---
abstract: 'The $SL(2)$-type of any smooth, irreducible and unitarizable representation of $GL_n$ over a $p$-adic field was defined by Venkatesh. We provide a natural way to extend the definition to all smooth and irreducible representations. For unitarizable representations we show that the $SL(2)$-type of a representation is preserved under base change with respect to any finite extension. The Klyachko model of a smooth, irreducible and unitarizable representation $\pi$ of $GL_n$ depends only on the $SL(2)$-type of $\pi.$ As a consequence we observe that the Klyachko model of $\pi$ and of its base-change are of the same type.'
author:
- Omer Offen and Eitan Sayag
date:
- 
- 
- 
title: 'The $SL(2)$-type and base change'
---

introduction
============

Let $F$ be a finite extension of ${\mathbb{Q}}_{p}$. In [@MR2133760], Venkatesh assigned a partition of $n,$ the *$SL(2)$-type* of $\pi,$ to any smooth, irreducible and unitarizable representation $\pi$ of $GL_n(F).$ For a representation of Arthur type the $SL(2)$-type encodes the combinatorial data in the Arthur parameter. In general, the $SL(2)$-type is defined in terms of Tadic’s classification of the unitary dual.

The reciprocity map for $GL_n(F)$ is a bijection from the set of isomorphism classes of smooth irreducible representations of $GL_n(F)$ to the set of isomorphism classes of $n$-dimensional Weil-Deligne representations (cf. [@MR1876802] and [@MR1738446]). Applying the reciprocity map we observe that there is a natural way to extend the definition of the $SL(2)$-type to all smooth and irreducible representations of $GL_n(F)$ (see Theorem \[thm: SL(2)-type\] and Remark \[rmk: SL(2)-type\]). The reciprocity map also allows the definition of base change with respect to any finite extension $E$ of $F.$ It is a map ${\operatorname{bc}}_{E/F}$ from isomorphism classes of smooth irreducible representation of $GL_n(F)$ to isomorphism classes of smooth irreducible representation of $GL_n(E)$ that is the ‘mirror image’ of restriction with respect to $E/F$ of Weil-Deligne representations. The content of Theorem \[thm: main\], our main result, is that for any smooth, irreducible and unitarizable representation $\pi$ of $GL_n(F)$ the representations $\pi$ and ${\operatorname{bc}}(\pi)$ have the same $SL(2)$-type.

In [@MR2332593], [@os], [@OS3] we studied the Klyachko models of smooth irreducible representations of $GL_{n}(F),$ that is, distinction of a representation with respect to certain subgroups that are a semi direct product of a unipotent and a symplectic group. Our results are also described in terms of Tadic’s classification and depend, in fact, only on the $SL(2)$-type of a representation. For example, a smooth, irreducible and unitarizable representation $\pi$ of $GL_{2n}(F)$ is $Sp_{2n}(F)$-distinguished, i.e. it satisfies ${\operatorname{Hom}}_{Sp_{2n}(F)}(\pi,{\mathbb{C}}) \ne 0,$ if and only if the $SL(2)$-type of $\pi$ consists entirely of even parts (and in this case ${\operatorname{Hom}}_{Sp_{2n}(F)}(\pi,{\mathbb{C}})$ is one dimensional [@MR1078382 Theorem 2.4.2]). For unitarizable representations, our results on Klyatchko models are reinterpreted here in terms of the $SL(2)$-type. As a consequence we show that Klyachko models are preserved under base-change with respect to any finite extension. In particular, we have

\[thm: symplectic main\] Let $E/F$ be a finite extension of $p$-adic fields. A smooth, irreducible and unitarizable representation $\pi$ of $GL_{2n}(F)$ is $Sp_{2n}(F)$-distinguished if and only if ${\operatorname{bc}}_{E/F}(\pi)$ is $Sp_{2n}(E)$-distinguished.

