2.1 Notation

Throughout this article, we fix a non-Archimedean locally compact field $F$ of characteristic zero with normalized absolute value $|\cdot |$. Let $G$ be the group of $F$-points of a connected reductive group defined over $F$, with the usual topology. We will only consider smooth representations of $G$, that is, representations such that the stabilizer of every vector is an open subgroup of $G$, and we write $\mathrm {Rep}(G)$ for the category of smooth complex representations of $G$ of finite length. Denote by $\mathrm {Irr}(G)$ the set of equivalence classes of irreducible objects of $\mathrm {Rep}(G)$. Let $\mathscr {R}(G)$ be the Grothendieck group of $\mathrm {Rep}(G)$. The canonical map from the objects of $\mathrm {Rep}(G)$ to $\mathscr {R} (G)$ will be denoted by $\pi \mapsto [\pi ]$.

For $\pi, \pi ' \in \mathrm {Rep}(G)$ we write $\pi \hookrightarrow \pi '$ (respectively $\pi \twoheadrightarrow \pi '$) if there exists an injective (respectively surjective) morphism from $\pi$ to $\pi '$.

Fix a minimal $F$-parabolic subgroup $P_0$ of $G$. A parabolic subgroup $P$ of $G$ is said to be standard if it contains $P_0$. Henceforth, the letter $P$ will always denote a standard parabolic subgroup of $G$ with an implicit standard Levi decomposition $P=MU$. Let $\Sigma$ denote the set of roots of $G$ with respect to $P_0$, and let $\Delta$ be a basis of $\Sigma$. For $\Theta \subset \Delta$ let $P_\Theta$ denote the standard parabolic subgroup of $G$ corresponding to $\Theta$ and let $M_\Theta$ be a corresponding standard Levi subgroup. Let $W$ be the Weyl group of $G$.

Let $\tau$ be a representation of $M$, regarded as a representation of $P$ on which $U$ acts trivially. We denote by $\mathrm {Ind}_P^G\tau$ the representation of $G$ parabolically induced from $\tau$. (We will always mean the normalized induction.) We view $\mathrm {Ind}_P^G$ as a functor. Its left adjoint, the Jacquet functor with respect to $P$, will be denoted by $\mathrm {Jac}_P^G$.

An irreducible representation $\pi$ of $G$ is said to be supercuspidal if it is not a composition factor of any representation of the form $\mathrm {Ind}^{G}_P(\tau )$ with $P$ a proper parabolic subgroup of $G$ and $\tau$ a representation of $M$. We write $\mathscr {C}(G)$ for the subset of $\mathrm {Irr}(G)$ consisting of supercuspidal representations. For any $\pi \in \mathrm {Rep}(G)$, we denote by $\pi ^\vee$ the contragredient of $\pi$. (The sets $\mathrm {Irr}(G)$ and $\mathscr {C}(G)$ are invariant under $^\vee$.)

Let $\Pi$ be a smooth representation of $G$ of finite length. The socle of $\Pi$ is the largest semisimple subrepresentation of $\Pi$. It is denoted by $\mathrm {soc}(\Pi )$. We say that $\Pi$ is socle irreducible (SI) if $\mathrm {soc}(\Pi )$ is irreducible and occurs with multiplicity one in $[\Pi ]$.

2.2 The Zelevinsky–Aubert duality

We consider the map

\begin{align*} D_{G} \colon \mathscr{R}(G) & \longrightarrow \mathscr{R}(G)\\ \pi &\longmapsto \sum_{P}(-1)^{\dim A_M} \big[\mathrm{Ind}_P^{G}(\mathrm{Jac}_{P}^G(\pi))\big], \end{align*}

where $P = MN$ runs over all standard parabolic subgroups of $G$. Aubert [Reference AubertAub95] showed that if $\pi$ is irreducible, then there exists a sign $\epsilon \in \{\pm 1\}$ such that $\hat {\pi } = \epsilon \cdot D_{G}(\pi )$ is also an irreducible representation. We call the map

\begin{align*} \mathrm{Irr}(G) & \rightarrow \mathrm{Irr}(G) \\ \pi& \mapsto \hat\pi \end{align*}

the Zelevinsky–Aubert duality.

It has the following important properties.

1. (1) For any $\pi \in \mathrm {Irr}(G)$, the dual of $\hat \pi$ is equal to $\pi$, that is, the map $\pi \mapsto \hat \pi$ is an involution [Reference AubertAub95, Théorème 1.7(3)].

2. (2) If $\pi \in \mathscr {C}(G)$, then $\hat \pi = \pi$ [Reference AubertAub95, Théorème 1.7(4)].

3. (3) Let $\Theta \subset \Delta$ and consider the standard parabolic subgroup $P=P_\Theta$ with Levi decomposition $P=MN$. Let $w_0$ be the longest element in the set $\{w\in W\mid w^{-1}(\Theta )>0\}$ and let $P'$ be the standard parabolic subgroup with Levi subgroup $M'=w^{-1}(M)$. Then we have (cf. [Reference AubertAub95, Théorème 1.7(2)])

   (2.1)\begin{equation} \mathrm{Jac}_P^G \circ D_G= {\rm Ad}(w_0)\circ D_{M'} \circ \mathrm{Jac}_{P'}^G. \end{equation}

2.3 Representations of general linear groups

Set $\mathrm {Irr}^\mathrm {GL} := \bigcup _{n\ge 0}\mathrm {Irr} (\mathrm {GL}_n(F))$, and let $\mathscr {C}^\mathrm {GL} \subset \mathrm {Irr}^\mathrm {GL}$ be the subset of supercuspidal representations of $\mathrm {GL}_n(F)$ for every $n>0$. We write $\mathscr {R}^\mathrm {GL} := \bigoplus _{n \geq 0} \mathscr {R}(\mathrm {GL}_n(F))$.

Let $d_1, \ldots, d_r$ be some positive integers. Let $\tau _i \in \mathrm {Rep}(\mathrm {GL}_{d_i}(F))$ for $1 \leq i \leq r$. It is customary to denote the normalized parabolically induced representation by

\[ \tau_1 \times \cdots \times \tau_r := \mathrm{Ind}_{P}^{\mathrm{GL}_{d_1+\cdots+d_r}(F)}(\tau_1 \boxtimes \cdots \boxtimes \tau_r). \]

This product induces a $\mathbb {Z}$-graded ring structure on $\mathscr {R}^\mathrm {GL}$. We denote the multiplication by $m$. If $\tau _1 = \cdots = \tau _r = \tau$, we will write $\tau ^r = \tau \times \cdots \times \tau$ ($r$ times).

The Jacquet functor for $\mathrm {GL}_{m}(F)$ along the maximal parabolic subgroup $P_{(d,m-d)}$ with Levi subgroup isomorphic to $\mathrm {GL}_{d}(F) \times \mathrm {GL}_{m-d}(F)$ is denoted by $\mathrm {Jac}_{(d,m-d)} = \mathrm {Jac}_{P_{(d,m-d)}}^{\mathrm {GL}_m(F)}$. It induces a co-multiplication, that is, a ring homomorphism

\begin{align*} m^\ast \colon \mathscr{R}^\mathrm{GL} &\longrightarrow \mathscr{R}^\mathrm{GL}\otimes \mathscr{R}^\mathrm{GL} \nonumber\\ \tau & \longmapsto \sum_{n \geq 0} \biggl(\sum_{n_1+n_2=n}\bigl[\mathrm{Jac}_{(n_1,n_2)}(\tau)\bigr]\biggr). \end{align*}

We finally take

\[ M^\ast \colon \mathscr{R}^\mathrm{GL} \longrightarrow \mathscr{R}^\mathrm{GL}\otimes \mathscr{R}^\mathrm{GL} \]

to be the composition $M^\ast = (m\otimes 1) \circ (\cdot ^\vee \otimes m^\ast ) \circ s \circ m^\ast$, where $s \colon \mathscr {R}^\mathrm {GL}\otimes \mathscr {R}^\mathrm {GL} \to \mathscr {R}^\mathrm {GL}\otimes \mathscr {R}^\mathrm {GL}$ denotes the transposition $s(\sum _i \tau _i\otimes \tau '_i)=\sum _i \tau '_i\otimes \tau _i$.

If $\tau \in \mathrm {Irr}^\mathrm {GL}$, there exist $\rho _1, \ldots, \rho _r \in \mathscr {C}^\mathrm {GL}$ such that $\tau$ is a subrepresentation of $\rho _1 \times \cdots \times \rho _r$. The set $\mathrm {scusp}(\pi ) := \{\rho _1, \ldots, \rho _r \}$ is uniquely determined by $\pi$ and is called the supercuspidal support of $\tau$.

For $\pi \in \mathrm {Rep}(\mathrm {GL}_n(F))$ and a character $\chi$ of $F^\times$, we denote by $\pi \cdot \chi$ the representation obtained from $\pi$ by twisting by the character $\chi \circ \det$. If $\rho \in \mathscr {C}^\mathrm {GL}$, we denote by $\mathbb {Z}_\rho = \{\rho |\cdot |^a \mid a\in \mathbb {Z}\}$ the line of $\rho$.

A segment $[x,y]_\rho$ is a sequence of supercuspidal representations of the form

\[ \rho|\cdot|^{x} ,\,\rho|\cdot|^{x-1} ,\ldots ,\rho|\cdot|^{y}, \]

where $\rho \in \mathscr {C}^\mathrm {GL}$ and $x,y \in \mathbb {R}$ with $x-y \in \mathbb {Z}$ and $x \geq y$.

One can associate with a segment $[x,y]_\rho$ two irreducible representations of $\mathrm {GL}_{d(x-y+1)}(F)$. We denote by $\Delta _{\rho }[x,y]$ the Steinberg representation of $\mathrm {GL}_{d(x-y+1)}(F)$, i.e. the unique irreducible subrepresentation of

\[ \rho|\cdot|^{x} \times \rho|\cdot|^{x-1} \times \cdots \times \rho|\cdot|^{y}, \]

and we also write $Z_\rho [y,x]$ for its unique irreducible quotient. For example, when $\rho = \mathbf {1}_{\mathrm {GL}_1(F)}$, we have $Z_\rho [-(n-1)/2,(n-1)/2] = \mathbf {1}_{\mathrm {GL}_n(F)}$.

The Steinberg representation $\Delta _{\rho }[x,y]$ is an essentially discrete series, and all essentially discrete series are of this form [Reference ZelevinskyZel80, Theorem 9.3]. By convention, we take $\Delta _\rho [x,x+1]= Z_\rho [x+1,x]$ to be the trivial representation of the trivial group $\mathrm {GL}_0(F)$.

If the segments $[x_1,y_1]_{\rho _1}, \ldots, [x_r,y_r]_{\rho _r}$ are such that $x_i \geq y_i$ and $x_1+y_1 \leq \cdots \leq x_r+y_r$, then the socle (Langlands subrepresentation)

\[ L(\Delta_{\rho_1}[x_1,y_1], \ldots, \Delta_{\rho_r}[x_r,y_r]) := \mathrm{soc}(\Delta_{\rho_1}[x_1,y_1] \times \cdots \times \Delta_{\rho_r}[x_r,y_r]) \]

is irreducible. When $\rho _1 = \cdots = \rho _r$, $x_1 < \cdots < x_r$, $y_1 < \cdots < y_r$ and $x_1 \equiv \cdots \equiv x_r \bmod \mathbb {Z}$, we call it a ladder representation. As a special case, when $x_i = x_1+i-1$ and $y_i = y_1+i-1$ for $1 \leq i \leq r$, the ladder representation $L(\Delta _\rho [x_1,y_1], \ldots, \Delta _\rho [x_r,y_r])$ is also called a Speh representation.

