula (\[comb\]) is true under the Euler specialization $(z,w)\mapsto\left(\sqrt{q},1/\sqrt{q}\right)$; namely, we have $$\H_{(n-1,1)}(z,z^{-1})
=\H^{[n]}(z,z^{-1}).
\label{CV=HS1}$$ Equivalently, the two varieties $\M_{(n-1,1)}$ and $X^{[n]}$ have the same $E$-polynomial.

Consider the generating function $$F:=(1-z)(1-w)\sum_\lambda\phi_\lambda(z,w)T^{|\lambda|}.$$

It is straightforward to see that for $\lambda\neq 0$ we have $$\begin{aligned}
(1-z)(1-w)\phi_\lambda(z,w)&=1+\sum_{i=1}^{l(\lambda)}(w^i-w^{i-1})z^{\lambda_i}
-w^{l(\lambda)}\\
&=1+\sum_{i\geq 1}(w^i-w^{i-1})z^{\lambda_i}.\end{aligned}$$ Interchanging summations we find $$F=\sum_{i\geq 1}(w^i-w^{i-1})\sum_{\lambda\neq
  0}z^{\lambda_i}T^{|\lambda|}+\sum_{\lambda\neq 0}T^{|\lambda|}.$$ To compute the sum over $\lambda$ for a fixed $i$ we break the partitions as follows: $$\lambda_1\geq\lambda_2\geq\cdots\geq
\lambda_{i-1}\geq\underbrace{\lambda_i\geq\lambda_{i+1}\geq\cdots}_{\rho}$$ and we put $$\begin{aligned}
&\rho:=(\lambda_i,\lambda_{i+1},\dots)\\
&\mu:=(\lambda_1-\lambda_i,\lambda_2-\lambda_i,\dots,\lambda_{i-1}-\lambda_i)\end{aligned}$$ Notice that $\mu_1'=l(\mu)<i$, $\rho_1=l(\rho')=\lambda_i$ and $|\lambda|=|\mu|+|\rho|+l(\rho')(i-1)$.

We then have $$\sum_\lambda
z^{\lambda_i}T^{|\lambda|}=\sum_{\mu_1<i}T^{|\mu|}\sum_\rho
z^{l(\rho)}T^{|\rho|+(i-1)l(\rho)}$$(changing $\rho$ to $\rho'$ and $\mu$ to $\mu'$). Each sum can be written as an infinite product, namely $$\sum_\lambda
z^{\lambda_i}T^{|\lambda|}=\prod_{k=1}^{i-1}(1-T^k)^{-1}\prod_{n\geq
  1}(1-zT^{n+i-1})^{-1}.$$

So $$\begin{aligned}
F&=\sum_{\lambda\neq 0}T^{|\lambda|}+\sum_{i\geq
    1}(w^i-w^{i-1})\left(\prod_{k=1}^{i-1}(1-T^k)^{-1}\prod_{n\geq
      1}(1-zT^{n+i-1})^{-1}-1\right)\\&=\sum_{\lambda\neq
    0}T^{|\lambda|}+\prod_{n\geq 1}(1-zT^n)^{-1}\sum_{i\geq
    1}(w^i-w^{i-1})\prod_{k=1}^{i-1}\frac{(1-zT^k)}{(1-T^k)}-\sum_{i\geq
    1}(w^i-w^{i-1}).\end{aligned}$$ The last sum telescopes to $1$ and we find $$F=\sum_\lambda T^{|\lambda|}+\prod_{n\geq
  1}(1-zT^n)^{-1}(w-1)\sum_{i\geq 1}w^{i-1}\prod_{k=1}^{i-1} 
\frac{(1-zT^k)}{(1-T^k)}.
\label{F}$$

By the Cauchy $q$-binomial theorem the sum equals $$\frac{1}{(1-w)}\prod_{n\geq
  1}\frac{(1-wzT^n)}{(1-wT^n)}.$$Also$$\sum_\lambda
T^{|\lambda|}=\prod_{n\geq 1}(1-T^n)^{-1}.$$If we divide Formula (\[F\]) by this we finally get

$$1-(1-z)(1-w)\prod_{n\geq
    1}(1-T^n)\sum_\lambda\phi_\lambda(z,w)T^{|\lambda|}=\prod_{n\geq
    1}\frac{(1-wzT^n)(1-T^n)}{(1-zT^n)(1-wT^n)}.$$

Putting now $(z,w)=(q,1/q)$ we find that

$$1-(1-q)(1-1/q)\prod_{n\geq
  1}(1-T^n)\sum_\lambda\phi_\lambda(q,1/q)T^{|\lambda|}=\prod_{n\geq
  1}\frac{(1-T^n)^2}{(1-qT^n)(1-q^{-1}T^n)}.\label{phi}$$

From Formula (\[H-specializ\]) we have $\calH_\lambda(\sqrt{q},1/\sqrt{q})=1$ and so $$\begin{aligned}
&A_1\left(\sqrt{q},\frac{1}{\sqrt{q}};T\right)
=\sum_\lambda\phi_\lambda\left(q,\frac{1}{q}\right)T^{|\lambda|}\\
&A_0\left(\sqrt{q},\frac{1}{\sqrt{q}};T\right)=\sum_\lambda 
  T^{|\lambda|}=\prod_{n\geq 1}(1-T^n)^{-1}\end{aligned}$$ Hence, under the specialization $(z,w)\mapsto(\sqrt{q},1/\sqrt{q})$ , the left hand side of Formula (\[comb\]) agrees with the left hand side of Formula (\[phi\]).

Finally, it is straightforward to see that if we put $(z,w)=(\sqrt{q},1/\sqrt{q})$, then the right hand side of Formula (\[comb\]) agrees with the right hand side of Formula (\[phi\]), hence the theorem.

Connection with modular forms
-----------------------------

For a positive, even integer $k$ let $G_k$ be the standard Eisenstein series for $SL_2(\Z)$ $$\label{Eisenstein-defn}
G_k(T)=\frac{-B_k}{2k}+\sum_{n\geq 1}\sum_{d\,|\, n} d^{k-1}T^n,$$ where $B_k$ is the $k$-th Bernoulli number.

For $k>2$ the $G_k$’s are modular forms of weight $k$; i.e., they are holomorphic (including at infinity) and satisfy $$\label{quasi-mod-2}
\begin{split}
&G_k \left( \frac{a\tau +b}{c\tau +d}\right) 
= (c\tau +d)^k G_k (\tau)\\
\noalign{\vskip6pt}
& \text{for }\ \begin{pmatrix}a&b\\ c&d\end{pmatrix} \in SL_2(\Z)\ ,
\quad T = e^{2\pi i\tau}\ ,\quad \Im \tau >0\ .
\end{split}$$ For $k=2$ we have a similar transformation up to an additive term. $$\label{transform}
  G_2 \left( \frac{a\tau +b}{c\tau +d}\right) 
  = (c\tau +d)^2 G_2 (\tau) - \frac{c}{4\pi i} (c\tau +d).$$ The ring $\Q [G_2,\ G_4,\ G_6]$ is called the ring of *quasi-modular* forms (see [@kaneko-zagier]).

We have $$1+\sum_{n\geq
  1}\H_{(n-1,1)}\left(e^{u/2},e^{-u/2}\right)T^n
=\frac{1}{u}\left(e^{u/2}-e^{-u/2}\right)\exp\left(2\sum_{k\geq
    2}G_k(T)\frac{u^k}{k!}\right).$$ In particular, the coefficient of any power of $u$ on the left hand side is in the ring of quasi-modular forms.