The rest of this note is organized as follows. After setting some general notation in Section \[sec: notation\], in Section \[sec: bc\] we recall the definition of the reciprocity map. In Section \[sec: SL(2)-type\] we recall the definition of Venkatesh for the $SL(2)$-type of a unitarizable representation and extend it to all smooth irreducible representations. We recall (and reformulate in terms of the $SL(2)$-type) our results on symplectic (and more generally on Klyachko) models in Section \[sec: Klyachko\]. Our main observation Theorem \[thm: main\] and its application to Klyachko models Corollary \[cor: main\] are stated in Section \[sec: statements\] and proved in Section \[sec: proofs\]. The main theorem says that base change respects $SL(2)$-types and its corollary says that base change respects Klyachko types. Theorem \[thm: symplectic main\] is a special case where the Klyachko type is purely symplectic.

Notation {#sec: notation}
========

Let $F$ be a finite extension of ${\mathbb{Q}}_p$ for some prime number $p$ and let ${\left|{\,\cdot}\right|}_F:F^\times \to {\mathbb{C}}^\times$ denote the standard absolute value normalized so that the inverse of uniformizers are mapped to the size of the residual field. Denote by $W_F$ the Weil group of $F$ and by $I_F$ the inertia subgroup of $W_F.$ We normalize the reciprocity map $T_F:W_F \to F^\times,$ given by local class field theory, so that geometric Frobenius elements are mapped to uniformizers. The map $T_F$ defines an isomorphism from the abelianization $W_F^{ab}$ of $W_F$ to $F^\times$ (this is the inverse of the Artin map). Let ${\left|{\,\cdot}\right|}_{W_F}={\left|{\,\cdot}\right|}_{F}\circ T_F$ denote the associated absolute value on $W_F.$

Denote by ${{\bf 1}}_\Omega$ the characteristic function of a set $\Omega.$ Let ${\operatorname{MS}}_{{\operatorname{fin}}}(\Omega)$ be the set of finite multisets of elements in $\Omega,$ that is, the set of functions $f:\Omega \to
{\mathbb{Z}}_{\ge 0}$ of finite support. When convenient we will also denote $f$ by $\{\omega_1,\dots,\omega_1,\omega_2,\dots,\omega_2,\dots\}$ where $\omega\in \Omega$ is repeated $f(\omega)$ times. Let ${\mathcal{P}}={\operatorname{MS}}_{{\operatorname{fin}}}({\mathbb{Z}}_{>0})$ be the set of partitions of positive integers and let $${\mathcal{P}}(n)=\{f \in {\mathcal{P}}: \sum_{k=1}^\infty k\,f(k)=n\}$$ denote the subset of partitions of $n.$ For $n,\,m \in {\mathbb{Z}}_{>0}$ let $(n)_m=m\,{{\bf 1}}_n=\{n,\dots,n\}$ be the partition of $nm$ with ‘$m$ parts of size $n$’. Let ${\operatorname{odd}}:{\mathcal{P}}\to {\mathbb{Z}}_{\ge 0}$ be defined by $${\operatorname{odd}}(f)=\sum_{k=0}^\infty f(2k+1),$$ i.e. ${\operatorname{odd}}(f)$ is the number of odd parts of the partition $f.$

Reciprocity and base-change for $GL_n(F)$ {#sec: bc}
=========================================