The Jacquet modules of $\Delta _\rho [x,y]$ and $Z_\rho [y,x]$ are given by

\begin{align*} \mathrm{Jac}_{(d,d(x-y))}(\Delta_{\rho}[x,y]) &= \rho|\cdot|^x \boxtimes \Delta_\rho[x-1,y],\\ \mathrm{Jac}_{(d,d(x-y))}(Z_{\rho}[y,x]) &= \rho|\cdot|^y \boxtimes Z_\rho[y+1,x], \end{align*}

respectively (see [Reference ZelevinskyZel80, Propositions 3.4 and 9.5]). For Jacquet modules of ladder representations, see [Reference Kret and LapidKL12, Theorem 2.1].

2.4 Representations of classical groups

In this paper, we let $G_n$ be either the split special orthogonal group $\mathrm {SO}_{2n+1}(F)$ or the symplectic group $\mathrm {Sp}_{2n}(F)$ of rank $n$. Set $\mathrm {Irr}^{G} := \bigcup _{n \geq 0} \mathrm {Irr} (G_n)$ and $\mathscr {R}^G := \bigoplus _{n \geq 0} \mathscr {R}(G_{n})$, where the union and the direct sum are taken over groups of the same type. Let $\mathscr {C}^G \subset \mathrm {Irr}^G$ be the subset of supercuspidal representations of $G_n$ for every $n \geq 0$ of the same type.

Fix a rational Borel subgroup of $G_n$. Let $P$ be the standard parabolic subgroup of $G_n$ with Levi subgroup isomorphic to $\mathrm {GL}_{d_1}(F) \times \cdots \times \mathrm {GL}_{d_r}(F) \times G_{n_0}$. Let $\pi \in \mathrm {Rep}(G_{n_0})$ and let $\tau _i \in \mathrm {Rep}(\mathrm {GL}_{d_i}(F))$ for $1 \leq i \leq r$. We denote the normalized parabolically induced representation by

\[ \tau_1 \times \cdots \times \tau_r \rtimes \pi := \mathrm{Ind}_{P}^{G_n}(\tau_1 \boxtimes \cdots \boxtimes \tau_r \boxtimes \pi). \]

As in the case of general linear groups, the Jacquet functors give rise, at the level of Grothendieck groups, to a map

\begin{align*} \mu^\ast \colon \mathscr{R}^G &\longrightarrow \mathscr{R}^\mathrm{GL}\otimes \mathscr{R}^G \\ \mathscr{R}(G_n) \ni \pi &\longmapsto \sum_{k=0}^{n} \bigl[\mathrm{Jac}_{P_k}^{G_n}(\pi)\bigr], \end{align*}

where $P_k$ is the standard parabolic subgroup of $G_n$ with Levi subgroup isomorphic to $\mathrm {GL}_{k}(F) \times G_{n-k}$. The geometric lemma at the level of Grothendieck groups is commonly known in this case as Tadić's formula.

Proposition 2.1 (Tadić's formula [Reference TadićTad95])

For $\tau \in \mathscr {R}^{\mathrm {GL}}$ and $\pi \in \mathscr {R}^G$, we have

\[ \mu^\ast(\tau \rtimes \pi)=M^\ast(\tau)\rtimes\mu^\ast(\pi). \]

We will also use the MVW-functor [Reference Mœglin, Vignéras and WaldspurgerMVW87]. It is a covariant functor

\begin{align*} \mathrm{MVW} \colon \mathrm{Rep}(G_n) &\longrightarrow \mathrm{Rep}(G_n) \\ \Pi &\longmapsto \Pi^\mathrm{MVW} \end{align*}

satisfying the following properties:

• • if $\pi \in \mathrm {Irr} (G_n)$, then $\pi ^\mathrm {MVW}$ is isomorphic to $\pi ^\vee$;

• • $(\tau \rtimes \pi )^\mathrm {MVW} \cong \tau \rtimes \pi ^\mathrm {MVW}$ for any $\pi \in \mathrm {Rep}(G_{n_0})$ and any $\tau \in \mathrm {Rep}(\mathrm {GL}_{d}(F))$ with $n=n_0+d$.

The Zelevinsky–Aubert duality extends by linearity to a map $D^G: \mathscr {R}^G \rightarrow \mathscr {R}^G$. With this notation, the compatibility of the duality with Jacquet functors in (2.1) stands:

(2.2)\begin{equation} \mu^\ast \circ D^G = d^G \circ \mu^\ast, \end{equation}

where

\begin{align*} d^G \colon \mathscr{R}^\mathrm{GL} \otimes \mathscr{R}^G &\longrightarrow \mathscr{R}^\mathrm{GL}\otimes \mathscr{R}^G \\ \sum_i \tau_i \otimes \pi_i &\longmapsto \sum_i \hat{\tau}_i^\vee \otimes \hat{\pi}_i. \end{align*}

Let $[x_1,y_1]_{\rho _1}, \ldots, [x_r,y_r]_{\rho _r}$ be some segments with $\rho _i \in \mathscr {C}(\mathrm {GL}_{d_i}(F))$ being unitary for $1 \leq i \leq r$, and let $\pi _\mathrm {temp}$ be an irreducible tempered representation of $G_{n_0}$. A parabolically induced representation of the form

\[ \Delta_{\rho_1}[x_1,y_1] \times \cdots \times \Delta_{\rho_r}[x_r,y_r] \rtimes \pi_\mathrm{temp} \]

is called a standard module if $x_1+y_1 \leq \cdots \leq x_r+y_r < 0$.

The Langlands classification says that any standard module is SI, and that any irreducible representation $\pi$ of $G_n$ is the unique irreducible subrepresentation (Langlands subrepresentation) of a standard module $\Delta _{\rho _1}[x_1,y_1] \times \cdots \times \Delta _{\rho _r}[x_r,y_r] \rtimes \pi _\mathrm {temp}$ with $n=n_0 + \sum _{i=1}^r d_i(x_i-y_i+1)$, which is unique up to isomorphism. For more details, see [Reference KonnoKon03]. In this case, we write $\pi = L(\Delta _{\rho _1}[x_1,y_1], \ldots, \Delta _{\rho _r}[x_r,y_r]; \pi _\mathrm {temp})$ and refer to $(\Delta _{\rho _1}[x_1,y_1], \ldots, \Delta _{\rho _r}[x_r,y_r]; \pi _\mathrm {temp})$ as the Langlands data of $\pi$.

2.5 The Jantzen decomposition

If $\pi \in \mathrm {Irr}(G_n)$, there exist $\rho _1, \ldots, \rho _r \in \mathscr {C}^\mathrm {GL}$ and $\sigma \in \mathscr {C}^G$ such that $\pi$ is a subrepresentation of $\rho _1 \times \cdots \times \rho _r \rtimes \sigma$. The set

\[ \mathrm{scusp}(\pi) := \{\rho_1, \ldots, \rho_r, \rho_1^\vee, \ldots, \rho_r^\vee ,\sigma\} \]

is uniquely determined by $\pi$ and is called the supercuspidal support of $\pi$. For $\sigma \in \mathscr {C}^G$, we put $\mathrm {Irr}_\sigma := \{\pi \in \mathrm {Irr}^G \mid \sigma \in \mathrm {scusp}(\pi )\}$.

In this paragraph, we fix a supercuspidal representation $\sigma \in \mathscr {C}^G$.

Every supercuspidal representation is either good, bad or ugly.

Ugly representations are easy to deal with owing to the following proposition, which reduces every problem to a similar problem for general linear groups.

Proof. We can write

\[ \pi \hookrightarrow \rho |\cdot |^{x_1} \times \cdots \times \rho |\cdot |^{x_r} \times \rho ^\vee |\cdot |^{-y_1} \times \cdots \times \rho ^\vee |\cdot |^{-y_s} \rtimes \sigma \]

for some $x_i, y_j \in \mathbb {Z}$. There exist irreducible subquotients $\tau _1$ of $\rho |\cdot |^{x_1} \times \cdots \times \rho |\cdot |^{x_r}$ and $\tau _2$ of $\rho ^\vee |\cdot |^{-y_1} \times \cdots \times \rho ^\vee |\cdot |^{-y_s}$ such that this inclusion factors through $\pi \hookrightarrow \tau _1 \times \tau _2 \rtimes \sigma$. As $\rho$ is ugly, we can apply [Reference Lapid and TadićLT20, Lemma 6.2] to $\tau _2 \rtimes \sigma$ and see that $\tau _2 \rtimes \sigma$ is irreducible. Hence $\pi \hookrightarrow \tau _1 \times \tau _2^\vee \rtimes \sigma$. Take an irreducible subquotient $\tau$ of $\tau _1 \times \tau _2^\vee$ such that $\pi \hookrightarrow \tau \rtimes \sigma$. Then by [Reference Lapid and TadićLT20, Lemma 6.2] again, we conclude that $\tau \rtimes \sigma$ is irreducible.

Remark 2.7 More precisely, by the Langlands classification, one can take $\tau _1$ and $\tau _2$ in the proof of this proposition so that

\[ \tau_1 = L(\Delta_\rho[x'_1, y'_1], \ldots, \Delta_\rho[x'_{r'},y'_{r'}]), \quad \tau_2 = L(\Delta_{\rho^\vee}[x_1'',y_1''], \ldots, \Delta_{\rho^\vee}[x''_{r''},y''_{r''}]) \]

with $x'_1+y'_1 \leq \cdots \leq x'_{r'}+y'_{r'} \leq 0$ and $x''_1+y''_1 \leq \cdots \leq x''_{r''}+y''_{r''} \leq 0$. Then since $\tau _2^\vee = L(\Delta _{\rho }[-y''_{r''},-x''_{r''}], \ldots, \Delta _{\rho }[-y''_{1},-x''_{1}])$ and $\pi = \mathrm {soc}(\tau _1 \times \tau _2^\vee \rtimes \sigma ) \hookrightarrow \mathrm {soc}(\tau _1 \times \tau _2^\vee ) \rtimes \sigma$, one can take $\tau$ to be

\[ \tau := \mathrm{soc}(\tau_1 \times \tau_2^\vee) = L(\Delta_\rho[x'_1, y'_1], \ldots, \Delta_\rho[x'_{r'},y'_{r'}], \Delta_{\rho}[-y''_{r''},-x''_{r''}], \ldots, \Delta_{\rho}[-y''_{1},-x''_{1}]). \]

Let $\pi \in \mathrm {Irr}_\sigma$. Then Jantzen [Reference JantzenJan97] defined representations $\pi ^{\rm good} \in \mathrm {Irr}_\sigma ^{\rm good}$, $\pi ^{\rho {\rm -bad}} \in \mathrm {Irr}_\sigma ^{\rho {\rm -bad}}$ and $\pi ^{\rho {\rm -ugly}} \in \mathrm {Irr}_\sigma ^{\rho {\rm -ugly}}$ as follows:

• • $\pi ^{\rm good}$ is the unique representation in $\mathrm {Irr}_\sigma ^{\rm good}$ such that $\pi \hookrightarrow \tau \times \pi ^{\rm good}$ with no good representations in $\mathrm {scusp}(\tau )$;

• • if $\rho$ is a bad supercuspidal representation, then $\pi ^{\rho {\rm -bad}}$ is the unique representation in $\mathrm {Irr}_\sigma ^{\rho {\rm -bad}}$ such that $\pi \hookrightarrow \tau \times \pi ^{\rho {\rm -bad} }$ with $\mathrm {scusp}(\tau ) \cap \mathbb {Z}_\rho =\emptyset$;

• • if $\rho$ is an ugly supercuspidal representation, then $\pi ^{\rho {\rm -ugly}}$ is the unique representation in $\mathrm {Irr}_\sigma ^{\rho {\rm -ugly}}$ such that $\pi \hookrightarrow \tau \times \pi ^{\rho {\rm -ugly}}$ with $\mathrm {scusp}(\tau ) \cap (\mathbb {Z}_\rho \cup \mathbb {Z}_{\rho ^\vee }) = \emptyset$.

