Typical representations for level zero Bernstein components of ${\rm GL}_n(F)$

TL;DR

This paper classifies typical representations of ${ m GL}_n( ext{integers }F)$ for level zero Bernstein components, extending prior results to all characteristics and clarifying their role in the structure of smooth irreducible representations.

Contribution

It provides a classification of typical representations for level zero Bernstein components of ${ m GL}_n(F)$, generalizing previous work to all characteristics.

Findings

Classified typical representations for level zero Bernstein components.

Extended results of Henniart and Paskunas to all characteristics.

Clarified the role of typical representations in the cuspidal support.

Abstract

Let $F$ be a non-discrete non-Archimedean locally compact field. In this article for a level zero Bernstein component $s$, we classify those irreducible smooth representations of ${\rm GL}_n{\integers{F}}$ (called typical representations) whose appearance in a smooth irreducible representation $\pi$ of ${\rm GL}_n{F}$ implies that the cuspidal support of $\pi$ is $s$. These results extend, for level zero representations, the results of Henniart and Pa\v{s}k\={u}nas on cuspidal representations. The results are independent of the characteristic of the base field.

Equations

$${\mathcal{A}}_n=\prod _{s\in {\mathcal{B}}_n}{\mathcal{A}}_n(s),$$

$${\mathrm{Hom}}_{{\mathrm{GL}}_n({\mathcal{O}}_F)}({\tau }_s,\pi )\ne 0\Rightarrow \pi \in {\mathcal{A}}_n(s).$$

$${\mathrm{Hom}}_{J_s}({\lambda }_s,\pi )\ne 0\iff \pi \in {\mathcal{A}}_n(s).$$

$${\mathrm{ind}}_{J_s}^{{\mathrm{GL}}_n({\mathcal{O}}_F)}{\lambda }_s$$

$${\mathrm{ind}}_{J_s}^{{\mathrm{GL}}_n({\mathcal{O}}_F)}({\lambda }_s).$$

$$(M_1,{\sigma }_1)\sim (M_2,{\sigma }_2)$$

$${\mathrm{Hom}}_{J_s}(\pi ,{\lambda }_s)\ne 0\iff \pi \in {\mathcal{A}}_n(s).$$

$${\mathrm{ind}}_{J_s}^{{\mathrm{GL}}_n({\mathcal{O}}_F)}({\lambda }_s)$$

$${\mathrm{ind}}_{J_s}^{K_n}({\lambda }_s)$$

$${\mathrm{res}}_{K_n}(i_{P_I}^{G_n}({\sigma }_I)),$$

$${\mathrm{res}}_{K_n}(i_{P_I}^{G_n}({\sigma }_I))\simeq {\mathrm{ind}}_{P_I\cap K_n}^{K_n}({\sigma }_I).$$

$${\mathrm{ind}}_{P_I\cap K_n}^{K_n}({\tau }_I)$$

$${\mathrm{res}}_{K_{n_i}}{\sigma }_i={\tau }_i^0\oplus {\tau }_i^1$$

$$[M_1\times M_2\times \cdots \times M_r,{\lambda }_1⊠{\lambda }_2⊠\cdots ⊠{\lambda }_r]$$

$${\mathrm{ind}}_{P_I\cap K_n}^{K_n}({\tau }_I^0)$$

$${\mathrm{ind}}_{P_I\cap K_n}^{K_n}({⊠}_{i=1}^r{\gamma }_i)$$

$$i_{P_I}^{G_n}\{i_{P_1}^{{\mathrm{GL}}_{n_1}(F)}({\lambda }_1)⊠\cdots ⊠i_{P_N^{\prime }}^{{\mathrm{GL}}_{n_N}(F)}({\lambda }_N^{\prime })⊠\cdots ⊠i_{P_r}^{{\mathrm{GL}}_{n_r}(F)}({\lambda }_r)\}$$

$$[M_1\times \cdots \times M_N^{\prime }\times \cdots \times M_r,{\lambda }_1⊠\cdots ⊠{\lambda }_N^{\prime }⊠\cdots ⊠{\lambda }_r].$$

$$[M_N=\prod _{j=1}^p{\mathrm{GL}}_{m_j}(F),{\lambda }_N={⊠}_{j=1}^p{\zeta }_j]\ne [M_N^{\prime }=\prod _{j=1}^{p^{\prime }}{\mathrm{GL}}_{m_j^{\prime }}(F),{\lambda }_N^{\prime }={⊠}_{j=1}^{p^{\prime }}{\zeta }_j^{\prime }]$$

$$\{[{\mathrm{GL}}_{m_1}(F),{\zeta }_1],[{\mathrm{GL}}_{m_2}(F),{\zeta }_2],\dots ,[{\mathrm{GL}}_{m_p}(F),{\zeta }_p]\}$$

$$\{[{\mathrm{GL}}_{m_1^{\prime }}(F),{\zeta }_1^{\prime }],[{\mathrm{GL}}_{m_2^{\prime }}(F),{\zeta }_2^{\prime }],\dots ,[{\mathrm{GL}}_{m_{p^{\prime }}}(F),{\zeta }_{p^{\prime }}^{\prime }]\}.$$

$${\mathrm{ind}}_{P_I\cap K_n}^{K_n}({\tau }_I)$$

$${\mathrm{ind}}_{P_I\cap K_n}^{K_n}({\tau }_I).$$

$$(H_i\cap \overline{U_I})\times (H_i\cap M_I)\times (H_i\cap U_I)\to H_i$$

$${\mathrm{ind}}_{H_i}^{K_n}(\tau )$$

$${\mathrm{ind}}_{K_n\cap P_I}^{K_n}(\tau ).$$

$${\mathrm{ind}}_{K_n\cap P_I}^{K_n}(\tau )$$

$${\mathrm{res}}_{J_1\cap M}{\mathrm{ind}}_{J_2}^{J_1}(\lambda )\simeq {\mathrm{ind}}_{J_2\cap M}^{J_1\cap M}(\lambda ).$$

$${\mathrm{res}}_{J_1\cap P}{\mathrm{ind}}_{J_2}^{J_1}(\lambda )\simeq {\mathrm{ind}}_{J_2\cap P}^{J_1\cap P}(\lambda ).$$

$$\frac{J_1\cap P}{J_2\cap P}$$

Full text

1. Introduction

Let $F$ be a non-discrete non-Archimedean locally compact field. The isomorphism classes of irreducible smooth complex representations of $\operatorname{GL}_{n}(F)$, denoted by $\mathcal{A}_{n}$, can be decomposed as a disjoint union

[TABLE]

where $\mathcal{A}_{n}(s)$ is defined in terms of parabolic induction and the parameter $s$ is called the Bernstein component or inertial support. In the context of the local Langlands correspondence, the parameter $s$ determines the isomorphism class of the restriction to the inertia subgroup $I_{F}$ of the Weil–Deligne representation associated by the classical local Langlands correspondence.

The Local class field theory gives a natural isomorphism between $I_{F}$ and $\mathcal{O}_{F}^{\times}$, the group of units of the ring of integers of $F$. It is natural to ask for a relation between the representations of $I_{F}$ which can be extended to a Weil–Deligne representation and the representations of the maximal compact subgroup $\operatorname{GL}_{n}(\operatorname{\mathcal{O}_{{\it F}}})$. One natural way would be to understand the cuspidal support of a smooth irreducible representation from its restriction to $\operatorname{GL}_{n}(\operatorname{\mathcal{O}_{{\it F}}})$. Indeed in several arithmetic applications (see [BM02], [EG14]) it is desired to construct irreducible smooth representations $\tau_{s}$ of the maximal compact subgroup $\operatorname{GL}_{n}(\operatorname{\mathcal{O}_{{\it F}}})$ such that for any irreducible smooth representation $\pi$ of $\operatorname{GL}_{n}(F)$,

[TABLE]

Such a representation $\tau_{s}$ is called a typical representation for $s$. In this article we completely classify typical representations $\tau_{s}$ for all level-zero Bernstein components (see section 3) of $\operatorname{GL}_{n}(F)$.

The existence of typical representation, for any $s$, follows from the theory of types developed by Bushnell and Kutzko. For all $s\in\mathcal{B}_{n}$, Bushnell and Kutzko constructed pairs $(J_{s},\lambda_{s})$ such that for any irreducible smooth representation $\pi$ of $\operatorname{GL}_{n}(F)$,

[TABLE]

It follows from Frobenius reciprocity that any irreducible subrepresentation of

[TABLE]

is a typical representation for $s$. In general the representation (1) is not irreducible and it is not known if there are any other typical representations which are not subrepresentations of (1).

For $n=2$, Henniart (see [BM02]) classified typical representations for all inertial classes. Later Paškūnas (see [Pas05]) classified typical representations occurring in cuspidal representation of $\operatorname{GL}_{n}(F)$ for $n\geq 3$. It turns out that there exists a unique typical representation occurring in a cuspidal representation. For a general Bernstein component $s$, typical representations may not be unique. In this article we classify typical representations for a level zero Bernstein component $s$.

We now describe the main result of this article. Let $s$ be a level zero Bernstein component and $(J_{s},\lambda_{s})$ (see section 3 for a complete description of the type $(J_{s},\lambda_{s})$) be the Bushnell–Kutzko type for the component $s$. We will prove that

Theorem 1.1**.**

Any typical representation $\tau_{s}$ for a level-zero Bernstein component $s$ occurs as a subrepresentation of

[TABLE]

In our analysis we will also obtain a certain multiplicity result on the typical representations $\tau_{s}$ (see Corollary 3.3).

This article is based on chapter $3$ of my thesis. In my thesis typical representations are classified for several Bernstein components. I would like to thank my thesis advisor Guy Henniart for suggesting this problem and numerous discussions. I thank Corinne Blondel for pointing out several corrections and improvements. I would like to express my deep gratitude to the referee for very useful suggestions and comments.

