Jacquet modules and local Langlands correspondence

Abstract

In this paper, we explicitly compute the semisimplifications of all Jacquet modules of irreducible representations with generic L-parameters of p-adic split odd special orthogonal groups or symplectic groups. Our computation represents them in terms of linear combinations of standard modules with rational coefficients. The main ingredient of this computation is to apply Mœglin’s explicit construction of local A-packets to tempered L-packets. Finally, we study the derivatives introduced by Mínguez.

1 Introduction

When G is a p-adic reductive group and $P=MN$ is a parabolic subgroup, there is the normalized parabolic induction functor

$$\begin{aligned} \mathrm {Ind}_P^G :\mathrm {Rep}(M) \rightarrow \mathrm {Rep}(G), \end{aligned}$$

where $\mathrm {Rep}(G)$ is the category of smooth admissible representations of G. The (normalized) Jacquet functor

$$\begin{aligned} \mathrm {Jac}_P :\mathrm {Rep}(G) \rightarrow \mathrm {Rep}(M) \end{aligned}$$

is the left adjoint functor of $\mathrm {Ind}_P^G$. For $\pi \in \mathrm {Rep}(G)$, the object $\mathrm {Jac}_P(\pi ) \in \mathrm {Rep}(M)$ is called the Jacquet module of $\pi$ with respect to P. In the representation theory of p-adic reductive groups, the parabolic induction functors and the Jacquet functors are ones of the most basic and important terminologies. One of the reasons why they are so important is that they are both exact functors.

The Jacquet modules have many applications. For example:

• Looking at the Jacquet modules of irreducible representation $\pi$ of G, one can take a parabolic subgroup $P=MN$ and an irreducible supercuspidal representation $\rho _M$ of M such that $\pi \hookrightarrow \mathrm {Ind}_P^G(\rho _M)$. Such a $\rho _M$ is called the cuspidal support of $\pi$.

• Casselman’s criterion says that the growth of matrix coefficients of an irreducible representation $\pi$ is determined by exponents of the Jacquet modules of $\pi$.

• Mœglin explicitly constructed the localA-packets, which are the “local factors of Arthur’s global classification”, by taking Jacquet functors intelligently.

In this paper, we shall give an explicit description of the semisimplifications of Jacquet modules of tempered representations of split odd special orthogonal groups $\mathrm {SO}_{2n+1}(F)$ or symplectic groups $\mathrm {Sp}_{2n}(F)$, where F is a non-archimedean local field of characteristic zero. To do this, it is necessary to have some sort of classification of irreducible representations of these groups. We use the local Langlands correspondence established by Arthur [1] for such a classification.

The local Langlands correspondence attaches each irreducible representation $\pi$ of $G = \mathrm {SO}_{2n+1}(F)$ (resp. $G = \mathrm {Sp}_{2n}(F)$) to its L-parameter $(\phi , \eta )$, where

$$\begin{aligned} \phi :W_F \times \mathrm {SL}_2(\mathbb {C}) \rightarrow {\widehat{G}} \end{aligned}$$

is a homomorphism from the Weil–Deligne group $W_F \times \mathrm {SL}_2(\mathbb {C})$ of F to the Langlands dual group ${\widehat{G}} = \mathrm {Sp}_{2n}(\mathbb {C})$ (resp. ${\widehat{G}} = \mathrm {SO}_{2n+1}(\mathbb {C})$) of G, and

$$\begin{aligned} \eta \in \mathrm {Irr}(A_\phi ) \end{aligned}$$

is an irreducible character of the component group $A_{\phi } = \pi _0(\mathrm {Cent}_{{\widehat{G}}}(\mathrm {Im}(\phi )))$ associated to $\phi$ which is trivial on the image of the center $Z({\widehat{G}})$ of ${\widehat{G}}$.

The Jacquet modules will be computed by two main theorems (Theorems 4.2 and 4.3) and Tadić’s formula (Theorem 2.8) together with Lemma 2.7. Fix an irreducible unitary supercuspidal representation $\rho$ of $\mathrm {GL}_d(F)$. By abuse of notation, we denote by the same notation $\rho$ the irreducible representation of $W_F$ corresponding to $\rho$ by the local Langlands correspondence. Let $P_k = M_kN_k$ be the standard parabolic subgroup of G with Levi subgroup $M_k \cong \mathrm {GL}_k(F) \times G_0$ for some classical group $G_0$ of the same type as G. When $k = d$, for an irreducible representation $\pi$ of G, if the semisimplification of $\mathrm {Jac}_{P_d}(\pi )$ is of the form

$$\begin{aligned} \mathrm {s.s.}\mathrm {Jac}_{P_d}(\pi ) = \bigoplus _{i \in I} \tau _i \boxtimes \pi _i \end{aligned}$$

with $\tau _i$ (resp. $\pi _i$) being an irreducible representation of $\mathrm {GL}_d(F)$ (resp. $G_0$), we define a partial Jacquet module$\mathrm {Jac}_{\rho |\cdot |^x}(\pi )$ for $x \in \mathbb {R}$ by

$$\begin{aligned} \mathrm {Jac}_{\rho |\cdot |^x}(\pi ) = \bigoplus _{\begin{array}{c} i \in I \\ \tau _i \cong \rho |\cdot |^x \end{array}} \pi _i. \end{aligned}$$

Here, $|\cdot | :W_F \rightarrow \mathbb {R}^\times$ is the norm map normalized so that $|\mathrm {Frob}| = q^{-1}$, where $\mathrm {Frob}\in W_F$ is a fixed (geometric) Frobenius element, and q is the cardinality of the residual field of F. The first main theorem is the description of the partial Jacquet module $\mathrm {Jac}_{\rho |\cdot |^x}(\pi )$ for tempered $\pi$ (Theorem 4.2).

For discrete series $\pi$, Theorem 4.2 has been proven by Xu [16, Lemma 7.3] to describe the cuspidal support of $\pi$ in terms of its L-parameter. As related works, Aubert–Moussaoui–Solleveld [2,3,4] defined the “cuspidality” of L-parameters $(\phi , \eta )$ by a geometric way, and compared this notion with the cuspidal supports or the Bernstein components of corresponding $\pi$. Theorem 4.2 gives us more information for $\pi$ than its cuspidal support. The main ingredient for the proof of Theorem 4.2 is Mœglin’s explicit construction of tempered L-packets (Theorem 3.5).

The second main theorem (Theorem 4.3) is a reduction of the computation of the Jacquet module $\mathrm {s.s.}\mathrm {Jac}_{P_k}(\pi )$ with respect to any maximal parabolic subgroup $P_k$ to the ones of partial Jacquet modules $\mathrm {Jac}_{\rho |\cdot |^x}(\pi )$. Using Theorems 4.2 and 4.3 (together with Lemma 2.7), we can explicitly compute the semisimplifications of all Jacquet modules of irreducible tempered representations $\pi$. In fact, using a generalization of the standard module conjecture by Mœglin–Waldspurger (Theorem 3.4) and Tadić’s formula (Theorem 2.8), we can apply this explicit computation to any irreducible representation $\pi$ with generic L-parameter $(\phi , \eta )$.

This paper is organized as follows. In Sect. 2, we review some basic results on parabolically induced representations and Jacquet modules for classical groups. In particular, Tadić’s formula, which computes the Jacquet modules of parabolically induced representations, is stated in Sect. 2.2. In Sect. 3, we explain the local Langlands correspondence and Mœglin’s explicit construction of tempered L-packets. In Sect. 4, we state the main theorems (Theorems 4.2 and 4.3) and give some examples. Finally, we prove the main theorems in Sect. 5. Some complements are in Sect. 6. In Sect. 6.1, we give an irreducibility condition for standard modules. In Sect. 6.2, we study derivatives of irreducible tempered representations introduced by Mínguez. It was a key notion for the proof of the Howe duality conjecture by Gan–Takeda [7].

1.1 Notation

Let F be a non-archimedean local field of characteristic zero. We denote by $W_F$ the Weil group of F. The norm map $|\cdot | :W_F \rightarrow \mathbb {R}^\times$ is normalized so that $|\mathrm {Frob}| = q^{-1}$, where $\mathrm {Frob}\in W_F$ is a fixed (geometric) Frobenius element, and $q = q_F$ is the cardinality of the residual field of F.

Each irreducible supercuspidal unitary representation $\rho$ of $\mathrm {GL}_d(F)$ is identified with the irreducible bounded representation of $W_F$ of dimension d via the local Langlands correspondence for $\mathrm {GL}_d(F)$. Through this paper, we fix such a $\rho$. For each positive integer a, the unique irreducible algebraic representation of $\mathrm {SL}_2(\mathbb {C})$ of dimension a is denoted by $S_a$.

For a p-adic group G, we denote by $\mathrm {Rep}(G)$ (resp. $\mathrm {Irr}(G)$) the set of equivalence classes of smooth admissible (resp. irreducible) representations of G. For $\Pi \in \mathrm {Rep}(G)$, we write $\mathrm {s.s.}(\Pi )$ for the semisimplification of $\Pi$.

2 Induced representations and Jacquet modules

In this section, we recall some results on parabolically induced representations and Jacquet modules.

2.1 Representations of $\mathrm {GL}_k(F)$

Let $P=MN$ be a standard parabolic subgroup of $\mathrm {GL}_k(F)$, i.e., P contains the Borel subgroup consisting of upper half triangular matrices. Then the Levi subgroup M is isomorphic to $\mathrm {GL}_{k_1}(F) \times \dots \times \mathrm {GL}_{k_r}(F)$ with $k_1 + \dots + k_r = k$. For smooth representations $\tau _1, \dots , \tau _r$ of $\mathrm {GL}_{k_1}(F), \dots , \mathrm {GL}_{k_r}(F)$, respectively, we denote the normalized parabolically induced representation by

$$\begin{aligned} \tau _1 \times \dots \times \tau _r :=\mathrm {Ind}_{P}^{\mathrm {GL}_k(F)}(\tau _1 \boxtimes \dots \boxtimes \tau _r). \end{aligned}$$

A segment is a symbol [x, y], where $x,y \in \mathbb {R}$ with $x-y \in \mathbb {Z}$ and $x \ge y$. We identify [x, y] with the set $\{x, x-1, \dots , y\}$ so that $\#[x,y] = x-y+1$. Then the normalized parabolically induced representation

$$\begin{aligned} \rho |\cdot |^{x} \times \dots \times \rho |\cdot |^y \end{aligned}$$

of $\mathrm {GL}_{d(x-y+1)}(F)$ has a unique irreducible subrepresentation, which is denoted by

$$\begin{aligned} \left\langle \rho ; x, \dots , y \right\rangle . \end{aligned}$$

If $y = -x \le 0$, this is called a Steinberg representation and is denoted by

$$\begin{aligned} \mathrm {St}(\rho , 2x+1) = \left\langle \rho ; x, \dots , -x \right\rangle , \end{aligned}$$

which is a discrete series representation of $\mathrm {GL}_{d(2x+1)}(F)$. In general, $\left\langle \rho ; x, \dots , y \right\rangle$ is the twist $|\cdot |^{\frac{x+y}{2}} \mathrm {St}(\rho , x-y+1)$. We say that two segments [x, y] and $[x', y']$ are linked if $[x,y] \not \subset [x',y']$, $[x',y'] \not \subset [x,y]$ as sets, and $[x,y] \cup [x,y']$ is also a segment. The linked-ness gives an irreducibility criterion for parabolically induced representations.

Theorem 2.1

(Zelevinsky [18, Theorem 9.7]) Let [x, y] and $[x',y']$ be segments, and let $\rho$ and $\rho '$ be irreducible unitary supercuspidal representations of $\mathrm {GL}_d(F)$ and $\mathrm {GL}_{d'}(F)$, respectively. Then the parabolically induced representation

$$\begin{aligned} \left\langle \rho ; x, \dots , y \right\rangle \times \left\langle \rho '; x', \dots , y' \right\rangle \end{aligned}$$

is irreducible unless [x, y] are $[x',y']$ are linked, and $\rho \cong \rho '$.

Let $\mathrm {Irr}_\rho (\mathrm {GL}_{dm}(F))$ be the subset of $\mathrm {Irr}(\mathrm {GL}_{dm}(F))$ consisting of $\tau$ with cuspidal support of the form $\rho |\cdot |^{x_1} \times \dots \times \rho |\cdot |^{x_m}$, i.e.,

$$\begin{aligned} \tau \hookrightarrow \rho |\cdot |^{x_1} \times \dots \times \rho |\cdot |^{x_m} \end{aligned}$$

for some $x_1, \dots , x_m \in \mathbb {R}$. We understand that $\mathbf {1}_{\mathrm {GL}_0(F)} \in \mathrm {Irr}_\rho (\mathrm {GL}_0(F))$. It is easy to see that

• for pairwise distinct irreducible unitary supercuspidal representations $\rho _1, \dots , \rho _r$, if $\tau _i \in \mathrm {Irr}_{\rho _i}(\mathrm {GL}_{d_im_i}(F))$ for $i = 1, \dots , r$, then the parabolically induced representation $\tau _1 \times \dots \times \tau _r$ is irreducible;

• any irreducible representation of $\mathrm {GL}_k(F)$ is of the above form for some $\tau _i \in \mathrm {Irr}_{\rho _i}(\mathrm {GL}_{d_im_i}(F))$.

Lemma 2.2

Let $\Omega _m$ be the subset of $\mathbb {R}^m$ consisting of elements

$$\begin{aligned} \underline{x} = (\underbrace{x_1, x_1-1, \dots , y_1}_{x_1-y_1+1}, \underbrace{x_2, x_2-1, \dots , y_2}_{x_2-y_2+1}, \dots , \underbrace{x_t, x_t-1, \dots , y_t}_{x_t-y_t+1} ) \end{aligned}$$

such that

• $x_i \ge y_i$ and $x_i-y_i \in \mathbb {Z}$ for $1 \le i \le t$;

• $x_{1} \le x_{2} \le \dots \le x_t$;

• $y_{i-1} \le y_{i}$ if $x_{i-1} = x_i$.

Let $\Delta _i = \left\langle \rho ; x_i, x_i-1, \dots , y_i \right\rangle$ be the essentially discrete series representation of $\mathrm {GL}_{d(x_i-y_i+1)}(F)$ corresponding to the segment $[x_i,y_i]$. Then the parabolically induced representation $\Delta _{\underline{x}} :=\Delta _1 \times \dots \times \Delta _t$ has a unique irreducible subrepresentation $\tau _{\underline{x}}$. The map $\underline{x} \mapsto \tau _{\underline{x}}$ gives a bijection

$$\begin{aligned} \Omega _m \rightarrow \mathrm {Irr}_\rho (\mathrm {GL}_{dm}(F)). \end{aligned}$$

Proof

This follows from the Langlands classification and Theorem 2.1, See also [18, Proposition 9.6]. $\square$

For a partition $(k_1,\dots , k_r)$ of k, we denote by $\mathrm {Jac}_{(k_1, \dots , k_r)}$ the normalized Jacquet functor on $\mathrm {Rep}(\mathrm {GL}_k(F))$ with respect to the standard parabolic subgroup $P = MN$ with $M \cong \mathrm {GL}_{k_1}(F) \times \dots \times \mathrm {GL}_{k_r}(F)$. The Jacquet module of $\left\langle \rho ; x, \dots , y \right\rangle$ with respect to a maximal parabolic subgroup is computed by Zelevinsky.

Proposition 2.3

([18, Proposition 9.5]) Suppose that $x \not = y$ and set $k = d(x-y+1)$. Then $\mathrm {Jac}_{(k_1, k_2)}(\left\langle \rho ; x, \dots , y \right\rangle ) = 0$ unless $k_1 \equiv 0 \bmod d$. If $k_1 = dm$ with $1 \le m \le x-y$, we have

$$\begin{aligned} \mathrm {Jac}_{(k_1, k_2)}(\left\langle \rho ; x, \dots , y \right\rangle ) = \left\langle \rho ; x, \dots , x-(m-1) \right\rangle \boxtimes \left\langle \rho ; x - m, \dots , y \right\rangle . \end{aligned}$$

If

$$\begin{aligned} \mathrm {s.s.}\mathrm {Jac}_{(d, k-d)}(\tau ) = \bigoplus _{i \in I} \tau _i \boxtimes \tau '_i, \end{aligned}$$

for $x \in \mathbb {R}$, we define a partial Jacquet module$\mathrm {Jac}_{\rho |\cdot |^x}(\tau )$ by

$$\begin{aligned} \mathrm {Jac}_{\rho |\cdot |^x}(\tau ) = \bigoplus _{\begin{array}{c} i \in I\\ \tau _i \cong \rho |\cdot |^x \end{array}}\tau '_i. \end{aligned}$$

For $\underline{x} = (x_1, \dots , x_r) \in \mathbb {R}^r$, we also set

$$\begin{aligned} \mathrm {Jac}_{\rho |\cdot |^{\underline{x}}} = \mathrm {Jac}_{\rho |\cdot |^{x_r}} \circ \dots \circ \mathrm {Jac}_{\rho |\cdot |^{x_1}}. \end{aligned}$$

This is a functor

$$\begin{aligned} \mathrm {Jac}_{\rho |\cdot |^{\underline{x}}} :\mathrm {Rep}(\mathrm {GL}_k(F)) \rightarrow \mathrm {Rep}(\mathrm {GL}_{k-dr}(F)). \end{aligned}$$

In particular, when $\tau \in \mathrm {Rep}(\mathrm {GL}_{dm}(F))$ is of finite length, for $\underline{x} = (x_1, \dots , x_m) \in \mathbb {R}^m$, the partial Jacquet module $\mathrm {Jac}_{\rho |\cdot |^{\underline{x}}}(\tau )$ is a representation of the trivial group $\mathrm {GL}_0(F)$ of finite length so that it is a finite dimensional $\mathbb {C}$-vector space.

Lemma 2.4

Let $\underline{x} = (x_1, \dots , y_1, \dots , x_t, \dots , y_t) \in \Omega _m$ such that $x_{i-1} \le x_{i}$ for $1 < i \le t$, and $y_{i-1} \le y_{i}$ if $x_{i-1} = x_i$ as in Lemma 2.2. For $(x,y) \in \{(x_i,y_i)\}_i$, if we set $m_{(x,y)} = \#\{i \ |\ (x_i,y_i) = (x,y)\}$, then for $\underline{y} \in \Omega _m$, we have

$$\begin{aligned} \dim _{\mathbb {C}} \mathrm {Jac}_{\rho |\cdot |^{\underline{y}}}(\Delta _{\underline{x}}) = \left\{ \begin{aligned}&\prod _{(x,y) \in \{(x_i,y_i)\}_i} m_{(x,y)}!&\quad&\text {if }\underline{y} = \underline{x}, \\&0&\quad&\text {if }\underline{y} < \underline{x}. \end{aligned} \right. \end{aligned}$$

Here, we regard $\mathbb {R}^m$ as a totally ordered set with respect to the lexicographical order.