The following theorem is a special case of Jantzen's decomposition.

Theorem 2.8 [Reference JantzenJan97, Theorem 9.3]

The map

\begin{align*} \Psi \colon \mathrm{Irr}_\sigma &\longrightarrow \mathrm{Irr}_\sigma^{\rm good} \sqcup \biggl (\,\bigsqcup_{\rho\in \mathscr{C}^{\rm bad}} \mathrm{Irr}_\sigma^{\rho{\rm -bad}} \biggr) \sqcup \biggl(\,\bigsqcup_{\rho\in \mathscr{C}^{\rm ugly}} \mathrm{Irr}_\sigma^{\rho{\rm -ugly}} \biggr) \\ \pi &\longmapsto \bigl( \pi^{\rm good}, \{\pi^{\rho{\rm -bad}}\}_\rho, \{\pi^{\rho{\rm -ugly}}\}_\rho \bigr) \end{align*}

is bijective. Moreover, it commutes with the Zelevinsky–Aubert duality in the sense that

\[ \Psi(\hat\pi)= \bigl( \widehat{\pi^{\rm good}}, \{\widehat{\pi^{\rho{\rm -bad}}}\}_\rho, \{ \widehat{\pi^{\rho{\rm -ugly}}}\}_\rho \bigr). \]

In practice, this theorem enables us to reduce the problem of making the Zelevinsky–Aubert duality explicit to the case where the representation is either ugly or of good or bad parity.

3.1 Definitions

We treat first the case of general linear groups. For $\tau \in \mathrm {Rep}(\mathrm {GL}_n(F))$, define semisimple representations $L_\rho ^{(k)}(\tau )$ and $R_\rho ^{(k)}(\tau )$ of $\mathrm {GL}_{n-dk}(F)$ so that

\begin{align*} \big[\mathrm{Jac}_{(dk,n-dk)}(\tau)\big] &= \rho^k \boxtimes L_{\rho}^{(k)}(\tau) + \sum_i \tau_i \boxtimes \sigma_i,\\ \big[\mathrm{Jac}_{(n-dk,dk)}(\tau)\big] &= R_{\rho}^{(k)}(\tau) \boxtimes \rho^k + \sum_i \sigma'_i \boxtimes \tau'_i, \end{align*}

where $\tau _i$ and $\tau '_i$ are irreducible representations of $\mathrm {GL}_{dk}(F)$ which are not isomorphic to $\rho ^k$. We call $L_\rho ^{(k)}(\tau )$ (respectively $R_\rho ^{(k)}(\tau )$) the $k$th left $\rho$-derivative (respectively the $k$th right $\rho$-derivative) of $\tau$.

Similarly we now treat the case of $G_n$. Again let $k \geq 0$, and now let $P_{dk}$ be the standard parabolic subgroup of $G_n$ with Levi subgroup of the form $\mathrm {GL}_{dk}(F) \times G_{n-dk}$. For $\Pi \in \mathrm {Rep}(G_n)$, define a semisimple representation $D_\rho ^{(k)}(\Pi )$ of $G_{n-dk}$ so that

\[ \big[\mathrm{Jac}_{P_{dk}}^{G_n}(\Pi)\big] = \rho^k \boxtimes D_{\rho}^{(k)}(\Pi) + \sum_i \tau_i \boxtimes \Pi_i, \]

where $\tau _i$ is an irreducible representation of $\mathrm {GL}_{dk}(F)$ which is not isomorphic to $\rho ^k$. We call $D_\rho ^{(k)}(\Pi )$ the $k$th $\rho$-derivative of $\Pi$.

3.2 The non-self-dual case

If $\pi$ is irreducible and $\rho$ is not self-dual, then the highest $\rho$-derivative $D_\rho ^{(k)}(\pi )$ is irreducible and $\pi$ is isomorphic to the unique irreducible subrepresentation of $\rho ^k \rtimes D_{\rho }^{(k)}(\pi )$ (see [Reference JantzenJan14, Lemma 3.1.3] and [Reference AtobeAto22b, Proposition 2.7]). Using these properties, we can show the following.

Proof. Consider the highest $\rho$-derivative $D_{\rho }^{(k)}(\pi )$. If $\pi ' \hookrightarrow \rho ^r \rtimes \pi$, then $\pi ' \hookrightarrow \rho ^{k+r} \rtimes D_{\rho }^{(k)}(\pi )$. In particular, $D_{\rho }^{(k+r)}(\pi ') = D_{\rho }^{(k)}(\pi )$. However, since

\[ D_{\rho}^{(k+r)}\bigl( \rho^{k+r} \rtimes D_{\rho}^{(k)}(\pi) \bigr) = D_{\rho}^{(k)}(\pi) \]

by Tadić's formula (Proposition 2.1), we see that $\pi '$ is determined uniquely. Hence $\mathrm {soc}(\rho ^r \rtimes \pi )$ is irreducible and satisfies

\[ D_{\rho}^{(k+r)}\big( \mathrm{soc}(\rho^r \rtimes \pi) \big) = D_{\rho}^{(k+r)}\big( \rho^r \rtimes \pi \big) = D_{\rho}^{(k)}(\pi). \]

These equations imply that $\mathrm {soc}(\rho ^r \rtimes \pi )$ appears with multiplicity one in $[\rho ^r \rtimes \pi ]$.

We set

\[ S_\rho^{(r)}(\pi) = \underbrace{S_\rho^{(1)} \circ \cdots \circ S_\rho^{(1)}}_{r\text{ times}}(\pi) = \mathrm{soc}(\rho^r \rtimes \pi) \]

for any $\pi \in \mathrm {Irr}(G_n)$.

3.3 The self-dual case

Recall from [Reference AtobeAto22b, Proposition 2.7] that the highest $\rho$-derivative $D_\rho ^{(k)}(\pi )$ of an irreducible representation is isotypic, i.e. $D_\rho ^{(k)}(\pi ) = m \cdot \pi _0$ with some irreducible representation $\pi _0$ and a certain multiplicity $m > 0$. In this case, we have $\pi \hookrightarrow \rho ^k \rtimes \pi _0$, but $\mathrm {soc}(\rho ^k \rtimes \pi _0)$ can be reducible.

We give a criterion for $\rho ^r \rtimes \pi$ being SI.

3.4 $\Delta _\rho [0,-1]$-derivatives and $Z_\rho [0,1]$-derivatives

In the case where $\rho$ is self-dual, $\rho$-derivatives are difficult. Therefore, we define some other derivatives in this paragraph. These will be key ingredients in making the Zelevinsky–Aubert duality explicit. In this subsection we assume that $\rho \in \mathscr {C}(\mathrm {GL}_d(F))$ is self-dual.

Let $\Pi \in \mathrm {Rep}(G_n)$. Define the $\Delta _\rho [0,-1]$-derivative $D_{\Delta _\rho [0,-1]}^{(k)}(\Pi )$ and the $Z_\rho [0,1]$-derivative $D_{Z_\rho [0,1]}^{(k)}(\Pi )$ by the semisimple representations of $G_{n-2dk}$ satisfying

\[ \big[\mathrm{Jac}_{P_{2dk}}^{G_n}(\pi)\big] = \Delta_\rho[0,-1]^k \boxtimes D_{\Delta_\rho[0,-1]}^{(k)}(\pi) +Z_\rho[0,1]^k \boxtimes D_{Z_\rho[0,1]}^{(k)}(\pi) + \sum_i \tau_i \boxtimes \pi_i, \]

where $\tau _i \in \mathrm {Irr}(\mathrm {GL}_{2dk}(F))$ such that $\tau _i \not \cong \Delta _\rho [0,-1]^k, Z_\rho [0,1]^k$.

Typically, when the supercuspidal representation $\rho$ is clear from the context, we will write $[0,-1]$-derivative for short instead of $\Delta _\rho [0,-1]$-derivative, and $[0,1]$-derivative instead of $Z_\rho [0,1]$-derivative. We also write $D_{[0,-1]}^{(k)}(\Pi ) := D_{\Delta _\rho [0,-1]}^{(k)}(\Pi )$ and $D_{[0,1]}^{(k)}(\Pi ) := D_{Z_\rho [0,1]}^{(k)}(\Pi )$. Similar to Definition 3.2, we define the notion of highest $[0,-1]$-derivative (respectively highest $[0,1]$-derivative) and the property of being $\Delta _\rho [0,-1]$-reduced (respectively $Z_\rho [0,1]$-reduced).

Proof. Note that $\pi \hookrightarrow \rho ^{k_0} \times (\rho |\cdot |^{\epsilon })^{k_1} \rtimes \pi _1$. If $k_1 > k_0$, then no irreducible subquotient of $\rho ^{k_0} \times (\rho |\cdot |^{\epsilon })^{k_1}$ is left $\rho |\cdot |^{\epsilon }$-reduced. Since $\pi$ is $\rho |\cdot |^{\epsilon }$-reduced, we must have $k_0 \geq k_1$ and

\[ \pi \hookrightarrow \begin{cases} Z_\rho[0,1]^{k_1} \times \rho^{k_0-k_1} \rtimes \pi_1 & \text{if } \epsilon = 1, \\ \Delta_\rho[0,-1]^{k_1} \times \rho^{k_0-k_1} \rtimes \pi_1 & \text{if } \epsilon = -1. \end{cases} \]

Now we claim that $\pi _1$ is $\rho$-reduced. This is trivial when $k_1 = 0$. If $k_1 > 0$ and $\pi _1$ is not $\rho$-reduced, since $\pi _0$ is $\rho$-reduced, we can find a representation $\pi _1' \not = 0$ such that

\[ \pi_0 \hookrightarrow \begin{cases} \Delta_\rho[1,0] \rtimes \pi_1' & \text{if } \epsilon = 1, \\ Z_\rho[-1,0] \rtimes \pi_1' & \text{if } \epsilon = -1. \end{cases} \]

Since $\pi \hookrightarrow \rho ^{k_0} \rtimes \pi _0$, this implies that $D_{\rho |\cdot |^{\epsilon }}^{(1)}(\pi ) \not = 0$, which is a contradiction, so we obtain the claim.