The relation between the $E$-polynomial of the Hilbert scheme of points on a surface and theta functions goes back to Göttsche [@Gottsche].

Consider the classical theta function $$\label{classical-theta}
\theta (w) = (1-w) \prod_{n\ge1} 
\frac{(1-q^nw)(1-q^nw^{-1})}{(1-q^n)^2},$$ with simple zeros at $q^n$, $n\in \Z$ and functional equations $$\label{functional-equation}
\begin{split}
\text{i)}\qquad& \theta (qw) = - w^{-1}\theta (w)\\
\text{ii)}\qquad& \theta (w^{-1}) = - w^{-1}\theta (w)
\end{split}$$ We have the following expansion $$\label{expansion}
\frac1{\theta (w)} = \frac1{1-w} + 
\sum_{\substack{n,m >0 \\ n\not\equiv m\mod 2}} 
(-1)^n\ q^{\frac{nm}2}\ w^{\frac{m-n-1}2}$$ This is classical but not that well known. For a proof see, for example,  [@jordan Chap.VI, p. 453], where it is deduced from a more general expansion due to Kronecker. Namely, $$\frac{\theta(uv)}{\theta(u)\theta(v)}=\sum_{m,n\geq
    0}q^{mn}u^mv^n-\sum_{m,n\geq 1}q^{mn}u^{-m}v^{-n}.$$ (To see this set $v=u^{-\tfrac12}$ and use the functional equation to get $$\frac1{\theta (w)} = \frac1{1-w} + \sum_{m,n\geq 1}q^{mn}
(w^{m-\tfrac12(n+1)}-w^{m+\tfrac12(n-1)}),$$ which is equivalent to .) It is not hard, as was shown to us by J. Tate, to give a direct proof using .

From we deduce, switching $q$ to $T$ and $w$ to $q$, that $$\label{expansion2}
\prod_{n\ge1} \frac{(1-T^n)^2}{(1-qT^n)(1-q^{-1}T^n)} 
= 1 + \sum_{\substack{r,s>0\\ r\not\equiv s \mod 2}} 
(-1)^r T^{\frac{rs}2} \left( q^{\frac{s-r-1}2} - q^{\frac{2-r+1}2}\right)$$ which combined with Theorem \[euler-spec\] gives $$\label{expansion-combo}
\H_{(n-1,1)} \left(\sqrt{q},\frac1{\sqrt{q}}\right) 
= \sum_{\substack{rs=2n\\ r\not\equiv s\mod2}} 
(-1)^r \left( q^{\frac{s-r-1}2} - q^{\frac{2-r+1}2}\right)$$

We compute the logarithm of the left hand side of and get $$\sum_{m,n\ge1} (q^m + q^{-m} -2) \frac{T^{mn}}{m}$$ Applying $(q\frac{d}{dq})^k$ and then setting $q=1$ we obtain $$\sum_{m,n\ge1} (m^k + (-m)^k) \frac{T^{mn}}{m},$$ which vanishes identically if $k$ is odd. For $k$ even, it equals $$2 \sum_{n\ge1} \sum_{d\mid n} d^{k-1}\ T^n.$$ Comparing with  we see that this series equals $2G_k$, up to the constant term.

Note that if $q=e^u$ then $$q\frac{d}{dq} = \frac{d}{du}\ ,\qquad q=1 \leftrightarrow u=0.$$ Hence , $$\log \bigg( 1+\sum_{n\ge1} \H_{(n-1,1)} 
( e^{u/2}, e^{-u/2})T^n\bigg) = \sum_{\substack{k\ge2 \\ \text{even}}}
\left( 2G_k + \frac{B_k}{k}\right) \frac{u^k}{k!}.$$ On the other hand, it is easy to check that $$u\exp \bigg( \sum_{k\ge2}\frac{B_k}{k}\ \frac{u^k}{k!}\bigg) 
= e^{u/2} - e^{-u/2}$$ ($B_k=0$ if $k>1$ is odd.) This proves the claim.

Connectedness of character varieties
====================================

The main result
---------------

Let $\muhat$ be a multi-partition $(\mu^1,\dots,\mu^k)$ of $n$ and let $\M_{\muhat}$ be a genus $g$ generic character variety of type $\muhat$ as in §\[char\].

The character variety $\M_\muhat$ is connected (if not empty). \[connectedness\]

Let us now explain the strategy of the proof.

We first need the following lemma.

If $\M_\muhat$ is not empty, its number of connected components equals the constant term in $E(\M_\muhat;q)$.

The number of connected components of $\M_\muhat$ is ${\rm dim}\hspace{.05cm}H^0(\M_\muhat,\C)$ which is also equal to the mixed Hodge number $h^{0,0;0}(\M_\muhat)$.

Poincaré duality implies that

$$h^{i,j;k}(\M_\muhat)=h_c^{d_\muhat-i,d_\muhat-j;2d_\muhat-k}(\M_\muhat).$$ From Formula (\[curious\]) we thus have

$$E(\M_\muhat;q)=\sum_i\left(\sum_k(-1)^kh^{i,i;k}(\M_\muhat)\right)q^i,$$

On the other hand the mixed Hodge numbers $h^{i,j;k}(X)$ of any complex non-singular variety $X$ are zero if $(i,j,k)\notin
\{(i,j,k)|\hspace{.05cm} i\leq k,j\leq k,k\leq i+j\}$, see [@Del1]. Hence $h^{0,0;k}(\M_\muhat)=0$ if $k>0$.

We thus deduce that the constant term of $E(\M_\muhat;q)$ is $h^{0,0;0}(\M_\muhat)$.

From the above lemma and Formula (\[mainresult\]) we are reduced to prove that the coefficient of the lowest power $q^{-\frac{d_\muhat}{2}}$ of $q$ in $\mathbb{H}_\muhat(\sqrt{q},1/\sqrt{q})$ is equal to $1$.

The strategy to prove this goes in two steps. First, \[step-1\] we analyze the lowest power of $q$ in $\calA_{\lambda\muhat}(q)$, where $$\Omega\left(\sqrt{q},1/\sqrt{q}\right) = \sum_{\lambda, \muhat}
\calA_{\lambda\muhat}(q)\, m_\muhat.$$ Then \[step-2\] we see how these combine in $\Log
\left(\Omega\left(\sqrt{q},1/\sqrt{q}\right)\right)$. In both case, Lemma \[nrm-ineq\] and Lemma \[ineq-1\], we will use in an essential way the inequality of §\[appendix\]. Though very similar, the relation between the partitions $\nu^p$ in these lemmas and the matrix of numbers $x_{i,j}$ in §\[appendix\] is dual to each other (the $\nu^p$ appear as rows in one and columns in the other).

Preliminaries {#delta-non-neg}
-------------

For a multi-partition $\muhat\in \left(\calP_n\right)^k$ we define $$\label{Delta-defn}
\Delta(\muhat):=\tfrac12 d_\muhat-1= \tfrac12(2g-2+k)n^2- \tfrac12
\sum_{i,j} \left(\mu_j^i\right)^2.$$

Note that when $g=0$ the quantity $-2\Delta(\muhat)$ is Katz’s [ *index of rigidity*]{} of a solution to $X_1\cdots X_k=I$ with $X_i\in
  \calC_i$ (see for example [@kostov2]\[p. 91\]).

