Restriction of $p$-modular representations of $U(2, 1)$ to a Borel subgroup

TL;DR

This paper investigates how irreducible smooth mod p representations of the unramified unitary group U(2,1) restrict to its Borel subgroup, extending known results from GL(2) to a more complex group.

Contribution

It extends the understanding of restriction problems from GL(2) to the unramified unitary group U(2,1), providing new analogous results.

Findings

Results on restriction of representations to Borel subgroup

Analogies with Paškūnas' work on GL(2)

New insights into mod p representations of U(2,1)

Abstract

Let $G$ be the unramified unitary group $U(2, 1)(E/F)$ defined over a non-archimedean local field $F$ of odd residue characteristic $p$, and $B$ be the standard Borel subgroup of $G$. In this note, we study the problem of the restriction of irreducible smooth $\overline{\mathbf{F}}_p$-representations of $G$ to $B$, and we obtain various results which are analogous to that of Pa$\check{\text{s}}$k$\bar{\text{u}}$nas on $GL_2 (F)$ (\cite{Pas07}).

Equations

$$\begin{array}{c}\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)\end{array}.$$

$$\beta n(x,y)=n({\bar{y}}^{-1}x,y^{-1})\cdot h({\bar{y}}^{-1})\cdot n^{\prime }(-{\bar{y}}^{-1}\bar{x},y^{-1}).$$

$$\begin{array}{c}\left(\begin{array}{ccc}{\varpi }_E^{-1}& 0& 0\\ 0& 1& 0\\ 0& 0& {\varpi }_E\end{array}\right)\end{array},$$

$$T{\hat{f}}_{v_0}=\sum _{u\in N_{n_K}/N_{n_K+2}}u{\alpha }^{-1}\cdot {\hat{f}}_{v_0}+{\lambda }_{{\beta }_K,\sigma }\sum _{u\in N_{n_K+1}/N_{n_K+2}}{\beta }_{K}u{\alpha }^{-1}\cdot {\hat{f}}_{v_0}$$

$$\varphi (g_2)=\epsilon (\alpha {)}^{-1}{S}_{+}\varphi (g_2)$$

Full text

1 Introduction

Let $G$ be the unitary group $U(2,1)(E/F)$ defined over a non-archimedean local field $F$ of odd residue characteristic $p$, and $B$ be the standard Borel subgroup of $G$. In this note, we investigate the restriction of irreducible smooth $\overline{\mathbf{F}}_{p}$-representations of $G$ to $B$. For representations arisen from principal series, we have the following at first:

Theorem 1.1**.**

([Vig08, Theorem 5])

Let $\varepsilon$ be a character of $B$. Then,

$1)$.  $\textnormal{Ind}^{G}_{B}\varepsilon\mid_{B}$ is of length two.

$2)$.  $St\mid_{B}$ is irreducible. Here $St$ is the Steinberg representation of $G$.

Remark 1.2**.**

In [Vig08], only split groups are considered but the arguments in loc.cit could be slightly modified to our case. For a crucial ingredient we reproduce the proof in full (Lemma 4.1).

Our first main result is a complement to the above, that is we deal with those representations called supersingular.

Theorem 1.3**.**

(Theorem 3.8) Let $\pi$ be a supersingular representation of $G$. Then

$\pi\mid_{B}$* is irreducible*

Our second main result is about non-supersingular representations:

Theorem 1.4**.**

(Corollary 4.5, 4.6) Let $\pi$ be a smooth representation of $G$. We have

$(1)$.  Let $\varepsilon$ be a character $B$ such that $\varepsilon\neq\eta\circ\textnormal{det}$ for any $\eta$. Then,

$\textnormal{Hom}_{G}(\textnormal{Ind}^{G}_{B}\varepsilon,\pi)\cong\textnormal{Hom}_{B}(\textnormal{Ind}^{G}_{B}\varepsilon,\pi)$**

$(2)$. For the trivial character $1$ of $B$, we have

$\textnormal{Hom}_{G}(\textnormal{Ind}^{G}_{B}1,\pi)\cong\textnormal{Hom}_{B}(St,\pi)$**

Remark 1.5**.**

Our results are in a representation theoretic nature, but we expect they would have some potential arithmetic applications. They are analogous to results of Pa$\check{\text{s}}$k$\bar{\text{u}}$nas on $GL_{2}(F)$ ([Paš07]), and we follow his strategy closely. When $F=\mathbf{Q}_{p}$, Pa$\check{\text{s}}$k$\bar{\text{u}}$nas' results were firstly discovered by Berger ([Ber10]), where he proved them by using the theory of $(\varphi,\Gamma)$-modules and classification of supersingular representations. In the work of Colmez on $p$-adic local Langlands correspondence of $GL_{2}(\mathbf{Q}_{p})$ ([Col10]), the restriction to a Borel subgroup plays a prominent role.