Weil-Deligne representations
----------------------------

An $n$-dimensional *Weil-Deligne* representation is a pair $((\rho,V),N)$ where $(\rho,V)$ is an $n$-dimensional representation of $W_{F}$ that decomposes as a direct sum of irreducible representations and $N:V \to V$ is a linear operator such that $${\left|{w}\right|}_{W_F}\,N \circ  \rho(w)=\rho(w)\circ N,\ w \in
    W_F.$$ The map $((\rho,V),N) \mapsto ([\rho],f),$ where $[\rho]$ denotes the isomorphism class of the $n$-dimensional representation $(\rho,V)$ of $W_F$ and $f \in {\mathcal{P}}(n)$ is the partition of $n$ associated to the Jordan decomposition of $N,$ defines an injective map on isomorphism classes of Weil-Deligne representations. Denote its image by ${\mathcal{G}}_{F}(n).$ In this way we identify the set ${\mathcal{G}}_F(n)$ with the set of isomorphism classes of $n$-dimensional Weil-Deligne representations. Let $P_{F,n}: {\mathcal{G}}_F(n) \to {\mathcal{P}}(n)$ be the projection to the second coordinate. Let ${\mathcal{G}}_F=\cup_{n=1}^\infty {\mathcal{G}}_F(n)$ be the set of isomorphism classes of all finite dimensional Weil-Deligne representations and let $P_F:{\mathcal{G}}_F \to {\mathcal{P}}$ be the map such that ${P_F}_{|{\mathcal{G}}_F(n)}=P_{F,n}.$

The local Langlands correspondence
----------------------------------

Let ${\mathcal{A}}_F(n)$ be the set of isomorphism classes of smooth and irreducible representations of $GL_n(F)$ and set ${\mathcal{A}}_F=\cup_{n=1}^\infty {\mathcal{A}}_F(n).$ For every $\pi \in {\mathcal{A}}_F$ we denote by $\omega_\pi$ the central character of (any representation in the isomorphism class of) $\pi.$ Fix a non trivial additive character $\psi$ of $F.$ Due to Harris-Taylor [@MR1876802] and independently to Henniart [@MR1738446] there exists a unique sequence of bijections $${\operatorname{rec}}_{F,n}: {\mathcal{A}}_{F}(n) \to {\mathcal{G}}_{F}(n)$$ for all $n\ge 1$ satisfying the following properties: $$\begin{aligned}
\label{eq: LCFT}& &{\operatorname{rec}}_{F}(\chi)=\chi \circ T_{F};\\& & \label{eq:
L-id}L(\pi_{1} \times \pi_{2},s)=L({\operatorname{rec}}_{F}(\pi_{1})\otimes
{\operatorname{rec}}_{F}(\pi_{2}),s);\\& & \label{eq: epsilon-id}\epsilon(\pi_{1}
\times \pi_{2},s,\psi)=\epsilon({\operatorname{rec}}_{F}(\pi_{1})\otimes
{\operatorname{rec}}_{F}(\pi_{2}),s,\psi);\\& & \label{eq: central char}\det \circ
{\operatorname{rec}}_{F}(\pi)=rec_{F}(\omega_\pi);\\& &\label{eq: dual-id}
{\operatorname{rec}}_{F}(\pi^\vee)={\operatorname{rec}}_{F}(\pi)^\vee.\end{aligned}$$ Here $\chi \in {\mathcal{A}}_F(1),\,\pi,\,\pi_1,\,\pi_2 \in {\mathcal{A}}_F,$ $\pi^\vee$ is the contragredient of $\pi,$ ${\operatorname{rec}}_{F}(\pi)^\vee$ is the dual of ${\operatorname{rec}}_{F}(\pi)$ and ${\operatorname{rec}}_F:{\mathcal{A}}_F \to {\mathcal{G}}_F$ is such that ${{\operatorname{rec}}_F}_{|{\mathcal{A}}_F(n)}={\operatorname{rec}}_{F,n}.$

Expressing ${\operatorname{rec}}_F$ in terms of ${\operatorname{rec}}_F^\circ$
------------------------------------------------------------------------------