Since $\pi _1$ is $\rho$-reduced and $\rho |\cdot |^{\epsilon }$-reduced, we see that $D_{[0,\epsilon ]}^{(1)}(\rho ^{k_0-k_1} \rtimes \pi _1) = 0$ by Tadić's formula (Proposition 2.1). Hence $D_{[0,\epsilon ]}^{(k_1)}(\pi )$ is the highest $[0,\epsilon ]$-derivative. Since it is a subrepresentation of $[\rho ^{k_0-k_1} \rtimes \pi _1]$, we see that $D_{[0, \epsilon ]}^{(k_1)}(\pi )$ is $\rho |\cdot |^{\epsilon }$-reduced.

In the next proposition, we will use the following simple lemma on representations of general linear groups.

Now we can prove the irreducibility of the highest $[0,\pm 1]$-derivatives of $\rho |\cdot |^{\pm 1}$-reduced irreducible representations.

Proof. We prove the assertions only for $[0,1]$. By the previous lemma, there exists an irreducible subrepresentation of $\pi _{[0,1]}$ of the highest $[0,1]$-derivative $D_{[0,1]}^{(k)}(\pi )$ such that

\[ \mathrm{Jac}_{P_{2dk}}^{G_n}(\pi) \twoheadrightarrow Z_\rho[0,1]^k \boxtimes \pi_0 \]

or, equivalently,

\[ \pi \hookrightarrow Z_\rho[0,1]^k \rtimes \pi_0. \]

Since $\pi$ is $\rho |\cdot |^1$-reduced, so is $\pi _0$. Hence, by Tadić's formula (Proposition 2.1) for

\[ [\mathrm{Jac}_{P_{2dk}}^{G_n}(Z_\rho[0,1]^k \rtimes \pi_0)], \]

we see that

\[ D_{[0,1]}^{(k)}(Z_\rho[0,1]^k \rtimes \pi_0) = \pi_0. \]

Therefore, $0 \not = D_{[0,1]}^{(k)}(\pi ) \subset \pi _0$ so that $D_{[0,1]}^{(k)}(\pi ) = \pi _0$. Moreover, this implies that $Z_\rho [0,1]^k \rtimes \pi _0$ is SI.

When $\pi '$ is an irreducible subrepresentation of $Z_\rho [0,1]^r \rtimes \pi$, we have $\pi ' \subset \mathrm {soc}(Z_\rho [0,1]^{k+r} \rtimes \pi _0)$. In particular, $\pi '$ is unique and appears with multiplicity one in $[Z_\rho [0,1]^{k+r} \rtimes \pi _0]$ and hence in $[Z_\rho [0,1]^r \rtimes \pi ]$. Therefore, $Z_\rho [0,1]^r \rtimes \pi$ is SI.

For simplicity, we set

\[ S_{[0,1]}^{(r)}(\pi) = S_{Z_\rho[0,1]}^{(r)}(\pi) := \mathrm{soc}(Z_\rho[0,1]^r \rtimes \pi) \]

for an irreducible representation $\pi$ of $G_n$ which is $\rho |\cdot |^1$-reduced.

The highest $[0,-1]$-derivatives are easy in a special case.

Proposition 3.8 Let $\pi = L(\Delta _{\rho _1}[x_1,y_1], \ldots, \Delta _{\rho _r}[x_r,y_r]; \pi _\mathrm {temp})$ be an irreducible representation of $G_n$. Suppose that $\pi$ is $\rho |\cdot |^z$-reduced for all $z \not = 0$ and that there exists $i \in \{1, \ldots, r\}$ such that $\rho _i \cong \rho$. Then $\min \{x_i \mid \rho _i \cong \rho \} = 0$, and the highest $[0,-1]$-derivative $D_{[0,-1]}^{(k)}(\pi )$ of $\pi$ is given by

\[ D_{[0,-1]}^{(k)}(\pi) = L(\Delta_{\rho_1}[z_1,y_1], \ldots, \Delta_{\rho_r}[z_r,y_r]; \pi_\mathrm{temp}) \]

with

\[ z_i = \begin{cases} -2 & \text{if } \rho_i \cong \rho, \, x_i = 0, \\ x_i & \text{otherwise}. \end{cases} \]

In particular,

\[ k = \bigl|\{i \in \{1, \ldots, r\} \mid \rho_i \cong \rho,\ x_i = 0\}\bigr| \geq 1. \]

Proof. With $x := \min \{x_i \mid \rho _i \cong \rho \}$, we see that $\pi$ is not $\rho |\cdot |^x$-reduced. Hence we must have $x = 0$. Moreover, we note that if $\rho _i \cong \rho$ and $x_i = 0$, then $y_i \leq -1$ since $x_i+y_i < 0$.

We remark that $D_\rho ^{(l)}(\pi _\mathrm {temp})$ is tempered since $\rho$ is self-dual (see [Reference AtobeAto20, Theorem 4.2(1) and (4)]), so $D_\rho ^{(l)}(\pi _\mathrm {temp})$ is $\rho |\cdot |^{-1}$-reduced by Casselman's criterion (see e.g. [Reference KonnoKon03, Lemma 2.4]). Hence by Lemma 3.5, with $k$ as in the statement, $D_{[0,-1]}^{(k)}(\pi )$ is the highest $[0,-1]$-derivative.

Set $\tau := L(\Delta _{\rho _1}[x_1,y_1], \ldots, \Delta _{\rho _r}[x_r,y_r])$. Then $\pi \hookrightarrow \tau \rtimes \pi _\mathrm {temp}$. Since $\min \{x_i \mid \rho _i \cong \rho \} = 0$ and $y_i < 0$, we see that $\tau \hookrightarrow \Delta _{\rho }[0,-1]^k \times \tau '$ with $\tau ' := L(\Delta _{\rho _1}[z_1,y_1], \ldots, \Delta _{\rho _r}[z_r,y_r])$. Hence

\[ \pi \hookrightarrow \Delta_{\rho}[0,-1]^k \times \tau' \rtimes \pi_\mathrm{temp}. \]

By the Frobenius reciprocity, we have a non-zero map

\[ \mathrm{Jac}_{P_{2dk}}^{G_n}(\pi) \rightarrow \Delta_{\rho}[0,-1]^k \boxtimes (\tau' \rtimes \pi_\mathrm{temp}), \]

which must factor through a non-zero map

\[ \Delta_{\rho}[0,-1]^k \boxtimes D_{[0,-1]}^{(k)}(\pi) \rightarrow \Delta_{\rho}[0,-1]^k \boxtimes (\tau' \rtimes \pi_\mathrm{temp}). \]

Since $D_{[0,-1]}^{(k)}(\pi )$ is irreducible by Proposition 3.7 and since $\tau ' \rtimes \pi _\mathrm {temp}$ is SI, we deduce that

\[ D_{[0,-1]}^{(k)}(\pi) = \mathrm{soc}(\tau' \rtimes \pi_\mathrm{temp}). \]

This completes the proof.

3.5 The Zelevinsky–Aubert duality and derivatives

We deduce the following compatibility between derivatives and duality.

5.1 $A$-parameters

We denote by $W_F$ the Weil group of $F$. A homomorphism

\[ \psi \colon W_F \times \mathrm{SL}_2(\mathbb{C}) \times \mathrm{SL}_2(\mathbb{C}) \rightarrow \mathrm{GL}_n(\mathbb{C}) \]

is called an $A$-parameter for $\mathrm {GL}_n(F)$ if

• • $\psi (\mathrm {Frob}) \in \mathrm {GL}_n(\mathbb {C})$ is semisimple and all its eigenvalues have absolute value $1$, where $\mathrm {Frob}$ is a fixed (geometric) Frobenius element;

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

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

The local Langlands correspondence for $\mathrm {GL}_d(F)$ asserts that there is a canonical bijection between the set of irreducible unitary supercuspidal representations of $\mathrm {GL}_d(F)$ and the set of irreducible $d$-dimensional representations of $W_F$ of bounded image. We identify these two sets and use the symbol $\rho$ for their elements.

Any such irreducible representation of $W_F \times \mathrm {SL}_2(\mathbb {C}) \times \mathrm {SL}_2(\mathbb {C})$ is of the form $\rho \boxtimes S_a \boxtimes S_b$, where $S_a$ is the unique irreducible algebraic representation of $\mathrm {SL}_2(\mathbb {C})$ of dimension $a$. We write $\rho \boxtimes S_a = \rho \boxtimes S_a \boxtimes S_1$ and $\rho = \rho \boxtimes S_1 \boxtimes S_1$ for short. For an $A$-parameter $\psi$, the multiplicity of $\rho \boxtimes S_a \boxtimes S_b$ in $\psi$ is denoted by $m_\psi (\rho \boxtimes S_a \boxtimes S_b)$. When $\psi = \bigoplus _{i \in I} \rho _i \boxtimes S_{a_i} \boxtimes S_{b_i}$ is an $A$-parameter of $\mathrm {GL}_n(F)$, we define $\tau _\psi$ by the product of Speh representations (see § 2.3)

\[ \tau_{\psi} := {\mathop \times\limits_{i \in I}} L\biggl( \Delta_{\rho_i}\biggl[ \frac{a_i-b_i}{2}, -\frac{a_i+b_i}{2}+1 \biggr], \ldots, \Delta_{\rho_i}\biggl[\frac{a_i+b_i}{2}-1, -\frac{a_i-b_i}{2} \biggr] \biggr). \]

Now we consider a split odd special orthogonal group $\mathrm {SO}_{2n+1}(F)$ or a symplectic group $\mathrm {Sp}_{2n}(F)$. We call $\psi$ an $A$-parameter for $\mathrm {SO}_{2n+1}(F)$ if it is an $A$-parameter for $\mathrm {GL}_{2n}(F)$ of symplectic type, i.e.

\[ \psi \colon W_F \times \mathrm{SL}_2(\mathbb{C}) \times \mathrm{SL}_2(\mathbb{C}) \rightarrow \mathrm{Sp}_{2n}(\mathbb{C}). \]

Similarly, $\psi$ is called an $A$-parameter for $\mathrm {Sp}_{2n}(F)$ if it is an $A$-parameter for $\mathrm {GL}_{2n+1}(F)$ of orthogonal type with the trivial determinant, i.e.

\[ \psi \colon W_F \times \mathrm{SL}_2(\mathbb{C}) \times \mathrm{SL}_2(\mathbb{C}) \rightarrow \mathrm{SO}_{2n+1}(\mathbb{C}). \]

For $G_n = \mathrm {SO}_{2n+1}(F)$ (respectively $G_n = \mathrm {Sp}_{2n}(F)$), we let $\Psi (G_n)$ be the set of $\widehat {G_n}$-conjugacy classes of $A$-parameters for $G_n$, where $\widehat {G_n} = \mathrm {Sp}_{2n}(\mathbb {C})$ (respectively $\widehat {G_n} = \mathrm {SO}_{2n+1}(\mathbb {C})$). We say that

• • $\psi \in \Psi (G_n)$ is tempered if the restriction of $\psi$ to the second $\mathrm {SL}_2(\mathbb {C})$ is trivial;

• • $\psi \in \Psi (G_n)$ is of good parity if $\psi$ is a sum of irreducible self-dual representations of the same type as $\psi$.