Remark 1.6**.**

Due to certain technical difficulty, at this moment we don't have an analogue of last Theorem for supersingular representations.

2 Notations and Preliminaries

2.1 General notations

Let $E/F$ be a unramified quadratic extension of non-archimedean local fields of odd residue characteristic $p$. Let $\mathfrak{o}_{E}$ be the ring of integers of $E$, $\mathfrak{p}_{E}$ be the maximal ideal of $\mathfrak{o}_{E}$, and $k_{E}=\mathfrak{o}_{E}/\mathfrak{p}_{E}$ be the residue field. Fix a uniformizer $\varpi_{E}$ in $E$. Equip $E^{3}$ with the Hermitian form h:

$\text{h}:~{}E^{3}\times E^{3}\rightarrow E$, $(v_{1},v_{2})\mapsto~{}v_{1}^{\text{T}}\beta\overline{v_{2}},v_{1},v_{2}\in E^{3}$.

Here, $-$ is a generator of $\text{Gal}(E/F)$, and $\beta$ is the matrix

[TABLE]

The unitary group $G$ is defined as:

$G=\{g\in\text{GL}(3,E)\mid\text{h}(gv_{1},gv_{2})=\text{h}(v_{1},v_{2}),\forall~{}v_{1},v_{2}~{}\in E^{3}\}.$

Let $B=HN$ (resp, $B^{\prime}=HN^{\prime}$) be the subgroup of upper (resp, lower) triangular matrices of $G$, with $N$ (resp, $N^{\prime}$) the unipotent radical of $B$ (resp, $B^{\prime}$) and $H$ the diagonal subgroup of $G$. Denote an element of the following form in $N$ and $N^{\prime}$ by $n(x,y)$ and $n^{\prime}(x,y)$ respectively:

$\begin{pmatrix}1&x&y\\ 0&1&-\bar{x}\\ 0&0&1\end{pmatrix}$,   $\begin{pmatrix}1&0&0\\ x&1&0\\ y&-\bar{x}&1\end{pmatrix},$

where $(x,y)\in E^{2}$ satisfies $x\bar{x}+y+\bar{y}=0$. For any $k\in\mathbb{Z}$, denote by $N_{k}$ (resp, $N^{\prime}_{k}$) the subgroup of $N$ (resp, $N^{\prime}$) consisting of $n(x,y)$ (resp, $n^{\prime}(x,y)$) with $y\in\mathfrak{p}^{k}_{E}$. For $x\in E^{\times}$, denote by $h(x)$ an element in $H$ of the following form:

$\begin{pmatrix}x&0&0\\ 0&-\bar{x}x^{-1}&0\\ 0&0&\bar{x}^{-1}\end{pmatrix}.$

We record a useful identity in $G$: for $y\neq 0$,

[TABLE]

Up to conjugacy, the group $G$ has two maximal compact open subgroups $K_{0}$ and $K_{1}$, given by:

$K_{0}=\begin{pmatrix}\mathfrak{o}_{E}&\mathfrak{o}_{E}&\mathfrak{o}_{E}\\ \mathfrak{o}_{E}&\mathfrak{o}_{E}&\mathfrak{o}_{E}\\ \mathfrak{o}_{E}&\mathfrak{o}_{E}&\mathfrak{o}_{E}\end{pmatrix}\cap G,~{}K_{1}=\begin{pmatrix}\mathfrak{o}_{E}&\mathfrak{o}_{E}&\mathfrak{p}^{-1}_{E}\\ \mathfrak{p}_{E}&\mathfrak{o}_{E}&\mathfrak{o}_{E}\\ \mathfrak{p}_{E}&\mathfrak{p}_{E}&\mathfrak{o}_{E}\end{pmatrix}\cap G.$

The maximal normal pro-$p$ subgroups of $K_{0}$ and $K_{1}$ are respectively:

$K^{1}_{0}=1+\varpi_{E}M_{3}(\mathfrak{o}_{E})\cap G,~{}K^{1}_{1}=\begin{pmatrix}1+\mathfrak{p}_{E}&\mathfrak{o}_{E}&\mathfrak{o}_{E}\\ \mathfrak{p}_{E}&1+\mathfrak{p}_{E}&\mathfrak{o}_{E}\\ \mathfrak{p}^{2}_{E}&\mathfrak{p}_{E}&1+\mathfrak{p}_{E}\end{pmatrix}\cap G.$

Let $\alpha$ be the following diagonal matrix in $G$:

[TABLE]

and put $\beta^{\prime}=\beta\alpha^{-1}$. Note that $\beta\in K_{0}$ and $\beta^{\prime}\in K_{1}$. We use $\beta_{K}$ to denote the unique element in $K\cap\{\beta,\beta^{\prime}\}$.