From $\muhat$ we define as above Theorem \[MA\] a comet-shaped quiver $\Gamma=\Gamma_\muhat$ as well as a dimension vector $\v=\v_\muhat$ of $\Gamma$. We denote by $I$ the set of vertices of $\Gamma$ and by $\Omega$ the set of arrows. Recall that $\muhat$ and $\v$ are linearly related ($v_0=n$ and $v_{[i,j]}=n-\sum_{r=1}^j \mu^i_r$ for $j>1$ and conversely, $\mu_1^i=n-v_{[i,1]}$ and $\mu_j^i=v_{[i,j-1]}-v_{[i,j]}$ for $j>1$). Hence $\Delta$ yields an integral-valued quadratic from on $\Z^I$. Let $(\cdot,\cdot)$ be the associated bilinear form on $\Z^I$ so that $$\label{pairing-defn}
(\v,\v)=2\Delta(\muhat).$$ Let $\e_0$ and $\e_{[i,j]}$ be the fundamental roots of $\Gamma$ (vectors in $\Z^I$ with all zero coordinates except for a $1$ at the indicated vertex). We find that $$(\e_0,\e_0)= 2g-2, \qquad (\e_{[i,j]},\e_{[i,j]})=-2, \qquad
(\e_0,\e_{[i,1]})=1 \qquad (\e_{[i,j]},\e_{[i,j+1]})=1,$$ for $ i=1,2,\ldots,k,  j=1,2,\ldots, s_i-1$ and all other pairings are zero. In other words, $\Delta$ is the negative of the Tits quadratic form of $\Gamma$ (with the natural orientation of all edges pointing away from the central vertex).

With this notation we define $$\label{delta-defn}
\delta=\delta(\muhat):=(\e_0,\v)=(2g-2+k)n-\sum_{i=1}^k\mu_1^i.$$

In the case of $g=0$ the quantity $\delta$ is called the [*defect*]{} by Simpson (see [@simpson1 p.12]).

Note that $\delta\geq (2g-2)n$ is non-negative unless $g=0$. On the other hand, $$\label{root-ineq}
(\e_{[i,j]},\v)=\mu_j^i-\mu_{j+1}^i\geq 0.$$ We now follow the terminology of [@kacconj].

\[fund-set\] The dimension vector $\v$ is in the fundamental set of imaginary roots of $\Gamma$ if and only if $\delta(\muhat) \geq 0$.

Note that $v_{[i,j]}>0$ if $j<l(\mu^i)$ and $v_{[i,j]}=0$ for $j\geq
  l(\mu^i)$; since $n>0$ the support of $\v$ is then connected. We already have $(\e_{[i,j]},\v)\geq 0$ by , hence $\v$ is in the fundamental set of imaginary roots of $\Gamma$ if and only if $\delta \geq 0$ (see [@kacconj]).

For a partition $\mu\in \calP_n$ we define $$\sigma(\mu):=n\mu_1-\sum_j\mu_j^2$$ and extend to a multipartition $\muhat \in \left(\calP_n\right)^k$ by $$\sigma(\muhat):=\sum_{i=1}^k\sigma(\mu^i).$$

Again for $g=0$ this is called the [*superdefect*]{} by Simpson.

We say that $\mu\in \calP_n$ is [*rectangular*]{} if and only if all of its (non-zero) parts are equal, i.e., $\mu=(t^{n/t})$ for some $t\mid n$. We extend this to multi-partitions: $\muhat=
(\mu^1,\ldots,\mu^k) \in \left(\calP_n\right)^k$ is rectangular if each $\mu^i$ is (the $\mu^i$’s are not required to be of the same length). Note that $\muhat$ is rectangular if and only if the associated dimension vector $\v$ satisfies $(\e_{[i,j]},\v)= 0$ for all $[i,j]$ by .

\[sigma-ineq\] For $\muhat \in \left(\calP_n\right)^k$ we have $$\sigma(\muhat) \geq 0$$ with equality if and only if $\muhat$ is rectangular.

For any $\mu\in \calP_n$ we have $n\mu_1=\mu_1\sum_j\mu_j \geq
\sum_j\mu_j^2$ and equality holds if and only if $\mu_1=\mu_j$.

Since $$\label{Delta-sigma}
2\Delta(\muhat)=n\,\delta(\muhat)+\sigma(\muhat)$$ we find that $$\label{dim-ineq}
d_\muhat  \geq n\,\delta(\muhat)+2$$ and in particular $d_\muhat \geq 2$ if $\delta(\muhat)\geq 0$.

If $\Gamma$ is affine it is known that the positive imaginary roots are of the form $t\v^*$ for an integer $t\geq 1$ and some $\v^*$. We will call $\v^*$ the [*basic positive imaginary root*]{} of $\Gamma$. The affine star-shaped quivers are given in the table below; their basic positive imaginary root is the dimension vector associated to the indicated multi-partition $\muhat^*$. These $\muhat^*$, and hence also any scaled version $t\muhat^*$ for $t\geq
1$, are rectangular. Moreover, $\Delta(\muhat^*)=0$ and in fact, $\muhat^*$ generates the one-dimensional radical of the quadratic form $\Delta$ so that $\Delta(\muhat^*,\nuhat)=0$ for all $\nuhat$.

\[affine-descrip\] Suppose that $\muhat = (\mu^1,\ldots,\mu^k) \in
\left(\calP_n\right)^k$ has $\delta(\muhat)\geq 0$. Then $d_\muhat=2$ if and only if $\Gamma$ is of affine type, i.e., $\Gamma$ is either the Jordan quiver $J$ (one loop on one vertex), $\tilde D_4,\tilde
E_6,\tilde E_7$ or $\tilde E_8$, and $\muhat=t\muhat^*$ (all parts scaled by $t$) for some $t\geq 1$, where $\muhat^*$, given in the table below, corresponds to the basic imaginary root of $\Gamma$.

By and Lemma \[sigma-ineq\] $d_\muhat=2$ when $\delta(\muhat)\geq 0$ if and only if $\delta(\muhat)=0$ and $\muhat$ is rectangular. As we observed above $\delta(\muhat)\geq (2g-2)n$. Hence if $\delta(\muhat)=0$ then $g=1$ or $g=0$. If $g=1$ then necessarily $\mu^i=(n)$ and $\Gamma$ is the Jordan quiver $J$.

If $g=0$ then $\delta=0$ is equivalent to the equation $$\label{affine-eqn}
\sum_{i=1}^k\frac{1}{l_i}=k-2,$$ where $l_i:=n/t_i$ is the length of $\mu^i=(t_i^{n/t_i})$. In solving this equation, any term with $l_i=1$ can be ignored. It is elementary to find all of its solutions; they correspond to the cases $\Gamma=\tilde D_4,\tilde E_6,\tilde E_7$ or $\tilde E_8$.

We summarize the results in the following table $$\label{table}
\begin{array}{|c|c|c|c|}
\hline
\Gamma & l_i& n & \muhat^* \\
\hline
J & (1) & 1& (1) \\
\tilde{D}_4 &  (2,2,2,2) & 2 &(1,1),\quad (1,1),\quad (1,1),\quad (1,1)\\
\tilde{E}_6&  (3,3,3) &  3 & (1,1,1),\quad (1,1,1),\quad (1,1,1)\\
\tilde{E}_7 & (2,4,4) & 4 &(2,2),\quad (1,1,1,1),\quad (1,1,1,1)\\
\tilde{E}_8 & (2,3,6) & 6 & (3,3),\quad (2,2,2),\quad (1,1,1,1,1,1)\\
\hline
\end{array}$$ where we listed the cases with smallest possible positive values of $n$ and $k$ and the corresponding multi-partition $\muhat^*$.

Proposition \[affine-descrip\] is due to Kostov, see for example [@simpson1 p.14].