Proof

Fix $z \in \mathbb {R}$. We note that $\mathrm {Jac}_{\rho |\cdot |^z}(\Delta _{\underline{x}}) \not = 0$ if and only if $z \in \{x_1, \dots , x_t\}$. Let $\underline{x'_1}, \dots , \underline{x'_{l}} \in \Omega _{m-1}$ be the elements obtained by removing z from a component of $\underline{x}$, and rearranging it (so that $l = \#\{i\ |\ x_i = z\}$). Then $\mathrm {Jac}_{\rho |\cdot |^z}(\Delta _{\underline{x}}) = \sum _{i=1}^l \Delta _{\underline{x'_i}}$. Using this, we obtain the lemma by induction on m. $\square$

When $\underline{y} > \underline{x}$, one can also compute $\dim _{\mathbb {C}} \mathrm {Jac}_{\rho |\cdot |^{\underline{y}}}(\Delta _{\underline{x}})$ inductively.

Let $\mathcal {R}_k$ be the Grothendieck group of the category of smooth representations of $\mathrm {GL}_k(F)$ of finite length. By the semisimplification, we identify the objects in this category with elements in $\mathcal {R}_k$. Elements in $\mathrm {Irr}(\mathrm {GL}_k(F))$ form a $\mathbb {Z}$-basis of $\mathcal {R}_k$. Set $\mathcal {R}= \oplus _{k \ge 0} \mathcal {R}_k$. The parabolic induction functor gives a product

$$\begin{aligned} m :\mathcal {R}\otimes \mathcal {R}\rightarrow \mathcal {R}, \ \tau _1 \otimes \tau _2 \mapsto \mathrm {s.s.}(\tau _1 \times \tau _2). \end{aligned}$$

This product makes $\mathcal {R}$ an associative commutative ring. On the other hand, the Jacquet functor gives a coproduct

$$\begin{aligned} m^* :&\mathcal {R}\rightarrow \mathcal {R}\otimes \mathcal {R}\end{aligned}$$

which is defined by the $\mathbb {Z}$-linear extension of

$$\begin{aligned} \mathrm {Irr}(\mathrm {GL}_k(F)) \ni \tau \mapsto \sum _{k_1 = 0}^k \mathrm {s.s.}\mathrm {Jac}_{(k_1, k-k_1)}(\tau ). \end{aligned}$$

Then m and $m^*$ make $\mathcal {R}$ a graded Hopf algebra, i.e., $m^* :\mathcal {R}\rightarrow \mathcal {R}\otimes \mathcal {R}$ is a ring homomorphism.

2.2 Representations of $\mathrm {SO}_{2n+1}(F)$ and $\mathrm {Sp}_{2n}(F)$

We set G to be split $\mathrm {SO}_{2n+1}(F)$ or $\mathrm {Sp}_{2n}(F)$, i.e., G is the group of F-points of the split algebraic group of type $B_n$ or $C_n$. Fix a Borel subgroup of G, and let $P=MN$ be a standard parabolic subgroup of G. Then the Levi part M is of the form $\mathrm {GL}_{k_1}(F) \times \dots \times \mathrm {GL}_{k_r}(F) \times G_{0}$ with $G_0 = \mathrm {SO}_{2n_0+1}(F)$ or $G_0 = \mathrm {Sp}_{2n_0}(F)$ such that $k_1+\dots +k_r+n_0=n$. For a smooth representation $\tau _1 \boxtimes \dots \boxtimes \tau _r \boxtimes \pi _0$ of M, we denote the normalized parabolically induced representation by

$$\begin{aligned} \tau _1 \times \dots \times \tau _r \rtimes \pi _0 = \mathrm {Ind}_P^{G}(\tau _1 \boxtimes \dots \boxtimes \tau _r \boxtimes \pi _0 ). \end{aligned}$$

The functor $\mathrm {Ind}_{P}^{G} :\mathrm {Rep}(M) \rightarrow \mathrm {Rep}(G)$ is exact.

On the other hand, for a smooth representation $\pi$ of G, we denote the normalized Jacquet module with respect to P by

$$\begin{aligned} \mathrm {Jac}_P(\pi ), \end{aligned}$$

and its semisimplification by $\mathrm {s.s.}\mathrm {Jac}_P(\pi )$. The functor $\mathrm {Jac}_{P} :\mathrm {Rep}(G) \rightarrow \mathrm {Rep}(M)$ is exact. The Frobenius reciprocity asserts that

$$\begin{aligned} \mathrm {Hom}_{G}(\pi , \mathrm {Ind}_P^{G}(\sigma )) \cong \mathrm {Hom}_{M}(\mathrm {Jac}_P(\pi ), \sigma ) \end{aligned}$$

for $\pi \in \mathrm {Rep}(G)$ and $\sigma \in \mathrm {Rep}(M)$.

The maximal standard parabolic subgroup with Levi $\mathrm {GL}_{k}(F) \times G_{0}$ is denoted by $P_k = M_k N_k$ for $0 \le k \le n$.

Definition 2.5

Let $\pi$ be a smooth representation of G. Consider $\mathrm {s.s.}\mathrm {Jac}_{P_d}(\pi )$ (and a fixed irreducible supercuspidal unitary representation $\rho$ of $\mathrm {GL}_d(F)$). If

$$\begin{aligned} \mathrm {s.s.}\mathrm {Jac}_{P_d}(\pi ) = \bigoplus _{i \in I} \tau _i \boxtimes \pi _i \end{aligned}$$

with $\tau _i$ (resp. $\pi _i$) being an irreducible representation of $\mathrm {GL}_d(F)$ (resp. $G_0$), for $x \in \mathbb {R}$, we define a partial Jacquet module$\mathrm {Jac}_{\rho |\cdot |^x}(\pi )$ by

$$\begin{aligned} \mathrm {Jac}_{\rho |\cdot |^x}(\pi ) = \bigoplus _{\begin{array}{c} i \in I\\ \tau _i \cong \rho |\cdot |^x \end{array}}\pi _i. \end{aligned}$$

This is a representation of $G_{0}$, which is $\mathrm {SO}_{2(n-d)+1}(F)$ or $\mathrm {Sp}_{2(n-d)}(F)$.

Also, for $\underline{x} = (x_1, \dots , x_r) \in \mathbb {R}^r$, we set

$$\begin{aligned} \mathrm {Jac}_{\rho |\cdot |^{\underline{x}}}(\pi ) = \mathrm {Jac}_{\rho |\cdot |^{x_1}, \dots , \rho |\cdot |^{x_r}}(\pi ) = \mathrm {Jac}_{\rho |\cdot |^{x_r}} \circ \dots \circ \mathrm {Jac}_{\rho |\cdot |^{x_1}}(\pi ). \end{aligned}$$

We recall some properties of $\mathrm {Jac}_{\rho |\cdot |^{x_1}, \dots , \rho |\cdot |^{x_r}}$.

Lemma 2.6

([16, Lemmas 5.3, 5.6]) Let $\pi$ be an irreducible representation of G.

• (1) Suppose that $\mathrm {Jac}_{\rho |\cdot |^{x_1}, \dots , \rho |\cdot |^{x_r}}(\pi )$ is nonzero. Then there exists an irreducible constituent $\sigma$ of $\mathrm {Jac}_{\rho |\cdot |^{x_1}, \dots , \rho |\cdot |^{x_r}}(\pi )$ such that we have an inclusion

  $$\begin{aligned} \pi \hookrightarrow \rho |\cdot |^{x_1} \times \dots \times \rho |\cdot |^{x_r} \rtimes \sigma . \end{aligned}$$

• (2) If $|x-y| \not = 1$, then $\mathrm {Jac}_{\rho |\cdot |^{x}, \rho |\cdot |^{y}}(\pi ) = \mathrm {Jac}_{\rho |\cdot |^{y}, \rho |\cdot |^{x}}(\pi )$.

• (3) If $\rho \not \cong \rho '$, then $\mathrm {Jac}_{\rho |\cdot |^{x}} \circ \mathrm {Jac}_{\rho '|\cdot |^{x'}} = \mathrm {Jac}_{\rho '|\cdot |^{x'}} \circ \mathrm {Jac}_{\rho |\cdot |^{x}}$ for any $x,x' \in \mathbb {R}$.

Let $\mathcal {R}_n(G)$ be the Grothendieck group of the category of smooth representations of G of finite length, where $n = \mathrm {rank}(G)$, i.e., $G = \mathrm {SO}_{2n+1}(F)$ or $G = \mathrm {Sp}_{2n}(F)$. Set $\mathcal {R}(G) = \oplus _{n \ge 0} \mathcal {R}_n(G)$. The parabolic induction defines a module structure

$$\begin{aligned} \rtimes :\mathcal {R}\otimes \mathcal {R}(G) \rightarrow \mathcal {R}(G), \ \tau \otimes \pi \mapsto \mathrm {s.s.}(\tau \rtimes \pi ), \end{aligned}$$

and the Jacquet functor defines a comodule structure

$$\begin{aligned} \mu ^* :\mathcal {R}(G) \rightarrow \mathcal {R}\otimes \mathcal {R}(G) \end{aligned}$$

by

$$\begin{aligned} \mathrm {Irr}(G) \ni \pi \mapsto \sum _{k=0}^{\mathrm {rank}(G)} \mathrm {s.s.}\mathrm {Jac}_{P_k}(\pi ). \end{aligned}$$

When

$$\begin{aligned} \mu ^*(\pi ) = \sum _{i \in I} \tau _i \otimes \pi _i, \end{aligned}$$

we define $\mu ^*_\rho (\pi )$ by

$$\begin{aligned} \mu ^*_\rho (\pi ) = \sum _{\begin{array}{c} i \in I \\ \tau _i \in \mathrm {Irr}_{\rho }(\mathrm {GL}_{dm}(F)) \end{array}} \tau _i \otimes \pi _i. \end{aligned}$$

Lemma 2.7

If we define $\iota :\mathcal {R}(G) \rightarrow \mathcal {R}\otimes \mathcal {R}(G)$ by $\pi \mapsto \mathbf {1}_{\mathrm {GL}_0(F)} \otimes \pi$, we have

$$\begin{aligned} \mu ^* = \circ _{\rho } \left( (m \otimes \mathrm {id}) \circ (\mathrm {id}\otimes \mu ^*_\rho ) \right) \circ \iota , \end{aligned}$$

where $\rho$ runs over all irreducible unitary supercuspidal representations of $\mathrm {GL}_d(F)$ for $d > 0$. Namely, for $\pi \in \mathrm {Irr}(G)$, if $\{\rho _1, \dots , \rho _t\}$ is the finite set of irreducible unitary supercuspidal representations $\rho$ of some $\mathrm {GL}_d(F)$ such that $\mu ^*_{\rho }(\pi ) \not = 0$, then

$$\begin{aligned} \mu ^*(\pi ) = \left( (m \otimes \mathrm {id}) \circ (\mathrm {id}\otimes \mu ^*_{\rho _t}) \right) \circ \dots \circ \left( (m \otimes \mathrm {id}) \circ (\mathrm {id}\otimes \mu ^*_{\rho _1}) \right) (\mathbf {1}_{\mathrm {GL}_0(F)} \otimes \pi ). \end{aligned}$$

Proof

Fix an irreducible representation $\pi$ of G. First, we note that there are only finitely many $\rho$ such that $\mu ^*_\rho (\pi ) \not = 0$. Second, we claim that

$$\begin{aligned} (m \otimes \mathrm {id}) \circ (\mathrm {id}\otimes \mu ^*_{\rho '}) \circ \mu ^*_{\rho }(\pi ) = (m \otimes \mathrm {id}) \circ (\mathrm {id}\otimes \mu ^*_{\rho }) \circ \mu ^*_{\rho '}(\pi ) \end{aligned}$$

for distinct $\rho$ and $\rho '$. In fact, this is the sum of subrepresentations appearing $\mu ^*(\pi )$ of the form

$$\begin{aligned} (\tau \times \tau ') \otimes \pi _0 = (\tau ' \times \tau ) \otimes \pi _0, \end{aligned}$$

where $\tau \in \mathrm {Irr}_{\rho }(\mathrm {GL}_{dm}(F))$ and $\tau ' \in \mathrm {Irr}_{\rho '}(\mathrm {GL}_{d'm'}(F))$ for various m and $m'$. By the same argument, we have

$$\begin{aligned} \mu ^*(\pi ) = \circ _{\rho } \left( (m \otimes \mathrm {id}) \circ (\mathrm {id}\otimes \mu ^*_\rho ) \right) \circ \iota (\pi ), \end{aligned}$$

as desired. $\square$

Tadić established a formula to compute $\mu ^*$ for parabolically induced representations. The contragredient functor $\tau \mapsto \tau ^\vee$ defines an automorphism $\vee :\mathcal {R}\rightarrow \mathcal {R}$ in a natural way. Let $s :\mathcal {R}\otimes \mathcal {R}\rightarrow \mathcal {R}\otimes \mathcal {R}$ be the homomorphism defined by $\sum _i\tau _i \otimes \tau '_i \mapsto \sum _i \tau '_i \otimes \tau _i$.

Theorem 2.8

(Tadić [15]) Consider the composition

$$\begin{aligned} M^* = (m \otimes \mathrm {id}) \circ (\vee \otimes m^*) \circ s \circ m^* :\mathcal {R}\rightarrow \mathcal {R}\otimes \mathcal {R}. \end{aligned}$$

Then for the maximal parabolic subgroup $P_k=M_kN_k$ of G and for an admissible representation $\tau \boxtimes \pi$ of $M_k$, we have

$$\begin{aligned} \mu ^*(\tau \rtimes \pi ) = M^*(\tau ) \rtimes \mu ^*(\pi ). \end{aligned}$$

In particular, we have the following.

Corollary 2.9

For a segment [x, y], we have

$$\begin{aligned}&\mu ^*(\left\langle \rho ; x, \dots , y \right\rangle \rtimes \pi )= \sum _{ \begin{array}{c} k,l \ge 0 \\ k+l \le x-y+1 \end{array}}\\&\quad \left( \left\langle \rho ^\vee ; \underbrace{-y, \dots , -y-k+1}_{k} \right\rangle \times \left\langle \rho ; \underbrace{x,\dots ,x-l+1}_{l} \right\rangle \right) \\&\qquad \otimes \left\langle \rho ; x-l, \dots , y+k \right\rangle \rtimes \mu ^*(\pi ). \end{aligned}$$

3 Local Langlands correspondence

In this section, we review the local Langlands correspondence for split $\mathrm {SO}_{2n+1}(F)$ or $\mathrm {Sp}_{2n}(F)$.

3.1 L-parameters

A homomorphism

$$\begin{aligned} \phi :W_F \times \mathrm {SL}_2(\mathbb {C}) \rightarrow \mathrm {GL}_k(\mathbb {C}) \end{aligned}$$

is called a representation of the Weil–Deligne group $W_F \times \mathrm {SL}_2(\mathbb {C})$ if

• $\phi (\mathrm {Frob}) \in \mathrm {GL}_k(\mathbb {C})$ is semisimple;

• $\phi |W_F$ is smooth, i.e., has an open kernel;

• $\phi |\mathrm {SL}_2(\mathbb {C})$ is algebraic.

We say that a representation $\phi$ of $W_F \times \mathrm {SL}_2(\mathbb {C})$ is tempered if $\phi (W_F)$ is bounded. The local Langlands correspondence for $\mathrm {GL}_k(F)$ asserts that there is a canonical bijection

$$\begin{aligned} \mathrm {Irr}(\mathrm {GL}_k(F)) \longleftrightarrow \{\phi :W_F \times \mathrm {SL}_2(\mathbb {C}) \rightarrow \mathrm {GL}_k(\mathbb {C})\}/\cong , \end{aligned}$$

which preserves the tempered-ness.

We say that a representation $\phi :W_F \times \mathrm {SL}_2(\mathbb {C}) \rightarrow \mathrm {GL}_k(\mathbb {C})$ is symplectic or of symplectic type (resp. orthogonal or of orthogonal type) if the image of $\phi$ is in $\mathrm {Sp}_k(\mathbb {C})$ (so that k is even) (resp. in $\mathrm {O}_k(\mathbb {C})$). An L-parameter for $\mathrm {SO}_{2n+1}(F)$ is a 2n-dimensional symplectic representation

$$\begin{aligned} \phi :W_F \times \mathrm {SL}_2(\mathbb {C}) \rightarrow \mathrm {Sp}_{2n}(\mathbb {C}). \end{aligned}$$

Similarly, an L-parameter for $\mathrm {Sp}_{2n}(F)$ is a $(2n+1)$-dimensional orthogonal representation

$$\begin{aligned} \phi :W_F \times \mathrm {SL}_2(\mathbb {C}) \rightarrow \mathrm {SO}_{2n+1}(\mathbb {C}) \end{aligned}$$

with trivial determinant. For $G = \mathrm {SO}_{2n+1}(F)$ or $G = \mathrm {Sp}_{2n}(F)$, we let $\Phi (G)$ be the set of equivalence classes of L-parameters for G. We say that

• $\phi \in \Phi (G)$ is discrete if $\phi$ is a multiplicity-free sum of irreducible self-dual representations of the same type as $\phi$;

• $\phi \in \Phi (G)$ is of good parity if $\phi$ is a sum of irreducible self-dual representations of the same type as $\phi$;

• $\phi \in \Phi (G)$ is tempered if $\phi (W_F)$ is bounded;

• $\phi \in \Phi (G)$ is generic if the adjoint L-function $L(s,\phi , \mathrm {Ad})$ is regular at $s=1$. Here, $L(s, \phi , \mathrm {Ad}) = L(s, \mathrm {Ad}\circ \phi )$ is the L-function associated to the composition of $\phi$ and the adjoint representation $\mathrm {Ad}:\mathrm {Sp}_{2n}(\mathbb {C}) \rightarrow \mathrm {GL}(\mathrm {Lie}(\mathrm {Sp}_{2n}(\mathbb {C})))$ or $\mathrm {Ad}:\mathrm {SO}_{2n+1}(\mathbb {C}) \rightarrow \mathrm {GL}(\mathrm {Lie}(\mathrm {SO}_{2n+1}(\mathbb {C})))$.

We denote by $\Phi _\mathrm {disc}(G)$ (resp. $\Phi _\mathrm {gp}(G)$, $\Phi _\mathrm {temp}(G)$, and $\Phi _\mathrm {gen}(G)$) the subset of $\Phi (G)$ consisting of discrete L-parameters (resp. L-parameters of good parity, tempered L-parameters, and generic L-parameters). Then we have inclusions

$$\begin{aligned} \Phi _\mathrm {disc}(G) \subset \Phi _\mathrm {gp}(G) \subset \Phi _\mathrm {temp}(G) \subset \Phi _\mathrm {gen}(G) \subset \Phi (G). \end{aligned}$$

For $\phi \in \Phi (G)$, we can decompose

$$\begin{aligned} \phi = m_1\phi _1 \oplus \dots \oplus m_r\phi _r \oplus (\phi ' \oplus \phi '^\vee ), \end{aligned}$$

where $\phi _1, \dots , \phi _r$ are distinct irreducible self-dual representations of the same type as $\phi$, $m_i \ge 1$ is the multiplicity of $\phi _i$ in $\phi$, and $\phi '$ is a sum of irreducible representations which are not self-dual or self-dual of the opposite type to $\phi$. We define the component group$A_\phi$ of $\phi$ by

$$\begin{aligned} A_\phi = \bigoplus _{i=1}^r (\mathbb {Z}/2\mathbb {Z}) \alpha _{\phi _i} \cong (\mathbb {Z}/2\mathbb {Z})^r. \end{aligned}$$

Namely, $A_\phi$ is a free $\mathbb {Z}/2\mathbb {Z}$-module of rank r and $\{\alpha _{\phi _1}, \dots , \alpha _{\phi _r}\}$ is a basis of $A_\phi$ with $\alpha _{\phi _i}$ associated to $\phi _i$. We set

$$\begin{aligned} z_\phi = \sum _{i=1}^r m_i \alpha _{\phi _i} \in A_\phi \end{aligned}$$

and we call $z_\phi$ the central element in $A_\phi$.