Let ${\mathcal{A}}_F^\circ(n)\subseteq {\mathcal{A}}_F(n)$ be the subset of isomorphism classes of supercuspidal representations and let ${\mathcal{G}}_F^\circ(n)\subseteq {\mathcal{G}}_F(n)$ be the subset of isomorphism classes $([\rho],f)$ such that $\rho$ is irreducible and $f={{\bf 1}}_n=\{n\}.$ The set ${\mathcal{G}}_F^\circ(n)$ is identified with the set of isomorphism classes of irreducible and $n$-dimensional representations of $W_F.$ It follows from the work of Harris-Taylor and independently of Henniart that there exists a unique sequence of bijections $${{\operatorname{rec}}_{F,n}}_{|{\mathcal{A}}_F^\circ(n)}={\operatorname{rec}}_{F,n}^\circ:{\mathcal{A}}_F(n)\to {\mathcal{G}}_F^\circ(n)$$ satisfying , , , and . The work of Zelevinsky [@MR584084] allows the extention of ${\operatorname{rec}}_F^\circ$ to the map ${\operatorname{rec}}_F$ on ${\mathcal{A}}_F.$ This is also explained in [@MR902829] and we now recall the construction of ${\operatorname{rec}}_F$ in terms of ${\operatorname{rec}}^\circ_F.$

For $s \in {\mathbb{C}}$ and every isomorphism class $\varpi=[\pi] \in
{\mathcal{A}}_F$ (resp. $\varrho=([\rho],f) \in {\mathcal{G}}_F$) let $\varpi[s]=[\pi
\otimes {\left|{\det}\right|}_F^s]$ (resp. $\varrho[s]=([\rho\otimes
{\left|{\,\cdot}\right|}_{W_F}^s],f)$). A *segment* in ${\mathcal{A}}_F^\circ$ (resp. ${\mathcal{G}}_F^\circ$) is a set of the form $$\Delta[\sigma,r]=\{\sigma[\frac{1-r}2],\sigma[\frac{3-r}2],\dots,\sigma[\frac{r-1}2]\}$$ (resp. $$\Delta[\rho,r]=\{\rho[\frac{1-r}2],\rho[\frac{3-r}2],\dots,\rho[\frac{r-1}2]\})$$ for some $\sigma \in {\mathcal{A}}_F^\circ$ (resp. $\rho \in {\mathcal{G}}_F^\circ$) and $r\in {\mathbb{Z}}_{>0}.$ Let ${\mathcal{S}}$ (resp. ${\mathcal{S}}'$) denote the set of all segments in ${\mathcal{A}}_F^\circ$ (resp. ${\mathcal{G}}_F^\circ$) and let ${\mathcal{O}}={\operatorname{MS}}_{{\operatorname{fin}}}({\mathcal{S}})$ (resp. ${\mathcal{O}}'={\operatorname{MS}}_{{\operatorname{fin}}}({\mathcal{S}}')$). The bijection ${\operatorname{rec}}^\circ_F:{\mathcal{A}}_F^\circ \to {\mathcal{G}}_F^\circ$ defines a bijection ${\operatorname{rec}}_F^\circ:{\mathcal{S}}\to {\mathcal{S}}'$ given by ${\operatorname{rec}}_F^\circ(\Delta[\sigma,r])=\Delta[{\operatorname{rec}}_F^\circ(\sigma),r]$ and a bijection ${\operatorname{rec}}_F^\circ:{\mathcal{O}}\to {\mathcal{O}}'$ given by ${\operatorname{rec}}_F^\circ(a)({\operatorname{rec}}_F^\circ(\Delta))=a(\Delta),\,\Delta\in {\mathcal{S}}.$