We denote by $\Psi _\mathrm {temp}(G_n) := \Phi _\mathrm {temp}(G_n)$ (respectively $\Psi _\mathrm {gp}(G_n)$) the subset of $\Psi (G_n)$ consisting of tempered $A$-parameters (respectively $A$-parameters of good parity). Also, we put $\Phi _\mathrm {gp}(G_n) := \Phi _\mathrm {temp}(G_n) \cap \Psi _\mathrm {gp}(G_n)$. Set $\Psi _*(G) := \bigcup _{n \geq 0}\Psi _*(G_n)$ and $\Phi _*(G) := \bigcup _{n \geq 0}\Phi _*(G_n)$ for $* \in \{\emptyset, \mathrm {temp}, \mathrm {gp}\}$.

For $\psi \in \Psi (G)$, a component group $\mathcal {S}_\psi$ is defined. We recall the definition only in the case where $\psi \in \Psi _\mathrm {gp}(G)$. Hence we can write $\psi = \bigoplus _{i=1}^{r} \psi _i$, where $\psi _i$ is an irreducible self-dual representation of the same type as $\psi$. We define an enhanced component group $\mathcal {A}_\psi$ as

\[ \mathcal{A}_\psi := \bigoplus_{i=1}^r \, (\mathbb{Z}/2\mathbb{Z})\alpha_{\psi_i}. \]

Specifically, $\mathcal {A}_\psi$ is a free $\mathbb {Z}/2\mathbb {Z}$-module of rank $r$ with a basis $\{\alpha _{\psi _i}\}$ associated with the irreducible components $\{\psi _i\}$. Define the component group $\mathcal {S}_\psi$ as the quotient of $\mathcal {A}_\psi$ by the subgroup generated by the elements

• • $z_\psi := \sum _{i=1}^r \alpha _{\psi _i}$; and

• • $\alpha _{\psi _i} + \alpha _{\psi _{i'}}$ such that $\psi _i \cong \psi _{i'}$.

Let $\widehat {\mathcal {S}_\psi }$ and $\widehat {\mathcal {A}_\psi }$ be the Pontryagin duals of $\mathcal {S}_\psi$ and $\mathcal {A}_\psi$, respectively. Via the canonical surjection $\mathcal {A}_\psi \twoheadrightarrow \mathcal {S}_\psi$, we may regard $\widehat {\mathcal {S}_\psi }$ as a subgroup of $\widehat {\mathcal {A}_\psi }$. For $\eta \in \widehat {\mathcal {A}_\psi }$, we write $\eta (\alpha _{\psi _i}) = \eta (\psi _i)$.

Let $\mathrm {Irr}_\mathrm {unit}(G_n)$ (respectively $\mathrm {Irr}_\mathrm {temp}(G_n)$) be the set of equivalence classes of irreducible unitary (respectively tempered) representations of $G_n$. For $\psi \in \Psi (G_n)$, Arthur [Reference ArthurArt13, Theorem 2.2.1] defined a multiset $\Pi _\psi$ over $\mathrm {Irr}_\mathrm {unit}(G_n)$, which is called the $A$-packet for $G_n$ associated with $\psi$. It has the following properties.

• • The multiset $\Pi _\psi$ is actually a (multiplicity-free) subset of $\mathrm {Irr}_\mathrm {unit}(G_n)$ (Mœglin [Reference MœglinMœg11]).

• • There exists a map $\Pi _\psi \rightarrow \widehat {\mathcal {S}_\psi }$, $\pi \mapsto \left \langle \cdot \, ,\pi \right \rangle _\psi$. If $\phi \in \Phi _\mathrm {temp}(G)$, it is a bijection. When $\pi \in \Pi _\phi$ corresponds to $\eta \in \widehat {\mathcal {S}_\phi }$, we write $\pi = \pi (\phi, \eta )$.

• • There is a canonical decomposition into a disjoint union

  \[ \mathrm{Irr}_\mathrm{temp}(G_n) = \bigsqcup_{\phi \in \Phi_\mathrm{temp}(G_n)}\Pi_\phi. \]

• • If $\psi = \psi _1 \oplus \psi _0 \oplus \psi _1^\vee$ for some irreducible representation $\psi _1$, then there exists a canonical injection $\mathcal {S}_{\psi _0} \hookrightarrow \mathcal {S}_{\psi }$, and

  \[ \tau_{\psi_1} \rtimes \pi_0 \cong \bigoplus_{\substack{\pi \in \Pi_\psi \\ \left\langle \cdot\,, \pi \right\rangle_\psi|\mathcal{S}_{\psi_0} = \left\langle \cdot\,, \pi_0 \right\rangle_{\psi_0}}} \pi \]
  for every $\pi _0 \in \Pi _{\psi _0}$ (see [Reference ArthurArt13, Proposition 2.4.3]).

5.2 A special example

Now we consider a special $A$-parameter of the form

\[ \psi = \phi \oplus (\rho \boxtimes S_{2x} \boxtimes S_2)^{t} \]

for $t \geq 1$, $\phi \in \Phi _\mathrm {gp}(G)$ and $x \in (1/2)\mathbb {Z}$ with $x > 0$ such that $\rho \boxtimes S_{2x+1}$ is self-dual of the same type as $\phi$.

For $l \in \mathbb {Z}/2\mathbb {Z}$ and for $\eta$ in a certain subset $\widehat {\mathcal {S}_{\psi,l}}$ of $\widehat {\mathcal {S}_\psi }$ (depending on $l$), we define $\pi (\psi, l, \eta )$ as follows. When $l = 1$, we set $\widehat {\mathcal {S}_{\psi,1}} := \widehat {\mathcal {S}_\phi } = \{ \eta \in \widehat {\mathcal {S}_\psi } \mid \eta (\rho \boxtimes S_{2x} \boxtimes S_2) = 1\}$ and

\[ \pi(\psi, 1, \eta) := L(\Delta_\rho[x-1,-x]^t; \pi(\phi, \eta)). \]

When $l = 0$ and $x \geq 1$, we let $\widehat {\mathcal {S}_{\psi,0}}$ be the subset of $\widehat {\mathcal {S}_\psi }$ consisting of $\eta$ that satisfy

• • $\eta (\rho \boxtimes S_{2x} \boxtimes S_2) = \eta (\rho \boxtimes S_{2x-1})$ if $\rho \boxtimes S_{2x-1} \subset \phi$;

• • $\eta (\rho \boxtimes S_{2x} \boxtimes S_2) = (-1)^{t}\eta (\rho \boxtimes S_{2x+1})$ if $\rho \boxtimes S_{2x+1} \subset \phi$;

• • $\eta (z_\phi ) = (-1)^t$.

When $l = 0$ and $x = 1/2$, we let $\widehat {\mathcal {S}_{\psi,0}}$ be the subset of $\widehat {\mathcal {S}_\psi }$ consisting of $\eta$ that satisfy

• • $\eta (\rho \boxtimes S_{1} \boxtimes S_2) = -1$;

• • $\eta (\rho \boxtimes S_{2}) = (-1)^{t}$ if $\rho \boxtimes S_{2} \subset \phi$;

• • $\eta (z_\phi ) = (-1)^t$.

For $\eta \in \widehat {\mathcal {S}_{\psi,0}}$, we define

\[ \pi(\psi, 0, \eta) := L\bigl(\Delta_\rho[x-1,-x]^{t-1}; \pi(\phi + \rho \boxtimes (S_{2x-1}+S_{2x+1}), \eta)\bigr). \]

Here, we regard $\eta$ as a character of the component group of $\phi + \rho \boxtimes (S_{2x-1}+S_{2x+1})$ by setting

\[ \begin{cases} \eta(\rho \boxtimes S_{2x-1}) = (-1)^t\eta(\rho \boxtimes S_{2x+1}) = \eta(\rho \boxtimes S_{2x} \boxtimes S_2) & \text{if } x \geq 1,\\ \eta(\rho \boxtimes S_{2}) = (-1)^{t} & \text{if } x = 1/2. \end{cases} \]

By specifying Mœglin's construction of $\Pi _\psi$, we have the following.

Proposition 5.2 Let $\psi = \phi \oplus (\rho \boxtimes S_{2x} \boxtimes S_2)^{t} \in \Psi _\mathrm {gp}(G)$ with $t \geq 1$. Then

\[ \Pi_\psi = \bigl\{ \pi(\psi, l, \eta)\bigm| l \in \mathbb{Z}/2\mathbb{Z}, \: \eta \in \widehat{\mathcal{S}_{\psi,l}} \bigr\}. \]

Moreover, the map $\Pi _\psi \rightarrow \widehat {\mathcal {S}_\psi }$ is given by $\left \langle \cdot \,,\pi (\psi, l ,\eta ) \right \rangle _\psi = \varepsilon _{l,\eta }$, where

\begin{align*} \varepsilon_{l,\eta}(\rho \boxtimes S_d) &= \eta(\rho \boxtimes S_d), \\ \varepsilon_{l,\eta}(\rho \boxtimes S_{2x} \boxtimes S_2) &= \begin{cases} (-1)^{l-1} & \text{if } x \geq 1, \\ \eta(\rho \boxtimes S_{1} \boxtimes S_2) & \text{if } x = 1/2. \end{cases} \end{align*}

Using this description, we obtain the formula for the highest $\rho |\cdot |^x$-derivatives and socles.

Theorem 5.3 Fix $\phi \in \Phi _\mathrm {gp}(G)$ and write $m = m_{\phi }(\rho \boxtimes S_{2x+1})$ and $m' = m_{\phi }(\rho \boxtimes S_{2x-1})$. Consider $\psi = \phi \oplus (\rho \boxtimes S_{2x} \boxtimes S_2)^t \in \Psi _\mathrm {gp}(G)$ with $t \geq 0$. Let $\pi (\psi, l, \eta ) \in \Pi _{\psi }$ be such that $\eta (\rho \boxtimes S_{2x-1})\eta (\rho \boxtimes S_{2x+1}) = (-1)^t$ if $mm' \not = 0$. Here, if $x=1/2$, we formally understand that $m'=1$ and $\eta (\rho \boxtimes S_0) = 1$. Let $s$ be a non-negative integer such that $s=0$ if $x=1/2$. Then the highest $\rho |\cdot |^x$-derivative of $\mathrm {soc}((\rho |\cdot |^{-x})^{s} \rtimes \pi (\psi, l ,\eta ))$ is given by

\begin{align*} & D_{\rho|\cdot|^x}^{(m+\max\{s-m',0\})}\bigl(\mathrm{soc}\bigl((\rho|\cdot|^{-x})^{s} \rtimes \pi(\psi, l ,\eta)\bigr)\bigr)\\ & \quad = \mathrm{soc}\bigl( (\rho|\cdot|^{-x})^{\min\{s,m'\}} \rtimes \pi\bigl(\psi - (\rho \boxtimes S_{2x+1})^m + (\rho \boxtimes S_{2x-1})^m, l+m, \eta\bigr)\bigr), \end{align*}

where we set $\eta (\rho \boxtimes S_{2x-1}) = (-1)^t \eta (\rho \boxtimes S_{2x+1})$ if needed. In particular,

\begin{align*} &S_{\rho|\cdot|^x}^{(1)}\bigl(\mathrm{soc}\big((\rho|\cdot|^{-x})^{s} \rtimes \pi(\psi, l ,\eta)\big)\bigr)\\ &\quad = \begin{cases} \mathrm{soc}\bigl((\rho|\cdot|^{-x})^{s} \rtimes \pi(\psi - \rho \boxtimes S_{2x-1} + \rho \boxtimes S_{2x+1}, l-1, \eta) \bigr) & \text{if } s < m',\\ \mathrm{soc}\bigl((\rho|\cdot|^{-x})^{s+1} \rtimes \pi(\psi, l ,\eta)\bigr) & \text{if } s \geq m', \end{cases} \end{align*}

where we set $\eta (\rho \boxtimes S_{2x+1}) = (-1)^t \eta (\rho \boxtimes S_{2x-1})$.