2. Preliminaries

2.1. Basic notation

Let $F$ be a non-Archimedean local field with ring of integers $\operatorname{\mathcal{O}_{{\it F}}}$, maximal ideal $\operatorname{\mathfrak{P}_{{\it F}}}$ and a finite residue field $k_{F}$. All our representations are on vector spaces over $\mathbb{C}$.

Let $G$ be a locally pro-finite group and $H$ be a closed subgroup of $G$. For $(\tau,V)$ a smooth representation of $H$, we denote by $\operatorname{ind}_{H}^{G}(\tau)$ the induced smooth representation of $G$ and by c-$\operatorname{ind}_{H}^{G}(\tau)$ the compactly induced representation. When $G$ is the group of $F$-rational points of an algebraic reductive group, the group $G$ is equipped with a locally profinite topology induced from $F$. For $P$ the set of $F$-rational points of an $F$-parabolic subgroup of $G$ and $\sigma$ a smooth representation of a Levi subgroup $M$ of $P$, we denote by $i^{G}_{P}(\sigma)$ the normalized parabolically-induced representation.

For any two groups $H_{1}$ and $H_{2}$ such that $H_{2}\subset H_{1}$ and $\sigma$ a representation of $H_{1}$, we denote by $\operatorname{res}_{H_{2}}(\sigma)$ the restriction of $\sigma$ to $H_{2}$. We use $\boxtimes$ and $\otimes$ for the tensor product of representations of two different groups and the same group respectively. If $H_{2}$ is a subgroup of a group $H_{1}$, $\tau$ is a representation of $H_{2}$ and $h\in H_{1}$ then we denote by ${}^{h}\tau$ the representation of $hH_{2}h^{-1}$ given by $h^{\prime}\mapsto\tau(h^{-1}h^{\prime}h)$ for all $h^{\prime}\in hH_{2}h^{-1}$.

After recalling some general definitions we will restrict ourself to the case where $G=\operatorname{GL}_{n}(F)$ and the following notation will be used: We denote by $G_{n}$ the group $\operatorname{GL}_{n}(F)$ and by $K_{n}$ the maximal compact subgroup $\operatorname{GL}_{n}(\operatorname{\mathcal{O}_{{\it F}}})$. Let $K_{n}(m)$ the principal congruence subgroup of $\operatorname{GL}_{n}(\operatorname{\mathcal{O}_{{\it F}}})$ of level $m$.

Let $I=(n_{1},n_{2},n_{3},\dots,n_{r})$ be an ordered partition of a positive integer $n$. Let $R$ be an $\mathcal{O}_{F}$ algebra. Let $P_{I}(R)$ be the group of invertible block upper triangular matrices of type $(n_{1},n_{2},\dots,n_{r})$ with entries in $R$. We denote by $M_{I}(R)$ and $U_{I}(R)$ the subgroups of $P_{I}(R)$ consisting of block diagonal matrices of type $I$ and the unipotent unipotent matrices of type $I$ respectively. We use the notation $P_{I}$, $M_{I}$ and $U_{I}$ for $P_{I}(F)$, $M_{I}(F)$ and $U_{I}(F)$ respectively. We call $P_{I}$ and $M_{I}$ the standard parabolic subgroup and standard Levi subgroup of type $I$ respectively.

2.2. Bernstein decomposition and typical representations

Let $B(G)$ be the set of pairs $(M,\sigma)$ where $M$ is a Levi subgroup of an $F$-parabolic subgroup $P$ of $G$ and $\sigma$ is an irreducible supercuspidal representation of $M$. Recall that the inertial equivalence relation on $B(G)$ is defined by setting

[TABLE]

if and only if there exist an element $g\in G$ and an unramified character $\chi$ of $M_{2}$ such that $M_{1}=gM_{2}g^{-1}$ and $\sigma_{1}^{g}\simeq\sigma_{2}\otimes\chi$. We denote by $\mathcal{B}_{G}$ the set of such equivalence classes called inertial classes or Bernstein components. Any irreducible smooth representation $\pi$ of $G$ occurs as a sub-representation of a parabolic induction $i_{P}^{G}(\sigma)$ where $\sigma$ is an irreducible supercuspidal representation of a Levi subgroup $M$ of $P$. The pair $(M,\sigma)$ is well determined up to $G$-conjugation. We call the class $s=[M,\sigma]$ the inertial support of $\pi$. We will use the notation $I(\pi)$ for the inertial support of $\pi$.

Let $\mathcal{M}(G)$ be the category of all smooth representations of $G$. For an inertial class $s=[M,\sigma]$ we denote by $\mathcal{M}_{s}(G)$ the full sub-category consisting of smooth representations all of whose irreducible sub-quotients appear in the composition series of some $i_{P}^{G}(\sigma\otimes\chi)$, with $\chi$ an unramified character of $M$. It is shown by Bernstein (see [Ren10, VI.7.2, Theorem]) that the category $\mathcal{M}(G)$ decomposes as a direct product of $\mathcal{M}_{s}(G)$ in particular every smooth representation can be written as a direct sum of objects in the categories $\mathcal{M}_{s}(G)$. We denote by $\mathcal{A}_{G}(s)$ the set of isomorphism classes of simple objects in the category $\mathcal{M}_{s}(G)$. If $G=\operatorname{GL}_{n}(F)$ we use the notation $\mathcal{A}_{n}(s)$ for $\mathcal{A}_{G}(s)$ and $\mathcal{B}_{n}$ for $\mathcal{B}_{\operatorname{GL}_{n}(F)}$.

Given an irreducible smooth representation $\rho$ of a maximal compact subgroup $K$ of $G$ the compact induction $\pi:=\text{c-}\operatorname{ind}_{K}^{G}(\rho)$ is a finitely generated smooth representation of $G$ and hence there exists an irreducible $G$-quotient of $\pi$. By Frobenius reciprocity [BH06, Proposition 2.5] we get that $\rho$ occurs in a smooth irreducible representation of $G$. For a given inertial class, we are interested in the representations $\rho$ of $K$ which only occur in irreducible smooth representations with inertial support $s$.

Definition 2.1**.**

Let $s$ be an inertial class for $G$. An irreducible smooth representation $\tau$ of a maximal compact subgroup $K$ of $G$ is called $K$-typical representation for $s$* if, for any irreducible smooth representation $\pi$ of $G$, $\operatorname{Hom}_{K}(\tau,\pi)\neq 0$ implies that $\pi\in\mathcal{A}_{G}(s)$.*

In this article we will confine ourselves to the cases where $G=\operatorname{GL}_{n}(F)=G_{n}$, $K=\operatorname{GL}_{n}(\operatorname{\mathcal{O}_{{\it F}}})=K_{n}$ and $n\geq 2$ and in these cases we call a $K$-typical representation for $s$ a typical representation for $s$. An irreducible representation $\tau$ of $K_{n}$ is called atypical if $\tau$ occurs in two irreducible smooth representations $\pi_{1}$ and $\pi_{2}$ such that $I(\pi_{1})\neq I(\pi_{2})$.

For any component $s\in\mathcal{B}_{n}$, the existence of a typical representation can be deduced from the theory of types developed by Bushnell and Kutzko in the articles [BK99] and [BK93]. Bushnell and Kutzko constructed a pair $(J_{s},\lambda_{s})$, which we call a Bushnell–Kutzko type, where $J_{s}$ is a compact open subgroup of $\operatorname{GL}_{n}(F)$ and $\lambda_{s}$ is an irreducible representation of $J_{s}$ such that for every irreducible smooth representation $\pi$ of $G_{n}$,

[TABLE]

The group $J_{s}$ can be arranged to be a subgroup of $\operatorname{GL}_{n}(\operatorname{\mathcal{O}_{{\it F}}})$ by conjugating with an element of $\operatorname{GL}_{n}(F)$ and hence we assume that $J_{s}\subset\operatorname{GL}_{n}(\operatorname{\mathcal{O}_{{\it F}}})$. It follows from Frobenius reciprocity that any irreducible sub-representation of

[TABLE]

is a typical representation. The irreducible sub-representations of (3) are classified by Schneider and Zink in [SZ99, Section 6, $T_{K,\lambda}$ functor].

For $s=[G_{n},\sigma]$, Paškūnas in [Pas05, Theorem 8.1] showed that up to isomorphism there exists a unique typical representation for $s$. More precisely,

Theorem 2.2** (Paškūnas).**

Let $n$ be a positive integer greater than one and $\sigma$ be an irreducible supercuspidal representation of $G_{n}$. Let $(J_{s},\lambda_{s})$ be a Bushnell–Kutzko type for the component $s=[G_{n},\sigma]$ with $J_{s}\subset\operatorname{GL}_{n}(\operatorname{\mathcal{O}_{{\it F}}})$. The representation

[TABLE]

is the unique typical representation for the component $[G_{n},\sigma]$ and occurs with multiplicity one in $\sigma\otimes\chi$, for all unramified characters $\chi$ of $G_{n}$.

We will consider the classification of typical representations for components $[M,\sigma]$ where $M$ is a Levi subgroup of a proper parabolic subgroup of $G_{n}$.