In [@MR584084 Section 6.5] Zelevinsky defines a bijection $a
\mapsto {\langle{a}\rangle}$ from ${\mathcal{O}}$ to ${\mathcal{A}}_F.$ The Zelevinsky involution is defined in [@MR584084 Section 9.12] as an involution on the Grothendick group associated with ${\mathcal{A}}_F.$ It is proved by Aubert [@MR1285969], [@MR1390967b] and independently by Procter [@MR1625483] that the Zelevinsky involution restricts to a bijection from ${\mathcal{A}}_F$ to itself that we denote by $\pi \mapsto
\pi^t.$ In [@MR584084 Section 10.2] Zelevinsky defines a bijection $\tau:{\mathcal{O}}' \to {\mathcal{G}}_F$ as follows. For a segment $\Delta[\rho,r]\in {\mathcal{S}}'$ where $\rho \in {\mathcal{G}}_F^\circ(t)$ let $$\tau(\Delta[\rho,r])=(\oplus_{i=1}^r \,\rho, (r)_t)$$ and for $a' \in {\mathcal{O}}'$ set $$\tau(a')=\oplus_{\Delta' \in {\mathcal{O}}'} \tau(\Delta')$$ where for $([\rho_1],f_1),\dots,([\rho_m],f_m) \in {\mathcal{G}}_F$ the direct sum is given by $$([\rho_1],f_1)\oplus\cdots \oplus ([\rho_m],f_m)=([
    \rho_1 \oplus\cdots \oplus \rho_m],f_1+\cdots+f_m).$$ The reciprocity map ${\operatorname{rec}}_F$ is given by $${\operatorname{rec}}_F({\langle{a}\rangle}^t)=\tau({\operatorname{rec}}_F^\circ(a)),\ a \in {\mathcal{O}}.$$

The $SL(2)$-type of a representation {#sec: SL(2)-type}
====================================

Denote by ${\mathcal{A}}^u_F(n)$ the subset of ${\mathcal{A}}_F(n)$ consisting of all isomorphism classes of unitarizable representations and let ${\mathcal{A}}_F^u=\cup_{n=1}^\infty {\mathcal{A}}_F(n).$ For $[\pi_1],\dots,[\pi_m] \in
{\mathcal{A}}_F$ we denote by $\pi_1 \times \cdots \times \pi_m$ the representation parabolically induced from $\pi_1 \otimes \cdots
\otimes \pi_m$ and by $[\pi_1] \times \cdots \times [\pi_m]$ its isomorphism class.

For $\sigma \in {\mathcal{A}}_F^\circ$ and integers $n,\,r >0$ let $$\delta[\sigma,n]={\langle{\Delta[\sigma,n]}\rangle}^t,$$ $$a(\sigma,n,r)=\{\Delta[\sigma[\frac{1-r}{2}],n],\Delta[\sigma[\frac{3-r}{2}],n],
    \cdots,\Delta[\sigma,n](\frac{r-1}{2})\}\in {\mathcal{O}}$$ and $$U(\delta[\sigma,n],r)={\langle{a(\sigma,n,r)}\rangle}.$$ Tadic’s classification of the unitary dual of $GL_n(F)$ [@MR870688] implies that if $\sigma \in {\mathcal{A}}_F^\circ
\cap {\mathcal{A}}_F^u$ then $U(\delta[\sigma,n],r) \in {\mathcal{A}}_F^u$ and that for any $\pi\in {\mathcal{A}}_F^u$ there exist $\sigma_1,\dots,\sigma_m \in
{\mathcal{A}}_F^\circ$ and integers $n_1,\dots,n_m,\,r_1,\dots,r_m >0$ such that $$\label{eq: general unitary}
    \pi=U(\delta[\sigma_1,n_1],r_1)\times \cdots \times
    U(\delta[\sigma_m,n_m],r_m).$$ It further follows from [@MR1359141 Lemma 3.3] that $$\label{eq: transpose of U}
    U(\delta[\sigma,n],r)^t=U(\delta[\sigma,r],n).$$ The $SL(2)$ of a representation $\pi\in {\mathcal{A}}_F^u$ of the form is defined in [@MR2133760 Definition 1] to be the partition $$\label{eq: explicit SL(2)-type}
\{(r_1)_{n_1},\dots,(r_m)_{n_m}\}.$$

\[thm: SL(2)-type\] The $SL(2)$-type of a representation $\pi\in {\mathcal{A}}_F^u$ equals $P_F({\operatorname{rec}}_F(\pi^t)).$

\[rmk: SL(2)-type\] Theorem \[thm: SL(2)-type\] allows us to define the $SL(2)$-type of any $\pi \in {\mathcal{A}}_F$ by the formula $P_F({\operatorname{rec}}_F(\pi^t)).$ Note further that given a reciprocity map (local Langlands conjecture), this provides a recipe to define the $SL(2)$-type of an irreducible representation for any reductive group!