5.3 Zelevinsky–Aubert duals of certain tempered representations

The initial step of our algorithm for computing the Zelevinsky–Aubert duals (Algorithm 4.1(3)) is to compute $\hat \pi$ for tempered $\pi$ such that $\pi$ is $\rho '$-reduced for every non-self-dual $\rho '\in \mathscr {C}^\mathrm {GL}$. If $\pi = \pi (\phi, \eta )$ for $\phi \in \Phi _\mathrm {gp}(G)$, then $\pi$ satisfies this condition if and only if

1. $~(\ast )$ if $\rho \boxtimes S_{d} \subset \phi$ with $d \geq 2$, then $m_\phi (\rho \boxtimes S_{d}) =1$, $\rho \boxtimes S_{d-2} \subset \phi$ and $\eta (\rho \boxtimes S_{d}) \not = \eta (\rho \boxtimes S_{d-2})$.

See [Reference AtobeAto20, Theorem 4.2]. Here, we formally understand that $\rho \boxtimes S_0 \subset \phi$ and $\eta (\rho \boxtimes S_0) = +1$ if $\rho$ is self-dual of the opposite type to $\phi$.

Proposition 5.4 Let $\pi = \pi (\phi, \eta )$ with $\phi \in \Phi _\mathrm {gp}(G)$. Assume that $\pi$ satisfies the above condition $(\ast )$. Write

\[ \{\rho \mid m_\phi(\rho) >0,\, m_\phi(\rho) \equiv 0 \bmod 2\} = \{\rho_1, \ldots, \rho_r\} \]

and set

\[ y_i := \max\biggl\{\frac{d_i-1}{2}\biggm| \rho_i \boxtimes S_{d_i} \subset \phi \biggr\}. \]

Suppose that $y_1 \geq \cdots \geq y_t > 0 = y_{t+1} = \cdots = y_r$. Then

\[ \hat\pi = L\big(\Delta_{\rho_1}[0,-y_1], \ldots, \Delta_{\rho_t}[0,-y_t]; \pi(\phi', \eta')\big), \]

where

\[ \phi' = \phi - \bigoplus_{i=1}^t \rho_i \boxtimes (S_1+S_{2y_i+1}) \]

and

\[ \eta'(\rho \boxtimes S_{d}) = \begin{cases} -\eta(\rho \boxtimes S_d) & \text{if } \rho \in \{\rho_1, \ldots, \rho_r\}, \\ \eta(\rho \boxtimes S_d) & \text{otherwise}. \end{cases} \]

Proof. Set

\[ \{\rho \mid m_\phi(\rho) >0,\, m_\phi(\rho) \equiv 1 \bmod 2\} = \{\rho'_1, \ldots, \rho'_{r'}\}. \]

Write $m_\phi (\rho _i) = 2k_i > 0$ and $m_{\phi }(\rho '_j) = 2k'_j+1$. Then, by [Reference AtobeAto20, Theorem 4.2], we have

\[ \Bigl(\circ_{j=1}^{r'}D_{\rho_j'}^{(k'_j)}\Bigr)\circ \Bigl(\circ_{i=1}^{r}D_{\rho_i|\cdot|^{y_i}}^{(1)} \circ \cdots \circ D_{\rho_i|\cdot|^1}^{(1)} \circ D_{\rho_i}^{(k_i)}\Bigr)(\pi) \not= 0. \]

This is $\pi (\phi '', \eta '')$ up to multiplicity, where

\[ \phi'' = \phi - \biggl(\bigoplus_{j=1}^{r'}{\rho'_j}^{2k_j'}\biggr) - \biggl(\bigoplus_{i=1}^r\rho_i \boxtimes (S_1^{2k_i-1}+S_{2y_i+1})\biggr) \]

and

\[ \eta''(\rho \boxtimes S_{d}) = \begin{cases} -\eta(\rho \boxtimes S_d) & \text{if } \rho \in \{\rho_1, \ldots, \rho_t\}, \\ \eta(\rho \boxtimes S_d) & \text{if } \rho \not\in \{\rho_1, \ldots, \rho_r\}. \end{cases} \]

For $t < i \leq r$, we note that $\rho _i \not \subset \phi ''$ so that $\eta ''(\rho _i \boxtimes S_d)$ does not appear. In particular, $\pi (\phi '', \eta '')$ is supercuspidal. By [Reference AtobeAto22b, Theorem 2.13], with $\phi '$ as in the statement, we have

\[ \hat\pi = L(\Delta_{\rho_1}[0,-y_1], \ldots, \Delta_{\rho_t}[0,-y_t]; \pi(\phi', \eta')) \]

for some $\eta ' \in \mathcal {A}_{\phi '}$ such that $\eta '' = \eta '|\mathcal {A}_{\phi ''}$ via the canonical inclusion $\mathcal {A}_{\phi ''} \hookrightarrow \mathcal {A}_{\phi '}$. Since $\mathcal {S}_{\phi '}$ is generated by $\mathcal {S}_{\phi ''}$ and the image of $\{\alpha _{\rho _i} \mid i > t\}$, the remaining task is to determine $\eta '(\rho _{i_0})$ for $i_0 > t$. To do this, by replacing $\pi$ with

\[ \Bigl(\circ_{j=1}^{r'}D_{\rho_j'}^{(k'_j)}\Bigr) \circ \Bigl(\circ_{\substack{1 \leq i \leq r \\ i \not= i_0}} D_{\rho_i|\cdot|^{y_i}}^{(1)} \circ \cdots \circ D_{\rho_i|\cdot|^1}^{(1)} \circ D_{\rho_i}^{(k_i)}\Bigr)(\pi), \]

we may assume that $\pi \subset \rho ^k \rtimes \sigma$ with $\sigma$ supercuspidal such that $\rho \rtimes \sigma$ is semisimple of length two. If we write $\rho \rtimes \sigma = \pi _+ \oplus \pi _-$, then $\rho ^{k-1} \rtimes \pi _{\pm }$ is irreducible and its Zelevinsky–Aubert dual is given by $\rho ^{k-1} \rtimes \hat \pi _{\pm }$. By [Reference AubertAub95, Corollaire 1.10], we know that $\hat \pi _{\pm } = \pi _{\mp }$. Hence we see that $\eta '(\rho _{i_0}) = -\eta (\rho _{i_0})$, as desired.

If $\pi$ is tempered, of $\rho$-bad parity and $\rho |\cdot |^z$-reduced for all $z \not = 0$, then $\pi$ must be of the form $\pi = \rho ^m \rtimes \sigma$ for some $m \geq 0$ and $\sigma$ supercuspidal. In particular, we have $\hat \pi = \pi$. Similarly, if $\pi$ is tempered, ugly and $\rho '$-reduced for all non-self-dual $\rho '\in \mathscr {C}^\mathrm {GL}$, then $\pi$ must be supercuspidal so that $\hat \pi = \pi$.

6.1 Best matching functions

Let $A$ and $B$ be totally ordered finite sets with respect to $\geq _A$ and $\geq _B$, respectively. For $a \in A$, write $A_{>a} := \{a' \in A \mid a' >_A a\}$. We consider a relation $\rightsquigarrow$ between $B$ and $A$ such that

\begin{align*} &\forall\, a_1 \geq_A a_2 \in A\quad \text{and} \quad \forall\, b_1 \geq_B b_2 \in B,\\ &b_1 \rightsquigarrow a_1 \text{ and } b_2 \rightsquigarrow a_1 \text{ and } b_2 \rightsquigarrow a_2 \implies b_1 \rightsquigarrow a_2. \end{align*}

We say that such a relation is traversable. In this case, we define a subset $A^0$ of $A$ and an injective map $f \colon A^0 \rightarrow B$ recursively by

\begin{align*} & a \in A^0 \iff \exists\, b \in B\setminus f(A^0 \cap A_{>a})\text{ such that }b \rightsquigarrow a, \\ &\text{in which case }f(a) := \min\{b \in B \setminus f(A^0 \cap A_{>a}) \mid b \rightsquigarrow a\}. \end{align*}

Let $B^0 := f(A^0)$ be the image of $f$. We call the bijection $f \colon A^0 \rightarrow B^0$ the best matching function between $A$ and $B$. By [Reference Lapid and MínguezLM16, Lemma 5.7], the domain $A^0$ is equal to $A$ if and only if Hall's criterion is satisfied, i.e. for any subset $A' \subset A$,

\[ \bigl|\{b \in B \mid b \rightsquigarrow a\text{ for some }a \in A'\}\bigr| \geq |A'|. \]

When one of $A$ or $B$ is the empty set, note that we have $A^0 = B^0 = \emptyset$. We set $A^{\rm c}=A\setminus A^0$ and $B^{\rm c} = B \setminus B^0$.

6.2 Derivatives and socles in the ugly and negative cases

Fix $\rho \in \mathscr {C}^\mathrm {GL}$ and $x \in \mathbb {R}$. In this subsection, we give explicit formulas using the best matching functions for the highest $\rho |\cdot |^x$-derivatives $D_{\rho |\cdot |^x}^{(k)}(\pi )$ and the socles $S_{\rho |\cdot |^x}^{(1)}(\pi ) = \mathrm {soc}(\rho |\cdot |^x \rtimes \pi )$ in the case where $\rho |\cdot |^x$ is ugly or where $\rho$ is self-dual and $x$ is negative.

Let $\pi \in \mathrm {Irr}(G_n)$. By Remark 2.7 and the Langlands classification, we can write $\pi = \mathrm {soc}(L(\Delta _{\rho _1}[x_1,y_1], \ldots, \Delta _{\rho _r}[x_r,y_r]) \rtimes \pi _\mathrm {temp})$, where

• • if $\rho |\cdot |^x$ is ugly, then $\rho _i = \rho$ for all $i = 1, \ldots, r$, $x_1+y_1 \leq \cdots \leq x_r+y_r$ and $\pi _\mathrm {temp} = \sigma$ is supercuspidal;

• • if $\rho$ is self-dual and $x$ is negative, then $x_1+y_1 \leq \cdots \leq x_r+y_r < 0$ and $\pi _\mathrm {temp}$ is tempered.