Let $s=[M,\sigma]$ be an inertial class of $G_{n}$. We will choose a representative for $s$. Let $P$ be a parabolic subgroup with $M$ as its Levi subgroup. There exists a $g\in G_{n}$ such that $gPg^{-1}=P_{I}$ for some ordered partition $I=(n_{1},n_{2},\dots,n_{r})$ of $n$. The groups $gMg^{-1}$ and $M_{I}$ are two Levi subgroups of $P_{I}$ hence we get an $u\in\operatorname{Rad}{P_{I}}$ such that $ugM(ug)^{-1}=M_{I}$. This shows that there exists an element $g^{\prime}\in G_{n}$ such that $g^{\prime}Mg^{\prime-1}=M_{I}$. Let $J$ be a permutation of the ordered partition $(n_{1},n_{2},\dots,n_{r})$. We can choose a $g^{\prime\prime}\in G_{n}$ such that $M_{I}$ and $M_{J}$ are conjugate so the two pairs $(M,\sigma)$ and $(M_{J},\sigma^{g^{\prime}g^{\prime\prime}})$ are inertially equivalent. In certain cases it is convenient to choose a particular permutation. For example in the proof of the main theorem in this article we choose $J=(n_{1}^{\prime},n_{2}^{\prime},\dots n_{r}^{\prime})$ such that $n_{i}^{\prime}\leq n_{j}^{\prime}$ for all $i\leq j$. We denote by $\sigma_{I}$ and $\sigma_{J}$ the representations $\sigma^{g^{\prime}}$ and $\sigma^{g^{\prime}g^{\prime\prime}}$ respectively and hence $s=[M_{I},\sigma_{I}]=[M_{J},\sigma_{J}]$.

Let $\tau$ be a typical representation for the component $s$. The representation $\tau$ occurs as a $K_{n}$ sub-representation of a $G_{n}$-irreducible smooth representation $\pi$ (see the reasoning given in the paragraph above Definition 2.1). From the above paragraph $\pi$ occurs in the composition series of $i_{P_{I}}^{G_{n}}(\sigma_{I})$ where $\sigma_{I}$ is a supercuspidal representation of $M_{I}$. Hence to classify typical representations we fix a pair $(M_{I},\sigma_{I})\sim(M,\sigma)$ and examine the $K_{n}$-irreducible sub-representations of

[TABLE]

looking for possible typical representations for $s$.

By the Iwasawa decomposition $G_{n}=K_{n}P_{I}$ we get that

[TABLE]

We write $\sigma_{I}$ as $\boxtimes_{i=1}^{r}\sigma_{i}$ where $\sigma_{i}$ is a supercuspidal representation of $G_{n_{i}}$ for $1\leq i\leq r$. We denote by $\tau_{i}$ the unique typical representation for the component $[G_{n_{i}},\sigma_{i}]$ for $1\leq i\leq r$ and let $\tau_{I}$ be the $M_{I}(\operatorname{\mathcal{O}_{{\it F}}})$-representation $\boxtimes_{i=1}^{r}\tau_{i}$. Will Conley observed in his thesis (see [Wil10, Theorem 2.28]) that the representation

[TABLE]

admits a complement in $\operatorname{ind}_{P_{I}\cap K_{n}}^{K_{n}}(\sigma_{I})$ whose irreducible sub-representations are atypical for $s$. We prove a mild generalisation which will be used later in proofs by induction.

Let $t_{i}=[M_{i},\lambda_{i}]$ be a Bernstein component of $G_{n_{i}}$ for $1\leq i\leq r$. Let $\sigma_{i}$ be a smooth representation from $\mathcal{M}_{t_{i}}({G_{n_{i}}})$. We suppose

[TABLE]

for $1\leq i\leq r$ such that irreducible subrepresentations of $\tau_{i}^{1}$ are atypical. We denote by $t$ the Bernstein component

[TABLE]

of $G_{n}$. The component $t$ is independent of the choice of representatives $(M_{i},\lambda_{i})$. Let $\tau_{I}^{0}=\boxtimes_{i=1}^{r}\tau_{i}^{0}$ and $\sigma_{I}=\boxtimes_{i=1}^{r}(\sigma_{i})$.

Proposition 2.3**.**

The representation

[TABLE]

admits a complement in $\operatorname{res}_{K_{n}}i_{P_{I}}^{G_{n}}(\sigma_{I})$ with all its irreducible subrepresentations atypical.

Proof.

Any $K_{n}$-irreducible subrepresentation of $\operatorname{res}_{K_{n}}i_{P_{I}}^{G_{n}}(\sigma_{I})$ occurs as a subrepresentation of

[TABLE]

where $\gamma_{i}$ is a $K_{n_{i}}$-irreducible subrepresentation of $\sigma_{i}$. Suppose there exists $N\leq r$ such that $\gamma_{N}$ occurs in $\tau_{N}^{1}$. Thus there exists a component $t_{N}^{\prime}\in\mathcal{B}_{n_{N}}$ such that $t_{N}^{\prime}$ is equal to $[M_{N}^{\prime},\lambda_{N}^{\prime}]\neq t_{N}$ and $\gamma_{N}$ occurs in the restriction $\operatorname{res}_{K_{N}}i_{P_{N}^{\prime}}^{\operatorname{GL}_{n_{N}}(F)}(\lambda_{N}^{\prime})$. Hence the representation (4) occurs as a $K_{n}$-subrepresentation of

[TABLE]

The inertial support $t^{\prime}$ of the above representation is

[TABLE]

We may assume that $M_{i}$ is a standard Levi subgroup for $1\leq i\leq r$. Now

[TABLE]

implies that there exists a cuspidal component $[\operatorname{GL}_{m_{k}}(F),\zeta_{k}]$ occurring in the multi-set

[TABLE]

which has a different multiplicity in

[TABLE]

Adding cuspidal components with the same multiplicity to the above two multi-sets cannot make the multiplicities of the component $[\operatorname{GL}_{k}(F),\zeta_{k}]$ the same. This shows that $t^{\prime}\neq t$ and hence the desired complement is the direct sum of the representations as in (4) such that $\gamma_{i}$ occur in $\tau_{i}^{1}$ for some $i\in\{1,2,\dots,r\}$. ∎

Lemma 2.4**.**

Let $t_{i}=[G_{n_{i}},\sigma_{i}]$ be a Bernstein component for $G_{n_{i}}$ and $\tau_{i}$ be a typical representation for $t_{i}$ and let $\tau_{I}$ be the representation $\tau_{1}\boxtimes\tau_{2}\boxtimes\dots\boxtimes\tau_{s}$. The representation

[TABLE]

admits a complement in $\operatorname{res}_{K_{n}}i_{P_{I}}^{G_{n}}(\sigma_{I})$ whose irreducible sub-representations are atypical.

Proof.

We use the uniqueness of typical representations for supercuspidal representations (see [Pas05]) to decompose $\operatorname{res}_{K_{n_{i}}}\sigma_{i}$ as $\tau_{i}\oplus\tau_{i}^{1}$ such that irreducible sub-representations of $\tau^{1}_{i}$ are atypical. The lemma follows as a consequence of Proposition 2.3. ∎

Given a component $s=[M_{I},\sigma_{I}]$ of $G_{n}$ the above lemma shows that typical representations only occur as sub-representations of

[TABLE]

The above representation is still an infinite dimensional representation of the compact group $K_{n}$. We write the above representation as an increasing union of finite-dimensional representations.

Let $\{H_{i}\}_{i\geq 1}$ be a decreasing sequence of compact open subgroups of the maximal compact subgroup $K_{n}$. Let $\bar{U}_{I}$ be the unipotent radical of the opposite parabolic subgroup $\bar{P}_{I}$ of $P_{I}$ with respect to the Levi subgroup $M_{I}$. We assume that $H_{i}$ has an Iwahori decomposition with respect to the parabolic subgroup $P_{I}$ and Levi subgroup $M_{I}$ for all $i\geq 1$ i.e. the product map

[TABLE]

is a homeomorphism for any ordering of the factors on the left hand side and that $\bigcap_{i\geq 1}H_{i}=K_{n}\cap P_{I}$. Let $\tau$ be a finite dimensional smooth representation of the group $M_{I}(\mathcal{O}_{F})$. We assume that $\tau$ extends to a representation of $H_{i}$ for all $i\geq 1$ such that $H_{i}\cap U_{I}$ and $H_{i}\cap\bar{U}_{I}$ are contained in the kernel of $\tau$. By definition the representation $\operatorname{ind}_{H_{i}}^{K_{n}}(\tau)$ is contained in $\operatorname{ind}_{K_{n}\cap P_{I}}^{K_{n}}(\tau)$.

Lemma 2.5**.**

The union of the representations

[TABLE]

for all $i\geq 1$ is equal to the representation

[TABLE]

Proof.

Let $W$ be the underlying space for the representation $\tau$. Any element $f$ in the space

[TABLE]

is a function $f:K_{n}\rightarrow W$ such that

1. (1)

$f(pk)=\tau(p)f(k)$ for all $p\in K_{n}\cap P_{I}$ and $k\in K_{n},$ 2. (2)

here exists a positive integer $m$ (depending on $f$) such that $f(gk)=f(g)$ for all $k\in K_{n}(m)$ and $g\in K_{n}$.

Now there exists a positive integer $i$ such that $H_{i}\cap\bar{U}_{I}\subset K_{n}(m)$. For such a choice of $i$ and $h\in H_{i}$ write $h=h^{-}h^{+}$ where $h^{+}\in{K_{n}}\cap P$, $h^{-}\in H_{i}\cap\bar{U}_{I}$ which we can do so by Iwahori decomposition of $H_{i}$. We observe that $f(hk)=f(h^{-}h^{+}k)=f(h^{+}k(h^{+}k)^{-1}h^{-}(h^{+}k))=f(h^{+}k)=\tau(h^{+})f(k)$ (since $(h^{+}k)^{-1}h^{-}(h^{+}k)\in K_{n}(m)$). Hence $f\in\operatorname{ind}_{H_{i}}^{K_{n}}(\tau)$. ∎

We shall need the following technical lemma for frequent reference. Let $P$ be any parabolic subgroup of $G_{n}$ with a Levi subgroup $M$ and $U$ be the unipotent radical of $P$. Let $J_{1}$ and $J_{2}$ be two compact open subgroups of $K_{n}$ such that $J_{1}$ contains $J_{2}$. Suppose $J_{1}$ and $J_{2}$ both satisfy Iwahori decomposition with respect to the Levi subgroup $M$. With $J_{1}\cap U=J_{2}\cap U$ and $J_{1}\cap\bar{U}=J_{2}\cap\bar{U}$. Let $\lambda$ be an irreducible smooth representation of $J_{2}$ which admits an Iwahori decomposition i.e. $J_{2}\cap U$ and $J_{2}\cap\bar{U}$ are contained in the kernel of $\lambda$.