Based on Tadic’s classification of the unitary dual of $GL_n(F),$ the proof of Theorem \[thm: SL(2)-type\] is merely a matter of following the definitions. For convenience, we provide the proof. The key is in the following simple observations.

\[lemma: rec of unitary\] Let $\pi\in {\mathcal{A}}_F^u$ be of the form . Then $$\label{eq: rec of unitary}
    {\operatorname{rec}}_F(\pi)=\oplus_{i=1}^m \oplus_{j=1}^{r_i}
    \tau(\Delta[\sigma_i[\frac{r_i+1}2-j],n_i])$$ and $$\label{eq: transpose of unitary}
    \pi^t=U(\delta[\sigma_1,r_1],n_1)\times \cdots \times
    U(\delta[\sigma_m,r_m],n_m)\in {\mathcal{A}}_F^u.$$

Let $a_i=a(\sigma_i,r_i,n_i).$ It follows from that $$\pi={\langle{a_1}\rangle}^t \times \cdots \times {\langle{a_m}\rangle}^t=
    ({\langle{a_1}\rangle} \times \cdots \times {\langle{a_m}\rangle})^t$$ and since $t$ is an involution on ${\mathcal{A}}_F$ that ${\langle{a_1}\rangle} \times
\cdots \times {\langle{a_m}\rangle} \in {\mathcal{A}}_F.$ Thus, it follows from [@MR584084 Proposition 8.4] that ${\langle{a_1}\rangle} \times \cdots
\times {\langle{a_m}\rangle}={\langle{a_1+\cdots+a_m}\rangle}.$ In other words $\pi={\langle{a_1+\cdots+a_m}\rangle}^t$ and therefore by definition $${\operatorname{rec}}_F(\pi)=\tau({\operatorname{rec}}_F^\circ(a_1+\cdots+a_m))=\oplus_{i=1}^m
    \tau({\operatorname{rec}}_F^\circ(a_i)).$$ The identity now follows from the definition of $\tau({\operatorname{rec}}_F^\circ(a_i)).$ Note that implies that $$\pi^t=U(\delta[\sigma_1,r_1],n_1)\times \cdots \times
    U(\delta[\sigma_m,r_m],n_m)$$ and the classification of Tadic therefore implies that $\pi^t \in
{\mathcal{A}}_F^u.$ Thus we get .

Applying to $\pi^t$ and comparing with Theorem \[thm: SL(2)-type\] follows from the definitions.

From now on for every $\pi \in {\mathcal{A}}_F$ we denote by $$\label{eq: def of vv}
    {\mathcal{V}}(\pi)=P_F({\operatorname{rec}}_F(\pi^t))$$ the $SL(2)$-type of $\pi.$

Klyachko models {#sec: Klyachko}
===============

For positive integers $r$ and $k$ denote by $U_r$ the subgroup of upper triangular unipotent matrices in $GL_r(F)$ and by $Sp_{2k}(F)$ the symplectic group in $GL_{2k}(F).$ Fix a decomposition $n=r+2k.$ Let $$H_{r,2k} =\{
\left(
\begin{array}{cc}
  u  & X   \\
  0  &  h  \\
\end{array}
\right): u \in U_r,\, X \in M_{r \times 2k}(F),\,h \in Sp_{2k}(F)\}.$$ Let $\psi$ be a non trivial character of $F.$ For $u=(u_{i,j}) \in
U_r$ let $$\psi_r(u)=\psi(u_{1,2}+\cdots+u_{r-1, r})$$ and let $\psi_{r,2k}$ be the character of $H_{r,2k}$ defined by $$\psi_{r,2k}
    \left(
        \begin{array}{cc}
        u  & X   \\
        0  &  h  \\
        \end{array}
    \right)=\psi_r(u).$$ We refer to the space $${\mathcal{M}}_{r,2k}=\mbox{Ind}_{H_{r,2k}}^{GL_{n}(F)}(\psi_{r,2k})$$ as a *Klyachko model* for $GL_n(F).$ Here ${\operatorname{Ind}}$ denotes the functor of non-compact smooth induction.