To unify the notation, let us call $(\Delta _{\rho _1}[x_1,y_1], \ldots, \Delta _{\rho _r}[x_r,y_r]; \pi _\mathrm {temp})$ the inducing data.

Define an ordered set $A_{\rho |\cdot |^x}$ by

\[ A_{\rho|\cdot|^x} := \{ i \in \{1, \ldots, r\} \mid \rho_i \cong \rho, \, x_i = x\} \]

with

\[ a \geq a' \iff y_a \geq y_{a'}. \]

We define a relation $\rightsquigarrow$ between $A_{\rho |\cdot |^x}$ and $A_{\rho |\cdot |^{x-1}}$ by

\[ A_{\rho|\cdot|^x} \ni a' \rightsquigarrow a \in A_{\rho|\cdot|^{x-1}} \iff y_{a'} > y_a. \]

Namely, $a' \rightsquigarrow a$ if and only if $L(\Delta _\rho [x_a,y_a], \Delta _\rho [x_{a'},y_{a'}])$ is a ladder representation. Note that this relation is traversable. Let $f \colon A_{\rho |\cdot |^{x-1}}^0 \rightarrow A_{\rho |\cdot |^{x}}^0$ be the best matching function. In the next proposition, we obtain explicit formulas for the highest $\rho |\cdot |^x$-derivative $D_{\rho |\cdot |^x}^{(k)}(\pi )$ and the socle $S_{\rho |\cdot |^x}^{(1)}(\pi )$.

Proposition 6.1 Suppose that $\rho |\cdot |^x$ is ugly or that $\rho$ is self-dual and $x$ is negative. With notation as above, the highest $\rho |\cdot |^x$-derivative $D_{\rho |\cdot |^x}^{(k)}(\pi )$ is the unique irreducible subrepresentation of $L(\Delta _{\rho _1}[x'_1,y_1], \ldots, \Delta _{\rho _r}[x'_r,y_r]) \rtimes \pi _\mathrm {temp}$, where

\[ x_i' = \begin{cases} x-1 & \text{if } i \in A_{\rho|\cdot|^{x}}^{\rm c},\\ x_i & \text{otherwise}. \end{cases} \]

In particular, $k = |A_{\rho |\cdot |^{x}}^{\rm c}|$. Moreover, the following hold.

1. (a) If $A_{\rho |\cdot |^{x-1}}^{\rm c}\neq \emptyset$, then the inducing data of $S_{\rho |\cdot |^x}^{(1)}(\pi )$ can be obtained from those of $\pi$ by replacing $x_a = x-1$ with $x$, where $a$ is the minimum element of $A_{\rho |\cdot |^{x-1}}^{\rm c}$.

2. (b) If $A_{\rho |\cdot |^{x-1}}^{\rm c} = \emptyset$, then the inducing data of $S_{\rho |\cdot |^x}^{(1)}(\pi )$ can be obtained from those of $\pi$ by inserting $\rho |\cdot |^x = \Delta _\rho [x,x]$.

Proof. Since $\rho |\cdot |^x$ is ugly or $\rho$ is self-dual and $x$ negative, we have

\begin{align*} D_{\rho|\cdot|^x}^{(k)}(\pi) &= \mathrm{soc}\bigl( L_{\rho|\cdot|^x}^{(k)}(L(\Delta_{\rho_1}[x_1,y_1], \ldots, \Delta_{\rho_r}[x_r,y_r])) \rtimes \pi_\mathrm{temp} \bigr),\\ S_{\rho|\cdot|^x}^{(1)}(\pi) &= \mathrm{soc}\bigl( \mathrm{soc}(\rho|\cdot|^x \times L(\Delta_{\rho_1}[x_1,y_1], \ldots, \Delta_{\rho_r}[x_r,y_r])) \rtimes \pi_\mathrm{temp} \bigr). \end{align*}

Therefore, the proposition is essentially a problem for general linear groups, which was treated in [Reference Lapid and MínguezLM16, Theorem 5.11].

7.1 The good-parity case

In this subsection, we assume that $\pi \in \mathrm {Irr}(G_n)$ is of good parity and that $\rho \boxtimes S_{2x+1}$ is self-dual of the same type as elements in $\Phi _\mathrm {gp}(G)$. Write $\pi = L(\Delta _{\rho _1}[x_1,y_1], \ldots, \Delta _{\rho _{r'}}[x_{r'},y_{r'}]; \pi (\phi, \eta ))$ as a Langlands subrepresentation so that $x_1+y_1 \leq \cdots \leq x_{r'}+y_{r'} <0$ and $\phi \in \Phi _\mathrm {gp}(G)$. Set

\[ t = \bigl|\{i \in \{1, \ldots, r'\}\mid \Delta_{\rho_i}[x_i,y_i] \cong \Delta_\rho[x-1,-x]\}\bigr| \]

and $r = r'-t$. Then we can rewrite

\[ \pi = \mathrm{soc}\bigl( L(\Delta_{\rho_1}[x_1,y_1], \ldots, \Delta_{\rho_r}[x_r,y_r]) \rtimes \pi_A \bigr), \]

where we set $\pi _A := L(\Delta _\rho [x-1,-x]^t; \pi (\phi, \eta ))$.

If $m_\phi (\rho \boxtimes S_{2x+1}) \not = 0$, $m_\phi (\rho \boxtimes S_{2x-1}) \not = 0$ and $\eta (\rho \boxtimes S_{2x+1})\eta (\rho \boxtimes S_{2x-1}) = (-1)^{t+1}$, set

\[ \psi := \phi - \rho \boxtimes (S_{2x+1}+S_{2x-1}) + (\rho \boxtimes S_{2x} \boxtimes S_2)^{t+1} \]

and $l := 0$. Otherwise, set $\psi := \phi + (\rho \boxtimes S_{2x} \boxtimes S_2)^t$ and $l := 1$. Then $\pi _A = \pi (\psi, l, \eta ) \in \Pi _\psi$ by Proposition 5.2. Set $m := m_\psi (\rho \boxtimes S_{2x+1})$ and $m' := m_\psi (\rho \boxtimes S_{2x-1})$. Then the highest $\rho |\cdot |^x$-derivative of $\mathrm {soc}((\rho |\cdot |^{-x})^s \rtimes \pi _A)$ is described in Theorem 5.3.

Note that $x_i \geq y_i$ for all $i = 1, \ldots, r$. Define ordered sets

\begin{align*} A_{\rho|\cdot|^{x}} &:= \{i \in \{1,\ldots,r\} \mid \rho_i \cong \rho,\, x_i = x\},\\ B_{\rho|\cdot|^x} &:= \{i \in \{1,\ldots,r\} \mid \rho_i \cong \rho,\, y_i = -x\} \end{align*}

with

\begin{alignat*}{4} a \geq a' & \iff y_a \geq y_{a'} & \quad & \text{for }a,a' \in A_{\rho|\cdot|^{x}},\\ b \geq b' & \iff x_b \leq x_{b'} & \quad & \text{for }b,b' \in B_{\rho|\cdot|^x}. \end{alignat*}

Notice that any two of $A_{\rho |\cdot |^{x-1}}, A_{\rho |\cdot |^{x}}, B_{\rho |\cdot |^{x-1}}$ and $B_{\rho |\cdot |^{x}}$ have no intersection. Define relations $\rightsquigarrow$ between $A_{\rho |\cdot |^{x}}$ and $A_{\rho |\cdot |^{x-1}}$ and between $B_{\rho |\cdot |^{x}}$ and $B_{\rho |\cdot |^{x-1}}$ by

\begin{alignat*}{4} A_{\rho|\cdot|^{x}} & \ni a' \rightsquigarrow a \in A_{\rho|\cdot|^{x-1}} & \iff & y_{a'} > y_a, \\ B_{\rho|\cdot|^{x}} & \ni b' \rightsquigarrow b \in B_{\rho|\cdot|^{x-1}} & \iff & x_{b'} < x_b, \end{alignat*}

respectively. Note that these relations are traversable. Let $f \colon A_{\rho |\cdot |^{x-1}}^0 \rightarrow A_{\rho |\cdot |^{x}}^0$ and $g \colon B_{\rho |\cdot |^{x-1}}^0 \rightarrow B_{\rho |\cdot |^{x}}^0$ be the best matching functions. Write $B_{\rho |\cdot |^{x}}^{\rm c}= \{i_1, \ldots, i_{s}\}$ with $i_1 < \cdots < i_{s}$. Notice that $s > 0$ only if $x > 1$.

Theorem 7.1 With notation as above, suppose that $x > 0$, $x \in (1/2)\mathbb {Z}$ and $\rho \boxtimes S_{2x+1}$ is self-dual of the same type as $\phi$. Then the highest $\rho |\cdot |^x$-derivative $D_{\rho |\cdot |^x}^{(k)}(\pi )$ is the unique irreducible subrepresentation of $L(\Delta _{\rho _1}[x'_1,y'_1], \ldots, \Delta _{\rho _r}[x'_r,y'_r]) \rtimes \pi _A'$, where

\begin{align*} x_i' &= \begin{cases} -1 & \text{if } i \in A_{\rho|\cdot|^{x}}^{\rm c},\\ x_i & \text{otherwise}, \end{cases} \\ y_i' &= \begin{cases} -(x-1) & \text{if } i=i_j, \: j > m'+\max\{|A_{\rho|\cdot|^{x-1}}^{\rm c}|-m,0\}, \\ y_i & \text{otherwise}, \end{cases} \end{align*}

and $\pi _A' = \pi (\psi ', l', \eta )$ with

\[ \psi' = \psi - (\rho \boxtimes S_{2x+1})^{\max\{m-|A_{\rho|\cdot|^{x-1}}^{\rm c}|,\,0\}} + (\rho \boxtimes S_{2x-1})^{\max\{m-|A_{\rho|\cdot|^{x-1}}^{\rm c}|,\,0\}} \]

and

\[ l' = l + \max\{m-|A_{\rho|\cdot|^{x-1}}^{\rm c}|,0\}. \]

In particular,

\[ k = |A_{\rho|\cdot|^{x}}^{\rm c}| + \max\bigl\{ m+\max\{|B_{\rho|\cdot|^{x}}^{\rm c}|-m',0\} -|A_{\rho|\cdot|^{x-1}}^{\rm c}|,0 \bigr\}. \]

Moreover, the following hold.

1. (a) If $m+\max \{|B_{\rho |\cdot |^{x}}^{\rm c}|-m',0\} < |A_{\rho |\cdot |^{x-1}}^{\rm c}|$, then the Langlands data of $S_{\rho |\cdot |^x}^{(1)}(\pi )$ can be obtained from those of $\pi$ by replacing $x_a = x-1$ with $x$, where $a$ is the minimum element of $A_{\rho |\cdot |^{x-1}}^{\rm c}$.