Lemma 2.6**.**

The representation $\operatorname{ind}_{J_{2}}^{J_{1}}(\lambda)$ is the extension of the representation $\operatorname{ind}_{J_{2}\cap M}^{J_{1}\cap M}(\lambda)$ such that $J_{1}\cap U$ and $J_{1}\cap\bar{U}$ are contained in the kernel of the extension.

Proof.

From the Iwahori decomposition we get that $(J_{1}\cap M)J_{2}=J_{1}$ and from the Mackey decomposition we get that

[TABLE]

We now verify that $J_{1}\cap U$ and $J_{1}\cap\bar{U}$ act trivially on $\operatorname{ind}_{J_{2}}^{J_{1}}(\lambda)$. Observe that

[TABLE]

Since the double coset representatives for

[TABLE]

can be chosen from $M\cap J_{1}$ the group $J_{1}\cap U$ acts trivially on $\operatorname{ind}_{J_{2}}^{J_{1}}(\lambda)$. Similarly $J_{1}\cap\bar{U}$ acts trivially on $\operatorname{ind}_{J_{2}}^{J_{1}}(\lambda)$. This concludes the lemma. ∎

Lemma 2.7**.**

Let $G$ be the $F$-rational points of an algebraic reductive group and $\chi$ be a character of $G$. Let $\tau$ be a $K$-typical representation for the component $s=[M,\sigma]$. The representation $\tau\otimes\chi$ is a typical representation for the component $[M,\sigma\otimes\chi]$.

Proof.

Let $\operatorname{Hom}_{K}(\tau\otimes\chi,\pi)\neq 0$ for some irreducible smooth representation $\pi$ of $G$. We now have $\operatorname{Hom}_{K}(\tau,\pi\otimes\chi^{-1})\neq 0$. This implies that $\pi\otimes\chi^{-1}$ occurs in the composition series of

[TABLE]

for some parabolic subgroup $P$ containing $M$ as a Levi subgroup and $\eta$ an unramified character of $M$. Now $\pi$ occurs in the composition series for the representation

[TABLE]

hence $\tau\otimes\chi$ is a $K$-typical representation for the component $[M,\sigma\otimes\chi]$. ∎

3. Level-Zero Bernstein components

Definition 3.1**.**

Let $I=(n_{1},n_{2},\dots,n_{r})$ be an ordered partition of $n$. An inertial class $s=[M_{I},\boxtimes_{i=1}^{r}\sigma_{i}]$ is called a level-zero inertial class if the $K_{n_{i}}(1)$ invariants of $\sigma_{i}$ is non trivial, for $1\leq i\leq r$.

We fix a level-zero inertial class $s=[M_{I},\sigma_{I}]$. The subgroup $K_{n_{i}}$ acts on the $K_{n_{i}}(1)$ invariants of $\sigma_{i}$ and $\tau_{i}$ be this representation of $K_{n_{i}}$ on $K_{n_{i}}(1)$ invariants of $\sigma_{i}$. The representation is $\tau_{i}$ is the inflation of a cuspidal representation of $\operatorname{GL}_{n_{i}}(k_{F})$. The pair $(K_{n_{i}},\tau_{i})$ is the Bushnell–Kutzko type for the inertial class $[G_{n_{i}},\sigma_{i}]$.

Let $m$ be a positive integer and $P_{I}(m)$ be the inverse image of $P_{I}(\mathcal{O}_{F}/\mathfrak{P}_{F}^{m})$ under the mod-$\mathfrak{P}_{F}^{m}$ reduction map

[TABLE]

The representation $\boxtimes_{i=1}^{r}\tau_{i}$ of $M_{I}(k_{F})$ can be viewed as a representation of $P_{I}(k_{F})$ by inflation via the quotient map

[TABLE]

The representation $\boxtimes_{i=1}^{r}\tau_{i}$ of $P_{I}(k_{F})$ is also a representation of $P_{I}(1)$ by inflation via the map $\pi_{1}$. We note that $P_{I}(1)\cap U_{I}$ and $P_{I}(1)\cap\bar{U}_{I}$ are contained in the kernel of this extension. The pair $(P_{I}(1),\tau_{I})$ is the Bushnell–Kutzko type for the component $s$ (see [BK99, Section 8.3.1]). The irreducible sub-representations of

[TABLE]

are thus typical for $s$.

We note that the groups $P_{I}(m)$ have Iwahori decomposition with respect to $P_{I}$ and $M_{I}$. The representation $\tau_{I}$ of $M_{I}(\operatorname{\mathcal{O}_{{\it F}}})$ extends to a representation of $P_{I}(m)$ such that $P_{I}(m)\cap U_{I}$ and $P_{i}(m)\cap\bar{U}_{I}$ are contained in the kernel of the extension. This shows that the sequence of groups $\{P_{I}(m)\ |\ m\geq 1\}$ and $\tau_{I}$ satisfy the hypothesis for the groups $\{H_{m}\ |m\geq 1\}$ and $\tau$ in Lemma 2.5 hence we have the isomorphism

[TABLE]

We recall that the Lemma 2.4 shows that typical representations for the component $s$ can only occur in the above representation.

Using Frobenius reciprocity we get that the representation $\tau_{I}$ occurs in $\operatorname{ind}_{P_{I}(m)}^{P_{I}(1)}(\tau_{I})$ with multiplicity one. Let $m\geq 1$ and $U_{m}^{0}(\tau_{I})$ be the $P_{I}(1)$-stable complement of the representation $\tau_{I}$ in $\operatorname{ind}_{P_{I}(m)}^{P_{I}(1)}(\tau_{I})$. Let $U_{m}(\tau_{I})$ be the representation

[TABLE]

We note that

[TABLE]

We will show that irreducible sub-representations of $U_{m}(\tau_{I})$ are atypical.

Theorem 3.2** (Main).**

Let $m\geq 1$. The $K_{n}$-irreducible subrepresentations of $U_{m}(\tau_{I})$ are atypical.

Using this, the classification of typical representations for the inertial class $s$ is given by the following corollary.

Corollary 3.3**.**

The irreducible sub-representations of $\operatorname{ind}_{P_{I}(1)}^{K_{n}}(\tau_{I})$ are precisely the typical representations for the level-zero inertial class $[M_{I},\sigma_{I}]$. Moreover if $\Gamma$ is a typical representation then

[TABLE]

Proof.

Given a typical representation $\Gamma$ for the inertial class $s$, the theorem shows that $\Gamma$ is a sub-representation of $\operatorname{ind}_{P_{I}(1)}^{K_{n}}(\tau_{I})$ and the multiplicity formula follows from Lemma 2.4 and the above theorem. Conversely if $\Gamma$ is a sub-representation of $\operatorname{ind}_{P_{I}(1)}^{K_{n}}(\tau_{I})$ then, by Frobenius reciprocity, we get that $\operatorname{Hom}_{P_{I}(1)}(\tau_{I},\Gamma)\neq 0$. If $\Gamma$ is contained as a $K_{n}$-irreducible sub-representation in an irreducible smooth representation $\pi$ of $G_{n}$ then the restriction of $\pi$ to $P_{I}(1)$ contains the representation $\tau_{I}$. The pair $(P_{I}(1),\tau_{I})$ is the Bushnell–Kutzko type for the inertial class $s=[M_{I},\sigma_{I}]$ hence the inertial support of $\pi$ is $s$. Hence $\Gamma$ is a typical representation and this proves the corollary. ∎

3.1. Decomposition of an auxiliary representation

We will need a few lemmas regarding the splitting of a certain representation for the proof of the main theorem. Let $I$ be the ordered partition $(n_{1},n_{2},\dots,n_{r})$ of the positive integer $n$ as fixed at the beginning of this chapter. Until the beginning of the section 4 we assume that $r>1$ in other words $M_{I}$ is a proper Levi subgroup. We denote by $I^{\prime}$ the ordered partition $(n_{1},n_{2},\dots,n_{r-1})$ of $n-n_{r}$. Let $m$ be a positive integer and $P_{I}(1,m)$ be the following set

[TABLE]

Here $tr$ denotes transpose. Note that $P_{I}(1,1)=P_{I}(1)$.

Lemma 3.4**.**

The set $P_{I}(1,m)$ is a subgroup of $P_{I}(1)$.

Proof.

The group $K_{n}$ acts on the set of lattices of $F^{n}$ contained in the lattice $\operatorname{\mathcal{O}_{{\it F}}}^{n}$. If $r-1=1$ the set $P_{I}(1,m)$ is the $K_{n}$-stabilizer of the lattice $(\operatorname{\mathcal{O}_{{\it F}}})^{n_{1}}\oplus(\varpi_{F}^{m}\operatorname{\mathcal{O}_{{\it F}}})^{n_{2}}$. In the case $r-1>1$ the set $P_{I}(1,m)$ is the $K_{n}$-stabilizer of the set of lattices $\{L_{k}\mid 1<k\leq r-1\}$ defined by:

[TABLE]

This shows that $P_{I}(1,m)$ is a subgroup and is contained in $P_{I}(1)$ from the definition. ∎

The structure of the representation

[TABLE]

will be used in the proof of the main theorem. Using Clifford theory we decompose the above representation. Let $K_{I}(m)$ be the group $K_{n}(m)U_{(n-n_{r},n_{r})}(\operatorname{\mathcal{O}_{{\it F}}})$. We note that this group only depends on $n$ and $n_{r}$, rather than the whole partition $I$.