We shall introduce an enhanced component group$\mathcal {A}_\phi$ associated to $\phi \in \Phi (G)$. Write $\phi = \phi _\mathrm {gp}\oplus (\phi ' \oplus \phi '^\vee )$, where $\phi _\mathrm {gp}$ is the sum of irreducible self-dual subrepresentations of the same type as $\phi$, and $\phi '$ is a sum of irreducible representations which are not of the same type as $\phi$. We decompose

$$\begin{aligned} \phi _\mathrm {gp}= \bigoplus _{i \in I} \phi _i \end{aligned}$$

into the sum of irreducible representations.

Definition 3.1

With the notation above, we define the enhanced component group$\mathcal {A}_\phi$ associated to $\phi$ by

$$\begin{aligned} \mathcal {A}_\phi = \bigoplus _{i \in I} (\mathbb {Z}/2\mathbb {Z}) \alpha _i. \end{aligned}$$

Namely, $\mathcal {A}_\phi$ is a free $\mathbb {Z}/2\mathbb {Z}$-module whose rank is equal to the length of $\phi _\mathrm {gp}$.

By abuse of notation, we put $z_\phi = \sum _{i \in I} \alpha _i \in \mathcal {A}_\phi$ and call it the central element in $\mathcal {A}_\phi$. There is a canonical surjection

$$\begin{aligned} \mathcal {A}_\phi \twoheadrightarrow A_\phi ,\ \alpha _i \mapsto \alpha _{\phi _i}. \end{aligned}$$

This map preserves the central elements. The kernel of this map is generated by $\alpha _i + \alpha _j$ such that $\phi _i \cong \phi _j$. In particular, if $\phi$ is discrete, then this map is an isomorphism.

Remark 3.2

For the definition of this enhanced component group, we rely on the shape of $\phi$. A similar definition will work for $\mathrm {SO}_{2n}(F)$ and unitary groups as well, but we do not know a uniform definition for general (connected) reductive groups, nor whether this notion makes sense.

3.2 Local Langlands correspondence

We denote by $\mathrm {Irr}_\mathrm {disc}(G)$ (resp. $\mathrm {Irr}_\mathrm {temp}(G)$) the set of equivalence classes of irreducible discrete series (resp. tempered) representations of G. The local Langlands correspondence established by Arthur is as follows:

Theorem 3.3

([1, Theorem 2.2.1]) Let G be a split $\mathrm {SO}_{2n+1}(F)$ or $\mathrm {Sp}_{2n}(F)$.

• (1) There exists a canonical surjection

  $$\begin{aligned} \mathrm {Irr}(G) \rightarrow \Phi (G). \end{aligned}$$

  For $\phi \in \Phi (G)$, we denote by $\Pi _\phi$ the inverse image of $\phi$ under this map, and call $\Pi _\phi$ the L-packet associated to$\phi$.

• (2) There exists an injection

  $$\begin{aligned} \Pi _\phi \rightarrow \widehat{A_\phi }, \end{aligned}$$

  which satisfies certain endoscopic character identities. Here, $\widehat{A_\phi }$ is the Pontryagin dual of $A_\phi$. The image of this map is

  $$\begin{aligned} \{\eta \in \widehat{A_\phi }\ |\ \eta (z_\phi ) = 1\}. \end{aligned}$$

  When $\pi \in \Pi _\phi$ corresponds to $\eta \in \widehat{A_\phi }$, we write $\pi = \pi (\phi , \eta )$.

• (3) For $* \in \{\mathrm {disc}, \mathrm {temp}\}$,

  $$\begin{aligned} \mathrm {Irr}_*(G) = \bigsqcup _{\phi \in \Phi _*(G)}\Pi _\phi . \end{aligned}$$

• (4) Assume that $\phi =\phi _\tau \oplus \phi _0 \oplus \phi _\tau ^\vee \in \Phi _\mathrm {temp}(G)$, where

  • $\phi _0 \in \Phi _\mathrm {temp}(G_0)$ with a classical group $G_0$ of the same type as G;

  • $\phi _\tau$ is a tempered representation of $W_F \times \mathrm {SL}_2(\mathbb {C})$ of dimension k.

  Let $\tau$ be the irreducible tempered representation of $\mathrm {GL}_k(F)$ corresponding to $\phi _\tau$. Then for $\pi _0 \in \Pi _{\phi _0}$, the induced representation $\tau \rtimes \pi _0$ decomposes into a direct sum of irreducible tempered representations of G. The L-packet $\Pi _\phi$ is given by

  $$\begin{aligned} \Pi _\phi = \left\{ \pi \ \big |\ \pi \subset \tau \rtimes \pi _0,\ \pi _0 \in \Pi _{\phi _0} \right\} . \end{aligned}$$

  Moreover there is a canonical inclusion $A_{\phi _0} \hookrightarrow A_\phi$. If $\pi (\phi , \eta )$ is a direct summand of $\tau \rtimes \pi _0$ with $\pi _0 = \pi (\phi _0, \eta _0)$, then $\eta _0 = \eta |A_{\phi _0}$.

• (5) Assume that

  $$\begin{aligned} \phi = \phi _1|\cdot |^{s_1} \oplus \dots \oplus \phi _r|\cdot |^{s_r} \oplus \phi _0 \oplus \phi _r^\vee |\cdot |^{-s_r} \oplus \dots \oplus \phi _1^\vee |\cdot |^{-s_1}, \end{aligned}$$

  where

  • $\phi _0 \in \Phi _\mathrm {temp}(G_0)$ with a classical group $G_0$ of the same type as G;

  • $\phi _{i}$ is a tempered representation of $W_F \times \mathrm {SL}_2(\mathbb {C})$ of dimension $k_i$ for $1 \le i \le r$;

  • $s_i$ is a real number such that $s_1 \ge \dots \ge s_r>0$.

  Let $\tau _i$ be the irreducible tempered representation of $\mathrm {GL}_{k_i}(F)$ corresponding to $\phi _{i}$. Then the L-packet $\Pi _\phi$ consists of the unique irreducible quotients $\pi$ of the standard modules

  $$\begin{aligned} \tau _1|\cdot |^{s_1} \times \dots \times \tau _r|\cdot |^{s_r} \rtimes \pi _0, \end{aligned}$$

  where $\pi _0$ runs over $\Pi _{\phi _0}$. Moreover, there is a canonical inclusion $A_{\phi _0} \hookrightarrow A_\phi$, which is in fact bijective. If $\pi (\phi , \eta )$ is the unique irreducible quotient of the above standard module with $\pi _0 = \pi (\phi _0, \eta _0)$, then $\eta _0 = \eta |A_{\phi _0}$. In this case, we denote this standard module by $I(\phi , \eta )$.

The injection $\Pi _\phi \rightarrow \widehat{A_\phi }$ is not canonical when $G = \mathrm {Sp}_{2n}(F)$. To specify this, we implicitly fix an $F^{\times 2}$-orbit of non-trivial additive characters of F through this paper. Remark that our main results (Theorems 4.2 and 4.3 below) are independent of such a choice.

We have the following irreducibility criterion for standard modules.

Theorem 3.4

(Generalized standard module conjecture) For $\phi \in \Phi (G)$, the standard module $I(\phi , \mathbf {1})$ attached to $\pi (\phi , \mathbf {1})$ is irreducible if and only if $\phi$ is generic. Moreover, if $\phi$ is generic, then all standard modules $I(\phi , \eta )$, where $\eta \in \widehat{A_\phi }$ with $\eta (z_\phi ) = 1$, are irreducible.

Proof

The first assertion is the usual standard module conjecture proven in [5, 8, 9, 14]. The second assertion was proven by Mœglin–Waldspurger [13, Corollaire 2.14] for special orthogonal groups and symplectic groups. Heiermann [10] also proved the second assertion in a more general setting. Note that their definition of generic L-parameters might look different from ours. The equivalence of two definitions is called a conjecture of Gross–Prasad and Rallis, which was proven by Gan–Ichino [6, Proposition B.1]. $\square$

However, even if $\phi$ is not generic, there might exist an irreducible standard module $I(\phi , \eta )$. An example of such standard modules will be given by Corollary 6.1 and Example 6.2 below.

3.3 Extension to enhanced component groups

To describe Jacquet modules of $\pi (\phi , \eta )$ for $\phi \in \Phi _\mathrm {gp}(G)$, it is useful to extend $\pi (\phi , \eta )$ to the case where $\eta$ is a character of the enhanced component group $\mathcal {A}_\phi$ as follows. Recall that there exists a canonical surjection $\mathcal {A}_\phi \twoheadrightarrow A_\phi$ so that we have an injection $\widehat{A_\phi } \hookrightarrow \widehat{\mathcal {A}_\phi }$. For $\eta \in \widehat{\mathcal {A}_\phi }$, set

$$\begin{aligned} \pi (\phi , \eta ) = \left\{ \begin{aligned}&\pi (\phi , \eta )&\quad&\text {if }\eta \in \widehat{A_\phi },\ \eta (z_\phi ) = 1,\\&0&\quad&\text {otherwise}. \end{aligned} \right. \end{aligned}$$

In particular, $\pi (\phi , \eta )$ is irreducible or zero for any $\eta \in \widehat{\mathcal {A}_\phi }$.

More precisely, if $\phi = \oplus _{i \in I} \phi _i$ is the irreducible decomposition so that $\mathcal {A}_\phi = \oplus _{i \in I}(\mathbb {Z}/2\mathbb {Z})\alpha _i$, then $\pi (\phi , \eta )$ is nonzero if and only if $\eta (z_\phi ) = 1$ and $\eta (\alpha _i) = \eta (\alpha _j)$ whenever $\phi _i \cong \phi _j$.

3.4 Mœglin’s construction of tempered L-packets

The L-packets are used for a local classification. On the other hand, Arthur [1, Theorem 2.2.1] introduced the notion of A-packets for a global classification. Mœglin constructed the local A-packets in her consecutive works (e.g., [11, 12], etc.). For a detailed why Mœglin’s local A-packets agree with Arthur’s, one can see Xu’s paper [17] in addition to the original papers of Mœglin. Since the temperedA-packets are the same notion as the tempered L-packets, Mœglin’s construction can be applied to the tempered L-packets.

We explain Mœglin’s construction of $\Pi _\phi$ for $\phi \in \Phi _\mathrm {gp}(G)$. Assume that

$$\begin{aligned} \phi = \left( \bigoplus _{i=1}^{t} \rho \boxtimes S_{a_i} \right) \oplus \phi _e \end{aligned}$$

with $a_1 \le \dots \le a_t$ and $\rho \boxtimes S_a \not \subset \phi _e$ for any $a>0$. (In particular, $\rho$ must be self-dual if $t \ge 1$.) Take a new L-parameter

$$\begin{aligned} \phi _\gg = \left( \bigoplus _{i=1}^{t} \rho \boxtimes S_{a'_i} \right) \oplus \phi _e \end{aligned}$$

for a bigger group $G'$ of the same type as G such that

• $a'_1< \dots < a'_t$;

• $a_i' \ge a_i$ and $a'_i \equiv a_i \bmod 2$ for any i;

Then we can identify $\mathcal {A}_{\phi _\gg }$ with $\mathcal {A}_{\phi }$ canonically, i.e., $\mathcal {A}_{\phi _\gg } = \mathcal {A}_{\phi }$ and if $\mathcal {A}_{\phi _{\gg }} \ni \alpha \mapsto \alpha _{\rho \boxtimes S_{a_i'}} \in A_{\phi _\gg }$, then $\mathcal {A}_{\phi } \ni \alpha \mapsto \alpha _{\rho \boxtimes S_{a_i}} \in A_\phi$. Let $\eta _\gg \in \widehat{\mathcal {A}_{\phi _\gg }}$ be the character corresponding to $\eta \in \widehat{\mathcal {A}_{\phi }}$, i.e., $\eta _\gg = \eta$ via $\mathcal {A}_{\phi _\gg } = \mathcal {A}_\phi$.

Theorem 3.5

(Mœglin) With the notation above, we have

$$\begin{aligned} \pi (\phi , \eta ) = \mathrm {Jac}_{\rho |\cdot |^{\frac{a'_t-1}{2}}, \dots , \rho |\cdot |^\frac{a_t+1}{2}} \circ \dots \circ \mathrm {Jac}_{\rho |\cdot |^{\frac{a'_1-1}{2}}, \dots , \rho |\cdot |^\frac{a_1+1}{2}} (\pi (\phi _\gg , \eta _\gg )). \end{aligned}$$

For a proof, one can specialize [17, Corollary 8.10], which treats all local A-packets. Using this theorem repeatedly, we can construct the L-packets $\Pi _\phi$ for $\phi \in \Phi _\mathrm {gp}(G)$ from the L-packets associated to discrete L-parameters for bigger groups.

Example 3.6

We construct $\Pi _\phi$ for $\phi = S_2 \oplus S_4 \oplus S_4 \oplus S_6 \oplus S_6 \in \Phi _\mathrm {gp}(\mathrm {SO}_{23}(F))$. Note that $A_\phi = (\mathbb {Z}/2\mathbb {Z}) \alpha _{S_2} \oplus (\mathbb {Z}/2\mathbb {Z}) \alpha _{S_4} \oplus (\mathbb {Z}/2\mathbb {Z}) \alpha _{S_6}$ with $z_{\phi } = \alpha _{S_2}$. Let $\eta \in \widehat{A_\phi }$. We write $\eta (\alpha _{S_{2a}}) = \eta _{a} \in \{\pm 1\}$. If $\eta (z_\phi ) = 1$, then $\eta _1 = +1$.

Now we take new L-parameters

$$\begin{aligned} \phi _\gg&= S_2 \oplus S_4 \oplus S_6 \oplus S_8 \oplus S_{10} \in \Phi _\mathrm {disc}(\mathrm {SO}_{31}(F)), \\ \phi '&= S_2 \oplus S_4 \oplus S_4 \oplus S_8 \oplus S_{10} \in \Phi _\mathrm {disc}(\mathrm {SO}_{29}(F)), \\ \phi ''&= S_2 \oplus S_4 \oplus S_4 \oplus S_6 \oplus S_{10} \in \Phi _\mathrm {disc}(\mathrm {SO}_{27}(F)), \end{aligned}$$

and we consider $\eta _\gg \in \widehat{A_{\phi _\gg }}$, $\eta ' \in \widehat{A_{\phi '}}$ and $\eta '' \in \widehat{A_{\phi ''}}$ given by

• $\eta _\gg (\alpha _{S_2}) = \eta '(\alpha _{S_2}) = \eta ''(\alpha _{S_2}) = \eta _1 = +1$;

• $\eta _\gg (\alpha _{S_4}) = \eta _\gg (\alpha _{S_6}) = \eta '(\alpha _{S_4}) = \eta ''(\alpha _{S_4}) = \eta _2$;

• $\eta _\gg (\alpha _{S_8}) = \eta _\gg (\alpha _{S_{10}}) = \eta '(\alpha _{S_8}) = \eta '(\alpha _{S_{10}}) = \eta ''(\alpha _{S_6}) = \eta ''(\alpha _{S_{10}}) = \eta _3$.

Then Theorem 3.5 says that

$$\begin{aligned} \mathrm {Jac}_{|\cdot |^{\frac{5}{2}}} (\pi (\phi _\gg , \eta _\gg ))&= \pi (\phi ', \eta '), \\ \mathrm {Jac}_{|\cdot |^{\frac{7}{2}}} (\pi (\phi ', \eta '))&= \pi (\phi '', \eta ''), \\ \mathrm {Jac}_{|\cdot |^{\frac{9}{2}}, |\cdot |^{\frac{7}{2}}} (\pi (\phi '', \eta ''))&= \pi (\phi , \eta ). \end{aligned}$$

4 Description of Jacquet modules

In this section, we state the main theorems, which compute the semisimplifications of the Jacquet modules of $\pi (\phi , \eta )$ for $\phi \in \Phi _\mathrm {gp}(G)$.

4.1 Statements

Note that:

Lemma 4.1

For $\phi \in \Phi _\mathrm {gp}(G)$ and $x \in \mathbb {R}$, if $\mathrm {Jac}_{\rho |\cdot |^x}(\pi ) \not = 0$ for some $\pi \in \Pi _\phi$, then x is a non-negative half-integer and $\rho \boxtimes S_{2x+1} \subset \phi$.

Proof

When $\phi \in \Phi _\mathrm {disc}(G)$, it follows from [16, Lemma 7.3]. We may assume that $\phi \in \Phi _\mathrm {gp}(G) \setminus \Phi _\mathrm {disc}(G)$. Then there exists an irreducible representation $\rho ' \boxtimes S_{a}$ which $\phi$ contains at least multiplicity two. By Theorem 3.3 (4), we have

$$\begin{aligned} \pi \hookrightarrow \mathrm {St}(\rho ', a) \rtimes \pi _0 \end{aligned}$$

for some $\pi _0 \in \Pi _{\phi _0}$ with $\phi _0 = \phi - (\rho ' \boxtimes S_{a})^{\oplus 2}$. Then by Corollary 2.9, we have

$$\begin{aligned} \mathrm {Jac}_{\rho |\cdot |^{x}}(\pi )&\hookrightarrow (1 \otimes \mathrm {St}(\rho ', a)) \rtimes \mathrm {Jac}_{\rho |\cdot |^{x}}(\pi _0)\\&+ \left\{ \begin{aligned}&2\left\langle \rho ; x,\dots ,-(x-1) \right\rangle \rtimes \pi _0&\quad&\text {if }\rho ' \cong \rho , a = 2x+1, \\&0&\quad&\text {otherwise}. \end{aligned} \right. \end{aligned}$$

Hence if $\mathrm {Jac}_{\rho |\cdot |^{x}}(\pi ) \not = 0$, then $\mathrm {Jac}_{\rho |\cdot |^{x}}(\pi _0) \not = 0$ or $\rho \boxtimes S_{2x+1} \cong \rho ' \boxtimes S_{a} \subset \phi$. By induction, we conclude that $\rho \boxtimes S_{2x+1} \subset \phi$ as desired. $\square$

The following is the first main theorem, which is a description of $\mathrm {Jac}_{\rho |\cdot |^x}(\pi (\phi , \eta ))$.