Let $K\in\{K_{0},K_{1}\}$, and $K^{1}$ be the maximal normal pro-$p$ subgroup of $K$. We identify the finite group $\Gamma_{K}=K/K^{1}$ with the $k_{F}$-points of an algebraic group defined over $k_{F}$. Let $\mathbb{B}$ (resp, $\mathbb{B}^{\prime}$) be the upper (resp, lower) triangular subgroup of $\Gamma_{K}$, and $\mathbb{U}$ (resp, $\mathbb{U}^{\prime}$) be its unipotent radical. The Iwahori subgroup $I_{K}$ (resp, $I^{\prime}_{K}$) and pro-$p$ Iwahori subgroup $I_{1,K}$ (resp, $I^{\prime}_{1,K}$) in $K$ are the inverse images of $\mathbb{B}$ (resp, $\mathbb{B}^{\prime}$) and $\mathbb{U}$ (resp, $\mathbb{U}^{\prime}$) in $K$.

Denote by $n_{K}$ and $m_{K}$ the unique integers such that $N\cap I_{1,K}=N_{n_{K}}$ and $N^{\prime}\cap I_{1,K}=N^{\prime}_{m_{K}}$. We have $n_{K}+m_{K}=1$.

All representations in this note are smooth over $\overline{\mathbf{F}}_{p}$.

2.2 Weights

Let $\sigma$ be an irreducible smooth representation of $K$. As $K_{1}$ is pro-$p$ and normal in $K$, $\sigma$ factors through the finite group $\Gamma_{K}$, i.e., $\sigma$ is the inflation of an irreducible representation of $\Gamma_{K}$. Conversely, any irreducible representation of $\Gamma_{K}$ inflates to an irreducible smooth representation of $K$. We may therefore identify irreducible smooth representations of $K$ with irreducible representations of $\Gamma_{K}$, and we shall call them weights of $K$ or $\Gamma_{K}$ from now on.

For a weight $\sigma$ of $K$, it is well-known that $\sigma^{I_{1,K}}$ and $\sigma_{I^{\prime}_{1,K}}$ are both one-dimensional and that the natural composition map $\sigma^{I_{1,K}}\hookrightarrow\sigma\twoheadrightarrow\sigma_{I^{\prime}_{1,K}}$ is an isomorphism of vector spaces ([CE04, Theorem 6.12]). This implies there exists a unique $\lambda_{\beta_{K},\sigma}\in\overline{\mathbf{F}}_{p}$, such that $\beta_{K}\cdot v-\lambda_{\beta_{K},\sigma}v\in\sigma(I^{\prime}_{1,K})$, for $v\in\sigma^{I_{1,K}}$.

2.3 The Hecke operator $T$

Let $K\in\{K_{0},K_{1}\}$, and $\sigma$ be a weight of $K$. Let $\text{ind}_{K}^{G}\sigma$ be the maximal compact induction and $\mathcal{H}(K,\sigma):=\text{End}_{G}(\text{ind}_{K}^{G}\sigma)$ be the associate Hecke algebra. The algebra $\mathcal{H}(K,\sigma)$ is isomorphic to $\overline{\mathbf{F}}_{p}[T]$, for certain $T\in\mathcal{H}(K,\sigma)$ ([Her11, Corollary 1.3], [Xu19, Proposition 3.3]).

For a non-zero vector $v\in\sigma$, we follow [BL94, 2.1] to denote by $\hat{f}_{v}$ the function in $\text{ind}^{G}_{K}\sigma$ supported on $K$ and having value $v$ at $Id$. Recall the following formula of $T$ ([Xu19, Proposition 3.6]):

Proposition 2.1**.**

Assume $v_{0}\in\sigma^{I_{1,K}}$. Then, we have

[TABLE]

2.4 The $I_{1,K}$-invariant map $S_{+}$

For a smooth representation $\pi$ of $G$, we have introduced two partial linear maps $S_{K}$ and $S_{-}$ in [Xu17, subsection 4.3] as follows:

$S_{K}:\pi^{N^{\prime}_{m_{K}}}\rightarrow\pi^{N_{n_{K}}}$,

$v\mapsto\sum_{u\in N_{n_{K}}/N_{n_{K}+1}}~{}u\beta_{K}v$.