Lemma 3.5**.**

The group $K_{I}(m)$ is a normal subgroup of $P_{I}(1,m)$ and $K_{I}(m)\cap P_{I}(1,m+1)$ is a normal subgroup of $K_{I}(m)$.

Proof.

The groups $K_{I}(m)$ and $P_{I}(1,m)$ satisfy Iwahori decomposition with respect to $U_{(n-n_{r},n_{r})}$, $\bar{U}_{(n-n_{r},n_{r})}$ and $M_{(n-n_{r},n_{r})}$. We also note that

[TABLE]

and

[TABLE]

Hence $P_{I}(1,m)\cap U_{(n-n_{r},n_{r})}$ and $P_{I}(1,m)\cap\bar{U}_{(n-n_{r},n_{r})}$ normalize $K_{I}(m)$. Since $K_{I}(m)$ is a product of the group $K_{n}(m)$ and $U_{(n-n_{r})}(\operatorname{\mathcal{O}_{{\it F}}})$ the group

$P_{I}(1,m)\cap M_{(n-n_{r},n_{r})}$ normalizes the group $K_{I}(m)$. This shows the first part.

Notice that $K_{I}(m)\cap U_{(n-n_{r},n_{r})}$ is equal to $K_{I}(m)\cap P_{I}(1,m+1)\cap U_{(n-n_{r},n_{r})}$ and $K_{I}(m)\cap M_{(n-n_{r},n_{r})}$ is equal to $K_{I}(m)\cap P_{I}(1,m+1)\cap M_{(n-n_{r},n_{r})}$ hence it is enough to check that $K_{I}(m)\cap\bar{U}_{(n-n_{r},n_{r})}$ normalizes the group $K_{I}(m)\cap P_{I}(1,m+1)$. Since $K_{I}(m)\cap P_{I}(1,m+1)\cap\bar{U}_{(n-n_{r},n_{r})}$ is abelian and is contained in $K_{I}(m)\cap\bar{U}_{(n-n_{r},n_{r})}$ hence we need to check that $u^{-}j(u^{-})^{-1}$ and $u^{-}u^{+}(u^{-})^{-1}$ are contained in $K_{I}(m)\cap P_{I}(1,m+1)$ for all $u^{-}$, $j$ and $u^{+}$ in

[TABLE]

respectively. Let $u^{+}$, $u^{-}$ and $j$ be three elements from $U_{n-n_{r},n_{r}}(\operatorname{\mathcal{O}_{{\it F}}})$, $K_{I}(m)\cap\bar{U}_{(n-n_{r},n_{r})}$ and $K_{I}(m)\cap P_{I}(1,m+1)\cap M_{(n-n_{r},n_{r})}$ respectively. We write them in their block form as:

[TABLE]

where $B\in M_{(n-n_{r})\times n_{r}}(\operatorname{\mathcal{O}_{{\it F}}})$,

[TABLE]

where $C\in M_{n_{r}\times(n-n_{r})}(\operatorname{\mathcal{O}_{{\it F}}})$ and

[TABLE]

We observe that $u^{-}j(u^{-})^{-1}=j\{j^{-1}u^{-}j(u^{-})^{-1}\}$ and the commutator

$\{j^{-1}u^{-}j(u^{-})^{-1}\}$ in its block form is as follows:

[TABLE]

We note that $J_{2}\in K_{n_{r}}(m)$ and $J_{1}\in K_{n-n_{r}}(m)$ hence $J_{2}^{-1}(\varpi_{F}^{m}CJ_{1}^{-1}-\varpi_{F}^{m}C)$ belongs to

[TABLE]

This shows that

[TABLE]

Now the element $(u^{-})u^{+}(u^{-})^{-1}$ is of the form

[TABLE]

Since $2m\geq m+1$ the matrix in (5) is contained in the group $K_{I}(m)\cap P_{I}(1,m+1)$. ∎

We now observe that $K_{I}(m)P_{I}(1,m+1)=P_{I}(1,m)$. From Mackey decomposition we get that

[TABLE]

Hence the above restriction decomposes into a direct sum of representations of the group

[TABLE]

The inclusion map of $K_{I}(m)\cap\bar{U}_{(n-n_{r},n_{r})}$ in $K_{I}(m)$ induces the natural homomorphism

[TABLE]

Lemma 3.6**.**

The map $\tilde{\theta}_{I}$ is an $M_{(n-n_{r},n_{r})}\cap P_{I}(1,m)$ equivariant isomorphism.

Proof.

The map is clearly injective and surjectivity follows from the Iwahori decomposition of $K_{I}(m)$ with respect to the Levi subgroup $M_{I}$. The inclusion of $K_{I}(m)\cap\bar{U}_{(n-n_{r},n_{r})}$ in $K_{I}(m)$ is an $M_{n-n_{r},n_{r}}\cap P_{I}(1,m)$ equivariant map. ∎

Let $u^{-}$ be an element of the group $K_{I}(m)\cap\bar{U}_{(n-n_{r},n_{r})}$ and its block form be given by

[TABLE]

The map $u^{-}\mapsto\varpi_{F}^{-m}U^{-}$ induces an isomorphism between the groups

[TABLE]

and $M_{n_{r}\times(n-n_{r})}(\operatorname{\mathcal{O}_{{\it F}}})$. Let $\bar{U^{-}}$ be the image of $U^{-}$ in the mod-$\operatorname{\mathfrak{P}_{{\it F}}}$ reduction of $M_{n_{r}\times(n-n_{r})}(\operatorname{\mathcal{O}_{{\it F}}})$. The map $u^{-}\mapsto\overline{\varpi_{F}^{-m}U^{-}}$ induces an isomorphism of the quotient (6) with the group of matrices $M_{n_{r}\times(n-n_{r})}(k_{F})$. We note that $M_{(n-n_{r},n_{r})}(\operatorname{\mathcal{O}_{{\it F}}})=K_{n-n_{r}}\times K_{n_{r}}$ acts on the group $M_{n_{r}\times(n-n_{r})}(k_{F})$ through its mod-$\operatorname{\mathfrak{P}_{{\it F}}}$ reduction $\operatorname{GL}_{n-n_{r}}(k_{F})\times\operatorname{GL}_{n_{r}}(k_{F})$, the action is given by $(g_{1},g_{2})U=g_{2}Ug_{1}^{-1}$ for all $g_{1}$ in $\operatorname{GL}_{n-n_{r}}(k_{F})$, $g_{2}$ in $\operatorname{GL}_{n_{r}}(k_{F})$ and $U$ in $M_{n_{r}\times(n-n_{r})}(k_{F})$. The map $u^{-}\mapsto\overline{\varpi_{F}^{-m}U^{-}}$ is hence an $M_{(n-n_{r},n_{r})}(\operatorname{\mathcal{O}_{{\it F}}})$-equivariant map between the quotient (6) and $M_{n_{r}\times(n-n_{r})}(k_{F})$. Moreover the action of $M_{(n-n_{r},n_{r})}(\operatorname{\mathcal{O}_{{\it F}}})$ factors through its quotient $M_{(n-n_{r},n_{r}}(k_{F})$.

The space $M_{n\times m}(k_{F})$ is equipped with an action of $G:=\operatorname{GL}_{m}(k_{F})\times\operatorname{GL}_{n}(k_{F})$ given by $(g_{1},g_{2})U=g_{2}Ug_{1}^{-1}$. We also have a $G$ action on the set of matrices $M_{m\times n}(k_{F})$ by setting $(g_{1},g_{2})V=g_{1}Vg_{2}^{-1}$. Let $\psi$ be a non-trivial character of the additive group $k_{F}$. We define a pairing $B$ between $M_{m\times n}(k_{F})$ and $M_{n\times m}(k_{F})$ by defining $B(V,U)=\psi\circ\operatorname{tr}(VU)$. Let $T$ be the map from $M_{m\times n}(k_{F})$ and $M_{n\times m}(k_{F})^{\wedge}$ defined by

[TABLE]

Lemma 3.7**.**

The map $T$ is a $G$-equivariant isomorphism.

Proof.

That the map $T$ is $G$ equivariant can be verified from the identity

[TABLE]

It remains to show that $B$ is non-degenerate. Let $V_{ij}$ ($U_{ij}$) be a matrix whose $ij$-th entry is $v_{ij}$ ($u_{ij}$) and all other entries are zero. We observe that $B(U_{ij},V_{ij})$ is equal to $\psi(u_{ij}v_{ij})$. This shows that $B$ is non-degenerate. ∎

The above two lemmas gives an $M_{(n-n_{r},n_{r})}\cap P_{I}(1,m)$ equivariant isomorphism

[TABLE]

Since the group $K_{I}(m)$ is a normal subgroup of $P_{I}(1,m)$, we have an action of this group $P_{I}(1,m)$ on the set of characters of the abelian group

[TABLE]

If $\eta$ is one such character we denote by $Z(\eta)$ the $P_{I}(1,m)$-stabilizer of this character $\eta$. Clifford theory now gives the decomposition

[TABLE]

where $\eta$ runs over a set of representatives for the orbits under the action of $P_{I}(1,m)$ and $U_{\eta}$ is some irreducible representation of the group $Z(\eta)$. We also note that $Z(\operatorname{id})=P_{I}(1,m)$ and the identity character occurs with multiplicity one (which follows from Frobenius reciprocity) and hence

[TABLE]

Observe that

[TABLE]

Let $\theta_{I}(\eta)=A$. Since $\theta_{I}$ is $M_{(n-n_{r},n_{r})}\cap P_{I}(1,m)$ equivariant we get that

[TABLE]

for some matrix $A$ in $M_{(n-n_{r})\times n_{r}}(k_{F})$. The group $M_{(n-n_{r},n_{r})}\cap P_{I}(1,m)$ acts on the group of matrices $M_{(n-n_{r},n_{r})}(k_{F})$ through its mod-$\operatorname{\mathfrak{P}_{{\it F}}}$ reduction. The mod-$\operatorname{\mathfrak{P}_{{\it F}}}$ reduction of the group $P_{I}(1,m)\cap M_{(n-n_{r},n_{r})}$ is equal to the group $P_{I^{\prime}}(k_{F})\times\operatorname{GL}_{n_{r}}(k_{F})$. In the next lemma we will bound the mod $\operatorname{\mathfrak{P}_{{\it F}}}$ reduction of the group $Z(\eta)\cap M_{I}$ for the proof of the Main theorem. Let $\mathcal{O}_{A}$ be an orbit for the action of $P_{I^{\prime}}(k_{F})\times\operatorname{GL}_{n_{r}}(k_{F})$ on the set of matrices $M_{(n-n_{r})\times n_{r}}(k_{F})$. Let $p_{j}$ be the projection onto $j^{th}$ factor of $M_{I}(k_{F})=\prod_{i=1}^{r}\operatorname{GL}_{n_{i}}(k_{F})$.