Theorem 4.2

Let $\phi \in \Phi _\mathrm {gp}(G)$ and $\eta \in \widehat{\mathcal {A}_\phi }$ such that $\pi (\phi , \eta ) \not = 0$. Fix a non-negative half-integer $x \in (1/2)\mathbb {Z}$. Write

$$\begin{aligned} \phi = \phi _0 \oplus (\rho \boxtimes S_{2x+1})^{\oplus m} \end{aligned}$$

with $\rho \boxtimes S_{2x+1} \not \subset \phi _0$ and $m > 0$.

• (1) Assume that $m \ge 3$. Take $\delta \in \{1,2\}$ such that $\delta \equiv m \bmod 2$. Then

  $$\begin{aligned}&\mathrm {Jac}_{\rho |\cdot |^x} (\pi (\phi , \eta ))\\&= (m-\delta ) \cdot \left\langle \rho ; \underbrace{x, x-1, \dots ,-(x-1)}_{2x} \right\rangle \rtimes \pi \left( \phi - (\rho \boxtimes S_{2x+1})^{\oplus 2}, \eta \right) \\&+ \underbrace{\mathrm {St}(\rho , 2x+1) \times \dots \times \mathrm {St}(\rho , 2x+1)}_{(m-\delta )/2} \rtimes \mathrm {Jac}_{\rho |\cdot |^{x}} \left( \pi \left( \phi _0 \oplus (\rho \boxtimes S_{2x+1})^{\oplus \delta }, \eta \right) \right) . \end{aligned}$$

  Here, we canonically identify the (usual) component groups of $\phi - (\rho \boxtimes S_{2x+1})^{\oplus 2}$ and $\phi _0 \oplus (\rho \boxtimes S_{2x+1})^{\oplus \delta }$ with $A_\phi$, so that we regard $\eta$ as a character of these groups.

• (2) Assume that $x > 0$ and $m = 1$. Set

  $$\begin{aligned} \phi ' = \phi - (\rho \boxtimes S_{2x+1}) \oplus (\rho \boxtimes S_{2x-1}). \end{aligned}$$

  There is a canonical inclusion $\mathcal {A}_{\phi '} \hookrightarrow \mathcal {A}_\phi$, which is in fact bijective if $x > 1/2$. Let $\eta ' \in \widehat{\mathcal {A}_{\phi '}}$ be the character corresponding to $\eta \in \widehat{\mathcal {A}_\phi }$, i.e., $\eta ' = \eta |\mathcal {A}_{\phi '}$. Then

  $$\begin{aligned} \mathrm {Jac}_{\rho |\cdot |^x} (\pi (\phi , \eta )) = \pi (\phi ', \eta '). \end{aligned}$$

  In particular, $\mathrm {Jac}_{\rho |\cdot |^x} (\pi (\phi , \eta ))$ is irreducible or zero.

• (3) Assume that $x > 0$ and $m = 2$. Set $\eta _+ = \eta$, and take the unique character $\eta _- \in \widehat{\mathcal {A}_\phi }$ so that $\eta _-|\mathcal {A}_{\phi _0} = \eta _+|\mathcal {A}_{\phi _0}$, $\eta _-(z_\phi ) = \eta _+(z_\phi ) = 1$ but $\eta _- \not = \eta _+$. For $\phi '$ as in (2) and for $\epsilon \in \{\pm \}$, let $\eta '_\epsilon \in \widehat{\mathcal {A}_{\phi '}}$ be the character corresponding to $\eta _\epsilon \in \widehat{\mathcal {A}_\phi }$ via the canonical inclusion $\mathcal {A}_{\phi '} \hookrightarrow \mathcal {A}_\phi$. Then

  $$\begin{aligned} \mathrm {Jac}_{\rho |\cdot |^x} (\pi (\phi , \eta ))&=\left\langle \rho ; x, x-1, \dots , -(x-1) \right\rangle \rtimes \pi (\phi _0, \eta |\mathcal {A}_{\phi _0}) \\&+ \pi (\phi ', \eta '_+) - \pi (\phi ', \eta '_-). \end{aligned}$$

• (4) Assume that $x = 0$. If $m=1$, then $\mathrm {Jac}_{\rho }(\pi (\phi , \eta )) = 0$. If $m=2$, then $\mathrm {Jac}_{\rho }(\pi (\phi , \eta )) = \pi (\phi _0, \eta |\mathcal {A}_{\phi _0})$.

When $\phi \in \Phi _\mathrm {disc}(G)$, Theorem 4.2 has been already proven by Xu ([16, Lemma 7.3]). In (2) (resp. (3)), we note that $\pi (\phi ', \eta ')$ (resp. $\pi (\phi ', \eta '_\epsilon )$) can be zero even if $\pi (\phi , \eta ) \not = 0$. In (3), the character $\eta _-$ is characterized so that

$$\begin{aligned} \pi (\phi , \eta _+) \oplus \pi (\phi , \eta _-) = \mathrm {St}(\rho , 2x+1) \rtimes \pi (\phi _0, \eta |\mathcal {A}_{\phi _0}). \end{aligned}$$

The second main theorem concerns $\mu ^*_\rho (\pi )$.

Theorem 4.3

Let $\phi \in \Phi _\mathrm {gp}(G)$, and write

$$\begin{aligned} \phi = \left( \bigoplus _{i=1}^{t} \rho \boxtimes S_{a_i} \right) \oplus \phi _e \end{aligned}$$

with $a_1 \le \dots \le a_t$ and $\rho \boxtimes S_a \not \subset \phi _e$ for any $a>0$. Set $x_i = \frac{a_i-1}{2}$. For $0 \le m \le (2d)^{-1} \cdot \dim (\phi )$, we denote by $K_\phi ^{(m)}$ the set of tuples of integers $\underline{k} = (k_1, \dots , k_t)$ such that

• $0 \le k_i \le a_i$ for any i;

• $k_{i-1} \ge k_{i}$ if $a_{i-1} = a_i$;

• $k_1+\dots +k_t = m$.

For $\underline{k} \in K_\phi ^{(m)}$, set

$$\begin{aligned} \underline{x}(\underline{k}) = (\underbrace{x_1, \dots , x_1-k_1+1}_{k_1}, \dots , \underbrace{x_t, \dots , x_t-k_t+1}_{k_t}) \in \Omega _{m}. \end{aligned}$$

For $\underline{k}, \underline{l} \in K_\phi ^{(m)}$, we set

$$\begin{aligned} m_{\underline{k}, \underline{l}} = \dim _\mathbb {C}\mathrm {Jac}_{\rho |\cdot |^{\underline{x}(\underline{k})}}(\Delta _{\underline{x}(\underline{l})}), \end{aligned}$$

and define $(m'_{\underline{k}, \underline{l}})_{\underline{k}, \underline{l} \in K_\phi ^{(m)}}$ to be the inverse matrix of $(m_{\underline{k}, \underline{l}})_{\underline{k}, \underline{l} \in K_\phi ^{(m)}}$, i.e.,

$$\begin{aligned} \sum _{\underline{k'} \in K_\phi ^{(m)}} m'_{\underline{k}, \underline{k'}} m_{\underline{k'}, \underline{l}} = \left\{ \begin{aligned}&1&\quad&\text {if }\underline{k} = \underline{l}, \\&0&\quad&\text {if }\underline{k} \not = \underline{l}. \end{aligned} \right. \end{aligned}$$

Then for $\pi \in \Pi _\phi$, we have

$$\begin{aligned} \mu ^*_\rho (\pi )&= \sum _{m=0}^{(2d)^{-1}\dim (\phi )} \sum _{\underline{k}, \underline{l} \in K_\phi ^{(m)}} m'_{\underline{k}, \underline{l}} \cdot \Delta _{\underline{x}(\underline{k})} \otimes \mathrm {Jac}_{\rho |\cdot |^{\underline{x}(\underline{l})}}(\pi ). \end{aligned}$$

When we formally regard $(\Delta _{\underline{x}(\underline{k})})_{\underline{k} \in K_\phi ^{(m)}}$ and $(\otimes \mathrm {Jac}_{\rho |\cdot |^{\underline{x}(\underline{l})}}(\pi ))_{\underline{l} \in K_\phi ^{(m)}}$ as column vectors, we have

$$\begin{aligned} \sum _{\underline{k}, \underline{l} \in K_\phi ^{(m)}} m'_{\underline{k}, \underline{l}} \cdot \Delta _{\underline{x}(\underline{k})} \otimes \mathrm {Jac}_{\rho |\cdot |^{\underline{x}(\underline{l})}}(\pi ) = {}^t (\Delta _{\underline{x}(\underline{k})}) \cdot (m_{\underline{k}, \underline{l}})^{-1} \cdot (\otimes \mathrm {Jac}_{\rho |\cdot |^{\underline{x}(\underline{l})}}(\pi )). \end{aligned}$$

By Lemma 2.4, $(m_{\underline{k}, \underline{l}})_{\underline{k}, \underline{l} \in K_\phi ^{(m)}}$ is a “triangular matrix”, which can be computed inductively. Here, we regard $K_\phi ^{(m)}$ as a totally ordered set with respect to the lexicographical order. The diagonal entries $m_{\underline{k}, \underline{k}}$ are given in Lemma 2.4 explicitly.

By Tadić’s formula (Theorem 2.8), Lemma 2.7, and Theorems 4.2, 4.3, we can deduce the following corollary.

Corollary 4.4

We can compute $\mu ^*(\pi )$ explicitly for any $\pi \in \Pi _\phi$ with $\phi \in \Phi _\mathrm {gen}(G)$.

4.2 Examples

We shall give some examples.

Example 4.5

Fix two positive integers a, b such that $a \equiv b \mod 2$ and $a < b$, and consider

$$\begin{aligned} \phi = \rho \boxtimes (S_a \oplus S_b) \in \Phi _\mathrm {disc}(\mathrm {SO}_{d(a+b)+1}(F)), \end{aligned}$$

where $\rho$ is symplectic (resp. orthogonal) if $a \equiv 1 \bmod 2$ (resp. if $a \equiv 0 \bmod 2$). Then $\Pi _\phi = \{\pi _+(a,b), \pi _-(a,b)\}$ with generic $\pi _+(a,b)$ and non-generic $\pi _-(a,b)$. Note that both $\pi _+(a,b)$ and $\pi _-(a,b)$ are discrete series. We compute $\mu ^*(\pi _\epsilon (a,b))$ for $\epsilon \in \{\pm \}$.

Note that $K_\phi ^{(m)} = \{(k_1,k_2) \in \mathbb {Z}^2\ |\ 0 \le k_1 \le a,\ 0 \le k_2 \le b, \ k_1+k_2 = m\}$. For $(k_1,k_2) \in K_\phi ^{(m)}$, since $x_1 = \frac{a-1}{2}$ and $x_2 = \frac{b-1}{2}$,

$$\begin{aligned} \Delta _{\underline{x}(k_1,k_2)} = \left\langle \rho ; \frac{a-1}{2}, \dots , \frac{a+1}{2}-k_1 \right\rangle \times \left\langle \rho ; \frac{b-1}{2}, \dots , \frac{b+1}{2}-k_2 \right\rangle . \end{aligned}$$

This induced representation is irreducible unless $(a+3)/2-k_1 \le (b+1)/2-k_2 \le (a+1)/2$, i.e., $(b-a)/2 \le k_2 \le (b-a)/2 + k_1-1$. Moreover, one can easy to see that for $(l_1, l_2) \in K_\phi ^{(m)}$, the virtual representation

$$\begin{aligned} \sum _{(k_1,k_2) \in K_\phi ^{(m)}} m'_{(k_1,k_2), (l_1, l_2)} \cdot \Delta _{\underline{x}(k_1,k_2)} \end{aligned}$$

is the unique irreducible subrepresentation $\tau _{\underline{x}(l_1,l_2)}$ of $\Delta _{\underline{x}(l_1,l_2)}$ (cf. see [18, Proposition 4.6]).

Note that

$$\begin{aligned} \mathrm {Jac}_{\rho |\cdot |^{\frac{a-1}{2}}, \dots , \rho |\cdot |^{\frac{a+1}{2}-k_1}}(\pi _{\epsilon }(a,b)) = \left\{ \begin{aligned}&\pi _{\epsilon }(a-2k_1, b)&\quad&\text {if }k_1 \le a/2, \\&0&\quad&\text {otherwise}. \end{aligned} \right. \end{aligned}$$

Here, when $k_1 = a/2$, we understand that $\pi _+(0,b)$ is the unique element in $\Pi _{\rho \boxtimes S_b}$, and $\pi _-(0,b) = 0$. Moreover, for $(k_1, k_2) \in K_\phi ^{(m)}$, when $k_1 \le a/2$ and $k_2 \le (b-a)/2+k_1$, we have

$$\begin{aligned} \mathrm {Jac}_{\rho |\cdot |^{\underline{x}(k_1,k_2)}}(\pi _\epsilon (a,b)) = \pi _\epsilon (a-2k_1, b-2k_2). \end{aligned}$$

When $k_1 < a/2$ and $(b-a)/2+k_1+1 \le k_2 \le b/2$, we have

$$\begin{aligned} \mathrm {Jac}_{\rho |\cdot |^{\underline{x}(k_1,k_2)}}(\pi _\epsilon (a,b))&=\left\langle \rho ; \frac{a-1}{2}-k_1, \dots , -\frac{b-1}{2}+k_2 \right\rangle \rtimes \mathbf {1}_{\mathrm {SO}_1(F)}\\ {}&+ \pi _\epsilon (b-2k_2, a-2k_1) - \pi _{-\epsilon }(b-2k_2, a-2k_1). \end{aligned}$$

When $k_1 < a/2$ and $b/2 < k_2 \le (a+b)/2-k_1$, we have

$$\begin{aligned} \mathrm {Jac}_{\rho |\cdot |^{\underline{x}(k_1,k_2)}}(\pi _\epsilon (a,b)) =&\left\langle \rho ; \frac{a-1}{2}-k_1, \dots , -\frac{b-1}{2}+k_2 \right\rangle \rtimes \mathbf {1}_{\mathrm {SO}_1(F)}. \end{aligned}$$

In particular, if $k_1+k_2 = (a+b)/2$, then $\mathrm {Jac}_{\rho |\cdot |^{\underline{x}(k_1,k_2)}}(\pi _\epsilon (a,b)) = \mathbf {1}_{\mathrm {SO}_1(F)}$. Hence $\mathrm {s.s.}\mathrm {Jac}_{P_{d(a+b)/2}}(\pi _\epsilon (a,b))$ contains the irreducible representation

$$\begin{aligned}&\left\langle \rho ; \frac{a-1}{2}, \dots , \frac{a+1}{2}-k_1 \right\rangle \times \left\langle \rho ; \frac{b-1}{2}, \dots , -\left( \frac{a-1}{2}-k_1\right) \right\rangle \otimes \mathbf {1}_{\mathrm {SO}_1(F)} \end{aligned}$$

with multiplicity one if $k_1 < a/2$, or if $k_1 = a/2$ and $\epsilon = +1$.

Example 4.6

Consider the L-parameter $\phi = S_2 \oplus S_4 \oplus S_4 \in \Phi _\mathrm {gp}(\mathrm {SO}_{11}(F))$. Then $\Pi _\phi$ has two elements $\pi _+(2,4,4)$ and $\pi _-(2,4,4)$ with generic $\pi _+(2,4,4)$ and non-generic $\pi _-(2,4,4)$. Then

$$\begin{aligned} K_\phi ^{(m)} {=} \{(k_1,k_2,k_3) \in \mathbb {Z}^3\ |\ 0 \le k_1 \le 2,\ 0 {\le } k_3 {\le } k_2 {\le } 4, k_1+k_2+k_3 = m\} \end{aligned}$$

for $0 \le m \le 5$. Write $\Pi _{\underline{x}(\underline{k})}^{\epsilon } = \mathrm {Jac}_{\rho |\cdot |^{\underline{x}(\underline{k})}}(\pi _\epsilon (2,4,4))$ for $\epsilon \in \{\pm \}$, and $\mathrm {St}_a = \mathrm {St}(\mathbf {1}_{\mathrm {GL}_1(F)}, a)$. We denote by ${\det }_a$ the determinant character of $\mathrm {GL}_a(F)$.

• (1) When $m = 1$, we have $K_\phi ^{(1)} = \{(1,0,0) > (0,1,0)\}$. Since

  $$\begin{aligned} \begin{pmatrix} \Delta _{\underline{x}(1,0,0)}&\Delta _{\underline{x}(0,1,0)} \end{pmatrix}= \begin{pmatrix} |\cdot |^{\frac{1}{2}}&|\cdot |^{\frac{3}{2}} \end{pmatrix}, \end{aligned}$$

  we have

  $$\begin{aligned} (m_{\underline{k}, \underline{k'}})_{\underline{k}, \underline{k'}}^{-1} = \begin{pmatrix} 1 &{} 0 \\ 0 &{} 1 \end{pmatrix}^{-1}= \begin{pmatrix} 1 &{} 0 \\ 0 &{} 1 \end{pmatrix}. \end{aligned}$$

  Moreover

  $$\begin{aligned} \begin{pmatrix} \Pi _{\underline{x}(1,0,0)}^\epsilon \\ \Pi _{\underline{x}(0,1,0)}^\epsilon \end{pmatrix}= \begin{pmatrix} \pi _\epsilon (4,4) \\ |\cdot |^{\frac{1}{2}}\mathrm {St}_3 \rtimes \pi _+(2) + \epsilon \cdot \pi _+(2,2,4) \end{pmatrix}. \end{aligned}$$

  Hence

  $$\begin{aligned} \mathrm {s.s.}\mathrm {Jac}_{P_1}(\pi _\epsilon (2,4,4))&=|\cdot |^{\frac{1}{2}} \otimes \pi _\epsilon (4,4)\\&+|\cdot |^{\frac{3}{2}} \otimes \left( |\cdot |^{\frac{1}{2}}\mathrm {St}_3 \rtimes \pi _+(2) + \epsilon \cdot \pi _+(2,2,4)\right) . \end{aligned}$$

• (2) When $m = 2$, we have $K_\phi ^{(2)} = \{(2,0,0)> (1,1,0)> (0,2,0) > (0,1,1)\}$. Since

  $$\begin{aligned}&\begin{pmatrix} \Delta _{\underline{x}(2,0,0)}&\Delta _{\underline{x}(1,1,0)}&\Delta _{\underline{x}(0,2,0)}&\Delta _{\underline{x}(0,1,1)} \end{pmatrix} \\ {}&= \begin{pmatrix} \mathrm {St}_2&|\cdot |^{\frac{1}{2}} \times |\cdot |^{\frac{3}{2}}&|\cdot |^1 \mathrm {St}_2&|\cdot |^{\frac{3}{2}} \times |\cdot |^{\frac{3}{2}} \end{pmatrix}, \end{aligned}$$

  we have

  $$\begin{aligned} (m_{\underline{k}, \underline{k'}})_{\underline{k}, \underline{k'}}^{-1} = \begin{pmatrix} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 1 &{}\quad 1 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 2 \end{pmatrix}^{-1} =\begin{pmatrix} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad -1 &{}\quad 1 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad \frac{1}{2} \end{pmatrix}. \end{aligned}$$