$S_{-}:\pi^{N_{n_{K}}}\rightarrow\pi^{N^{\prime}_{m_{K}}}$,

$v\mapsto\sum_{u^{\prime}\in N^{\prime}_{m_{K}}/N^{\prime}_{m_{K}+1}}~{}u^{\prime}\beta_{K}\alpha^{-1}v$

Proposition 2.2**.**

If $v\in\pi^{I_{1,K}}$, then it is the same for $S_{K}v$ and $S_{-}v$.

Proof.

This is [Xu17, Appendix, Proposition 10.2]. ∎

We consider the composition $S_{+}=S_{K}\circ S_{-}$. By definition it lies in $\text{End}_{\overline{\mathbf{F}}_{p}}(\pi^{N_{n_{K}}})$ and preserves the $I_{1,K}$-invariants of a smooth representation (Proposition 2.2).

3 Supersingular representations

3.1 Definition

Definition 3.1**.**

An irreducible smooth $\overline{\mathbf{F}}_{p}$-representation $\pi$ of $G$ is called supersingular if it is a quotient of $\textnormal{ind}^{G}_{K}\sigma/(T_{\sigma})$, for some weight $\sigma$ of the hyperspecial $K$.

Remark 3.2**.**

Here, $T_{\sigma}$ is a specific Hecke operator modified from $T$, see [Xu18b, 4.1].

3.2 A key property

Let $\pi$ be an irreducible smooth representation of $G$, and $\sigma$ be a weight of $K$ contained in $\pi$. By [Xu18a], $\pi$ admits Hecke eigenvalues for the spherical Hecke algebra $\mathcal{H}(K,\sigma)$. Thus, $\pi$ is a quotient of $\text{ind}^{G}_{K}\sigma/(T_{\sigma}-\lambda)$, for some scalar $\lambda$. By [Xu18b], $\pi$ is supersingular if $K$ is hyperspecial and $\lambda=0$.

Lemma 3.3**.**

Let $\pi$ be a supersingular representation of $G$, and assume $\phi$ is a non-zero $G$-map from $\textnormal{ind}^{G}_{K}\sigma$ to $\pi$. Then, for large enough $k\geq 1$, we have

$\phi\circ T_{\sigma}^{k}=0.$**

Proof.

By [Xu18a, Corollary 4.2], there is a non-constant polynomial $P(X)$ such that $\phi\circ P(T_{\sigma})=0$. Assume $P(X)$ is such a polynomial of minimal degree. Take a root $\lambda$ of $P(X)$, and write $P(X)=(X-\lambda)P_{1}(X)$. Put $\phi^{\prime}=\phi\circ P_{1}(T_{\sigma})$. Note that $\phi^{\prime}$ is still a $G$-map from $\textnormal{ind}^{G}_{K}\sigma$ to $\pi$. By our assumption, the map $\phi^{\prime}$ is non-zero and factors through $\textnormal{ind}^{G}_{K}\sigma/(T_{\sigma}-\lambda)$. As $\pi$ is supersingular, we have $\lambda=0$ ([Xu18b, Theorem 1.1]) and hence $P(X)=X^{n}$ for some $n\geq 1$. ∎

Lemma 3.4**.**

Let $\pi$ be a smooth representation of $G$. Assume $v$ is a non-zero vector in $\pi^{I_{1,K}}$, such that $I_{K}$ acts on $v$ as a character. Then, either $S_{+}v=0$, or $S_{+}v$ generates a weight of $K$ of dimension greater than one.

Proof.

Assume $S_{+}v\neq 0$. We put $v^{\prime}:=S_{-}v$. By definition,

$S_{+}v=S_{K}v^{\prime}$,

and we see $v^{\prime}$ must be non-zero. Consider the $K$-representation $\kappa=\langle K\cdot v^{\prime}\rangle$. As $I_{K}$ acts on $v$ by a character $\chi$, $I_{K}$ acts on $v^{\prime}$ by $\chi^{s}$. By Frobenius reciprocity, there is a surjective $K$-map from $\text{Ind}^{K}_{I_{K}}\chi^{s}$ to $\kappa$, sending $\varphi_{\chi^{s}}$ to $v^{\prime}$. Here, $\varphi_{\chi^{s}}$ is the function in $\text{Ind}^{K}_{I_{K}}\chi^{s}$ supported on $I_{K}$ and having value $1$ at $Id$.