We will need the following result about $\Delta$.

\[Delta-ineq\] Let $\muhat\in \left(\calP_n\right)^k$ and $\nuhat^p=(\nu^{1,p},\ldots,\nu^{k,p})\in \left(\calP_{n_p}\right)^k$ for $p=1,\ldots, s$ be non-zero multi-partitions such that up to permutations of the parts of $\nu^{i,p}$ we have $$\mu^i=\sum_{p=1}^s \nu^{i,p}, \qquad \qquad i=1,\ldots,k.$$ Assume that $\delta(\muhat)\geq 0$. Then $$\sum_{p=1}^s\Delta(\nuhat^p)\leq\Delta(\muhat).$$ Equality holds if and only if

\(i) $s=1$ and $\muhat=\nuhat^1$.

or

\(ii) $\Gamma$ is affine and $\muhat,\nuhat^i,\ldots,\nuhat^s$ correspond to positive imaginary roots.

We start with the following. For partitions $\mu,\nu$ define $$\nrm_\mu(\nu):=\mu_1 |\nu|^2 -|\mu|\sum_i \nu_i^2.$$ Note that $\nrm_\mu(\mu)=|\mu|\,\sigma(\mu)$.

\[nrm-ineq\] Let $\nu^1,\ldots,\nu^s$ and $\mu$ be non-zero partitions such that up to permutation of the parts of each $\nu^p$ we have $\sum_{p=1}^s\nu^p=\mu.$ Then $$\sum_{p=1}^s \nrm_\mu(\nu^p)\leq \nrm_\mu(\mu).$$ Equality holds if and only if:

\(i) $s=1$ and $\mu=\nu^1$.

or

\(ii) $\nu^1,\ldots,\nu^s$ and $\mu$ all are rectangular of the same length.

This is just a restatement of the inequality of §\[appendix\] with $x_{i,k}=\nu_{\sigma_k(i)}^k$, for the appropriate permutations $\sigma_k$, where $1\leq i \leq l(\mu),1\leq k \leq s$.

\[sigma-rectang\] If the partitions $\mu,\nu$ are rectangular of the same length then $$\sigma_\mu(\nu)=0.$$

Direct calculation.

From the definition we get $$2n\Delta(\muhat)= \delta(\muhat)n^2+\sum_{i=1}^k\nrm_{\mu^i}(\mu^i)$$ and similarly $$2n\Delta(\nuhat^p)=\delta(\muhat)n_p^2 +
\sum_{i=1}^k\nrm_{\mu^i}(\nu^{i,p}), \qquad \qquad p=1,\ldots,s$$ hence $$2n\sum_{p=1}^s\Delta(\nuhat^p)=\delta(\muhat)\sum_{p=1}^sn_p^2 +
\sum_{i=1}^k\sum_{p=1}^s\nrm_{\mu^i}(\nu^{i,p}).$$ Since $n=\sum_{p=1}^sn_p$ and $\delta(\muhat)\geq 0$ we get from Lemma \[nrm-ineq\] that $$\sum_{p=1}^s\Delta(\nuhat^p)\leq\Delta(\muhat)$$ as claimed.

Clearly, equality cannot occur if $\delta(\muhat)>0$ and $s>1$. If $\delta(\muhat)=0$ and $s>1$ it follows from Lemmas \[nrm-ineq\], \[sigma-rectang\] and that $\Delta(\muhat)=\Delta(\nuhat^p)=0$ for $p=1,2,\ldots,s$. Now (ii) is a consequence of Proposition \[affine-descrip\].

Proof of Theorem \[connectedness\]
----------------------------------

### Step I {#step-1}

Let $$\label{calA-defn}
\calA_{\lambda\muhat}(q):= 
q^{(1-g)|\lambda|}\left(q^{-n(\lambda)} 
  H_{\lambda}(q)\right)^{2g+k-2}
\prod_{i=1}^k\left\langle
  h_{\mu^i}(\x_i),
  s_\lambda(\x_i\y)\right\rangle,$$ so that by Lemma \[specializ\] $$\Omega\left(\sqrt{q},1/\sqrt{q}\right) = \sum_{\lambda, \muhat}
\calA_{\lambda\muhat}(q)\, m_\muhat.$$It is easy to verify that $\calA_{\lambda\muhat}$ is in $\Q(q)$.

For a non-zero rational function $\calA\in \Q(q)$ we let $v_q\left(\calA\right)\in \Z$ be its valuation at $q$. We will see shortly that $\calA_{\lambda\muhat}$ is nonzero for all $\lambda,
\muhat$; let $v(\lambda) :=v_q\left(\calA_{\lambda\muhat}(q)\right)$. The first main step toward the proof of the connectedness is the following theorem.

\[minim\] Let $\muhat=(\mu^1,\mu^2,\ldots,\mu^k)\in {\P_n}^k$ with $\delta(\muhat) \geq 0$. Then

i\) The minimum value of $v(\lambda)$ as $\lambda$ runs over the set of partitions of size $n$, is $$v((1^n))=-\Delta(\muhat).$$

ii\) There are two cases as to where this minimum occurs.

Case I: The quiver $\Gamma$ is affine and the dimension vector associated to $\muhat$ is a positive imaginary root $t\v^*$ for some $t\mid n$. In this case, the minimum is reached at all partitions $\lambda$ which are the union of $n/t$ copies of any $\lambda_0\in\calP_t$.

Case II: Otherwise, the minimum occurs only at $\lambda=(1^n)$. \[step1\]

Before proving the theorem we need some preliminary results.

$\langle h_{\mu}(\x),s_\lambda(\x\y)\rangle$ is non-zero for all $\lambda$ and $\mu$.

We have $s_\lambda(\x\y)=\sum_\nu K_{\lambda\nu}m_\nu(\x\y)$ [@macdonald I 6 p.101] and $m_\nu(\x\y)=\sum_\mu
  C_{\nu\mu}(\y)\,m_\mu(\x)$ for some $C_{\nu\mu}(\y)$. Hence $$\label{m-coeff}
\langle
  h_\mu(\x),s_\lambda(\x\y)\rangle=\sum_\nu
  K_{\lambda\nu}C_{\nu\mu}(\y).$$ For any set of variables $\x\y=\{x_iy_j\}_{1\leq i,1\leq 
  j}$ we have $$\label{C-fmla}
C_{\nu\mu}(\y)=\sum m_{\rho^1}(\y)\cdots m_{\rho^r}(\y),$$ where the sum is over all partitions $\rho^1,\ldots, \rho^r$ such that $|\rho^p|=\mu_p$ and $\rho^1\cup \cdots \cup \rho^r=\nu$. In particular the coefficients of $C_{\nu\mu}(\y)$ as power series in $q$ are non-negative. We can take, for example, $\rho^p=(1^{\mu_p})$ and then $\nu=(1^n)$. Since $K_{\lambda\nu}\geq 0$ [@macdonald I (6.4)] for any $\lambda,\nu$ and $K_{\lambda,(1^n)}=n!/h_\lambda$ [@macdonald I 6 ex. 2], with $h_\lambda=\prod_{s\in \lambda}h(s)$ the product of the hook lengths, we see that $\langle h_\mu(\x),s_\lambda(\x\y)\rangle$ is non-zero and our claim follows.