  Moreover

  $$\begin{aligned} \begin{pmatrix} \Pi _{\underline{x}(2,0,0)}^\epsilon \\ \Pi _{\underline{x}(1,1,0)}^\epsilon \\ \Pi _{\underline{x}(0,2,0)}^\epsilon \\ \\ \Pi _{\underline{x}(0,1,1)}^\epsilon \end{pmatrix} = \begin{pmatrix} 0 \\ |\cdot |^{\frac{1}{2}} \mathrm {St}_3 \rtimes \mathbf {1}_{\mathrm {SO}_1(F)} + \pi _\epsilon (2,4) - \pi _{-\epsilon }(2,4) \\ |\cdot |^1 \mathrm {St}_2 \rtimes \pi _+(2) + |\cdot |^{\frac{1}{2}} \mathrm {St}_3 \rtimes \mathbf {1}_{\mathrm {SO}_1(F)} \\ + \epsilon \left( |\cdot |^{\frac{1}{2}} \rtimes \pi _+(4) + \pi _+(2,4) \right) \\ (1+\epsilon ) \cdot \pi _+(2,2,2) \end{pmatrix}. \end{aligned}$$

  Hence

  $$\begin{aligned} \mathrm {s.s.}\mathrm {Jac}_{P_2}(\pi _\epsilon (2,4,4))&= |{\det }_2|^{1} \otimes \left( |\cdot |^{\frac{1}{2}} \mathrm {St}_3 \rtimes \mathbf {1}_{\mathrm {SO}_1(F)} + \pi _\epsilon (2,4) - \pi _{-\epsilon }(2,4) \right) \\ {}&+ |\cdot |^{1}\mathrm {St}_2 \otimes \Big ( |\cdot |^1 \mathrm {St}_2 \rtimes \pi _+(2) + |\cdot |^{\frac{1}{2}} \mathrm {St}_3 \rtimes \mathbf {1}_{\mathrm {SO}_1(F)} \\ {}&\phantom {|\cdot |^{\frac{1}{2}}\mathrm {St}_2 \otimes \otimes } + \epsilon \left( |\cdot |^{\frac{1}{2}} \rtimes \pi _+(4) + \pi _+(2,4) \right) \Big )\\ \\ {}&+ \left( |\cdot |^{\frac{3}{2}} \times |\cdot |^{\frac{3}{2}} \right) \otimes \frac{1+\epsilon }{2} \cdot \pi _+(2,2,2). \end{aligned}$$

• (3) When $m = 3$, we have $K_\phi ^{(3)} = \{(2,1,0)> (1,2,0)> (1,1,1)> (0,3,0) > (0,2,1)\}$. Since

  $$\begin{aligned}&\begin{pmatrix} \Delta _{\underline{x}(2,1,0)}&\Delta _{\underline{x}(1,2,0)}&\Delta _{\underline{x}(1,1,1)}&\Delta _{\underline{x}(0,3,0)}&\Delta _{\underline{x}(0,2,1)} \end{pmatrix}\\&=\begin{pmatrix} \mathrm {St}_2 \times |\cdot |^{\frac{3}{2}}&|\cdot |^{\frac{1}{2}} \times |\cdot |^{1}\mathrm {St}_2&|\cdot |^{\frac{1}{2}} \times |\cdot |^{\frac{3}{2}} \times |\cdot |^{\frac{3}{2}}&|\cdot |^{\frac{1}{2}} \mathrm {St}_3&|\cdot |^{1}\mathrm {St}_2 \times |\cdot |^{\frac{3}{2}} \end{pmatrix}, \end{aligned}$$

  we have

  $$\begin{aligned} (m_{\underline{k}, \underline{k'}})_{\underline{k}, \underline{k'}}^{-1} = \begin{pmatrix} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 2 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 2 &{}\quad 0 &{}\quad 1 \end{pmatrix}^{-1}= \begin{pmatrix} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad \frac{1}{2} &{}\quad 0 &{}\quad 0\\ -1 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad -1 &{}\quad 0 &{}\quad 1\end{pmatrix}. \end{aligned}$$

  Moreover

  $$\begin{aligned} \begin{pmatrix} \Pi _{\underline{x}(2,1,0)}^\epsilon \\ \Pi _{\underline{x}(1,2,0)}^\epsilon \\ \Pi _{\underline{x}(1,1,1)}^\epsilon \\ \Pi _{\underline{x}(0,3,0)}^\epsilon \\ \Pi _{\underline{x}(0,2,1)}^\epsilon \end{pmatrix}= \begin{pmatrix} 0 \\ |\cdot |^1 \mathrm {St}_2 \rtimes \mathbf {1}_{\mathrm {SO}_1(F)} + \epsilon \cdot \pi _+(4) \\ 2 \cdot \pi _\epsilon (2,2) \\ |\cdot |^{\frac{3}{2}} \rtimes \pi _+(2) + \epsilon \cdot \pi _+(4) \\ (1+\epsilon ) \left( |\cdot |^{\frac{1}{2}} \rtimes \pi _+(2) + \pi _+(2,2) \right) + \pi _-(2,2) \end{pmatrix}. \end{aligned}$$

  Hence

  $$\begin{aligned} \mathrm {s.s.}&\mathrm {Jac}_{P_3}(\pi _\epsilon (2,4,4))\\&= \left( |\cdot |^{\frac{1}{2}} \times |\cdot |^{1}\mathrm {St}_2 \right) \otimes \left( |\cdot |^1 \mathrm {St}_2 \rtimes \mathbf {1}_{\mathrm {SO}_1(F)} + \epsilon \cdot \pi _+(4) \right) \\&+\left( |{\det }_2|^{1} \times |\cdot |^{\frac{3}{2}} \right) \otimes \pi _\epsilon (2,2)\\ {}&+|\cdot |^{\frac{1}{2}} \mathrm {St}_3 \otimes \left( |\cdot |^{\frac{3}{2}} \rtimes \pi _+(2) + \epsilon \cdot \pi _+(4) \right) \\ {}&+ \left( |\cdot |^{1}\mathrm {St}_2 \times |\cdot |^{\frac{3}{2}} \right) \otimes \left( (1+\epsilon ) |\cdot |^{\frac{1}{2}} \rtimes \pi _+(2) +\pi _+(2,2) + \epsilon \cdot \pi _{-\epsilon }(2,2) \right) . \end{aligned}$$

• (4) When $m = 4$, we have $K_\phi ^{(4)} = \{(2,2,0)> (2,1,1)> (1,3,0)> (1,2,1)> (0,4,0)> (0,3,1) > (0,2,2)\}$. Since

  $$\begin{aligned}&\begin{pmatrix} \Delta _{\underline{x}(2,2,0)} \\ \Delta _{\underline{x}(2,1,1)} \\ \Delta _{\underline{x}(1,3,0)} \\ \Delta _{\underline{x}(1,2,1)} \\ \Delta _{\underline{x}(0,4,0)} \\ \Delta _{\underline{x}(0,3,1)} \\ \Delta _{\underline{x}(0,2,2)} \end{pmatrix} = \begin{pmatrix} \mathrm {St}_2 \times |\cdot |^1 \mathrm {St}_2 \\ \mathrm {St}_2 \times |\cdot |^{\frac{3}{2}} \times |\cdot |^{\frac{3}{2}} \\ |\cdot |^{\frac{1}{2}} \times |\cdot |^{\frac{1}{2}} \mathrm {St}_3 \\ |\cdot |^{\frac{1}{2}} \times |\cdot |^{1} \mathrm {St}_2 \times |\cdot |^{\frac{3}{2}} \\ \mathrm {St}_4 \\ |\cdot |^{\frac{1}{2}}\mathrm {St}_3 \times |\cdot |^{\frac{3}{2}} \\ |\cdot |^1 \mathrm {St}_2 \times |\cdot |^1 \mathrm {St}_2 \end{pmatrix}, \end{aligned}$$

  we have

  $$\begin{aligned}&(m_{\underline{k}, \underline{k'}})_{\underline{k}, \underline{k'}}^{-1} = \begin{pmatrix} 1 &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad 0 \\ 0 &{}{}\quad 2 &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad 0 \\ 1 &{}{}\quad 0 &{}{}\quad 1 &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad 0 \\ 0 &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad 1 &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad 0 \\ 0 &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad 1 &{}{}\quad 0 &{}{}\quad 0 \\ 0 &{}{}\quad 2 &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad 1 &{}{}\quad 0 \\ 0 &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad 3 &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad 2 \end{pmatrix}^{-1}\\&= \begin{pmatrix} 1 &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad 0 \\ 0 &{}{}\quad \frac{1}{2} &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad 0 \\ -1 &{}{}\quad 0 &{}{}\quad 1 &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad 0 \\ 0 &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad 1 &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad 0 \\ 0 &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad 1 &{}{}\quad 0 &{}{}\quad 0 \\ 0 &{}{}\quad -1 &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad 1 &{}{}\quad 0 \\ 0 &{}{}\quad 0 &{}{}\quad 0 &{}{}\quad -\frac{3}{2} &{}{}\quad 0 &{}{}\quad 0&{}{}\quad \frac{1}{2} \end{pmatrix}. \end{aligned}$$

  Moreover

  $$\begin{aligned} \begin{pmatrix} \Pi _{\underline{x}(2,2,0)}^\epsilon \\ \Pi _{\underline{x}(2,1,1)}^\epsilon \\ \Pi _{\underline{x}(1,3,0)}^\epsilon \\ \Pi _{\underline{x}(1,2,1)}^\epsilon \\ \Pi _{\underline{x}(0,4,0)}^\epsilon \\ \Pi _{\underline{x}(0,3,1)}^\epsilon \\ \Pi _{\underline{x}(0,2,2)}^\epsilon \end{pmatrix}= \begin{pmatrix} 0 \\ 0 \\ |\cdot |^{\frac{3}{2}} \rtimes \mathbf {1}_{\mathrm {SO}_1(F)} \\ |\cdot |^{\frac{1}{2}} \rtimes \mathbf {1}_{\mathrm {SO}_1(F)} + \epsilon \cdot \pi _+(2) \\ \pi _+(2) \\ (1+\epsilon ) \cdot \pi _+(2) \\ (3+2\epsilon ) |\cdot |^{\frac{1}{2}} \rtimes \mathbf {1}_{\mathrm {SO}_1(F)} + (1+2\epsilon ) \cdot \pi _+(2) \end{pmatrix}. \end{aligned}$$

  Hence

  $$\begin{aligned} \mathrm {s.s.}&\mathrm {Jac}_{P_4}(\pi _\epsilon (2,4,4))\\ {}&= \left( |\cdot |^{\frac{1}{2}} \times |\cdot |^{\frac{1}{2}} \mathrm {St}_3 \right) \otimes |\cdot |^{\frac{3}{2}} \rtimes \mathbf {1}_{\mathrm {SO}_1(F)}\\ {}&+\left( |\cdot |^1 \mathrm {St}_2 \times |{\det }_2|^1 \right) \otimes \left( |\cdot |^{\frac{1}{2}} \rtimes \mathbf {1}_{\mathrm {SO}_1(F)} + \epsilon \cdot \pi _+(2) \right) \\ {}&+\mathrm {St}_4 \otimes \pi _+(2)\\ {}&+\left( |\cdot |^{\frac{1}{2}}\mathrm {St}_3 \times |\cdot |^{\frac{3}{2}} \right) \otimes (1+\epsilon ) \cdot \pi _+(2)\\ {}&+\left( |\cdot |^1 \mathrm {St}_2 \times |\cdot |^1 \mathrm {St}_2 \right) \otimes \left( (1+\epsilon ) |\cdot |^{\frac{1}{2}} \rtimes \mathbf {1}_{\mathrm {SO}_1(F)} + \frac{1+\epsilon }{2} \cdot \pi _+(2) \right) . \end{aligned}$$

• (5) When $m = 5$, we have $K_\phi ^{(5)} = \{(2,3,0)> (2,2,1)> (1,4,0)> (1,3,1)> (1,2,2)> (0,4,1) > (0,3,2) \}$. Since

  $$\begin{aligned}&\begin{pmatrix} \Delta _{\underline{x}(2,3,0)} \\ \Delta _{\underline{x}(2,2,1)} \\ \Delta _{\underline{x}(1,4,0)} \\ \Delta _{\underline{x}(1,3,1)} \\ \Delta _{\underline{x}(1,2,2)} \\ \Delta _{\underline{x}(0,4,1)} \\ \Delta _{\underline{x}(0,3,2)} \end{pmatrix}= \begin{pmatrix} \mathrm {St}_2 \times |\cdot |^{\frac{1}{2}} \mathrm {St}_3 \\ \mathrm {St}_2 \times |\cdot |^{1}\mathrm {St}_2 \times |\cdot |^{\frac{3}{2}} \\ |\cdot |^{\frac{1}{2}} \times \mathrm {St}_4 \\ |\cdot |^{\frac{1}{2}} \times |\cdot |^{\frac{1}{2}} \mathrm {St}_3 \times |\cdot |^{\frac{3}{2}} \\ |\cdot |^{\frac{1}{2}} \times |\cdot |^1 \mathrm {St}_2 \times |\cdot |^1 \mathrm {St}_2 \\ \mathrm {St}_4 \times |\cdot |^{\frac{3}{2}} \\ |\cdot |^{\frac{1}{2}}\mathrm {St}_3 \times |\cdot |^{1}\mathrm {St}_2 \end{pmatrix}, \end{aligned}$$

  we have

  $$\begin{aligned} (m_{\underline{k}, \underline{k'}})_{\underline{k}, \underline{k'}}^{-1} = \begin{pmatrix} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 2 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 \\ 0 &{}\quad 2 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 1 \end{pmatrix}^{-1}= \begin{pmatrix} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad -1 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \frac{1}{2} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 \\ 0 &{}\quad -1 &{}\quad 0 &{}\quad -1 &{}\quad 0 &{}\quad 0 &{}\quad 1 \end{pmatrix}. \end{aligned}$$

  Moreover

  $$\begin{aligned} \begin{pmatrix} \Pi _{\underline{x}(2,3,0)}^\epsilon \\ \Pi _{\underline{x}(2,2,1)}^\epsilon \\ \Pi _{\underline{x}(1,4,0)}^\epsilon \\ \Pi _{\underline{x}(1,3,1)}^\epsilon \\ \Pi _{\underline{x}(1,2,2)}^\epsilon \\ \Pi _{\underline{x}(0,4,1)}^\epsilon \\ \Pi _{\underline{x}(0,3,2)}^\epsilon \end{pmatrix}= \begin{pmatrix} 0 \\ 0 \\ \mathbf {1}_{\mathrm {SO}_1(F)} \\ \mathbf {1}_{\mathrm {SO}_1(F)} \\ (1+\epsilon ) \cdot \mathbf {1}_{\mathrm {SO}_1(F)} \\ 0 \\ (1+\epsilon ) \cdot \mathbf {1}_{\mathrm {SO}_1(F)} \end{pmatrix}. \end{aligned}$$

  Hence

  $$\begin{aligned} \mathrm {s.s.}\mathrm {Jac}_{P_5}(\pi _\epsilon (2,4,4))&= \left( |\cdot |^{\frac{1}{2}} \times \mathrm {St}_4 \right) \otimes \mathbf {1}_{\mathrm {SO}_1(F)} \\ {}&+ \left( |\cdot |^{\frac{1}{2}} \mathrm {St}_3 \times |{\det }_2|^1 \right) \otimes \mathbf {1}_{\mathrm {SO}_1(F)} \\ {}&+ \left( |\cdot |^{\frac{1}{2}} \times |\cdot |^1 \mathrm {St}_2 \times |\cdot |^1 \mathrm {St}_2 \right) \otimes \frac{1+\epsilon }{2} \cdot \mathbf {1}_{\mathrm {SO}_1(F)} \\ {}&+ \left( |\cdot |^{\frac{1}{2}}\mathrm {St}_3 \times |\cdot |^{1}\mathrm {St}_2 \right) \otimes (1+\epsilon ) \cdot \mathbf {1}_{\mathrm {SO}_1(F)}. \end{aligned}$$

Remark 4.7

In Theorem 4.3, one can replace $\Delta _{\underline{x}(\underline{k})}$ with its unique irreducible subrepresentation $\tau _{\underline{x}(\underline{k})}$. Then one should consider the matrix $(M_{\underline{k}, \underline{l}}) = (\dim _\mathbb {C}\mathrm {Jac}_{\rho |\cdot |^{\underline{x}(\underline{k})}}(\tau _{\underline{x}(\underline{l})}))$. One might seem that $(M_{\underline{k}, \underline{l}})$ is easier than $(m_{\underline{k}, \underline{l}})$. For instance, if $\phi$ is in Example 4.5, all $(M_{\underline{k}, \underline{l}})$ are the identity matrix, but not so is some $(m_{\underline{k}, \underline{l}})$. However, in general, $(M_{\underline{k}, \underline{l}})$ is not always diagonal. In Example 4.6 (3) and (4), such non-diagonal $(M_{\underline{k}, \underline{l}})$ would appear.

5 Proof of the main theorems

In this section, we prove the main theorems (Theorems 4.2 and 4.3).

5.1 The case of higher multiplicity

We prove Theorem 4.2 (1) in this subsection. It immediately follows from Tadić’s formula (Corollary 2.9).