2. (b) If $|B_{\rho |\cdot |^{x}}^{\rm c}| < m'$ and $m \geq |A_{\rho |\cdot |^{x-1}}^{\rm c}|$, the Langlands data of $S_{\rho |\cdot |^x}^{(1)}(\pi )$ can be obtained from those of $\pi$ by replacing $\pi _A = \pi (\psi, l, \eta )$ with

   \[ S_{\rho|\cdot|^x}^{(1)}(\pi_A) = \pi\bigl(\psi - (\rho \boxtimes S_{2x-1}) + (\rho \boxtimes S_{2x+1}), l-1, \eta\bigr). \]

3. (c) If $|B_{\rho |\cdot |^{x}}^{\rm c}| \geq m'$, $m+|B_{\rho |\cdot |^{x}}^{\rm c}|-m' \geq |A_{\rho |\cdot |^{x-1}}^{\rm c}|$ and $B_{\rho |\cdot |^{x-1}}^{\rm c} \not = \emptyset$, the Langlands data of $S_{\rho |\cdot |^x}^{(1)}(\pi )$ can be obtained from those of $\pi$ by replacing $y_b = -(x-1)$ with $-x$, where $b$ is the minimum element of $B_{\rho |\cdot |^{x-1}}^{\rm c}$.

4. (d) If $|B_{\rho |\cdot |^{x}}^{\rm c}| \geq m'$, $m+|B_{\rho |\cdot |^{x}}^{\rm c}|-m' \geq |A_{\rho |\cdot |^{x-1}}^{\rm c}|$ and $B_{\rho |\cdot |^{x-1}}^{\rm c} = \emptyset$, then the Langlands data of $S_{\rho |\cdot |^x}^{(1)}(\pi )$ can be obtained from those of $\pi$ by inserting $\rho |\cdot |^{-x} = \Delta _{\rho }[-x,-x]$.

Proof. To obtain the formula for the highest derivative, we use Jantzen's algorithm [Reference JantzenJan18a, § 3.3] together with [Reference Lapid and MínguezLM16, Theorem 5.11] and Theorem 5.3.

(1) Recall that

\[ \pi = \mathrm{soc}\big( L(\Delta_{\rho_1}[x_1,y_1], \ldots, \Delta_{\rho_r}[x_r,y_r]) \rtimes \pi_A \big) \]

with $\pi _A = L(\Delta _\rho [x-1,-x]^t; \pi (\phi, \eta ))$ and $\Delta _{\rho _i}[x_i,y_i] \not \cong \Delta _\rho [x-1,-x]$ for all $i = 1, \ldots,r$.

(2) By [Reference Lapid and MínguezLM16, Theorem 5.11], we can compute the highest right $\rho |\cdot |^{-x}$-derivative

\[ R_{\rho|\cdot|^{-x}}^{(s)}(L(\Delta_{\rho_1}[x_1,y_1], \ldots, \Delta_{\rho_r}[x_r,y_r])) = L(\Delta_{\rho_1}[x_1,y_1''], \ldots, \Delta_{\rho_r}[x_r,y_r'']), \]

where

\[ y_i'' = \begin{cases} -(x-1) & \text{if } i \in B_{\rho|\cdot|^{x}}^{\rm c}, \\ y_i & \text{otherwise}. \end{cases} \]

In particular, $s = |B_{\rho |\cdot |^{x}}^{\rm c}|$. Claim  1 in [Reference JantzenJan18a, § 3.3] says that

\[ \pi = \mathrm{soc}\big( L(\Delta_{\rho_1}[x_1,y_1''], \ldots, \Delta_{\rho_r}[x_r,y_r'']) \rtimes \pi_1 \big) \]

with $\pi _1 := \mathrm {soc}((\rho |\cdot |^{-x})^s \rtimes \pi _A)$.

(3) By Theorem 5.3, the highest $\rho |\cdot |^x$-derivative $\pi _2 := D_{\rho |\cdot |^x}^{(k_1)}(\pi _1)$ of $\pi _1$ is

\[ \pi_2 = \mathrm{soc}\big( (\rho|\cdot|^{-x})^{\min\{s,m'\}} \rtimes \pi(\psi - (\rho \boxtimes S_{2x+1})^m + (\rho \boxtimes S_{2x-1})^m, l+m,\eta) \big) \]

with $k_1 = m + \max \{s-m', 0\}$. Claim 2 in [Reference JantzenJan18a, § 3.3] says that

\[ \pi = \mathrm{soc}\big( L(\Delta_{\rho_1}[x_1,y_1''], \ldots, \Delta_{\rho_r}[x_r,y_r''], (\rho|\cdot|^x)^{k_1}) \rtimes \pi_2 \big). \]

(4) We will apply [Reference Lapid and MínguezLM16, Theorem 5.11] to compute the highest left $\rho |\cdot |^x$-derivative of $L(\Delta _{\rho _1}[x_1,y_1''], \ldots, \Delta _{\rho _r}[x_r,y_r''], (\rho |\cdot |^x)^{k_1})$. To do this, we have to replace $A_{\rho |\cdot |^x}$ with $A_{\rho |\cdot |^x} \cup \{r+1, \ldots, r+k_1\}$, where we set $\Delta _{\rho _i}[x_i,y_i] = \rho |\cdot |^x$ for $i = r+1, \ldots, r+k_1$. Note that any $a' \in \{r+1, \ldots, r+k_1\}$ is bigger than any element of $A_{\rho |\cdot |^x}$ with respect to the order of $A_{\rho |\cdot |^x} \cup \{r+1, \ldots, r+k_1\}$, and $a' \rightsquigarrow a$ for every $a \in A_{\rho |\cdot |^{x-1}}$. Hence the image of the resulting best matching function is

\[ A_{\rho|\cdot|^x}^0 \cup \bigl\{ r+i \bigm|1 \leq i \leq \min\{k_1, |A_{\rho|\cdot|^{x-1}}^{\rm c}|\} \bigr\}. \]

Therefore, with $k_2 = \min \{k_1, |A_{\rho |\cdot |^{x-1}}^{\rm c}|\}$ and $k = |A_{\rho |\cdot |^{x}}^{\rm c}| + k_1-k_2$, the highest left $\rho |\cdot |^x$-derivative is

\begin{align*} &L_{\rho|\cdot|^x}^{(k)} \bigl(L(\Delta_{\rho_1}[x_1,y_1''], \ldots, \Delta_{\rho_r}[x_r,y_r''], (\rho|\cdot|^x)^{k_1})\bigr)\\ &\quad = L(\Delta_{\rho_1}[x_1',y_1''], \ldots, \Delta_{\rho_r}[x_r',y_r''], (\rho|\cdot|^x)^{k_2}), \end{align*}

where $x_i'$ is as in the statement of this theorem. Then the highest $\rho |\cdot |^x$-derivative of $\pi$ is

\[ D_{\rho|\cdot|^x}^{(k)}(\pi) = \mathrm{soc}\bigl( L(\Delta_{\rho_1}[x_1',y_1''], \ldots, \Delta_{\rho_r}[x_r',y_r''], (\rho|\cdot|^x)^{k_2}) \rtimes \pi_2 \bigr). \]

(5) Claim  3 in [Reference JantzenJan18a, § 3.3] says that

\[ D_{\rho|\cdot|^x}^{(k)}(\pi) = \mathrm{soc}\bigl( L(\Delta_{\rho_1}[x_1',y_1''], \ldots, \Delta_{\rho_r}[x_r',y_r'']) \rtimes S_{\rho|\cdot|^x}^{(k_2)}(\pi_2) \bigr). \]

By Theorem 5.3, we have

\[ S_{\rho|\cdot|^x}^{(k_2)}(\pi_2) = \mathrm{soc}((\rho|\cdot|^{-x})^{s'} \rtimes \pi_A'), \]

where $\pi _A'$ is as in the statement of this theorem and $s' = \min \{s,m'\} + \max \{k_2-m,0\}$. Note that $s' \leq s$.

(6) Finally, note that

• • if $s'=s$, then $m'+\max \{|A_{\rho |\cdot |^{x-1}}^{\rm c}|-m,0\} \geq s$, so that $y_i' = y_i$ for all $i=1, \ldots, r$;

• • if $s' < s$, then $s > m'$ and $k_1 = m+s-m' > k_2 = |A_{\rho |\cdot |^{x-1}}^{\rm c}|$, so that $s' = m'+\max \{|A_{\rho |\cdot |^{x-1}}^{\rm c}|-m,0\}$.

By [Reference Lapid and MínguezLM16, Theorem 5.11], we have

\begin{align*} &\mathrm{soc}\big( L(\Delta_{\rho_1}[x_1',y_1''], \ldots, \Delta_{\rho_r}[x_r',y_r'']) \times (\rho|\cdot|^{-x})^{s'} \big)\\ &\quad = L(\Delta_{\rho_1}[x_1',y_1'], \ldots, \Delta_{\rho_r}[x_r',y_r']), \end{align*}

where $y_i'$ is as in the statement of this theorem. Claim  4 in [Reference JantzenJan18a, § 3.3] says that

\[ D_{\rho|\cdot|^x}^{(k)}(\pi) = \mathrm{soc}\bigl( L(\Delta_{\rho_1}[x_1',y_1'], \ldots, \Delta_{\rho_r}[x_r',y_r']) \rtimes \pi_A' \bigr). \]

This gives the Langlands data of $D_{\rho |\cdot |^x}^{(k)}(\pi )$.

Recall that $S_{\rho |\cdot |^x}^{(1)}(\pi )$ is an irreducible representation determined by the relation

\[ D_{\rho|\cdot|^x}^{(k+1)} \bigl( S_{\rho|\cdot|^x}^{(1)}(\pi) \bigr) = D_{\rho|\cdot|^x}^{(k)}(\pi). \]

One can easily check this equation for the representations given in (a), (b), (c) and (d).

As an application of Proposition 6.1 and Theorem 7.1, we have a combinatorial irreducibility criterion for $\rho |\cdot |^x \rtimes \pi$ as follows.

As a special case, when $\pi = \pi (\phi, \eta )$ is tempered, since $A_{\rho ^\vee |\cdot |^{-x-1}}, A_{\rho |\cdot |^{x-1}}, A_{\rho |\cdot |^x}, B_{\rho |\cdot |^{x-1}}$ and $B_{\rho |\cdot |^x}$ are all the empty set, we see that $\rho |\cdot |^x \rtimes \pi$ if and only if $m_\psi (\rho \boxtimes S_{2x-1}) = 0$, which is equivalent to

• • $\phi \not \supset \rho \boxtimes S_{2x-1}$; or

• • $m_\phi (\rho \boxtimes S_{2x-1}) = 1$, $m_\phi (\rho \boxtimes S_{2x+1}) > 0$ and $\eta (\rho \boxtimes S_{2x-1}) \not = \eta (\rho \boxtimes S_{2x+1})$.

This special case was already known to Jantzen [Reference JantzenJan18b, Theorem 4.7].

7.2 The bad-parity case

We now treat the bad-parity case. Specifically, we assume that $\rho \boxtimes S_{2x+1}$ is self-dual of the opposite type to elements in $\Phi _\mathrm {gp}(G)$, and we take $\pi \in \mathrm {Irr}(G_n)$ such that $\mathrm {scusp}(\pi ) \subset \mathbb {Z}_{\rho |\cdot |^x} \cup \{\sigma \}$ for some $\sigma \in \mathscr {C}^G$.