In particular $\calA_{\lambda\muhat}$ is non-zero for all $\lambda$ and $\muhat$. Define $$\label{v-1-defn}
v(\lambda,\mu):=v_q\left(\langle
h_{\mu}(\x),s_\lambda(\x\y)\rangle\right).$$

\[v-lambda\] We have $$-v(\lambda)=(2g-2+k) n(\lambda)+(g-1)n
-\sum_{i=1}^kv(\lambda,\mu^i).$$

Straightforward.

\[v-fmla-lemma\] For $\mu=(\mu_1,\mu_2,\ldots,\mu_r)\in \calP_n$ we have $$\label{v-fmla}
v(\lambda,\mu)= \min
\{n(\rho^1)+\cdots+n(\rho^r)\,\,|\,\, |\rho^p|=\mu_p, \, 
\cup_p\rho^p \unlhd\lambda\}.$$

For $C_{\nu\mu}(\y)$ non-zero let $v_m(\nu,\mu):=v_q\left(C_{\nu\mu}(\y)\right)$. When $y_i=q^{i-1}$ we have $v_q(m_\rho(\y))= n(\rho)$ for any partition $\rho$. Hence by $$v_m(\nu,\mu)= \min \{n(\rho^1)+\cdots+n(\rho^r)\,\,|\,\, |\rho^p|=\mu_p, \, 
\cup_p\rho^p=\nu\}.$$ Since $K_{\lambda\nu}\geq 0$ for any $\lambda,\nu$, $K_{\lambda\nu}>0$ if and only if $\nu\unlhd \lambda$ [@fulton Ex 2, p.26], and the coefficients of $C_{\nu\mu}(\y)$ are non-negative, our claim follows from .

For example, if $\lambda=(1^n)$ then necessarily $\rho^p=(1^{\mu_p})$ and hence $\rho^1\cup \cdots \cup \rho^r=\lambda$. We have then $$\label{v-example}
v\left((1^n\right),\mu)=\sum_{p=1}^r\binom{\mu_p} 2 
= -\tfrac12 n +\tfrac12 \sum_{p=1}^r \mu_p^2.$$ Similarly, $$\label{v-example-1}
v(\lambda,(n))=n(\lambda)$$ by the next lemma.

\[n-ineq\] If $\beta\unlhd\alpha$ then $n(\alpha)\leq n(\beta)$ with equality if and only if $\alpha=\beta$.

We will use the raising operators $R_{ij}$ see [@macdonald I p.8]. Consider vectors $w$ with coefficients in $\Z$ and extend the function $n$ to them in the natural way $$n(w):=\sum_{i\geq 1}(i-1)w_i.$$ Applying a raising operator $R_{ij}$, where $i<j$, has the effect $$n(R_{ij}w)=n(w)+i-j.$$ Hence for any product $R$ of raising operators we have $n(Rw) < n(w)$ with equality if and only if $R$ is the identity operator. Now the claim follows from the fact that $\beta\unlhd\alpha$ implies there exist such and $R$ with $\alpha=R\beta$.

Recall [@macdonald (1.6)] that for any partition $\lambda$ we have $\langle\lambda,\lambda\rangle=2n(\lambda)+|\lambda|=\sum_i(\lambda'_i)^2$, where $\lambda'=(\lambda'_1,\lambda'_2,\dots)$ is the dual partition. Note also that $(\lambda\cup\mu)'=\lambda'+\mu'$. Define $$\|\lambda\|:=\sqrt{\langle\lambda',\lambda'\rangle}=\sqrt{\sum_i\lambda_i^2}.$$ The following inequality is a particular case of the theorem of §\[appendix\].

\[ineq-1\] Fix $\mu=(\mu_1,\mu_2,\ldots,\mu_r)\in \calP_n$. Then for every $(\nu^1,...,\nu^r)\in\calP_{\mu_1}\times\cdots\times\calP_{\mu_r}$ we have $$\mu_1\left\|\sum_p\nu^p\right\|^2-n\sum_p\left\|\nu^p\right\|^2\leq
\mu_1n^2-n\left\|\mu\right\|^2.
\label{cauchy-ineq}$$ Moreover, equality holds in if and only if either:

\(i) The partition $\mu$ is rectangular and all partitions $\nu^p$ are equal.

or

\(ii) For each $p=1,2,\ldots,r$ we have $\nu^p=(\mu_p)$.

Our claim is a consequence of the theorem of §\[appendix\]. Taking $x_{ps}=\nu^p_s$ we have $c_p:=\sum_sx_{ps}=\sum_s\nu^p_s=\mu_p$ and $c:=\max_pc_p=\mu_1$.

The following fact will be crucial for the proof of connectedness.

\[maxima\] For a fixed $\mu=(\mu_1,\mu_2,\ldots,\mu_r)\in \calP_n$ we have $$\mu_1n(\lambda)-nv(\lambda,\mu) \leq \mu_1n^2-n\left\|\mu\right\|^2,
\qquad \lambda \in \calP_n.$$ Equality holds only at $\lambda=(1^n)$ unless $\mu$ is rectangular $\mu=(t^{n/t})$, in which case it also holds when $\lambda$ is the union of $n/t$ copies of any $\lambda_0\in\calP_t$. \[opti\]

Given $\nu \unlhd \lambda$ write $\mu_1n(\lambda)-nv(\lambda,\mu)$ as $$\label{first-step}
\mu_1n(\lambda)-nv(\lambda,\mu)= 
\mu_1 (n(\lambda)-n(\nu)) +
\mu_1n(\nu)-nv(\lambda,\mu)$$ By Lemma \[n-ineq\] the first term is non-negative. Hence $$\mu_1n(\lambda)-nv(\lambda,\mu)\leq
\mu_1n(\nu)-nv(\lambda,\mu), \qquad \qquad \nu\unlhd \lambda.$$ Combinining this with yields $$\label{ineq}
\max_{|\lambda|=n}\left[\mu_1n(\lambda)-nv(\lambda,\mu)\right]\leq
\max_{|\rho^p|=\mu_p}\left[\mu_1n(\rho^1\cup\rho^2\cup\cdots\cup\rho^r)-
(n(\rho^1)+\cdots+n(\rho^r))n  \right].  
$$

Take $\nu^p$ to be the dual of $\rho^p$ for $p=1,2,\ldots,r$. Then the right hand side of is precisely $$\mu_1\left\|\sum_p\nu^p\right\|^2-n\sum_p\left\|\nu^p\right\|^2,$$ which by Lemma \[ineq-1\] is bounded above by $\mu_1n^2-n\left\|\mu\right\|^2$ with equality only where either $\rho^p=(1^{\mu_p})$ (case (ii)) or all $\rho^p$ are equal and $\mu=(t^{n/t})$ for some $t$ (case (i)).

Combining this with Lemma \[n-ineq\] we see that to obtain the maximum of the left hand side of we must also have $\rho^1\cup \cdots\cup\rho^r=\lambda$. In case (i) then, $\lambda$ is the union of $n/t$ copies of $\lambda_0$, the common value of $\rho^p$, and in case (ii), $\lambda=(1^n)$.

We first prove (ii). Using Lemma \[v-lambda\] we have $$-v(\lambda)=(2g-2+k) n(\lambda) +(g-1)n -\sum_{i=1}^kv(\lambda,\mu^i)=
   \frac{\delta}{n}n(\lambda)+(g -1)n
   +\frac1n\sum_{i=1}^k\left[\mu_1^in(\lambda)-nv(\lambda,\mu^i)\right].
\label{vlambda}$$ The terms $n(\lambda)$ and $\sum_{i=1}^n\left[\mu_1^in(\lambda)-nv(\lambda,\mu^i)\right]$ are all maximal at $\lambda=(1^n)$ (the last by Proposition \[opti\]). Hence $-v(\lambda)$ is also maximal at $(1^n)$, since $\delta\geq 0$. Now $n(\lambda)$ has a unique maximum at $(1^n)$ by Lemma \[n-ineq\], hence $-v(\lambda)$ reaches its maximum at other partitions if and only if $\delta=0$ and for each $i$ we have $\mu^i=(t_i^{n/t_i})$ for some positive integer $t_i\mid n$ (again by Proposition \[opti\]). In this case the maximum occurs only for $\lambda$ the union of $n/t$ copies of a partition $\lambda_0\in \calP_t$, where $t=\gcd t_i$. Now (ii) follows from Proposition \[affine-descrip\].