Via aforementioned map, we see $\langle K\cdot S_{+}v\rangle$ is the image of $\langle K\cdot S_{K}\varphi_{\chi^{s}}\rangle$. But the latter, by [KX15, Proposition 5.7], is an irreducible smooth representation of $K$ of dimension greater than one. The assertion follows. ∎

Proposition 3.5**.**

Assume $\pi$ is a supersingular representation of $G$, and $v$ is a non-zero vector in $\pi^{I_{1,K}}$. Then, for $k\gg 0$, we have $S^{k}_{+}v=0$.

Proof.

Assume firstly $I_{K}$ acts on $v$ as a character $\chi$.

Assume $S_{+}v\neq 0$. By Lemma 3.4 the $K$-subrepresentation generated by $S_{+}v$ is a weight of dimension greater than one, and denote it by $\sigma$. By Frobenius reciprocity, we have a $G$-map $\phi$ from $\textnormal{ind}^{G}_{K}\sigma$ to $\pi$, sending the function $\hat{f}_{S_{+}v}$ to $S_{+}v$. From Lemma 3.3, there is some $k\geq 1$ such that

$S_{+}^{k}v=0,$

and we are done in this special case.

Note that $I_{K}/I_{1,K}$ is an abelian group of finite order prime to $p$. For any non-zero $v\in\pi^{I_{1,K}}$, the $I_{K}$-representation $\langle I_{K}\cdot v\rangle$ generated by $v$ is a sum of characters, and we may write $v$ as $\sum v_{i}$ so that $I_{K}$ acts on $v_{i}$ by a character $\chi_{i}$ of $I_{K}/I_{1,K}$. We then apply the previous process to each $v_{i}$, and take the largest $k_{i}$. We are done. ∎

3.3 A criteria of Pa$\check{\text{s}}$k$\bar{\text{u}}$nas

Proposition 3.6**.**

Let $\pi$ be an irreducible smooth representation of $G$. If, for any non-zero vector $w\in\pi$, there is a non-zero vector $v\in\pi^{I_{1,K}}\cap\langle B\cdot w\rangle$ such that

$S_{+}v=0$,

then $\pi\mid_{B}$ is irreducible.

Proof.

Let $w$ be a non-zero vector in $\pi$. As $\pi$ is smooth, there exists a $k\geq 0$ such that $w$ is fixed by $N^{\prime}_{2k+m_{K}}$. Hence, the vector $w_{1}=\alpha^{-k}w$ is fixed by $N^{\prime}_{m_{K}}$. Since $I_{1,K}=(I_{1,K}\cap B)\cdot N^{\prime}_{m_{K}}$, we see

$\langle I_{1,K}\cdot w_{1}\rangle=\langle(I_{1,K}\cap B)\cdot w_{1}\rangle.$

As $I_{1,K}$ is pro-$p$, the space $\langle(I_{1,K}\cap B)\cdot w_{1}\rangle$ has non-zero $I_{1,K}$-invariant ([BL95, Lemma 1]). We conclude $\pi^{I_{1,K}}\cap\langle B\cdot w\rangle\neq 0$.

Lemma 3.7**.**

If $S_{+}v=0$, then $\beta_{K}v\in\langle B\cdot v\rangle.$

Proof.

By the assumption $S_{+}v=0$, we get

$v=-\alpha\cdot\sum_{u\in(N_{n_{K}}\setminus N_{n_{K}+2})/N_{n_{K}+2}}u\alpha^{-1}v$

or equivalently

$\beta_{K}v=-\sum_{u\in(N_{n_{K}}\setminus N_{n_{K}+2})/N_{n_{K}+2}}\beta_{K}\alpha u\alpha^{-1}v$.

Applying (1), we see $\beta_{K}\alpha u\alpha^{-1}\in BN^{\prime}_{m_{K}}$, for any $u\in(N_{n_{K}}\setminus N_{n_{K}+2})/N_{n_{K}+2}$, and that gives $\beta_{K}\alpha u\alpha^{-1}v\in\langle B\cdot v\rangle$ for $v\in\pi^{I_{1,K}}$. From the above identity we conclude $\beta_{K}v\in\langle B\cdot v\rangle$. ∎

We proceed to complete the proof of Proposition 3.6. Choose $0\neq v\in\pi^{I_{1,K}}\cap\langle B\cdot w\rangle$ such that $S_{+}v=0$. The above Lemma says $\beta_{K}v\in\langle B\cdot v\rangle$. As $\pi$ is irreducible, we have $\pi=\langle G\cdot v\rangle$. By the Bruhat decomposition $G=BI_{1,K}\cup B\beta_{K}I_{1,K}$, we see

$\pi\subseteq\langle B\cdot v\rangle\subseteq\langle B\cdot w\rangle$.