In [@OS3 Corollary 1] we showed that for any $\pi \in {\mathcal{A}}_F^u(n)$ there exists a unique decomposition $$n=r(\pi)+2k(\pi)$$ such that $${\operatorname{Hom}}_{GL_n(F)}(\pi,{\mathcal{M}}_{r(\pi),2k(\pi)})\ne 0$$ and that in fact $\dim_{\mathbb{C}}({\operatorname{Hom}}_{GL_n(F)}(\pi,{\mathcal{M}}_{r(\pi),2k(\pi)}))=1.$

For $\pi \in {\mathcal{A}}_F^u,$ the *Klyachko type* of $\pi$ is the ordered pair $(r(\pi),2k(\pi)).$

In fact, for ${\mathcal{A}}_F^u$ [@os Theorem 8] provides a receipt in order to read the Klyachko type off Tadic’s classification. Based on , our results can be reinterpreted by the formula $$\label{eq: kly in terms of SL(2)-type}
    r(\pi)={\operatorname{odd}}({\mathcal{V}}(\pi)),\ \pi \in {\mathcal{A}}_F^u.$$

Base change-The main results {#sec: statements}
============================

Let $E$ be a finite extension of $F.$ Denote by ${\operatorname{res}}_{E/F,n}:{\mathcal{G}}_F(n)\to {\mathcal{G}}_E(n)$ the map defined by ${\operatorname{res}}_{E/F,n}(([\rho],f))=([\rho_{|W_E}],f).$ For $n\ge 1$ the *base change* ${\operatorname{bc}}_{E/F}(\pi)\in {\mathcal{A}}_E(n)$ of $\pi \in
{\mathcal{A}}_F(n)$ is defined by $${\operatorname{rec}}_E({\operatorname{bc}}_{E/F}(\pi))={\operatorname{res}}_{E/F}({\operatorname{rec}}_F(\pi)).$$

\[thm: main\] Let $E/F$ be a finite extension of $p$-adic fields and let $\pi$ be a smooth, irreducible and unitarizable representation of $GL_n(F).$ Then ${\operatorname{bc}}_{E/F}(\pi)$ is a smooth, irreducible and unitarizable representation of $GL_n(E)$ and $${\mathcal{V}}(\pi)={\mathcal{V}}({\operatorname{bc}}_{E/F}(\pi)),$$ i.e. $\pi$ and ${\operatorname{bc}}_{E/F}(\pi)$ have the same $SL(2)$-type.

As a consequence we have the following.

\[cor: main\] Under the assumptions of Theorem \[thm: main\] we have $$r(\pi)=r({\operatorname{bc}}_{E/F}(\pi)),$$ i.e. $\pi$ and ${\operatorname{bc}}_{E/F}(\pi)$ have the same Klyachko type.

Corollary \[cor: main\] is straightforward from Theorem \[thm: main\] and .

Proof of the main result {#sec: proofs}
========================

\[lemma: bc for cuspidal\] Let $E/F$ be a finite extension. For $\sigma \in {\mathcal{A}}_F^\circ\cap {\mathcal{A}}_F^u$ there exist $\sigma_1,\dots,\sigma_m \in
{\mathcal{A}}_E^\circ\cap {\mathcal{A}}_E^u$ such that $${\operatorname{bc}}_{E/F}(\sigma)=\sigma_1\times \cdots \times \sigma_m.$$