Lemma 3.8**.**

Let $\mathcal{O}_{A}$ be an orbit consisting of non-zero matrices in

[TABLE]

We can choose a representative $A$ such that the $P_{I^{\prime}}(k_{F})\times\operatorname{GL}_{n_{r}}(k_{F})$-stabilizer of $A$,

[TABLE]

satisfies one of the following conditions.

1. (1)

There exists a positive integer $j$, $j\leq r$ such that the image of

[TABLE]

is contained in a proper parabolic subgroup of $\operatorname{GL}_{n_{j}}(k_{F})$. 2. (2)

There exists an $i$ with $1\leq i\leq r-1$ such that $p_{i}(g)=p_{r}(g)$ for all $g$ in

[TABLE]

Proof.

Let $A=[U_{1},U_{2},\dots,U_{(r-1)}]^{tr}$ be the block form ($U_{k}$ is a matrix of size $n_{r}\times n_{k}$ for $1\leq k\leq r-1$) of a representative $m$ for an orbit $\mathcal{O}_{m}$ consisting of non-zero matrices. If $((M_{ij}),B)\in Z_{P_{I^{\prime}}(k_{F})\times\operatorname{GL}_{n_{r}}(k_{F})}(A)$ ($M_{ii}$ is a matrix of size $n_{i}\times n_{i}$) then we have

[TABLE]

Since $(M_{ij})\in P_{I^{\prime}}(k_{F})$, we have $M_{ij}=0$ for all $i>j$. Let $l\leq r-1$ be the maximal positive integer such that $U_{l}$ is non-zero and such an $l$ exists since $m\neq 0$. From (9) we get that $M_{ll}U_{l}^{tr}t=U_{l}^{tr}B$ where. There exist matrices $P\in\operatorname{GL}_{n_{r}}(k_{F})$ and $Q\in\operatorname{GL}_{n_{l}}(k_{F})$ such that $PU_{l}^{tr}Q$ is a matrix of the form

[TABLE]

where $t$ is the rank of the matrix $U_{l}^{tr}$. Now we may change the representative $A$ to $A^{\prime}=[U_{1}^{\prime},U_{2}^{\prime},\dots,U_{r}^{\prime}]^{tr}$ by the action of the element

[TABLE]

in $P_{I^{\prime}}(k_{F})\times\operatorname{GL}_{n_{r}}(k_{F})$ such that $U_{l}^{\prime tr}$ is the matrix (10). If $t=n_{l}=n_{r}$ then condition $(2)$ is satisfied. Consider the maps $T_{1}:k_{F}^{n_{l}}\rightarrow k_{F}^{n_{r}}$ and $T_{2}:k_{F}^{n_{r}}\rightarrow k_{F}^{n_{l}}$ given by

[TABLE]

and

[TABLE]

respectively. If $t=n_{l}=n_{r}$ does not hold then either of $T_{1}$ or $T_{2}$ has a non-trivial proper kernel (since $U_{l}\neq 0$). If $T_{1}$ has a non-trivial proper kernel then $M_{ll}$ preserves this kernel and hence belongs to a proper parabolic subgroup of $\operatorname{GL}_{n_{r}}(k_{F})$. If $T_{2}$ has a non-trivial proper kernel then $B$ preserves this kernel and hence belongs to a proper parabolic subgroup of $\operatorname{GL}_{n_{l}}(k_{F})$. Hence if $t=n_{l}=n_{r}$ does not hold then condition $(1)$ is satisfied. ∎

The following lemma is due to Paškūnas but we give a mild modification for our applications (see [Pas05, Proposition 6.8]).

Lemma 3.9**.**

Let $G$ be a reductive algebraic group over $k_{F}$ and $U$ be the unipotent radical of a proper parabolic subgroup of $G$. For any subgroup $H$ of $G$ such that $H\cap U=\{\operatorname{id}\}$ and $\xi$ any irreducible subrepresentation of $H$, there exists an irreducible non-cuspidal representation $\sigma$ such that $\xi$ occurs in $\operatorname{res}_{H}\sigma$.

Proof.

Suppose the lemma is false then

[TABLE]

such that $\sigma_{k}$ is cuspidal representation for all $k\leq t$. Since $U\cap H=\{\operatorname{id}\}$, using Mackey decomposition we deduce that,

[TABLE]

Now by our assumption we have $\operatorname{Hom}_{U}(\operatorname{id},\sigma_{k})\neq 0$ for some $k\leq t$ and hence a contradiction. ∎

The following lemma is similar to Proposition 2.3. The lemma is just a modified version of the Proposition 2.3 for our present use.

Lemma 3.10**.**

Let $\Gamma$ be a $K_{n}$-irreducible sub-representation of

[TABLE]

If the irreducible sub-representations of $U_{m}(\tau_{I^{\prime}})$ are atypical for the component $s=[M_{I^{\prime}},\sigma_{I^{\prime}}]$, then the representation $\Gamma$ is atypical for the component $s=[M_{I},\sigma_{I}]$.

Proof.

Let $\rho$ be an irreducible sub-representation of $U_{m}(\tau_{I^{\prime}})$. If $\rho$ is not typical then, there exists another Bernstein component $[M_{J},\lambda_{J}]$ of $\operatorname{GL}_{n-n_{r}}(F)$ such that

[TABLE]

and $\rho$ is contained in

[TABLE]

where $J=(n_{1}^{\prime},n_{2}^{\prime},\dots,n^{\prime}_{r^{\prime}-1})$ and $\lambda_{J}=\boxtimes_{i=1}^{r^{\prime}-1}\lambda_{i}$. The representation

[TABLE]

is contained in

[TABLE]

The representation (11) is contained in the representation

[TABLE]

Since $[M_{I^{\prime}},\sigma_{I^{\prime}}]\neq[M_{J},\lambda_{J}]$ there exist an inertial class $[G_{p},\sigma]$ occurring in the multi-set

[TABLE]

with a multiplicity not equal to its multiplicity in the multi-set

[TABLE]

Hence the classes $[M_{I},\sigma_{I}]$ and $[M_{J}\times G_{n_{r}},\lambda_{J}\boxtimes\sigma_{r}]$ represent two distinct Bernstein components for the group $G_{n}$. ∎

4. Proof of the main theorem

Proof of theorem 3.2.

We prove the theorem by using induction on the positive integer $n$, the rank of $G_{n}$. The theorem is true for $n=1$ since $U_{m}(\tau_{I})$ is zero. We assume that the theorem is true for all positive integers less than $n+1$. We will show the theorem for the positive integer $n+1$. Let $s=[M_{I},\sigma_{I}]$ be a level-zero inertial class. We assume that the partition $I=(n_{1},n_{2},\dots,n_{r})$ of $n+1$ satisfies the hypothesis $n_{i}\leq n_{j}$ for all $1\leq i\leq j\leq r$. If $r=1$ we have $U_{m}(\tau_{I})=0$ and the theorem holds by default. We now assume that $r>1$ and let $I^{\prime}=(n_{1},n_{2},\dots,n_{r-1})$.

We now break the proof into two cases. The first case is $n_{r}=1$ and the second case is $n_{r}>1$.

The case where $n_{r}=1$

In this case $n_{i}=1$ for $1\leq i\leq r$ and $P_{I}=B_{n}$ where $B_{n}$ is the Borel subgroup of $\text{GL}_{n}$. We denote by $T_{n}$ and $U_{n}$ the maximal torus and the unipotent radical respectively. We also use the notation $B_{n}(m)$ for the subgroup $P_{I}(m)$ and $\chi_{I_{n}}$ for $\tau_{I}$ since $I=(1,1,\dots,1)$ is a tuple of length $n$. The proof is by induction on the integer $n$, the rank of $T_{n}$. The statement is immediate for $n=1$ and for $n=2$ we refer to [BM02, A.2.4] for a proof. (We will require the proof for later use and we will recall it at that stage.) So we prove the theorem for $n\geq 3$. Suppose the theorem is true for some positive integer $n\geq 2$. By the definition of $U_{m}(\chi_{I_{n+1}})$ we have

[TABLE]

We have the isomorphism

[TABLE]

We also have the decomposition

[TABLE]

By the inductive hypothesis and Lemma 3.10 irreducible sub-representations of

[TABLE]

are atypical representations. We now consider the irreducible factors of the representation

[TABLE]

We use induction on the integer $m$ to show that the representation

[TABLE]

has a complement say $U_{1,m}(\chi_{I{n+1}})$ in the representation (12) whose irreducible sub-representations are all atypical representations. This shows that irreducible sub-representations of $U_{m}(\chi_{I_{n+1}})$ are atypical. To reduce the notations we denote by $P(m)$ the subgroup $P_{(n,1)}(m)$. We note that $P_{n,1}(m)=P_{(n,1)}(1,m)$ and now applying the decomposition (8) to the parabolic subgroup $P_{(n,1)}$ we get that

[TABLE]

where $\eta$ is any non-trivial character of the group $K_{n+1}(m)U_{n,1}(\operatorname{\mathcal{O}_{{\it F}}})$ which is trivial on $K_{n+1}(m)U_{n,1}(\operatorname{\mathcal{O}_{{\it F}}})\cap P(m+1)$ and $K_{n+1}(m)$ is the principal congruence subgroup of level $m$ (in the present situation we just have one orbit consisting of non-trivial characters.) Let $\theta_{(n,1)}$ be the isomorphism as in equation (7). We choose $\eta$ such that $\theta_{(n,1)}(\eta)=[1,0\dots,0]$.