Proof of Theorem 4.2 (1)

We prove the assertion by induction on m. Since

$$\begin{aligned} \pi (\phi , \eta ) = \mathrm {St}(\rho , 2x+1) \rtimes \pi (\phi - (\rho \boxtimes S_{2x+1})^{\oplus 2}, \eta ) \end{aligned}$$

with $\mathrm {St}(\rho , 2x+1) = \left\langle \rho ; x,\dots ,-x \right\rangle$, by Corollary 2.9, $\mathrm {s.s.}\mathrm {Jac}_{P_d}(\pi (\phi , \eta ))$ is equal to

$$\begin{aligned}&(\rho ^\vee |\cdot |^x \otimes \left\langle \rho ; x,\dots ,-(x-1) \right\rangle ) \rtimes (\mathbf {1}_{\mathrm {GL}_0(F)} \otimes \pi (\phi - (\rho \boxtimes S_{2x+1})^{\oplus 2}, \eta )) \\ {}&+ (\rho |\cdot |^x \otimes \left\langle \rho ; x-1,\dots ,-x \right\rangle ) \rtimes (\mathbf {1}_{\mathrm {GL}_0(F)} \otimes \pi (\phi - (\rho \boxtimes S_{2x+1})^{\oplus 2}, \eta )) \\ {}&+ (\mathbf {1}_{\mathrm {GL}_0(F)} \otimes \mathrm {St}(\rho , 2x+1)) \rtimes \mathrm {s.s.}\mathrm {Jac}_{P_d}(\pi (\phi - (\rho \boxtimes S_{2x+1})^{\oplus 2}, \eta )). \end{aligned}$$

Note that $\mathrm {s.s.}(\left\langle \rho ; x-1,\dots ,-x \right\rangle \rtimes \pi _0) \cong \mathrm {s.s.}(\left\langle \rho ^\vee ; x,\dots ,-(x-1) \right\rangle \rtimes \pi _0)$ for any representation $\pi _0$. Since $\rho ^\vee \cong \rho$, we have

$$\begin{aligned} \mathrm {Jac}_{\rho |\cdot |^x} (\pi (\phi , \eta ))&= 2 \cdot \left\langle \rho ; x,\dots ,-(x-1) \right\rangle \rtimes \pi \left( \phi - (\rho \boxtimes S_{2x+1})^{\oplus 2}, \eta \right) \\&+\mathrm {St}(\rho , 2x+1) \rtimes \mathrm {Jac}_{\rho |\cdot |^{x}} \left( \pi \left( \phi - (\rho \boxtimes S_{2x+1})^{\oplus 2}, \eta \right) \right) . \end{aligned}$$

This proves the assertion when $m=3$ or $m=4$. When $m \ge 5$, since

$$\begin{aligned}&\mathrm {St}(\rho , 2x+1) \times \left\langle \rho ; x,\dots ,-(x-1) \right\rangle \rtimes \pi \left( \phi - (\rho \boxtimes S_{2x+1})^{\oplus 4}, \eta \right) \\&\cong \left\langle \rho ; x,\dots ,-(x-1) \right\rangle \times \mathrm {St}(\rho , 2x+1) \rtimes \pi \left( \phi - (\rho \boxtimes S_{2x+1})^{\oplus 4}, \eta \right) \\&\cong \left\langle \rho ; x,\dots ,-(x-1) \right\rangle \rtimes \pi \left( \phi - (\rho \boxtimes S_{2x+1})^{\oplus 2}, \eta \right) , \end{aligned}$$

we obtain the assertion by the induction hypothesis. $\square$

5.2 The case of multiplicity one

Next, we prove Theorem 4.2 (2). Let $\phi = \phi _0 \oplus (\rho \boxtimes S_{2x+1})$ with $\rho \boxtimes S_{2x+1} \not \subset \phi _0$, and $\eta \in \widehat{\mathcal {A}_\phi }$. Set

$$\begin{aligned} \phi ' = \phi - (\rho \boxtimes S_{2x+1}) \oplus (\rho \boxtimes S_{2x-1}). \end{aligned}$$

Proof of Theorem 4.2 (2)

First, we assume that $x > 1/2$ and $\pi (\phi ', \eta ') \not = 0$. We apply Mœglin’s construction to $\Pi _{\phi '}$. Write

$$\begin{aligned} \phi ' = \left( \bigoplus _{i=1}^{t} \rho \boxtimes S_{a_i} \right) \oplus \phi '_e \end{aligned}$$

with $a_1 \le \dots \le a_t$ and $\rho \boxtimes S_a \not \subset \phi '_e$ for any $a>0$. Set

$$\begin{aligned} t_0 = \max \{i \in \{1, \dots , t\}\ |\ a_i = 2x-1\}. \end{aligned}$$

Take a new L-parameter

$$\begin{aligned} \phi '_\gg = \left( \bigoplus _{i=1}^{t} \rho \boxtimes S_{a'_i} \right) \oplus \phi '_e \end{aligned}$$

such that

• $a'_1< \dots < a'_t$;

• $a_i' \ge a_i$ and $a'_i \equiv a_i \bmod 2$ for any i;

• $a'_{t_0} \ge 2x+1$.

We can identify $\mathcal {A}_{\phi '_\gg }$ with $\mathcal {A}_{\phi '}$ canonically. Let $\eta '_\gg \in \widehat{\mathcal {A}_{\phi '_\gg }}$ be the character corresponding to $\eta ' \in \widehat{\mathcal {A}_{\phi '}}$. Then Theorem 3.5 says that

$$\begin{aligned} \pi (\phi ', \eta ') = \mathrm {Jac}_{\rho |\cdot |^{\frac{a_t'-1}{2}}, \dots , \rho |\cdot |^\frac{a_t+1}{2}} \circ \dots \circ \mathrm {Jac}_{\rho |\cdot |^{\frac{a_1'-1}{2}}, \dots , \rho |\cdot |^\frac{a_1+1}{2}} (\pi (\phi '_\gg , \eta '_\gg )). \end{aligned}$$

When $i=t_0$, we note that

$$\begin{aligned} \mathrm {Jac}_{\rho |\cdot |^{\frac{a_{t_0}'-1}{2}}, \dots , \rho |\cdot |^\frac{a_{t_0}+1}{2}} = \mathrm {Jac}_{\rho |\cdot |^{x}} \circ \mathrm {Jac}_{\rho |\cdot |^{\frac{a_{t_0}'-1}{2}}, \dots , \rho |\cdot |^\frac{a_{t_0}+3}{2}}. \end{aligned}$$

Since $\phi '$ does not contain $\rho \boxtimes S_{2x+1}$, for $i>t_0$ and $a_i < 2x'+1 \le a'_i$ with $2x'+1 \equiv a_i \bmod 2$, we have $x'-x > 1$. By Lemma 2.6 (2), we see that $\pi (\phi ', \eta ')$ is the image of

$$\begin{aligned} \mathrm {Jac}_{\rho |\cdot |^{\frac{a_t'-1}{2}}, \dots , \rho |\cdot |^\frac{a_t+1}{2}} \circ \dots \circ \mathrm {Jac}_{\rho |\cdot |^{\frac{a_{t_0}'-1}{2}}, \dots , \rho |\cdot |^\frac{a_{t_0}+3}{2}} \circ \dots \circ \mathrm {Jac}_{\rho |\cdot |^{\frac{a_1'-1}{2}}, \dots , \rho |\cdot |^\frac{a_1+1}{2}} (\pi (\phi '_\gg , \eta '_\gg )) \end{aligned}$$

under $\mathrm {Jac}_{\rho |\cdot |^x}$. However, by applying Theorem 3.5 again, we see that this representation is isomorphic to $\pi (\phi , \eta )$. Therefore $\pi (\phi ', \eta ') = \mathrm {Jac}_{\rho |\cdot |^x} (\pi (\phi , \eta ))$, as desired.

Next, we consider the case where $x= 1/2$ and $\pi '(\phi ', \eta ') \not = 0$. We reduce this case to the case where $\phi$ is discrete ( [16, Lemma 7.3]) using Theorem 3.5 repeatedly. The argument is similar to the first case so that we omit the detail.

Finally, we assume that $\pi (\phi ', \eta ') = 0$. We claim that $\mathrm {Jac}_{\rho |\cdot |^x}(\pi (\phi , \eta )) = 0$. When $\phi \in \Phi _\mathrm {disc}(G)$, this was proven in [16, Lemma 7.3]. When $\phi \in \Phi _\mathrm {gp}(G) \setminus \Phi _\mathrm {disc}(G)$, there exists an irreducible representation $\phi _1$ which is contained in $\phi$ with multiplicity at least two. Then $\pi (\phi , \eta )$ is a subrepresentation of $\tau _1 \rtimes \pi (\phi -\phi _1^{\oplus 2}, \eta )$, where $\tau _1$ is the irreducible tempered representation of $\mathrm {GL}_k(F)$ corresponding to $\phi _1$. Since $\rho \boxtimes S_{2x+1}$ is contained in $\phi$ with multiplicity one, we have $\phi _1 \not \cong \rho \boxtimes S_{2x+1}$. This implies that

$$\begin{aligned} \mathrm {Jac}_{\rho |\cdot |^x}(\tau _1 \rtimes \pi (\phi -\phi _1^{\oplus 2}, \eta )) = \tau _1 \rtimes \mathrm {Jac}_{\rho |\cdot |^x}(\pi (\phi -\phi _1^{\oplus 2}, \eta )). \end{aligned}$$

By the induction hypothesis, $\mathrm {Jac}_{\rho |\cdot |^x}(\pi (\phi -\phi _1^{\oplus 2}, \eta )) = 0$ unless $\phi _1 = \rho \boxtimes S_{2x-1}$ and $\phi$ contains it with multiplicity exactly two. In this case, one can take $\eta _- \in \widehat{\mathcal {A}_\phi }$ such that $\pi (\phi , \eta _-) \not = 0$ and

$$\begin{aligned} \pi (\phi , \eta ) \oplus \pi (\phi , \eta _-) = \mathrm {St}(\rho , 2x-1) \rtimes \pi (\phi -\phi _1^{\oplus 2}, \eta ). \end{aligned}$$

Then by the first case, we see that $\mathrm {Jac}_{\rho |\cdot |^x}(\pi (\phi , \eta _-)) \not = 0$ and $\mathrm {St}(\rho , 2x-1) \rtimes \mathrm {Jac}_{\rho |\cdot |^x}(\pi (\phi -\phi _1^{\oplus 2}, \eta ))$ is irreducible. Hence $\mathrm {Jac}_{\rho |\cdot |^x}(\pi (\phi , \eta ))$ must be zero. This completes the proof of Theorem 4.2 (2).$\square$

By the same argument as the last part, one can prove that $\mathrm {Jac}_{\rho }(\pi (\phi , \eta )) = 0$ when $x = 0$ and $m = 1$.

5.3 Description of small standard modules

Before proving Theorem 4.2 (3), we describe the structures of some standard modules.

Lemma 5.1

Let $\phi \in \Phi _\mathrm {disc}(G)$. Suppose that $x > 0$, and $\phi \supset \rho \boxtimes S_{2x-1}$ but $\phi \not \supset \rho \boxtimes S_{2x+1}$. Let $\eta \in \widehat{\mathcal {A}_\phi }$ such that $\pi (\phi , \eta ) \not = 0$. We set

• $\Pi = \rho |\cdot |^x \rtimes \pi (\phi , \eta )$ to be a standard module;

• $\sigma$ to be the unique irreducible quotient of $\Pi$;

• $\phi ' = \phi - (\rho \boxtimes S_{2x-1}) \oplus (\rho \boxtimes S_{2x+1})$ so that there is a canonical injection $\mathcal {A}_\phi \hookrightarrow \mathcal {A}_{\phi '}$, which is bijection unless $x=1/2$;

• $\eta ' \in \widehat{\mathcal {A}_{\phi '}}$ to be the character satisfying $\eta '|\mathcal {A}_\phi = \eta$ (and $\pi (\phi ', \eta ') \not = 0$ if $x=1/2$).

Then there exists an exact sequence

In particular, $\Pi$ has length two.

Proof

We note that $\pi (\phi ', \eta ')$ is an irreducible subrepresentation of $\Pi$ by Theorem 4.2 (2) and Lemma 2.6 (1).

If $\sigma '$ is an irreducible subquotient of $\Pi$ which is non-tempered, by Tadić’s formula and Casselman’s criterion, there exists a maximal parabolic subgroup $P_k$ of G such that $\mathrm {s.s.}\mathrm {Jac}_{P_k}(\sigma ')$ contains an irreducible representation of the form $(\rho |\cdot |^{-x} \times \tau ) \boxtimes \sigma _0$. In particular, we have $\mathrm {Jac}_{\rho |\cdot |^{-x}}(\sigma ') \not = 0$. However, since $\mathrm {Jac}_{\rho |\cdot |^{-x}}(\Pi ) = \pi (\phi , \eta )$ is irreducible, we see that $\sigma ' = \sigma$, i.e., $\Pi$ has only one irreducible non-tempered subquotient.

Let $\Pi ^{\mathrm {sub}}$ be the maximal proper subrepresentation of $\Pi$, i.e., $\Pi /\Pi ^{\mathrm {sub}} \cong \sigma$. By the above argument, all irreducible subquotients of $\Pi ^\mathrm {sub}$ must be tempered. Since they have the same cuspidal support, they share the same Plancherel measure. This implies that all irreducible subquotients of $\Pi ^\mathrm {sub}$ belong to the same L-packet $\Pi _{\phi '}$ (see [6, Lemma A.6]), so that they are all discrete series. Hence $\Pi ^\mathrm {sub}$ is semisimple. In particular, any irreducible subquotient $\pi '$ of $\Pi ^\mathrm {sub}$ is a subrepresentation of $\Pi$, so that $\mathrm {Jac}_{\rho |\cdot |^x}(\pi ') \not = 0$. However, since $\mathrm {Jac}_{\rho |\cdot |^{x}}(\Pi ) = \pi (\phi , \eta )$ is irreducible, $\Pi$ has only one irreducible subrepresentation. Therefore $\Pi ^\mathrm {sub}= \pi (\phi ', \eta ')$. This completes the proof. $\square$

We describe the standard module appearing in Theorem 4.2 (3). When $x=1/2$, the standard module was described in Lemma 5.1. Hence we assume $x > 1/2$.

Proposition 5.2

Let $\phi \in \Phi _\mathrm {gp}(G)$. Suppose that $x > 1/2$ and $\phi \not \supset \rho \boxtimes S_{2x+1}$. Let $\eta \in \widehat{\mathcal {A}_\phi }$ such that $\pi (\phi , \eta ) \not = 0$. We set

• $\Pi = \left\langle \rho ; x, x-1, \dots , -(x-1) \right\rangle \rtimes \pi (\phi , \eta )$ to be a standard module;

• $\sigma$ to be the unique irreducible quotient of $\Pi$;

• $\phi ' = \phi \oplus (\rho \boxtimes S_{2x-1}) \oplus (\rho \boxtimes S_{2x+1})$;

• $\eta '_+$ and $\eta '_-$ to be the two distinct characters of $\widehat{\mathcal {A}_{\phi '}}$ such that $\eta '_\pm |\mathcal {A}_{\phi } = \eta$ and $\eta '_\pm (z_{\phi '}) = 1$.

Then there exists an exact sequence

In particular, $\Pi$ has length 2 or 3 according to $\phi \supset \rho \boxtimes S_{2x-1}$ or not.

Proof

First, we show that there is an inclusion $\pi (\phi ', \eta '_\epsilon ) \hookrightarrow \Pi$ for each $\epsilon \in \{\pm \}$. To do this, we may assume that $\pi (\phi ', \eta '_\epsilon ) \not = 0$. Note that $\phi '$ contains $\rho \boxtimes S_{2x+1}$ with multiplicity one. By Theorem 4.2 (2), we see that $\mathrm {Jac}_{\rho |\cdot |^x}(\pi (\phi ', \eta '_\epsilon ))$ is nonzero and is an irreducible subrepresentation of $\mathrm {St}(\rho , 2x-1) \rtimes \pi (\phi , \eta )$. By Lemma 2.6 (1), we have an inclusion

$$\begin{aligned} \pi (\phi ', \eta '_\epsilon ) \hookrightarrow \rho |\cdot |^x \times \mathrm {St}(\rho , 2x-1) \rtimes \pi (\phi , \eta ). \end{aligned}$$

Since $\Pi$ is a subrepresentation of $\rho |\cdot |^x \times \mathrm {St}(\rho , 2x-1) \rtimes \pi (\phi , \eta )$ such that

$$\begin{aligned} \mathrm {Jac}_{\rho |\cdot |^x}\left( \rho |\cdot |^x \times \mathrm {St}(\rho , 2x-1) \rtimes \pi (\phi , \eta ) \right) = \mathrm {Jac}_{\rho |\cdot |^x}(\Pi ), \end{aligned}$$

the above inclusion factors through $\pi (\phi ', \eta '_\epsilon ) \hookrightarrow \Pi$.

If $\mathrm {s.s.}\mathrm {Jac}_{P_k}(\Pi )$ contains an irreducible representation $\tau \boxtimes \pi _0$ such that the central character of $\tau$ is of the form $\chi |\cdot |^{s}$ with $\chi$ unitary and $s<0$, by Tadić’s formula (Theorem 2.8) and Casselman’s criterion, $\tau = |\cdot |^{-\frac{1}{2}}\mathrm {St}(\rho , 2x) \times (\times _{i =1}^r \tau _i)$, where $\tau _i$ is a discrete series representation of $\mathrm {GL}_{k_i}(F)$ such that the corresponding irreducible representation $\phi _i$ of $W_F \times \mathrm {SL}_2(\mathbb {C})$ is contained in $\phi$ with multiplicity at least two, and $\pi _0 = \pi (\phi _0, \eta _0)$ with $\phi _0 = \phi - (\oplus _{i=1}^r \phi _i)^{\oplus 2}$ and $\eta _0 = \eta |\mathcal {A}_{\phi _0}$. Since such an irreducible representation $\tau \boxtimes \pi _0$ is also contained in $\mathrm {s.s.}\mathrm {Jac}_{P_k}(\sigma )$, we see that $\sigma$ is the unique irreducible non-tempered subquotient of $\Pi$. Namely, if we let $\Pi ^\mathrm {sub}$ be the maximal proper subrepresentation of $\Pi$, i.e., $\Pi /\Pi ^\mathrm {sub}\cong \sigma$, then all irreducible subquotients of $\Pi ^\mathrm {sub}$ must be tempered. Moreover since these irreducible subquotients have the same cuspidal support so that they share the same Plancherel measure, they belong to the same L-packet $\Pi _{\phi '}$ (see [6, Lemma A.6]).

Now we show that $\Pi ^\mathrm {sub}$ is isomorphic to $\pi (\phi ', \eta '_+) \oplus \pi (\phi ', \eta '_-)$. We separate the cases as follows:

• $\phi$ is discrete and $\rho \boxtimes S_{2x-1} \not \subset \phi$;

• $\phi$ is discrete and $\rho \boxtimes S_{2x-1} \subset \phi$;

• $\phi$ is in general.

When $\phi$ is discrete and $\rho \boxtimes S_{2x-1} \not \subset \phi$, we note that $\phi ' = \phi \oplus (\rho \boxtimes S_{2x-1}) \oplus (\rho \boxtimes S_{2x+1})$ is also discrete. Then since all irreducible subquotients of $\Pi ^\mathrm {sub}$ are discrete series, $\Pi ^\mathrm {sub}$ is semisimple. In particular, any irreducible subquotient $\pi '$ of $\Pi ^\mathrm {sub}$ is a subrepresentation of $\Pi$ so that $\mathrm {Jac}_{\rho |\cdot |^x}(\pi ') \not = 0$. However, since $\mathrm {Jac}_{\rho |\cdot |^x}(\Pi ) = \mathrm {Jac}_{\rho |\cdot |^x}(\pi (\phi ', \eta '_+) \oplus \pi (\phi ', \eta '_-))$, we have $\Pi ^\mathrm {sub}\cong \pi (\phi ', \eta '_+) \oplus \pi (\phi ', \eta '_-)$.