Hence, we have proved $\pi=\langle B\cdot w\rangle$ for any $w\in\pi$, and the proposition follows. ∎

3.4 Proof of Theorem 1.3

Theorem 3.8**.**

Let $\pi$ be a supersingular representation of $G$. Then $\pi\mid_{B}$ is irreducible.

Proof.

Let $w$ be any non-zero vector in $\pi$. We already know that (by the argument of Proposition 3.6)

$\pi^{I_{1,K}}\cap\langle B\cdot w\rangle\neq 0$.

Take any non-zero vector $v$ in the above space. By Proposition 3.5, $v$ will be annihilated by $S^{k}_{+}$ for $k$ large enough, and let $m$ be the least integer satisfying that. Now the vector $v^{\prime}=S^{m-1}_{+}v$ is non-zero. By Proposition 2.2, $v^{\prime}$ is still $I_{1,K}$-invariant, and lies in $\langle B\cdot w\rangle$ by the form of $S_{+}$. It satisfies

$S_{+}v^{\prime}=0$.

We are done by Proposition 3.6. ∎

4 Non-supersingular representations

For a character $\varepsilon$ of $B$, consider the principal series $\text{Ind}^{G}_{B}\varepsilon$. Recall that it is reducible if and only if $\varepsilon=\eta\circ\text{det}$ for some $\eta$, and in this case it is of length two.

We sketch a proof of Theorem 1.1 in the introduction. Evaluating an $f\in\text{Ind}^{G}_{B}\varepsilon$ at the identity, we get a $B$-map from the principal series to the character $\varepsilon$. Denote the kernel by $\kappa_{\varepsilon}$. Then we have a short exact sequence of $B$-representations:

$0\rightarrow\kappa_{\varepsilon}\rightarrow\text{Ind}^{G}_{B}\varepsilon\rightarrow\varepsilon\rightarrow 0$

By almost the same argument of [Vig08, Theorem 5], the $B$-representation $\kappa_{\varepsilon}$ is irreducible (as is shown below). Indeed, one may prove the restriction to $\alpha^{\mathbb{Z}}N$ of $\kappa_{\varepsilon}$ is irreducible. But by the same map $\Phi$ below, one may verify that $St\mid_{B}\cong\kappa_{1}$. This gives irreducibility of $St\mid_{B}$.

Lemma 4.1**.**

The $B$-representation $\kappa_{\varepsilon}$ is irreducible.

Proof.

1).  We firstly identify the underlying space of $\kappa_{\varepsilon}$ with $C^{\infty}_{c}(N)$.

$\Phi:\kappa_{\varepsilon}\rightarrow C^{\infty}_{c}(N),f\mapsto\Phi(f),\Phi(f)(u)=f(\beta u),\forall u\in N.$

$\Psi:C^{\infty}_{c}(N)\rightarrow\kappa_{\varepsilon},f\mapsto\Psi(f),\Psi(f)(b\beta u)=\varepsilon(b)f(u),\forall b\in B,u\in N.$

One verifies easily that $\Psi$ and $\Phi$ are inverse to each other. This gives $C^{\infty}_{c}(N)$ a structure of $B$-representation.

2). We modify the argument of [Ly15, Proposition 5.2] to our case. Let $V$ be a non-zero $B$-stable subspace of $C^{\infty}_{c}(N)$, and $f$ be a non-zero function in $V$. As $f$ is compactly supported and $N$ has the decreasing open compact cover $(N_{k})_{k\in\mathbb{Z}}$, we may assume the support of $f$ is contained in $N_{k}$ for some integer $k$. Write $V_{k}$ the subspace of $V$ consisting of functions supported in $N_{k}$, we have $V_{k}\neq 0$. By [BL95, Lemma 1], we know $V^{N_{k}}_{k}\neq 0$. This shows that $V$ contains the characteristic function $1_{N_{k}}$ of $N_{k}$.