With the above choice for the character $\eta$ we have

[TABLE]

Since the representation $\operatorname{ind}_{B_{n}(1)}^{K_{n}}(\chi_{I_{n}})\boxtimes\chi_{n+1}$ is trivial on $K_{n}(1)\times K_{1}(1)$ we get that

[TABLE]

is isomorphic to the inflation of the representation

[TABLE]

where $\overline{Z(\eta)\cap M_{(n,1)}}$ is the mod-$\operatorname{\mathfrak{P}_{{\it F}}}$ reduction of the group $Z(\eta)\cap M_{(n,1)}$. The group $\overline{Z(\eta)\cap M_{(n,1)}}$ is contained in the following subgroup

[TABLE]

Let $\operatorname{Mir}_{k}$ be the mirabolic group

[TABLE]

Now we have to understand the restriction

[TABLE]

We begin with understanding the restriction

[TABLE]

We use the theory of derivatives (originally for $G_{n}$ due to Bernstein and Zelevinsky (see [BZ76])) to describe this restriction in a way sufficient for our application. We refer to [Zel81, Chapter 3, §13] for details of these constructions.

In the case of finite fields from Clifford theory one can define four exact functors and we recall the formalism here. The precise definitions are not required for our purpose except for one functor $\Psi^{+}$ which will be recalled latter:

\mathcal{M}(\operatorname{Mir}_{k-1})$$\mathcal{M}(\operatorname{Mir}_{k})$$\mathcal{M}(\operatorname{GL}_{k-1}(k_{F}))$$\Phi^{+}$$\Phi^{-}$$\Psi^{-}$$\Psi^{+}

The key results we use from Zelevinsky are summarised below (see [Zel81, Chapter 3, §13]).

Theorem 4.1** (Zelevinsky).**

The functors $\Psi^{+}$ and $\Phi^{-}$ are left adjoint to $\Psi^{-}$ and $\Phi^{+}$ respectively. The compositions $\Phi^{-}\Phi^{+}$ and $\Psi^{-}\Psi^{+}$ are naturally equivalent to identity. Moreover $\Phi^{+}\Psi^{-}$ and $\Phi^{-}\Psi^{+}$ are zero. The diagram

[TABLE]

obtained from these properties is exact.

Using this theorem and following Bernstein-Zelevinsky one can define a filtration $Fil$ on a finite dimensional representation $\tau$ of $\operatorname{Mir}_{n}$, for all $n>1$. The filtration $Fil$ is given by

[TABLE]

where $\tau_{k}=(\Phi^{+})^{k-1}(\Phi^{-})^{k-1}$ and $\tau_{k}/\tau_{k+1}=(\Phi^{+})^{k-1}\Psi^{+}\Psi^{-}(\Phi^{-})^{k-1}(\tau)$ for all $k\geq 1$. The representation $\tau^{(k)}:=\Psi^{-}(\Phi^{-})^{k-1}(\tau)$ for all $k\geq 0$ of $\operatorname{GL}_{n-k}(k_{F})$ is called the $k^{th}$-derivative of $\tau$ and by convention $\tau^{(0)}:=\tau$.

Let $R_{n}$ be the Grothendieck group of $\operatorname{GL}_{n}(k_{F})$ for all $n\geq 1$ and set $R_{0}=\mathbb{Z}$. Zelevinsky defined a ring structure on the group $R=\oplus_{n\geq 0}R_{n}$ by setting parabolic induction as the product rule. Recall that the ring $R$ has a $\mathbb{Z}$-linear map $D$ defined by setting $D(\pi)=\sum_{k\geq 0}(\pi|_{\operatorname{Mir}_{n}})^{(k)}$ for all $\pi$ in $R_{n}$. It follows from [Zel81, Chapter 3, §13] that $D$ is an endomorphism of the ring $R$. If $\pi$ is a supercuspidal representation of $\operatorname{GL}_{n}(k_{F})$ then by Gelfand-Kazhdan theory it follows that $\pi^{(n)}=1$, $\pi^{(0)}=\pi$ and all other derivatives are zero (see [Zel81, Chapter 3, §13]). Let $1_{R}\in R_{0}$ be the identity element of $R$.

In our present situation we have

[TABLE]

Let $X_{n-k}$ be the term of degree $(n-k)$ in the expansion of the above product. (It is an actual representation of $\operatorname{GL}_{n-k}(k_{F})$, since the coefficients of the above expansion are positive.) Then we have

[TABLE]

Observe that $P_{(n-1,1)}=\operatorname{Mir}_{(n-1)}k_{F}^{\times}$ (here $k_{F}^{\times}$ is the centre of $\operatorname{GL}_{n}(k_{F})$) and $\operatorname{Mir}_{(n-1)}\cap k_{F}^{\times}=\operatorname{id}$. The representation

[TABLE]

extends to a representation of $P_{(n-1,1)}$ by setting $\rho(a)=\chi(a)$ for all $a\in k_{F}^{\times}$ where $\chi$ is the central character of the representation

[TABLE]

Since the central character will play some role, we denote the extended representation by

[TABLE]

By inflation and (later restriction) we extend the $P_{(n-1,1)}(k_{F})\times k_{F}^{\times}$-representation

[TABLE]

to a representation of $Z(\eta)\cap M_{(n,1)}$. We continue to use the notation

[TABLE]

considered as a representation of $Z(\eta)\cap M_{(n,1)}$. We now have

[TABLE]

We will show that any irreducible sub-representation of

[TABLE]

is atypical for the component $[T_{n},\chi_{I_{n}}]$.

We first consider the case when $k\geq 2$. The representation $X_{n-k}$ is a direct sum of the representations:

[TABLE]

The above term also occurs in the expansion

[TABLE]

where $\lambda$ is a cuspidal representation of $\operatorname{GL}_{k}(k_{F})$. To shorten the notation we use the symbol $\times$ for the multiplication in the ring $R$. We get that the representation

[TABLE]

occurs in the representation

[TABLE]

Note that the mod-$\operatorname{\mathfrak{P}_{{\it F}}}$ reduction of the group $Z(\eta)\cap M_{(n,1)}$ is contained in the subgroup of the form (13) and recall that the $(n-1)^{th}$ diagonal entry of any element in (13) is the same as its $n^{th}$ diagonal entry. Let $\eta_{1}$ and $\eta_{2}$ be the central characters of $\chi_{1}\times\chi_{2}\times\ldots\times\chi_{n}$ and $(\chi_{i_{1}}\times\chi_{i_{2}}\times\ldots\times\chi_{i_{n-k}})\times\lambda$ respectively. Let $\chi_{n+1}^{\prime}$ be the character $\chi_{n+1}\eta_{2}^{-1}\eta_{1}$ and now the representation

[TABLE]

occurs in the representation

[TABLE]

Hence an irreducible sub-representation of

[TABLE]

occurs as a sub-representation of

[TABLE]

The above representation occurs as a sub-representation of

[TABLE]

Hence the sub-representation of (14) are not typical representations.

Now we are left with the term

[TABLE]

If we repeat the same strategy as for $k\geq 2$ then $\lambda$ is one-dimensional so the representations $(\ref{equation_case_4})$ and $\chi_{1}\times\chi_{2}\times\ldots\chi_{n+1}$ may not have distinct inertial support. In order to tackle the terms of the above representation we use a different technique. We now recall the definition of the representation $U_{\eta}$, the functor $\Psi^{+}$ and some facts due to Casselman regarding the restriction of an irreducible smooth representation to the maximal compact subgroup $\operatorname{GL}_{2}(\operatorname{\mathcal{O}_{{\it F}}})$.

The representation $U_{\eta}$ is a character on the group $Z(\eta)$. From $(\ref{equation_case_2})$ any element of the group $Z(\eta)$ is of the form

[TABLE]

where $A\in\operatorname{GL}_{n-1}(\operatorname{\mathcal{O}_{{\it F}}})$; $(X^{\prime}),X^{tr},B,C^{tr}\in M_{(n-1)\times 1}(\operatorname{\mathcal{O}_{{\it F}}})$; $e,d\in\mathcal{O}_{F}^{\times}$; $y,y^{\prime}\in\operatorname{\mathcal{O}_{{\it F}}}$ and $d\equiv e\pmod{\operatorname{\mathfrak{P}_{{\it F}}}}$. The character $U_{\eta}$ is given by

[TABLE]

The functor

[TABLE]

is the inflation functor via the quotient map of $\operatorname{Mir}_{k}$ modulo the unipotent radical of $\operatorname{Mir}_{k}$.