When $\phi$ is discrete and $\rho \boxtimes S_{2x-1} \subset \phi$, any irreducible subquotient $\pi '$ of $\Pi ^\mathrm {sub}$ belongs to $\Pi _{\phi '}$ with $\phi ' = \phi _0' \oplus (\rho \boxtimes S_{2x-1})^{\oplus 2}$, where $\phi _0' = \phi - (\rho \boxtimes S_{2x-1}) \oplus (\rho \boxtimes S_{2x+1})$ is discrete such that $\rho \boxtimes S_{2x-1} \not \subset \phi _0'$. Hence the Jacquet module $\mathrm {s.s.}\mathrm {Jac}_{P_{d(2x-1)}}(\pi ')$ contains an irreducible representation of the form $\mathrm {St}(\rho , 2x-1) \otimes \pi _0'$. By Tadić’s formula (Corollary 2.9), the sum of irreducible representations of this form appearing in $\mathrm {s.s.}\mathrm {Jac}_{P_{d(2x-1)}}(\Pi )$ is

$$\begin{aligned} \mathrm {St}(\rho , 2x-1) \otimes \mathrm {s.s.}( \rho |\cdot |^x \rtimes \pi (\phi , \eta ) ). \end{aligned}$$

By Lemma 5.1, we have an exact sequence

where $\sigma '$ is the unique irreducible quotient of $\rho |\cdot |^x \rtimes \pi (\phi , \eta )$, and $\eta _0' \in \widehat{\mathcal {A}_{\phi _0'}}$ is the character corresponding to $\eta \in \widehat{\mathcal {A}_\phi }$ via the identification $\mathcal {A}_\phi = \mathcal {A}_{\phi _0'}$. Now there exists $\epsilon \in \{\pm \}$ such that $\pi (\phi ', \eta '_{-\epsilon }) = 0$. Moreover, $\mathrm {s.s.}\mathrm {Jac}_{P_{d(2x-1)}}(\pi (\phi ', \eta '_\epsilon )) \supset \mathrm {St}(\rho , 2x-1) \otimes \pi (\phi _0', \eta _0')$ since

$$\begin{aligned} \eta '_\epsilon (\alpha _{\rho \boxtimes S_{2x+1}}) = \eta '_\epsilon (\alpha _{\rho \boxtimes S_{2x-1}}) = \eta (\alpha _{\rho \boxtimes S_{2x-1}}) = \eta _0'(\alpha _{\rho \boxtimes S_{2x+1}}). \end{aligned}$$

On the other hand, since $\sigma \hookrightarrow \mathrm {St}(\rho , 2x-1) \times \rho |\cdot |^{-x} \rtimes \pi (\phi , \eta )$, we see that $\mathrm {s.s.}\mathrm {Jac}_{P_{d(2x-1)}}(\sigma )$ is nonzero and contains $\mathrm {St}(\rho , 2x-1) \otimes \sigma '$. Hence

$$\begin{aligned} \mathrm {s.s.}\mathrm {Jac}_{P_{d(2x-1)}}(\Pi ) - \mathrm {s.s.}\mathrm {Jac}_{P_{d(2x-1)}}(\pi (\phi ', \eta '_\epsilon )) - \mathrm {s.s.}\mathrm {Jac}_{P_{d(2x-1)}}(\sigma ) \end{aligned}$$

has no irreducible representation of the form $\mathrm {St}(\rho , 2x-1) \otimes \pi _0'$. This shows that $\Pi ^\mathrm {sub}= \pi (\phi ', \eta '_\epsilon )$.

In general, we prove the claim by induction on the dimension of $\phi$. When $\phi$ is not discrete, there exists an irreducible representation $\phi _1$ of $W_F \times \mathrm {SL}_2(\mathbb {C})$ which $\phi$ contains with multiplicity at least two. Note that $\phi _1 \not \cong \rho \boxtimes S_{2x+1}$. Set $\phi _0 = \phi - \phi _1^{\oplus 2}$, and $\eta _0 = \eta |\mathcal {A}_{\phi _0}$. Take $\Pi _0$, $\sigma _0$, $\phi _0'$ and $\eta '_{0,\epsilon } \in \widehat{\mathcal {A}_{\phi _0'}}$ as in the statement of the proposition. By induction hypothesis, we have an exact sequence

Let $\tau$ be the irreducible discrete series representation of $\mathrm {GL}_k(F)$ corresponding to $\phi _1$. The above exact sequence remains exact after taking the parabolic induction functor $\pi _0 \mapsto \tau \rtimes \pi _0$. Note that $\tau \times \left\langle \rho ; x, x-1, \dots , -(x-1) \right\rangle \cong \left\langle \rho ; x, x-1, \dots , -(x-1) \right\rangle \times \tau$ by Theorem 2.1. Since $\sigma _0$ is unitary, the parabolic induction $\tau \rtimes \sigma _0$ is semisimple. In particular, any irreducible subquotient of $\tau \rtimes \sigma _0$ is non-tempered. Considering the cases where

• $\phi$ contains $\phi _1$ with multiplicity more than two;

• $\phi$ contains $\phi _1$ with multiplicity exactly two and $\phi _1 \not \cong \rho \boxtimes S_{2x-1}$;

• $\phi$ contains $\phi _1$ with multiplicity exactly two and $\phi _1 \cong \rho \boxtimes S_{2x-1}$

separately, we see that $\Pi ^\mathrm {sub}\cong \pi (\phi ', \eta '_+) \oplus \pi (\phi ', \eta '_-)$ in all cases. This completes the proof. $\square$

5.4 The case of multiplicity two

Finally, we prove Theorem 4.2 (3).

Lemma 5.3

Let $\phi \in \Phi _\mathrm {gp}(G)$, $\eta \in \widehat{\mathcal {A}_\phi }$ and $x > 0$. Suppose that $\phi$ contains both $\rho \boxtimes S_{2x+1}$ and $\rho \boxtimes S_{2x+3}$ with multiplicity one. Then we have

$$\begin{aligned} \mathrm {Jac}_{\rho |\cdot |^{x+1}, \rho |\cdot |^{x}, \rho |\cdot |^{x}}(\pi (\phi , \eta )) \subset 2 \cdot \mathrm {Jac}_{\rho |\cdot |^{x}, \rho |\cdot |^{x+1}, \rho |\cdot |^{x}}(\pi (\phi , \eta )). \end{aligned}$$

Proof

We may assume that $\mathrm {Jac}_{\rho |\cdot |^{x+1}, \rho |\cdot |^{x}, \rho |\cdot |^{x}}(\pi (\phi , \eta )) \not = 0$. By Lemma 2.6 (1), there exists an irreducible subquotient $\sigma$ of this Jacquet module such that

$$\begin{aligned} \pi (\phi , \eta ) \hookrightarrow \rho |\cdot |^{x+1} \times \rho |\cdot |^{x} \times \rho |\cdot |^{x} \rtimes \sigma . \end{aligned}$$

Since there exists an exact sequence

where $\left\langle \rho ; x,x+1 \right\rangle$ is the unique irreducible subrepresentation of $\rho |\cdot |^{x} \times \rho |\cdot |^{x+1}$, we see that $\pi (\phi , \eta )$ is a subrepresentation of $\left\langle \rho ; x+1,x \right\rangle \times \rho |\cdot |^{x} \rtimes \sigma$ or $\left\langle \rho ; x,x+1 \right\rangle \times \rho |\cdot |^{x} \rtimes \sigma$. Since $\left\langle \rho ; x+1,x \right\rangle \times \rho |\cdot |^{x} \cong \rho |\cdot |^x \times \left\langle \rho ; x+1,x \right\rangle$ and $\left\langle \rho ; x,x+1 \right\rangle \times \rho |\cdot |^{x} \cong \rho |\cdot |^x \times \left\langle \rho ; x,x+1 \right\rangle$, we have $\mathrm {Jac}_{\rho |\cdot |^x}(\pi (\phi , \eta )) \not = 0$.

By Theorem 4.2 (2), $\mathrm {Jac}_{\rho |\cdot |^{x+1}}(\pi (\phi , \eta )) \not = 0$ and $\mathrm {Jac}_{\rho |\cdot |^x}(\pi (\phi , \eta )) \not = 0$ imply that $\sigma ' = \mathrm {Jac}_{\rho |\cdot |^{x}, \rho |\cdot |^{x+1}, \rho |\cdot |^{x}}(\pi (\phi , \eta ))$ is nonzero and irreducible. Moreover, we have $\mathrm {Jac}_{\rho |\cdot |^{x}}(\sigma ') = 0$ and $\mathrm {Jac}_{\rho |\cdot |^{x+1}}(\sigma ') = 0$. By Lemma 2.6 (1), we have an inclusion

$$\begin{aligned} \pi (\phi , \eta ) \hookrightarrow \rho |\cdot |^{x} \times \rho |\cdot |^{x+1} \times \rho |\cdot |^{x} \rtimes \sigma '. \end{aligned}$$

Since

$$\begin{aligned} \mathrm {Jac}_{\rho |\cdot |^{x+1}, \rho |\cdot |^{x}, \rho |\cdot |^{x}}(\rho |\cdot |^{x} \times \rho |\cdot |^{x+1} \times \rho |\cdot |^{x} \rtimes \sigma ') = 2 \cdot \sigma ', \end{aligned}$$

we have $\mathrm {Jac}_{\rho |\cdot |^{x+1}, \rho |\cdot |^{x}, \rho |\cdot |^{x}}(\pi (\phi , \eta )) \subset 2 \cdot \sigma '$, as desired. $\square$

Suppose that $x > 0$. Let $\phi = \phi _0 \oplus (\rho \boxtimes S_{2x+1})^{\oplus 2}$ with $\rho \boxtimes S_{2x+1} \not \subset \phi _0$, and $\eta \in \widehat{\mathcal {A}_\phi }$.

Lemma 5.4

Set $\phi _1 = \phi - (\rho \boxtimes S_{2x+1})^{\oplus 2} \oplus (\rho \boxtimes S_{2x-1})^{\oplus 2}$ so that there is a canonical injection $\mathcal {A}_{\phi _1} \hookrightarrow \mathcal {A}_\phi$, which is bijective unless $x=1/2$. Define $\eta _1 \in \widehat{\mathcal {A}_{\phi _1}}$ by $\eta _1 = \eta |\mathcal {A}_{\phi _1}$. Then we have

$$\begin{aligned} \mathrm {Jac}_{\rho |\cdot |^x, \rho |\cdot |^x} (\pi (\phi , \eta )) = 2 \cdot \pi (\phi _1, \eta _1). \end{aligned}$$

Proof

Let $\eta _+ = \eta$ and $\eta _- \in \widehat{\mathcal {A}_\phi }$ be as in the statement of Theorem 4.2 (3). Define $\eta _{1,\pm } \in \widehat{\mathcal {A}_{\phi _1}}$ by $\eta _{1,\pm } = \eta _{\pm }|\mathcal {A}_\phi$ if $x > 1/2$. When $x=1/2$, we set $\eta _{1,+} = \eta |\mathcal {A}_{\phi _1}$ and $\pi (\phi _1, \eta _{1,-}) = 0$ formally. Then we have

• $\pi (\phi , \eta _+) \oplus \pi (\phi , \eta _-) = \mathrm {St}(\rho , 2x+1) \rtimes \pi (\phi _0, \eta |\mathcal {A}_{\phi _0})$;

• $\pi (\phi _1, \eta _{1,+}) \oplus \pi (\phi _1, \eta _{1,-}) = \mathrm {St}(\rho , 2x-1) \rtimes \pi (\phi _0, \eta |\mathcal {A}_{\phi _0})$;

• $\mathrm {Jac}_{\rho |\cdot |^x, \rho |\cdot |^x} (\mathrm {St}(\rho , 2x+1) \rtimes \pi (\phi _0, \eta |\mathcal {A}_{\phi _0})) \cong 2 \cdot \mathrm {St}(\rho , 2x-1) \rtimes \pi (\phi _0, \eta |\mathcal {A}_{\phi _0})$.

Therefore, it is enough to show that $\mathrm {Jac}_{\rho |\cdot |^x, \rho |\cdot |^x} (\pi (\phi , \eta )) \subset 2 \cdot \pi (\phi _1, \eta _1)$.

We apply Mœglin’s construction to $\Pi _{\phi }$. Write

$$\begin{aligned} \phi = \left( \bigoplus _{i=1}^{t} \rho \boxtimes S_{a_i} \right) \oplus \phi _e \end{aligned}$$

with $a_1 \le \dots \le a_t$ and $\rho \boxtimes S_a \not \subset \phi _e$ for any $a>0$. There exists $t_0 > 1$ such that $a_{t_0-1} = a_{t_0} = 2x+1$. Take a new L-parameter

$$\begin{aligned} \phi _\gg = \left( \bigoplus _{i=1}^{t} \rho \boxtimes S_{a'_i} \right) \oplus \phi _e \end{aligned}$$

such that

• $a'_1< \dots < a'_t$;

• $a_i' \ge a_i$ and $a'_i \equiv a_i \bmod 2$ for any i.

In particular, $a'_{t_0} \ge 2x+3$. We can identify $\mathcal {A}_{\phi _\gg }$ with $\mathcal {A}_{\phi }$ canonically. Let $\eta _\gg \in \widehat{\mathcal {A}_{\phi _\gg }}$ be the character corresponding to $\eta \in \widehat{\mathcal {A}_{\phi }}$. Then Theorem 3.5 says that

$$\begin{aligned} \pi (\phi , \eta ) = \mathrm {Jac}_{\rho |\cdot |^{\frac{a'_t-1}{2}}, \dots , \rho |\cdot |^\frac{a_t+1}{2}} \circ \dots \circ \mathrm {Jac}_{\rho |\cdot |^{\frac{a'_1-1}{2}}, \dots , \rho |\cdot |^\frac{a_1+1}{2}} (\pi (\phi _\gg , \eta _\gg )). \end{aligned}$$

Note that $(a_{t_0}+1)/2 = x+1$. By Lemma 2.6 (2), we see that

$$\begin{aligned}&\mathrm {Jac}_{\rho |\cdot |^x, \rho |\cdot |^x} (\pi (\phi , \eta ))\\&=\mathrm {Jac}_{\rho |\cdot |^{\frac{a'_t-1}{2}}, \dots , \rho |\cdot |^\frac{a_t+1}{2}} \circ \dots \circ \mathrm {Jac}_{\rho |\cdot |^{\frac{a'_{t_0+1}-1}{2}}, \dots , \rho |\cdot |^\frac{a_{t_0+1}+1}{2}}\\&\circ \mathrm {Jac}_{\rho |\cdot |^{x+1}, \rho |\cdot |^{x}, \rho |\cdot |^{x}} \left( \mathrm {Jac}_{\rho |\cdot |^{\frac{a'_{t_0}-1}{2}}, \dots , \rho |\cdot |^\frac{a_{t_0}+3}{2}} \circ \dots \circ \mathrm {Jac}_{\rho |\cdot |^{\frac{a'_1-1}{2}}, \dots , \rho |\cdot |^\frac{a_1+1}{2}} (\pi (\phi _\gg , \eta _\gg )) \right) . \end{aligned}$$

By Lemma 5.3, we have

$$\begin{aligned}&\mathrm {Jac}_{\rho |\cdot |^{x+1}, \rho |\cdot |^{x}, \rho |\cdot |^{x}} \left( \mathrm {Jac}_{\rho |\cdot |^{\frac{a'_{t_0}-1}{2}}, \dots , \rho |\cdot |^\frac{a_{t_0}+3}{2}} \circ \dots \circ \mathrm {Jac}_{\rho |\cdot |^{\frac{a'_1-1}{2}}, \dots , \rho |\cdot |^\frac{a_1+1}{2}} (\pi (\phi _\gg , \eta _\gg )) \right) \\&\subset 2 \cdot \mathrm {Jac}_{\rho |\cdot |^{x}, \rho |\cdot |^{x+1}, \rho |\cdot |^{x}} \left( \mathrm {Jac}_{\rho |\cdot |^{\frac{a'_{t_0}-1}{2}}, \dots , \rho |\cdot |^\frac{a_{t_0}+3}{2}} \circ \dots \circ \mathrm {Jac}_{\rho |\cdot |^{\frac{a'_1-1}{2}}, \dots , \rho |\cdot |^\frac{a_1+1}{2}} (\pi (\phi _\gg , \eta _\gg )) \right) . \end{aligned}$$

Since

$$\begin{aligned}&\mathrm {Jac}_{\rho |\cdot |^{\frac{a'_t-1}{2}}, \dots , \rho |\cdot |^\frac{a_t+1}{2}} \circ \dots \circ \mathrm {Jac}_{\rho |\cdot |^{\frac{a'_{t_0+1}-1}{2}}, \dots , \rho |\cdot |^\frac{a_{t_0+1}+1}{2}}\\&\circ \mathrm {Jac}_{\rho |\cdot |^{x}, \rho |\cdot |^{x+1}, \rho |\cdot |^{x}} \left( \mathrm {Jac}_{\rho |\cdot |^{\frac{a'_{t_0}-1}{2}}, \dots , \rho |\cdot |^\frac{a_{t_0}+3}{2}} \circ \dots \circ \mathrm {Jac}_{\rho |\cdot |^{\frac{a'_1-1}{2}}, \dots , \rho |\cdot |^\frac{a_1+1}{2}} (\pi (\phi _\gg , \eta _\gg )) \right) \\&= \pi (\phi _1, \eta _1) \end{aligned}$$

by Theorem 3.5, we have $\mathrm {Jac}_{\rho |\cdot |^x, \rho |\cdot |^x} (\pi (\phi , \eta )) \subset 2 \cdot \pi (\phi _1, \eta _1)$, as desired. $\square$

Now we can prove Theorem 4.2 (3).

Proof of Theorem 4.2 (3)

By Corollary 2.9, we have

$$\begin{aligned} \mathrm {Jac}_{\rho |\cdot |^x}( \pi (\phi , \eta _+) \oplus \pi (\phi , \eta _-) ) = 2 \cdot \left\langle \rho ; x, x-1, \dots , -(x-1) \right\rangle \rtimes \pi (\phi _0, \eta |\mathcal {A}_{\phi _0}). \end{aligned}$$

By Proposition 5.2, we have an exact sequence

where $\Pi = \left\langle \rho ; x, x-1, \dots , -(x-1) \right\rangle \rtimes \pi (\phi _0, \eta |\mathcal {A}_{\phi _0})$, and $\sigma$ is the unique irreducible quotient of $\Pi$. Fix $\epsilon \in \{\pm \}$. By Lemma 5.4, we see that $\mathrm {Jac}_{\rho |\cdot |^x}(\pi (\phi , \eta _\epsilon )) \supset 2 \cdot \pi (\phi ', \eta '_\epsilon )$. On the other hand, since $\mathrm {s.s.}\mathrm {Jac}_{P_{d(2x+1)}}(\pi (\phi , \eta _\epsilon )) \supset \mathrm {St}(\rho , 2x+1) \otimes \pi (\phi _0, \eta |\mathcal {A}_{\phi _0})$, we have

$$\begin{aligned} \mathrm {s.s.}\mathrm {Jac}_{P_{2dx}}(\mathrm {Jac}_{\rho |\cdot |^x}(\pi (\phi , \eta _\epsilon ))) \supset |\cdot |^{-\frac{1}{2}} \mathrm {St}(\rho , 2x) \otimes \pi (\phi _0, \eta |\mathcal {A}_{\phi _0}). \end{aligned}$$

This implies that $\mathrm {Jac}_{\rho |\cdot |^x}(\pi (\phi , \eta _\epsilon ))$ contains an irreducible non-tempered representation, which must be $\sigma$. Hence

$$\begin{aligned} \mathrm {Jac}_{\rho |\cdot |^x}(\pi (\phi , \eta _\epsilon )) \supset 2 \cdot \pi (\phi ', \eta _\epsilon ) + \sigma = \Pi + \pi (\phi ', \eta _\epsilon ) - \pi (\phi ', \eta _{-\epsilon }). \end{aligned}$$

Considering $\mathrm {Jac}_{\rho |\cdot |^x}(\pi (\phi , \eta _+) \oplus \pi (\phi , \eta _-))$, we see that this inclusion must be an equality. $\square$

If $x=0$ and $m=2$, we see that $\mathrm {Jac}_{\rho }(\pi (\phi , \eta )) \supset \pi (\phi _0, \eta |\mathcal {A}_{\phi _0})$. By the same argument, this inclusion must be an equality. This completes the proof of Theorem 4.2 (4), so that the ones of all statements of Theorem 4.2.