To prove (i) we use Lemma \[v-lambda\] and and find that $v((1^n))=-\Delta(\muhat)$ as claimed.

Let $\muhat=(\mu^1,\mu^2,\ldots,\mu^k)\in {\P_n}^k$ with $\delta(\muhat) \geq 0$. Suppose that $v(\lambda)$ is minimal. Then the coefficient of $q^{v(\lambda)}$ in $\calA_{\lambda \muhat}$ is $1$. \[lowest\]

We use the notation of the proof of Lemma \[v-fmla-lemma\]. Note that the coefficient of the lowest power of $q$ in $\calH_\lambda(\sqrt q,1/\sqrt q)
  \left(q^{-n(\lambda)}H_\lambda(q)\right)^k$ is $1$ (see ). Also, the coefficient of the lowest power of $q$ in each $m_{\lambda}(\y)$ is always $1$; hence so is the coefficient of the lowest power of $q$ in $C_{\nu\mu}(\y)$.

In the course of the proof of Proposition \[maxima\] we found that when $v(\lambda)$ is minimal, and $\rho^1,\ldots,\rho^r$ achieve the minimum in the right hand side of , then $\lambda=\rho^1\cup\cdots\cup\rho^r$. Hence by Lemma \[v-fmla-lemma\], the coefficient of the lowest power of $q$ in $\langle
  h_{\mu}(\x),s_\lambda(\x\y)\rangle=\sum_{\nu\unlhd\lambda}
  K_{\lambda\nu}C_{\nu\mu}(\y)$ equals the coefficient of the lowest power of $q$ in $K_{\lambda\lambda}C_{\lambda\mu}(\y)=C_{\lambda\mu}(\y)$ which we just saw is $1$. This completes the proof.

### Leading terms of $\Log\,\Omega$ {#step-2}

We now proceed to the second step in the proof of connectedness where we analyze the smallest power of $q$ in the coefficients of $\Log\left(\Omega\left(\sqrt{q},1/\sqrt{q}\right)\right)$. Write $$\Omega\left(\sqrt{q},1/\sqrt{q}\right)=\sum_\muhat
  P_\muhat(q)\,m_\muhat
\label{pmu}$$ with $P_\muhat(q):=\sum_\lambda\calA_{\lambda\muhat}$ and $\calA_{\lambda \muhat}$ as in .

Then by Lemma \[Log-w\] we have $$\Log\left(\Omega\left(\sqrt{q},1/\sqrt{q}\right)\right)
=\sum_\omhat C_\omhat^0 P_\omhat(q)\,m_\omhat(q)$$ where $\omhat$ runs over [*multi-types*]{} $(d_1,\omhat^1)\cdots(d_s,\omhat^s)$ with $\omhat^p\in(\calP_{n_p})^k$ and $P_\omhat(q):=\prod_pP_{\omhat^p}(q^{d_p}),
m_\omhat(\x):=\prod_pm_{\omhat^p}(\x^{d_p})$..

Now if we let $\gamma_{\muhat\omhat}:=\langle
m_\omhat,h_\muhat\rangle$ then we have $$\H_\muhat\left(\sqrt{q},1/\sqrt{q}\right)
=\frac{(q-1)^2}{q}\left(\sum_{\omhat\in\mathbf{T}^k}
  C_{\omhat}^0P_\omhat(q)\gamma_{\muhat\omhat}\right).$$ By Theorem \[step1\], $v_q\left(P_\omhat(q)\right)=-d\sum_{p=1}^s\Delta(\omhat^p)$ for a multi-type $\omhat=(d,\omhat^1)\cdots(d,\omhat^s)$.

\[perm-sum-lemma\] Let $\nu^1,\ldots,\nu^s$ be partitions. Then $$\langle m_{\nu^1}\cdots m_{\nu^s}, h_\mu\rangle  \neq 0$$ if and only if $\mu=\nu^1+\cdots+\nu^s$ up to permutation of the parts of each $\nu^p$ for $p=1,\ldots,s$.

It follows immediately from the definition of the monomial symmetric function.

Let $\v$ be the dimension vector associated to $\muhat$.

\[connectedness1\] If $\v$ is in the fundamental set of imaginary roots of $\Gamma$ then the character variety $\M_\muhat$ is non-empty and connected.

Assume $\v$ is in the fundamental set of roots of $\Gamma$. By Lemma \[fund-set\] this is equivalent to $\delta(\muhat)\geq 0$.

Note that $m_\nu(\x^d)=m_{d\nu}(\x)$ for any partition $\nu$ and positive integer $d$. Suppose $\omhat=(d,\omhat^1)\cdots(d,\omhat^s)$ is a multi-type for which $\gamma_{\muhat\omhat}$ is non-zero. Let $\nuhat^p=d\omhat^p$ for $p=1,\ldots, s$ (scale every part by $d$). These multi-partitions are then exactly in the hypothesis of Proposition \[Delta-ineq\] by Lemma \[perm-sum-lemma\]. Hence $$\label{conn-ineq}
 d\sum_{p=1}^s
  \Delta(\omhat^p)\leq d^2\sum_{p=1}^s
  \Delta(\omhat^p)=\sum_{p=1}^s\Delta(\nuhat^p) \leq\Delta(\muhat).$$ Suppose $\Gamma$ is not affine. Then by Proposition \[Delta-ineq\] we have equality of the endpoints in if and only if $s=1$, $\nuhat^1=\muhat$ and $d=1$, in other words, if and only if $\omhat=(1,\muhat)$. Hence, since $C_{(1,\,\muhat)}^0=1$, the coefficient of the lowest power of $q$ in $\H_\muhat\left(\sqrt{q},1/\sqrt{q}\right)$ equals the coefficient of the lowest power of $q$ in $P_\muhat(q)$ which is $1$ by Lemma \[lowest\] and Theorem \[minim\], Case II. This proves our claim in this case.

Suppose now $\Gamma$ is affine. Then by Proposition \[Delta-ineq\] we have equality of the endpoints in if and only if $\muhat=t\muhat^*$ and $\omhat=(1,t_1\muhat^*),\ldots,(1,t_s\muhat^*)$ for a partition $(t_1,t_2,\ldots, t_s)$ of $t$ and $d=1$. Combining this with Lemma \[lowest\] and Theorem \[minim\], Case I we see that the lowest order terms in $q$ in $\Log\left(\Omega\left(\sqrt{q},1/\sqrt{q}\right)\right)$ are $$L:=\sum C^0_\omhat  p(t_1)\cdots
p(t_s) \,m_{t\muhat^*},$$ where the sum is over types $\omhat$ as above. Comparison with Euler’s formula $$\Log\left(\sum_{n\geq 0}p(n)\,T^n\right)=\sum_{n\geq1}T^n,$$ shows that $L$ reduces to $\sum_{t\geq 1}
\,m_{t\muhat^*}$. Hence the coefficient of the lowest power of $q$ in $\H_\muhat\left(\sqrt{q},1/\sqrt{q}\right)$ is also $1$ in this case finishing the proof.