Let $(\pi,V_{\pi})$ be an irreducible smooth representation of $\operatorname{GL}_{2}(F)$. We denote by $c(\pi)$ and $\varpi_{\pi}$ the conductor (see [Cas73a, Theorem 1]) and central character of the representation $\pi$ respectively. Let $V^{N}$ be the space of all vectors fixed by the principal congruence subgroup of level $N$ for all $N\geq 1$. For all $i>c(\varpi_{\pi})$ we define the representation $U_{i}(\varpi_{\pi})$ as the complement of the representation $\operatorname{ind}_{B_{2}(i-1)}^{\operatorname{GL}_{2}(\operatorname{\mathcal{O}_{{\it F}}})}(\varpi_{\pi})$ in $\operatorname{ind}_{B_{2}(i)}^{\operatorname{GL}_{2}(\operatorname{\mathcal{O}_{{\it F}}})}(\varpi_{\pi})$. For $i=c(\varpi_{\pi})$ we set

[TABLE]

It follows from [Cas73b, Proposition 1] that the representation $U_{i}(\varpi_{\pi})$ is an irreducible representation of $\operatorname{GL}_{2}(\operatorname{\mathcal{O}_{{\it F}}})$. From the result [Cas73b, Proposition 2] we get that $c(\pi)\geq c(\varpi_{\pi})$. By [Cas73b, Theorem 1] we have

[TABLE]

We now describe the representation $U_{i}(\varpi_{\pi})$ in our language. Let $\eta$ be a non-trivial character of the group $K_{2}(m)U_{(1,1)}(\operatorname{\mathcal{O}_{{\it F}}})$ trivial modulo

[TABLE]

Let $Z(\eta)$ be the $B_{2}(m)$ stabilizer of $\eta$. Any element of the group $Z(\eta)$ is of the form

[TABLE]

where $a,d\in\mathcal{O}_{F}^{\times}$; $b\in\operatorname{\mathcal{O}_{{\it F}}}$, $c\in\mathfrak{P}_{F}^{m}$ and $d\equiv a$ modulo $\operatorname{\mathfrak{P}_{{\it F}}}$. We define a character $U_{\eta}$ by setting

[TABLE]

We then have

[TABLE]

Now let us resume the proof in the general case $n>2$ the representation

[TABLE]

is contained in the representation

[TABLE]

where $\chi$ is given by $\prod_{i=1}^{n}\chi_{i}$ of $\mathcal{O}_{F}^{\times}$. This representation, by the theorem of Casselman (see the decomposition (19)) is contained in the representation

[TABLE]

where $\sigma$ is a supercuspidal representation of level-zero with central character $\chi$ (see the remark below for the existence) and $X_{n-1}^{\prime}$ is the $(n-1)^{\rm th}$ derivative of the representation

[TABLE]

Hence irreducible sub-representations of (17) are atypical. This completes the proof that irreducible sub-representations of

[TABLE]

are atypical. From the decomposition

[TABLE]

we get the theorem for the case where $n_{r}=1$.

Remark 4.2**.**

The existence of the cuspidal representation of $\operatorname{GL}_{2}(k_{F})$ with a given central character can be deduced from the explicit formula for such representations, we refer to [BH06, Theorem section 6.4]. To be precise we begin with a quadratic extension $k$ of $k_{F}$ and $\theta$ a character of $k^{\times}$ such that $\theta^{q}\neq\theta$ where $q=\#k_{F}$. These characters are called regular characters and for any regular character one can define a supercuspidal representation $\pi_{\theta}$ and conversely all supercuspidal representations are of the form $\pi_{\theta}$ for some regular character $\theta$. The central character of $\pi_{\theta}$ is given by $\operatorname{res}_{k_{F}^{\times}}(\theta)$. Now to get a supercuspidal representation with a central character $\chi$ we begin with a character $\chi$ on $k_{F}^{\times}$, there are $\#k_{F}+1$ possible extensions to $k^{\times}$. The set of characters $\theta$ such that $\theta^{q}=\theta$ has cardinality $\#k_{F}-1$. Hence there exists at least one supercuspidal representation with a given central character $\chi$. This shows that irreducible sub-representations of $(\ref{equation_case_6})$ are not typical and this completes the proof of the theorem in this case.

The case where $n_{r}>1$

By transitivity of induction we have

[TABLE]

We note that $P_{I}(1,m)\cap U_{(n-n_{r}+1,n_{r})}$ is equal to $P_{I}(m)\cap U_{(n-n_{r}+1,n_{r})}$ and $P_{I}(1,m)\cap\bar{U}_{(n-n_{r}+1,n_{r})}$ is equal to $P_{I}(m)\cap\bar{U}_{(n-n_{r}+1,n_{r})}$ hence Lemma 2.6 gives the isomorphism

[TABLE]

Splitting the representation $\operatorname{ind}_{P_{I^{\prime}}(m)}^{P_{I^{\prime}}(1)}(\tau_{I^{\prime}})$ as $\tau_{I^{\prime}}\oplus U_{m}^{0}(\tau_{I^{\prime}})$ we get that

[TABLE]

From Frobenius reciprocity the representation $\tau_{I}$ occurs in $\operatorname{ind}_{P_{I}(1,m)}^{P_{I}(1)}(\tau_{I})$ with multiplicity one. Let $U_{(1,m)}^{0}(\tau_{I})$ be the complement of $\tau_{I}$ in $\operatorname{ind}_{P_{I}(1,m)}^{P_{I}(1)}(\tau_{I})$. With this we conclude that

[TABLE]

By definition $U_{m}(\tau_{I})=\operatorname{ind}_{P_{I}(1)}^{K_{n+1}}(U_{m}^{0}(\tau_{I}))$ which shows that

[TABLE]

We observe that

[TABLE]

and

[TABLE]

hence Lemma 2.6 applied to the groups $J_{2}=P_{I}(1,m)$ and $J_{1}=P_{(n-n_{r}+1,n_{r})}(m)$ and $\lambda=U^{0}_{m}(\tau_{I^{\prime}})\boxtimes\tau_{r}$ gives us the isomorphism

[TABLE]

With this we are in a place to use the induction hypothesis through the isomorphism

[TABLE]

By induction hypothesis $\operatorname{GL}_{n-n_{r}+1}(\operatorname{\mathcal{O}_{{\it F}}})$-irreducible sub-representations of $U_{m}(\tau_{I^{\prime}})$ are atypical for the component $[M_{I^{\prime}},\sigma_{I^{\prime}}]$. Now Lemma 3.10 and the equation (21) reduce the proof of the theorem to showing that irreducible sub-representations of $\operatorname{ind}_{P_{I}(1)}^{K_{n+1}}(U_{(1,m)}^{0}(\tau_{I}))$ are atypical representations.

Proposition 4.3**.**

The irreducible sub-representations of

[TABLE]

are atypical for $m\geq 1$.

Proof.

We observe that

[TABLE]

and the decomposition (8) gives us the isomorphism

[TABLE]

which gives the equality

[TABLE]

If we show that the irreducible sub-representations of

[TABLE]

(for $\eta_{k}\neq\operatorname{id}$) are atypical for $[M_{I},\sigma_{I}]$ then induction on the positive integer $m$ completes the proof of the proposition in this case. To begin with we note that

[TABLE]

The representation $\tau_{I}$ is trivial on $M_{I}\cap K_{n+1}(1)$. Hence $\operatorname{res}_{Z(\eta_{k})\cap M_{I}}\tau_{I}$ is isomorphic to the inflation of the representation $\operatorname{res}_{\overline{Z(\eta_{k})\cap M_{I}}}\tau_{I}$ where $\overline{Z(\eta_{k})\cap M_{I}}$ is mod-$\operatorname{\mathfrak{P}_{{\it F}}}$ reduction of ${Z(\eta_{k})\cap M_{I}}$. Let $A=\theta_{I}(\eta_{k})$ where $\theta_{I}$ is the map defined in equation (7). The mod-$\operatorname{\mathfrak{P}_{{\it F}}}$ reduction $\overline{Z(\eta_{k})\cap M_{I}}$ is contained in $Z_{P_{I^{\prime}}(k_{F})\times\operatorname{GL}_{n_{r}}(k_{F})}(A)$. If $\eta_{k}$ is a nontrivial character then $A\neq 0$. Applying Lemma 3.8 to the parabolic corresponding to the partition $I=(n_{1},n_{2},\dots,n_{r})$ and the assumption that $n_{r}>1$ shows that

[TABLE]

has trivial intersection with a unipotent subgroup $U$ of $M_{I}(k_{F})$. If $A$ satisfies condition $(1)$ of the Lemma 3.8 then let $U_{j}$ be the unipotent radical of the opposite parabolic subgroup containing the image of $p_{j}$ (see Lemma 3.8) and if $A$ satisfies condition $(2)$ of 3.8 then let $U_{r}$ be any unipotent radical of a parabolic subgroup of $\operatorname{GL}_{n_{r}}(k_{F})$. Now $U$ can be chosen to be the group

[TABLE]

if $A$ satisfies condition $(1)$ in Lemma 3.8 and $U$ to be

[TABLE]

if $A$ satisfies the condition $(2)$ in Lemma 3.8.

Now applying Lemma 3.9 to the reductive group $G=M_{I}$ and the representation $\tau=\tau_{I}$, we get that every irreducible subrepresentation $\Gamma$ of

[TABLE]

occurs as a subrepresentation of some representation of the form

[TABLE]

with $\tau_{I}^{\prime}$ some non-cuspidal representation of $M_{I}(k_{F})$. The representation 22 occurs as a subrepresentation of

[TABLE]

which occurs as a subrepresentation of

[TABLE]

where $\sigma_{I}^{\prime}$ is a non-cuspidal representation such that $\sigma_{I}^{\prime K_{n+1}(1)\cap M_{I}}\simeq\tau_{I}^{\prime}$. Hence the representation $\Gamma$ is not a typical representation for the component $[M_{I},\sigma_{I}]$.

This completes the proof of the proposition and also the proof of the theorem. ∎

∎