Now for any $n\in\mathbb{Z},u\in N$, we have $u\alpha^{n}\cdot 1_{N_{k}}=\varepsilon(\alpha^{-n})1_{N_{k-2n}u^{-1}}$, and as $V$ is $B$-stable we conclude $V$ contains $1_{N_{k-2n}u^{-1}}$. Note that $1_{N_{k-1}}=\sum_{u\in N_{k-1}/N_{k}}u^{-1}\cdot 1_{N_{k}}$, so we have $1_{N_{k-1}}\in V$. Playing the same game, we conclude $V$ contains $1_{N_{k-2n-1}u^{-1}}$. In all we have shown $V$ contains all the functions $1_{N_{k}u}$ for any $k\in\mathbb{Z}$ and $u\in N$. We are done, as all the functions $\{1_{N_{k}u}\mid k\in\mathbb{Z},u\in N\}$ span the underlying space of $C^{\infty}_{c}(N)$. ∎

We now come to the main input of this part.

Theorem 4.2**.**

The restriction map induces an isomorphism between the following spaces:

$\textnormal{Hom}_{G}(\textnormal{Ind}^{G}_{B}\varepsilon,\pi)\cong\textnormal{Hom}_{B}(\kappa_{\varepsilon},\pi)$**

Proof.

We show firstly that the restriction map is injective. Given two $\phi_{1}$ and $\phi_{2}$ in the first space, suppose that $\phi=\phi_{1}-\phi_{2}$ vanishes at the space $\kappa_{\varepsilon}$. By the proceeding remark, $\phi$ induces a $B$-map from the character $\varepsilon$ to $\pi$.

Lemma 4.3**.**

If $\varepsilon\neq\eta\circ\textnormal{det}$, then $\textnormal{Hom}_{B}(\varepsilon,\pi)=0$.

Proof.

Assume $\phi\neq 0$. As $\pi$ is smooth, the vector $\phi(1)$ is fixed by some $N^{\prime}_{m_{K}+2k}$ for large enough $k$. Using the following repeatedly

$\alpha N^{\prime}_{m_{K}+2k-2}\alpha^{-1}=N^{\prime}_{m_{K}+2k}$

and $\phi(\alpha\cdot 1)=\varepsilon(\alpha)\phi(1)=\alpha\cdot\phi(1)$, we see $\phi(1)$ is fixed by $N^{\prime}$. As the group $G$ is generated by $B$ and $N^{\prime}$, we see $\varepsilon$ extends uniquely to a character of $G$ (put $\varepsilon(N^{\prime})=1$). In such a situation, $\textnormal{Hom}_{B}(\varepsilon,\pi)\cong\textnormal{Hom}_{G}(\varepsilon,\pi)$. ∎

Remark 4.4**.**

Under the same assumption on $\varepsilon$, the Lemma implies that any non-zero map in $\textnormal{Hom}_{P}(\textnormal{Ind}^{G}_{B}\varepsilon,\pi)$ is an injection, from which one may deduce that $\textnormal{End}_{B}(\textnormal{Ind}^{G}_{B}\varepsilon)\cong\textnormal{End}_{G}(\textnormal{Ind}^{G}_{B}\varepsilon)$ (Note that the latter space is one-dimensional, as the representation is irreducible admissible and we may apply Schur's Lemma ([BL95])).

We are done if $\varepsilon\neq\eta\circ\text{det}$ for any $\eta$. Otherwise, $\varepsilon=\eta\circ\text{det}$ for some $\eta$. After a twist we may assume $\eta=1$. If $\phi\neq 0$, it induces a non-zero map in $\textnormal{Hom}_{B}(1,\pi)\cong\textnormal{Hom}_{G}(1,\pi)$. This in turn implies the map $\phi\in\textnormal{Hom}_{G}(\textnormal{Ind}^{G}_{B}1,\pi)$ realizes the trivial character of $G$ as a quotient of $\textnormal{Ind}^{G}_{B}1$ ($\phi$ is not injective by our assumption), which is certainly not true.

We proceed to prove the restriction map is surjective.

Recall that the space $(\textnormal{Ind}^{G}_{B}\varepsilon)^{I_{1,K}}$ is two dimensional with a basis of functions $g_{1}$ and $g_{2}$ characterized by: $g_{1}(Id)=1,g_{1}(\beta_{K})=0,g_{2}(Id)=0,g_{2}(\beta_{K})=1$. By Proposition 2.2, we may check that (by evaluating the function at $Id$ and $\beta_{K}$)

$S_{+}g_{2}=\varepsilon(\alpha)g_{2}$

Then, by Lemma 3.4 the representation $\langle K\cdot g_{2}\rangle$ is a weight, denoted by $\sigma$, of dimension greater than one (note that $I_{K}$ acts on $g_{2}$ by the character $\varepsilon^{s}_{0}$).