If $g\geq 1$, the dimension vector $\v$ is always in the fundamental set of imaginary roots of $\Gamma$. If $g=0$ the character variety if not empty if and only if $\v$ is a strict root of $\Gamma$ and if $\v$ is real then $\M_\muhat$ is a point [@crawley-par Theorem 8.3]. If $\v$ is imaginary then it can be taken by the Weyl group to some $\v'$ in the fundamental set and the two corresponding varieties $\M_\muhat$ and $\M_{\muhat'}$ are isomorphic for appropriate choices of conjugacy classes [@crawley-par Theorem 3.2, Lemma 4.3 (ii)], hence Theorem \[connectedness\].

Appendix by Gergely Harcos {#appendix}
==========================

Let $n,r$ be positive integers, and let $x_{ik}$ ($1\leq i\leq n$, $1\leq k\leq r$) be arbitrary nonnegative numbers. Let $c_i:=\sum_k x_{ik}$ and $c:=\max_i c_i$. Then we we have $$c\sum_k\biggl(\sum_i x_{ik}\biggr)^2-\biggl(\sum_i
c_i\biggr)\biggl(\sum_{i,k}x_{ik}^2\biggr)\leq c\biggl(\sum_i
c_i\biggr)^2-\biggl(\sum_i c_i\biggr)\biggl(\sum_i c_i^2\biggr).$$ Assuming $\min_i c_i>0$, equality holds if and only if we are in one of the following situations

\(i) $x_{ik}=x_{jk}$ for all $i,j,k$,

\(ii) there exists some $l$ such that $x_{ik}=0$ for all $i$ and all $k\neq l$. \[harcos\]

The assumption $\min_i c_i>0$ does not result in any loss of generality, because the values $i$ with $c_i=0$ can be omitted without altering any of the sums.

Without loss of generality we can assume $c=c_1\geq\dots\geq
c_n$, then the inequality can be rewritten as $$\biggl(\sum_i c_i\biggr)\biggl(\sum_j\sum_{k,l}x_{jk}x_{jl}-\sum_{j,k}x_{jk}^2\biggr)
\leq c\biggl(\sum_{i,j}\sum_{k,l}x_{ik}x_{jl}-\sum_{i,j}\sum_k
x_{ik}x_{jk}\biggr).$$ Here and later $i,j$ will take values from $\{1,\dots,n\}$ and $k,l,m$ will take values from $\{1,\dots,r\}$. We simplify the above as $$\biggl(\sum_i c_i\biggr)\biggl(\sum_j\sum_{\substack{{k,l}\\k\neq l}}x_{jk}x_{jl}\biggr)
\leq c\biggl(\sum_{i,j}\sum_{\substack{{k,l}\\k\neq
l}}x_{ik}x_{jl}\biggr),$$ then we factor out and also utilize the symmetry in $k,l$ to arrive at the equivalent form $$\sum_{i,j}c_i\sum_{\substack{{k,l}\\k<l}}x_{jk}x_{jl}\leq
\sum_{i,j}c\sum_{\substack{{k,l}\\k<l}}x_{ik}x_{jl}.$$ We distribute the terms in $i,j$ on both sides as follows: $$\sum_i c_i\sum_{\substack{{k,l}\\k<l}}x_{ik}x_{il}+
\sum_{\substack{{i,j}\\i<j}}\biggl(c_i\sum_{\substack{{k,l}\\k<l}}x_{jk}x_{jl}
+c_j\sum_{\substack{{k,l}\\k<l}}x_{ik}x_{il}\biggr)\leq \sum_i
c\sum_{\substack{{k,l}\\k<l}}x_{ik}x_{il}+
\sum_{\substack{{i,j}\\i<j}}c\sum_{\substack{{k,l}\\k<l}}(x_{ik}x_{jl}+x_{jk}x_{il}).$$ It is clear that $$c_i\sum_{\substack{{k,l}\\k<l}}x_{ik}x_{il}\leq c\sum_{\substack{{k,l}\\k<l}}x_{ik}x_{il},\quad 1\leq i\leq n,$$ therefore it suffices to show that $$c_i\sum_{\substack{{k,l}\\k<l}}x_{jk}x_{jl}+c_j\sum_{\substack{{k,l}\\k<l}}x_{ik}x_{il}
\leq c\sum_{\substack{{k,l}\\k<l}}(x_{ik}x_{jl}+x_{jk}x_{il}),\quad
1\leq i<j\leq n.$$ We will prove this in the stronger form $$c_i\sum_{\substack{{k,l}\\k<l}}x_{jk}x_{jl}+c_j\sum_{\substack{{k,l}\\k<l}}x_{ik}x_{il}
\leq
c_i\sum_{\substack{{k,l}\\k<l}}(x_{ik}x_{jl}+x_{jk}x_{il}),\quad
1\leq i<j\leq n.$$