Recall that a representation in $ {\mathcal{A}}_F^\circ$ is unitarizable if and only if its central character is unitary. Let $\rho$ be the irreducible representation of $W_F$ such that ${\operatorname{rec}}_F(\sigma)=([\rho],{{\bf 1}}_n).$ It follows from that $\rho$ has a unitary central character and therefore it has a unitary structure. Thus, the restriction $\rho_{|W_E}$ to $W_E$ also has a unitary structure and therefore each of its irreducible componencts has a unitary central character. The lemma follows by applying to ${\operatorname{res}}_{E/F}({\operatorname{rec}}_F(\sigma))$.

\[prop: bc of Zel inv\] Let $E/F$ be a finite extension and let $\pi\in {\mathcal{A}}_F^u$ then ${\operatorname{bc}}(\pi)\in {\mathcal{A}}_E^u$ and $$\label{eq: bc and transpose commute}
    {\operatorname{bc}}_{E/F}(\pi^t)={\operatorname{bc}}_{E/F}(\pi)^t.$$

Let $\pi \in {\mathcal{A}}_F^u$ be of the form . By Lemma \[lemma: bc for cuspidal\] there exist $\sigma_{i,k}\in
{\mathcal{A}}_E^\circ,\,i=1,\dots,m,\,k=1,\dots,t_i$ such that $${\operatorname{bc}}_{E/F}(\sigma_i)=\sigma_{i,1}\times \cdots \times \sigma_{i,t_i}.$$ Let $\rho_i={\operatorname{rec}}_F^\circ(\sigma_i)$ and $\rho_{i,k}={\operatorname{rec}}_E^\circ(\sigma_{i,k}).$ Thus, $${\operatorname{res}}_{E/F}(\rho_i)=\oplus_{k=1}^{t_i}\rho_{i,k}.$$ It follows from that $$\label{eq: res of rec}
    {\operatorname{res}}_{E/F}({\operatorname{rec}}_F(\pi))=\oplus_{i=1}^m
    \oplus_{j=1}^{r_i}\oplus_{k=1}^{t_i}
    \tau(\Delta[\sigma_{i,k}[\frac{r_i+1}2-j],n_i]).$$ On the other hand, let $$\Pi=\times_{i=1}^m \times_{k=1}^{t_i}U(\delta[\sigma_{i,k},n_i],r_i)$$ Since $\pi\in {\mathcal{A}}_F^u,$ the classification of Tadic implies that $\Pi \in {\mathcal{A}}_E^u$ and by applied to $E$ instead of $F$ we have $$\label{eq: rec_E id}
    {\operatorname{rec}}_E(\Pi)=\oplus_{i=1}^m
    \oplus_{j=1}^{r_i}\oplus_{k=1}^{t_i}
    \tau(\Delta[\sigma_{i,k}[\frac{r_i+1}2-j],n_i]).$$ Comparing with we obtain that $\Pi={\operatorname{bc}}_{E/F}(\pi)$ and in particular that ${\operatorname{bc}}_{E/F}(\pi)
\in{\mathcal{A}}_E^u.$ Applying this to $\pi^t$ expressed by gives $${\operatorname{bc}}_{E/F}(\pi^t)=\times_{i=1}^m
    \times_{k=1}^{t_i}U(\delta[\sigma_{i,k},r_i],n_i).$$ Applying now to ${\operatorname{bc}}_{E/F}(\pi)^t$ we obtain the identity .

It is straightforward from the definitions that $$\label{eq: trivial id}
    P_F({\operatorname{rec}}_F(\pi))=P_E({\operatorname{rec}}_E({\operatorname{bc}}_{E/F}(\pi)),\ \pi \in {\mathcal{A}}_F.$$ For $\pi \in {\mathcal{A}}_F^u,$ applying to $\pi^t$ and then we get that $$P_F({\operatorname{rec}}_F(\pi^t))=P_E({\operatorname{rec}}_E({\operatorname{bc}}_{E/F}(\pi)^t).$$ The identity ${\mathcal{V}}(\pi)={\mathcal{V}}({\operatorname{bc}}_{E/F}(\pi))$ is now immediate from . This completes the proof of Theorem \[thm: main\].

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