5.5 Description of $\mu ^*_\rho$

We prove Theorem 4.3 in this subsection. To do this, we need the following specious lemma.

Lemma 5.5

Let $\phi \in \Phi _\mathrm {gp}(G)$ and $\underline{x} = (x_1, \dots , x_m) \in \mathbb {R}^m$. Suppose that $\mathrm {Jac}_{\rho |\cdot |^{\underline{x}}}(\pi ) \not = 0$ for some $\pi \in \Pi _\phi$.

• (1) If $x_m < 0$, then $x_i = -x_m$ for some i.

• (2) Suppose that $\underline{x}$ is of the form

  $$\begin{aligned} \underline{x} = ( \underbrace{x_1^{(1)}, \dots , x_{m_1}^{(1)}}_{m_1}, \dots , \underbrace{x_1^{(k)}, \dots , x_{m_k}^{(k)}}_{m_k} ) \end{aligned}$$

  with $x_{i-1}^{(j)} > x_{i}^{(j)}$ for $1 \le j \le k$, $1 < i \le m_j$, and $x_1^{(1)} \le \dots \le x_1^{(k)}$. Then $x_1^{(j)} \ge 0$ for $j =1, \dots , k$, and

  $$\begin{aligned} \phi \supset \bigoplus _{j=1}^{k} \rho \boxtimes S_{2x_1^{(j)}+1}. \end{aligned}$$

Proof

We prove the lemma by induction on m. By Lemma 4.1, we see that $2x_1+1$ is a positive integer, and $\phi$ contains $\rho \boxtimes S_{2x_1+1}$. In particular, we obtain the lemma for $m=1$.

Suppose that $m \ge 2$ and put $\underline{x'} = (x_2, \dots , x_m) \in \mathbb {R}^{m-1}$. By Theorem 4.2, one of the following holds.

• $\mathrm {Jac}_{\rho |\cdot |^{\underline{x'}}}(\pi ') \not = 0$ for some $\pi ' \in \Pi _{\phi '}$ with $\phi ' = \phi - (\rho \boxtimes S_{2x_1+1}) \oplus (\rho \boxtimes S_{2x_1-1})$;

• $\mathrm {Jac}_{\rho |\cdot |^{\underline{x'}}}(\left\langle \rho ; x_1, x_1-1, \dots , -(x_1-1) \right\rangle \rtimes \pi _0) \not = 0$ for some $\pi _0 \in \Pi _{\phi _0}$ with $\phi _0 = \phi - (\rho \boxtimes S_{2x_1+1})^{\oplus 2}$.

The former case can occur only if $x_1 > 0$, and the latter case can occur only if $\phi \supset (\rho \boxtimes S_{2x_1+1})^{\oplus 2}$.

We consider the former case. Assume that $x_1 > 0$ and $\mathrm {Jac}_{\rho |\cdot |^{\underline{x'}}}(\pi ') \not = 0$ for some $\pi ' \in \Pi _{\phi '}$ with $\phi ' = \phi - (\rho \boxtimes S_{2x_1+1}) \oplus (\rho \boxtimes S_{2x_1-1})$. By the induction hypothesis, we have $x_i = -x_m$ for some $i \ge 2$ when $x_m < 0$, and

$$\begin{aligned} \phi ' \supset (\rho \boxtimes S_{2x_1^{(1)}-1}) \oplus \left( \bigoplus _{j=2}^{k} \rho \boxtimes S_{2x_1^{(j)}+1} \right) \end{aligned}$$

when $\underline{x}$ is of the form in (2) since $\underline{x'}$ is also of the form. This implies the assertion for $\phi$.

We consider the latter case. Assume that $\phi \supset (\rho \boxtimes S_{2x_1+1})^{\oplus 2}$, and that

$$\begin{aligned} \mathrm {Jac}_{\rho |\cdot |^{\underline{x'}}}(\left\langle \rho ; x_1, x_1-1, \dots , -(x_1-1) \right\rangle \rtimes \pi _0) \not = 0 \end{aligned}$$

for some $\pi _0 \in \Pi _{\phi _0}$ with $\phi _0 = \phi - (\rho \boxtimes S_{2x_1+1})^{\oplus 2}$. By Corollary 2.9, we can divide

$$\begin{aligned} \{2, \dots , m\} = \{i_1, \dots , i_{m_1}\} \sqcup \{j_{1}, \dots , j_{m_2}\} \sqcup \{k_1, \dots , k_{m_3}\} \end{aligned}$$

with $i_1< \dots < i_{m_1}$, $j_1< \dots < j_{m_2}$, $k_1< \dots < k_{m_3}$ and $m_2+m_3 \le 2x_1$ such that

• $\mathrm {Jac}_{\rho |\cdot |^{y_1}, \dots , \rho |\cdot |^{y_{m_1}}}(\pi _0) \not = 0$ with $y_t = x_{i_t}$;

• $x_{j_t} = x_1+1-t$ for $t = 1, \dots , m_2$;

• $x_{k_t} = x_1-t$ for $t = 1, \dots , m_3$.

Considering the following four cases, we can prove the existence $x_i$ satisfying $x_i = -x_m$ when $x_m < 0$.

• When $m = i_{m_1}$, by the induction hypothesis, we have $x_{i_t} = -x_m$ for some t.

• When $m = j_{m_2}$, we have $x_m = x_1+1-m_2 < 0$ so that $x_{j_t} = -x_m$ with $t = 2x_1+2-m_2$.

• When $m = k_{m_3}$ and $m_3 < 2x_1$, we have $x_m = x_1-m_3 < 0$ so that $x_{k_t} = -x_m$ with $t = 2x_1-m_3$.

• When $m = k_{m_3}$ and $m_3 = 2x_1$, we have $x_1 = -x_m$.

On the other hand, when $\underline{x}$ is of the form in (2), since $x_1^{(j)} \ge x_1^{(1)} = x_1$, there is at most one $j_0 \ge 2$ such that

$$\begin{aligned} x_1^{(j_0)} \in \{x_{j_t}\ |\ t = 1, \dots , m_2\} \cup \{x_{k_t}\ |\ t = 1, \dots , m_3\}, \end{aligned}$$

in which case, $x_1^{(j_0)} = x_1$. By the induction hypothesis, we have

$$\begin{aligned} \phi _0 \supset \bigoplus _{\begin{array}{c} 2 \le j \le k \\ j \not = j_0 \end{array}} \rho \boxtimes S_{2x_1^{(j)}+1}. \end{aligned}$$

This implies the assertion for $\phi$. This completes the proof.$\square$

Now we can prove Theorem 4.3.

Proof of Theorem 4.3

Since the subgroup of $\mathcal {R}_{dm}$ spanned by $\mathrm {Irr}_\rho (\mathrm {GL}_{dm}(F)) = \{\tau _{\underline{x}}\ |\ \underline{x} \in \Omega _m\}$ has another basis $\{\Delta _{\underline{x}}\ |\ \underline{x} \in \Omega _m\}$, we can write

$$\begin{aligned} \mu ^*_\rho (\pi ) = \sum _{m \ge 0}\sum _{\underline{x} \in \Omega _m} \Delta _{\underline{x}} \otimes \Pi _{\underline{x}}(\pi ) \end{aligned}$$

for some virtual representation $\Pi _{\underline{x}}(\pi )$. For $\underline{y} \in \Omega _m$, applying $\mathrm {Jac}_{\rho |\cdot |^{\underline{y}}}$ to $\mathrm {s.s.}\mathrm {Jac}_{P_{dm}}(\pi )$, we have

$$\begin{aligned} \mathrm {Jac}_{\rho |\cdot |^{\underline{y}}}(\pi ) = \sum _{\underline{x} \in \Omega _m} \mathrm {Jac}_{\rho |\cdot |^{\underline{y}}}(\Delta _{\underline{x}}) \otimes \Pi _{\underline{x}}(\pi ) = \sum _{\underline{x} \in \Omega _m} m(\underline{y}, \underline{x}) \cdot \Pi _{\underline{x}}(\pi ), \end{aligned}$$

where $m(\underline{y}, \underline{x}) = \dim _\mathbb {C}\mathrm {Jac}_{\rho |\cdot |^{\underline{y}}}(\Delta _{\underline{x}})$. If $m'(\underline{x'}, \underline{y}) \in \mathbb {Q}$ satisfies that

$$\begin{aligned} \sum _{\underline{y} \in \Omega _m} m'(\underline{x'}, \underline{y}) m(\underline{y}, \underline{x}) = \delta _{\underline{x'}, \underline{x}}, \end{aligned}$$

we have

$$\begin{aligned} \Pi _{\underline{x}}(\pi ) = \sum _{\underline{y} \in \Omega _m} m'(\underline{x}, \underline{y}) \cdot \mathrm {Jac}_{\rho |\cdot |^{\underline{y}}}(\pi ). \end{aligned}$$

Hence we have

$$\begin{aligned} \mu ^*_\rho (\pi ) = \sum _{m \ge 0}\sum _{\underline{x}, \underline{y} \in \Omega _m} m'(\underline{x}, \underline{y}) \cdot \Delta _{\underline{x}} \otimes \mathrm {Jac}_{\rho |\cdot |^{\underline{y}}}(\pi ). \end{aligned}$$

By Lemma 5.5, if $\mathrm {Jac}_{\rho |\cdot |^{\underline{y}}}(\pi ) \not = 0$ for $\underline{y} \in \Omega _m$, then $\underline{y} = \underline{x}(\underline{k})$ for some $\underline{k} \in K_\phi ^{(m)}$. If $m'(\underline{x}, \underline{y}) \not = 0$, then the image of $\underline{x}$ under the canonical map $\mathbb {R}^m \rightarrow \mathbb {R}^m/{\mathfrak {S}}_m$ coincides with the one of $\underline{y}$ since the same property holds for $m(\underline{x}, \underline{y})$. In particular, for fixed $\underline{k} \in K_\phi ^{(m)}$, if $m'(\underline{x}, \underline{x}(\underline{k})) \not = 0$, then $\underline{x} = \underline{x}(\underline{k'})$ for some $\underline{k'} \in K_\phi ^{(m)}$. Therefore, we have

$$\begin{aligned} \mu ^*_\rho (\pi ) = \sum _{m \ge 0}\sum _{\underline{k}, \underline{k'} \in K_\phi ^{(m)}} m'(\underline{x}(\underline{k'}), \underline{x}(\underline{k})) \cdot \Delta _{\underline{x}(\underline{k'})} \otimes \mathrm {Jac}_{\rho |\cdot |^{\underline{x}(\underline{k})}}(\pi ). \end{aligned}$$

This completes the proof of Theorem 4.3. $\square$

6 Complements

6.1 A remark on standard modules

As a consequence of Theorem 4.2, we can prove the irreducibility of certain standard modules.

Corollary 6.1

Let $\phi \in \Phi _\mathrm {gp}(G)$ and $\eta \in \widehat{\mathcal {A}_\phi }$ such that $\pi (\phi , \eta ) \not = 0$. Suppose that $\phi \supset \rho \boxtimes S_{2x+1}$ for $x>0$ but $\mathrm {Jac}_{\rho |\cdot |^x}(\pi (\phi , \eta )) = 0$. Then the standard module

$$\begin{aligned} \Pi = \left\langle \rho ; x, x-1, \dots , -(x-1) \right\rangle \rtimes \pi (\phi , \eta ) \end{aligned}$$

is irreducible.

Proof

Let $\sigma$ be the unique irreducible quotient of $\Pi$, which is non-tempered. By the same argument as the proof of Proposition 5.2, we see that $\sigma$ is the unique irreducible non-tempered subquotient of $\Pi$. Suppose that $\Pi$ is reducible. If $\pi '$ is another irreducible subquotient of $\Pi$, by considering its cuspidal support or its Plancherel measure, we see that $\pi ' \in \Pi _{\phi '}$ with $\phi ' = \phi \oplus (\rho \boxtimes S_{2x-1}) \oplus (\rho \boxtimes S_{2x+1})$. Since $\phi \supset \rho \boxtimes S_{2x+1}$, we see that $\phi ' \supset (\rho \boxtimes S_{2x+1})^{\oplus 2}$. By Theorem 4.2, $\mathrm {Jac}_{\rho |\cdot |^x}(\pi ')$ contains an irreducible non-tempered representation. However, $\mathrm {Jac}_{\rho |\cdot |^x}(\Pi ) = \mathrm {St}(\rho , 2x-1) \rtimes \pi (\phi , \eta )$ consists of tempered representations. This is a contradiction. $\square$

Example 6.2

Consider $\phi = S_2 \oplus S_4 \oplus S_6 \in \Phi _\mathrm {disc}(\mathrm {SO}_{13}(F))$ and $\eta \in \widehat{A_\phi }$ given by $\eta (\alpha _{S_{2a}}) = (-1)^{a}$ for $a=1,2,3$. Then $\pi (\phi , \eta )$ is an irreducible supercuspidal representation. Moreover, the standard module

$$\begin{aligned} \Pi = \left\langle \mathbf {1}_{\mathrm {GL}_1(F)}; \frac{5}{2}, \frac{3}{2}, \frac{1}{2}, -\frac{1}{2}, -\frac{3}{2} \right\rangle \rtimes \pi (\phi , \eta ) \end{aligned}$$

of $\mathrm {SO}_{23}(F)$ is irreducible. Note that the L-parameter $\phi ' = \phi \oplus |\cdot |^{\frac{1}{2}} S_5 \oplus |\cdot |^{-\frac{1}{2}} S_5$ of $\Pi$ is non-generic since

$$\begin{aligned} L(s, \phi ', \mathrm {Ad})&= {} \zeta _F(s-1)\zeta _F(s)^{3}\zeta _F(s+1)^{13} \zeta _F(s+2)^{10}\\&\times \zeta _F(s+3)^{12}\zeta _F(s+4)^{5}\zeta _F(s+5)^{3} \end{aligned}$$

has a pole at $s=1$, where $\zeta _F$ is the local zeta function associated to F. In particular, the standard module

$$\begin{aligned} \Pi _0 = \left\langle \mathbf {1}_{\mathrm {GL}_1(F)}; \frac{5}{2}, \frac{3}{2}, \frac{1}{2}, -\frac{1}{2}, -\frac{3}{2} \right\rangle \rtimes \pi (\phi , \mathbf {1}) \end{aligned}$$

is reducible by Theorem 3.4.

6.2 Mínguez’s derivatives

The following notion is introduced by Mínguez. Fix $x > 0$ such that $\rho \boxtimes S_{2x+1}$ is good with respect to G, i.e., we assume that there exists $\phi \in \Phi _\mathrm {gp}(G)$ such that $\phi \supset \rho \boxtimes S_{2x+1}$. For $\pi \in \mathrm {Irr}_\mathrm {temp}(G)$, we set

$$\begin{aligned} \mathrm {Jac}^{\rho |\cdot |^x}(\pi ) = \underbrace{\mathrm {Jac}_{\rho |\cdot |^x} \circ \dots \circ \mathrm {Jac}_{\rho |\cdot |^x}}_{k} (\pi ), \end{aligned}$$

where $k \ge 0$ is the maximal integer such that $\mathrm {Jac}^{\rho |\cdot |^x}(\pi ) \not = 0$. we call $\mathrm {Jac}^{\rho |\cdot |^x}(\pi )$ the $\rho |\cdot |^x$-derivative of$\pi$.

Proposition 6.3

Let $\phi \in \Phi _\mathrm {gp}(G)$ and $\eta \in \widehat{\mathcal {A}_\phi }$ such that $\pi (\phi , \eta ) \not = 0$. Suppose that $x > 0$ and $\mathrm {Jac}_{\rho |\cdot |^x}(\pi ) \not = 0$, so that the multiplicity m of $\rho \boxtimes S_{2x+1}$ in $\phi$ is positive. For $1 \le k \le m$, set

$$\begin{aligned} \phi ^{(k)} = \phi - (\rho \boxtimes S_{2x+1})^{\oplus k} \oplus (\rho \boxtimes S_{2x-1})^{\oplus k} \end{aligned}$$

and $\eta ^{(k)} \in \widehat{\mathcal {A}_{\phi ^{(k)}}}$ to be $\eta |\mathcal {A}_{\phi ^{(k)}}$ via the canonical inclusion $\mathcal {A}_{\phi ^{(k)}} \hookrightarrow \mathcal {A}_\phi$. Then

$$\begin{aligned} \mathrm {Jac}^{\rho |\cdot |^x}(\pi (\phi , \eta )) = k! \cdot \pi (\phi ^{(k)}, \eta ^{(k)}) \end{aligned}$$

with

$$\begin{aligned} k = \left\{ \begin{aligned}&m&\quad&\text {if }\pi (\phi ^{(m)}, \eta ^{(m)}) \not = 0, \\&m-1&\quad&\text {if }\pi (\phi ^{(m)}, \eta ^{(m)}) = 0. \end{aligned} \right. \end{aligned}$$

In particular, for the above k, we have an inclusion

$$\begin{aligned} \pi (\phi , \eta ) \hookrightarrow \underbrace{\rho |\cdot |^{x} \times \dots \times \rho |\cdot |^x}_{k} \rtimes \pi (\phi ^{(k)}, \eta ^{(k)}), \end{aligned}$$

and $\mathrm {Jac}_{\rho |\cdot |^x} (\pi (\phi ^{(k)}, \eta ^{(k)})) = 0$.

Proof

When $m = 1$ (resp. $m = 2$), it is Theorem 4.2 (2) (resp. Lemma 5.4). In general, by Theorem 4.2 (1), we can reduce the assertion to the case of $m \in \{1,2\}$. $\square$

Remark 6.4

• (1) Embedding an irreducible representation $\pi$ into $(\rho |\cdot |^{x})^{\times k} \rtimes \pi _0$ with k maximal is the key idea of the proof of the Howe duality conjecture by Gan–Takeda [7].

• (2) If $\rho$ is not self-dual, we should define $\mathrm {Jac}^{\rho |\cdot |^x}$ as the maximal nonzero iterated composition of $\mathrm {Jac}_{\rho ^\vee |\cdot |^{x}} \circ \mathrm {Jac}_{\rho |\cdot |^x}$. Then Proposition 6.3 can be extended to any $\phi \in \Phi _\mathrm {temp}(G)$ and any supercuspidal unitary representation $\rho$ of $\mathrm {GL}_d(F)$. We leave the detail for readers.

• (3) In particular, one can show that for any $\pi \in \mathrm {Irr}_\mathrm {temp}(G)$, the $\rho |\cdot |^x$-derivative $\mathrm {Jac}^{\rho |\cdot |^x}(\pi )$ is an isotypic of an irreducible tempered representation.