We now fix $1\leq i<j\leq n$ and introduce $x_k:=x_{ik}$, $x'_k:=x_{jk}$. Then the previous inequality reads $$\biggl(\sum_m
x_m\biggr)\biggl(\sum_{\substack{{k,l}\\k<l}}x'_kx'_l\biggr)+
\biggl(\sum_m
x'_m\biggr)\biggl(\sum_{\substack{{k,l}\\k<l}}x_kx_l\biggr)\leq
\biggl(\sum_m
x_m\biggr)\sum_{\substack{{k,l}\\k<l}}(x_kx'_l+x'_kx_l),$$ that is, $$\sum_{\substack{{k,l,m}\\k<l}}(x_mx'_kx'_l+x_kx_lx'_m)\leq
\sum_{\substack{{k,l,m}\\k<l}}(x_kx_mx'_l+x_lx_mx'_k).$$ The right hand side equals $$\begin{aligned}
\sum_{\substack{{k,l,m}\\k<l}}(x_kx_mx'_l+x_lx_mx'_k)
&=\sum_{\substack{{k,l,m}\\l\neq
k}}x_kx_mx'_l=\sum_{\substack{{k,l,m}\\m\neq k}}x_kx_lx'_m
=\sum_{\substack{{k,m}\\m\neq
k}}x_k^2x'_m+\sum_{\substack{{k,l,m}\\l\neq k\\m\neq k}}x_kx_lx'_m\\
&=\sum_{\substack{{k,m}\\m\neq
k}}x_k^2x'_m+\sum_{\substack{{k,m}\\m\neq k}}x_kx_mx'_m
+\sum_{\substack{{k,l,m}\\l\neq k\\m\neq k,l}}x_kx_lx'_m\\
&=\sum_{\substack{{k,m}\\m\neq
k}}x_k^2x'_m+\sum_{\substack{{k,m}\\k<m}}x_kx_mx'_m
+\sum_{\substack{{k,m}\\m<k}}x_kx_mx'_m
+2\sum_{\substack{{k,l,m}\\k<l\\m\neq k,l}}x_kx_lx'_m\\
&=\sum_{\substack{{k,m}\\m\neq
k}}x_k^2x'_m+\sum_{\substack{{k,l}\\k<l}}x_kx_lx'_l
+\sum_{\substack{{k,l}\\k<l}}x_kx_lx'_k
+2\sum_{\substack{{k,l,m}\\k<l\\m\neq k,l}}x_kx_lx'_m\\
&=\sum_{\substack{{k,m}\\m\neq
k}}x_k^2x'_m+\sum_{\substack{{k,l,m}\\k<l}}x_kx_lx'_m
+\sum_{\substack{{k,l,m}\\k<l\\m\neq k,l}}x_kx_lx'_m,\end{aligned}$$ therefore it suffices to prove $$\sum_{\substack{{k,l,m}\\k<l}}x_mx'_kx'_l\leq\sum_{\substack{{k,m}\\m\neq
k}}x_k^2x'_m+\sum_{\substack{{k,l,m}\\k<l\\m\neq k,l}}x_kx_lx'_m.$$ This is trivial if $x'_m=0$ for all $m$. Otherwise $\sum_m x'_m>0$, hence $c_i\geq c_j$ yields $$\lambda:=\biggl(\sum_m x_m\biggr)\biggl(\sum_m x'_m\biggr)^{-1}\geq 1.$$ Clearly, we are done if we can prove $$\lambda^2\sum_{\substack{{k,l,m}\\k<l}}x_mx'_kx'_l\leq\lambda\sum_{\substack{{k,m}\\m\neq
k}}x_k^2x'_m+\lambda\sum_{\substack{{k,l,m}\\k<l\\m\neq
k,l}}x_kx_lx'_m.$$ We introduce $\tilde x_m:=\lambda x'_m$, then $$\sum_m\tilde x_m=\sum_m x_m,$$ and the last inequality reads $$\sum_{\substack{{k,l,m}\\k<l}}x_m\tilde x_k\tilde x_l\leq
\sum_{\substack{{k,m}\\m\neq k}}x_k^2\tilde
x_m+\sum_{\substack{{k,l,m}\\k<l\\m\neq k,l}}x_kx_l\tilde x_m.$$ By adding equal sums to both sides this becomes $$\sum_{\substack{{k,l,m}\\k<l}}x_m\tilde x_k\tilde
x_l+\sum_{\substack{{k,l,m}\\k<l}}x_kx_l\tilde x_m\leq
\sum_{\substack{{k,m}\\m\neq k}}x_k^2\tilde
x_m+\sum_{\substack{{k,l,m}\\k<l\\m\neq k,l}}x_kx_l\tilde
x_m+\sum_{\substack{{k,l,m}\\k<l}}x_kx_l\tilde x_m,$$ which can also be written as $$\biggl(\sum_m x_m\biggr)\biggl(\sum_{\substack{{k,l}\\k<l}}\tilde x_k\tilde x_l\biggr)+
\biggl(\sum_m \tilde
x_m\biggr)\biggl(\sum_{\substack{{k,l}\\k<l}}x_k
x_l\biggr)\leq\sum_k x_k^2\biggl(\sum_{\substack{m\\m\neq k}}\tilde
x_m\biggr)+\sum_{\substack{{k,l}\\k<l}}x_kx_l\biggl(\sum_{\substack{m\\m\neq
k}}\tilde x_m+\sum_{\substack{m\\m\neq l}}\tilde x_m\biggr).$$ The right hand side equals $$\begin{aligned}
\sum_k x_k^2\biggl(\sum_{\substack{m\\m\neq k}}\tilde
x_m\biggr)+\sum_{\substack{{k,l}\\k<l}}x_kx_l\biggl(\sum_{\substack{m\\m\neq
k}}\tilde x_m+\sum_{\substack{m\\m\neq l}}\tilde x_m\biggr)&= \sum_k
x_k^2\biggl(\sum_{\substack{m\\m\neq k}}\tilde x_m\biggr)+
\sum_{\substack{{k,l}\\l<k}}x_kx_l\biggl(\sum_{\substack{m\\m\neq
l}}\tilde
x_m\biggr)+\sum_{\substack{{k,l}\\k<l}}x_kx_l\biggl(\sum_{\substack{m\\m\neq
l}}\tilde x_m\biggr)\\
&= \sum_k x_k^2\biggl(\sum_{\substack{m\\m\neq k}}\tilde x_m\biggr)+
\sum_{\substack{{k,l}\\k\neq
l}}x_kx_l\biggl(\sum_{\substack{m\\m\neq l}}\tilde x_m\biggr)\\
&=\sum_{k,l}x_kx_l\biggl(\sum_{\substack{m\\m\neq l}}\tilde
x_m\biggr)=\biggl(\sum_k
x_k\biggr)\biggl(\sum_{\substack{{m,l}\\m\neq l}}x_l\tilde
x_m\biggr),\end{aligned}$$ hence the previous inequality is the same as $$\biggl(\sum_m x_m\biggr)\biggl(\sum_{\substack{{k,l}\\k<l}}\tilde x_k\tilde x_l\biggr)+
\biggl(\sum_m \tilde
x_m\biggr)\biggl(\sum_{\substack{{k,l}\\k<l}}x_k x_l\biggr)\leq
\biggl(\sum_k x_k\biggr)\biggl(\sum_{\substack{{m,l}\\m\neq
l}}x_l\tilde x_m\biggr).$$ The first factors are equal and positive, hence after renaming $m,l$ to $k,l$ when $m<l$ and to $l,k$ when $m>l$ on the right hand side we are left with proving $$\sum_{\substack{{k,l}\\k<l}}(\tilde x_k\tilde x_l+x_k x_l)\leq
\sum_{\substack{{k,l}\\k<l}}(\tilde x_k x_l+x_k\tilde x_l).$$ This can be written in the elegant form $$\sum_{\substack{{k,l}\\k<l}}(\tilde x_k-x_k)(\tilde x_l-x_l)\leq
0.$$ However, $$0=\biggl(\sum_k (\tilde x_k-x_k)\biggr)^2=\sum_{k,l}(\tilde x_k-x_k)(\tilde x_l-x_l)
=\sum_k(\tilde x_k-x_k)^2 +2\sum_{\substack{{k,l}\\k<l}}(\tilde
x_k-x_k)(\tilde x_l-x_l),$$ so that $$\sum_{\substack{{k,l}\\k<l}}(\tilde x_k-x_k)(\tilde
x_l-x_l)=-\frac{1}{2}\sum_k(\tilde x_k-x_k)^2\leq 0$$ as required.

We now verify, under the assumption $\min_i c_i>0$, that equation in the theorem holds if and only if $x_{ik}=x_{jk}$ for all $i,j,k$ or there exists some $l$ such that $x_{ik}=0$ for all $i$ and all $k\neq l$. The “if" part is easy, so we focus on the “only if" part. Inspecting the above argument carefully, we can see that equation can hold only if for any $1\leq i<j\leq n$ the numbers $x_k:=x_{ik}$, $x'_k:=x_{jk}$ satisfy $$\lambda\sum_{\substack{{k,l,m}\\k<l}}x_mx'_kx'_l=
\sum_{\substack{{k,l,m}\\k<l}}x_mx'_kx'_l=\sum_{\substack{{k,m}\\m\neq
k}}x_k^2x'_m+\sum_{\substack{{k,l,m}\\k<l\\m\neq k,l}}x_kx_lx'_m,$$ where $\lambda$ is as before. If $x'_kx'_l=0$ for all $k<l$, then $x_k^2x'_m=0$ for all $k\neq m$, i.e. $x_kx'_l=0$ for all $k\neq l$. Otherwise $\lambda=1$ and $x_k=\tilde x_k=x'_k$ for all $k$ by the above argument. In other words, equation in the theorem can hold only if for any $i\neq j$ we have $x_{ik}x_{jl}=0$ for all $k\neq l$ or we have $x_{ik}=x_{jk}$ for all $k$. If there exist $j,l$ such that $x_{jk}=0$ for all $k\neq l$, then $x_{jl}>0$ and for any $i\neq j$ both alternatives imply $x_{ik}=0$ for all $k\neq l$, hence we are done. Otherwise the first alternative cannot hold for any $i\neq j$, so we are again done.

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