Let $\phi$ be a non-zero $B$-map from $\kappa_{\varepsilon}$ to $\pi$. The function $g_{2}$ by definition is supported on $B\beta_{K}I_{K}$ so it lies in $\kappa_{\varepsilon}$. As $\kappa_{\varepsilon}$ is irreducible, we have $\phi(g_{2})$ is non-zero. Since $\phi$ respects the action of $B$, the vector $\phi(g_{2})$ is fixed by $B\cap I_{1,K}$. Now we compute $\phi(S_{+}g_{2})$:

$\phi(S_{+}g_{2})=\varepsilon(\alpha)\phi(g_{2})=S_{+}\phi(g_{2})$

that is

[TABLE]

As $\pi$ is smooth, the vector $\phi(g_{2})$ is fixed by some $N^{\prime}_{m_{K}+2k}$ for $k$ large enough. Now by applying [Xu17, Lemma 10.3] (and its argument) to the above equality, we see $\phi(g_{2})$ is fixed by $N^{\prime}_{m_{K}+2k-2}$. Repeating such a process enough times, we prove that the vector $\phi(g_{2})$ is fixed by $N^{\prime}_{m_{K}}$. By Iwahori decomposition $I_{1,K}=N^{\prime}_{m_{K}}\cdot(B\cap I_{1,K})$ we conclude that $\phi(g_{2})$ is fixed by $I_{1,K}$.

By Lemma 3.4 again the representation $\langle K\cdot\phi(g_{2})\rangle$ is a weight $\sigma^{\prime}$ of dimension greater than one. We claim that $\sigma\cong\sigma^{\prime}$. The Iwahori group $I_{K}$ acts on the vector $\phi(g_{2})$ by $\varepsilon^{s}_{0}$. If $\varepsilon_{0}\neq\eta\circ\text{det}$ for $K=K_{0}$ (resp, $\varepsilon_{0}\neq\varepsilon^{s}_{0}$ for $K=K_{1}$), the claim follows as in this case a weight is determined by the character of $I_{K}$ on its $I_{1,K}$-invariants. In the other case, we are also done: neither $\sigma$ or $\sigma^{\prime}$ is a one-dimensional character, and as quotients of the principal series $\text{Ind}^{K}_{I_{K}}\varepsilon^{s}_{0}$ are both therefore isomorphic to $st\otimes\varepsilon_{0}$.

Now the main results in [Xu18b, Proposition 4.15, 4.16] give an isomorphism

$\text{Ind}^{G}_{B}\varepsilon\cong\text{ind}^{G}_{K}\sigma/(T_{\sigma}-\lambda)$,

where $\lambda$ is just $\varepsilon(\alpha)$. Now the representation $\langle G\cdot\phi(g_{2})\rangle$ contains the weight $\sigma$ and is thus a quotient of the above representation (by (3) and that $\sigma$ is not a character). We have shown the map $\phi$ extends to a $G$-map as required. ∎

Corollary 4.5**.**

Suppose $\varepsilon\neq\eta\circ\textnormal{det}$ for any $\eta$. Then

$\textnormal{Hom}_{G}(\textnormal{Ind}^{G}_{B}\varepsilon,\pi)\cong\textnormal{Hom}_{B}(\textnormal{Ind}^{G}_{B}\varepsilon,\pi)$**

Proof.

Let $\phi\in\textnormal{Hom}_{B}(\textnormal{Ind}^{G}_{B}\varepsilon,\pi)$ be non-zero. By Remark 4.4, we know $\phi$ is injective. The assertion follows from the following two points:

1. The image of $\phi$ is contained in $\langle G\cdot\phi(\kappa_{\varepsilon})\rangle$ (using Remark 4.4 again).

2. $\text{Ind}^{G}_{B}\varepsilon$ is isomorphic to $\langle G\cdot\phi(\kappa_{\varepsilon})\rangle$ (By Theorem 4.2 and irreducibility of $\textnormal{Ind}^{G}_{B}\varepsilon$). ∎

Corollary 4.6**.**

We have

$\textnormal{Hom}_{B}(St,\pi)\cong\textnormal{Hom}_{G}(\textnormal{Ind}^{G}_{B}1,\pi)$**

Proof.

As $St\mid_{B}\cong\kappa_{1}$ (remarks before Lemma 4.1). The assertion is then a special case of Theorem 4.2. ∎

Acknowledgements

The author was supported by a Postdoc grant from European Research Council project 669655, and he would like to thank Einstein Institute of Mathematics for the hospitality. Our debt owned to the works of Pa$\check{\text{s}}$k$\bar{\text{u}}$nas ([Paš07]) and Hu ([Hu12]) should be clear to the readers.