3.1

Let $\unicode[STIX]{x1D70C}$ be a cuspidal irreducible $\text{R}$ -representation of $\text{G}$ . Associated with $\unicode[STIX]{x1D70C}$ , there is a positive integer $s(\unicode[STIX]{x1D70C})$ defined in [Reference Mínguez and SécherreMS14b, § 3.4] (see also Remark 3.8). When $\text{R}$ is the field of complex numbers, $s(\unicode[STIX]{x1D70C})$ is the unique positive integer $k$ such that $\unicode[STIX]{x1D70C}\times \unicode[STIX]{x1D70C}\unicode[STIX]{x1D708}^{k}$ is reducible, and it is related to the parametric degree $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70C})$ defined in [Reference Bushnell and HenniartBH11a, § 2] by the formula $s(\unicode[STIX]{x1D70C})\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70C})=n$ . For the general case, see Remark 3.8.

In [Reference Mínguez and SécherreMS14a] we attach to $\unicode[STIX]{x1D70C}$ and any integer $r\geqslant 1$ an irreducible subrepresentation $\text{Z}(\unicode[STIX]{x1D70C},r)$ and an irreducible quotient $\text{L}(\unicode[STIX]{x1D70C},r)$ of the induced representation

(3.1) $$\begin{eqnarray}\unicode[STIX]{x1D70C}\times \unicode[STIX]{x1D70C}\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D70C}}^{}\times \cdots \times \unicode[STIX]{x1D70C}\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D70C}}^{r-1}\end{eqnarray}$$

(see [Reference Mínguez and SécherreMS14a, Paragraph 7.2 and Définition 7.5]), where $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D70C}}$ is the character $\unicode[STIX]{x1D708}^{s(\unicode[STIX]{x1D70C})}$ .

When $\text{R}$ is the field of complex numbers, $\text{Z}(\unicode[STIX]{x1D70C},r)$ and $\text{L}(\unicode[STIX]{x1D70C},r)$ are uniquely determined in this way, and all essentially square-integrable representations of $\text{G}$ are isomorphic to a representation of the form $\text{L}(\unicode[STIX]{x1D70C},r)$ for a unique pair $(\unicode[STIX]{x1D70C},r)$ .

For an arbitrary $\text{R}$ , the representation $\text{L}(\unicode[STIX]{x1D70C},r)$ is called a discrete series $\text{R}$ -representation of $\text{G}$ , and $\text{Z}(\unicode[STIX]{x1D70C},r)$ is called a Speh $\text{R}$ -representation. If $\unicode[STIX]{x1D70C}$ is supercuspidal, $\text{Z}(\unicode[STIX]{x1D70C},r)$ is called a super-Speh representation.

According to [Reference Mínguez and SécherreMS14a, § 8.1], where the notion of residually nondegenerate representation is defined, the induced representation (3.1) contains a unique residually nondegenerate irreducible subquotient, denoted by

$$\begin{eqnarray}\text{Sp}(\unicode[STIX]{x1D70C},r).\end{eqnarray}$$

When $\text{R}$ has characteristic $0$ , this is equal to $\text{L}(\unicode[STIX]{x1D70C},r)$ . When $\text{R}$ has characteristic $\ell >0$ , however, it may differ from $\text{L}(\unicode[STIX]{x1D70C},r)$ (see [Reference Mínguez and SécherreMS14a, Remark 8.14]).

Assume that $\text{R}$ has characteristic $\ell >0$ , and write $\unicode[STIX]{x1D714}(\unicode[STIX]{x1D70C})$ for the smallest positive integer $i\geqslant 1$ such that $\unicode[STIX]{x1D70C}\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D70C}}^{i}$ is isomorphic to $\unicode[STIX]{x1D70C}$ . Then the irreducible representation

(3.2) $$\begin{eqnarray}\text{Sp}(\unicode[STIX]{x1D70C},\unicode[STIX]{x1D714}(\unicode[STIX]{x1D70C})\ell ^{v})\end{eqnarray}$$

is cuspidal for any integer $v\geqslant 0$ . Moreover, any cuspidal non-supercuspidal irreducible representation is of the form (3.2) for a supercuspidal irreducible representation $\unicode[STIX]{x1D70C}$ and a unique integer $v\geqslant 0$ (see [Reference Mínguez and SécherreMS14a, Théorème 6.14]). We record this latter fact for future reference.

3.2

In this subsection, we assume that $\text{R}$ is an algebraic closure $\overline{\mathbf{Q}}_{\ell }$ of the field of $\ell$ -adic numbers. Recall (see [Reference VignérasVig96]) that an irreducible $\ell$ -adic representation of $\text{G}$ is integral if it contains a $\text{G}$ -stable $\overline{\mathbf{Z}}_{\ell }$ -lattice. Let $\widetilde{\unicode[STIX]{x1D70C}}$ be an $\ell$ -adic cuspidal irreducible representation of $\text{G}$ . By [Reference VignérasVig96, II.4.12], it is integral if and only if its central character has values in $\overline{\mathbf{Z}}_{\ell }$ . In particular, there is always an unramified twist of $\widetilde{\unicode[STIX]{x1D70C}}$ which is integral.

Assume that $\widetilde{\unicode[STIX]{x1D70C}}$ is integral and write $a=a(\widetilde{\unicode[STIX]{x1D70C}})$ for the length of its reduction mod $\ell$ , denoted by  $\text{}\text{r}_{\ell }(\widetilde{\unicode[STIX]{x1D70C}})$ .

Proposition 3.2 [Reference Mínguez and SécherreMS14b, Theorem 3.15].

Let $\unicode[STIX]{x1D70C}$ be an irreducible factor of $\text{}\text{r}_{\ell }(\widetilde{\unicode[STIX]{x1D70C}})$ . Then

$$\begin{eqnarray}\text{}\text{r}_{\ell }(\widetilde{\unicode[STIX]{x1D70C}})=\unicode[STIX]{x1D70C}+\unicode[STIX]{x1D70C}\unicode[STIX]{x1D708}+\cdots +\unicode[STIX]{x1D70C}\unicode[STIX]{x1D708}^{a-1},\end{eqnarray}$$

where $\unicode[STIX]{x1D708}$ denotes the unramified mod- $\ell$ character ‘absolute value of the reduced norm’.

3.3

We recall briefly the language of simple strata, though we do not require much of the detail of the constructions. For a detailed presentation, see [Reference SécherreSéc05b, Reference Mínguez and SécherreMS14b]. For simple strata, we use the simplified notation of [Reference Bushnell and HenniartBH14, ch. 2].

Let $[\mathfrak{a},\unicode[STIX]{x1D6FD}]$ be a simple stratum in the simple central $\text{F}$ -algebra $\text{M}_{m}(\text{D})$ . Rather than giving the precise definition, we simply recall that it is composed of an element $\unicode[STIX]{x1D6FD}\in \text{M}_{m}(\text{D})$ such that the $\text{F}$ -algebra $\text{F}[\unicode[STIX]{x1D6FD}]$ is a field and a hereditary order $\mathfrak{a}\subseteq \text{M}_{m}(\text{D})$ normalized by $\text{F}[\unicode[STIX]{x1D6FD}]^{\times }$ . The centralizer of $\unicode[STIX]{x1D6FD}$  in $\text{M}_{m}(\text{D})$ , denoted by $\text{B}$ , is a simple central $\text{F}[\unicode[STIX]{x1D6FD}]$ -algebra. There are an $\text{F}[\unicode[STIX]{x1D6FD}]$ -division algebra $\text{D}\text{sp}\prime$ and an integer $m\text{sp}\prime \geqslant 1$ such that

(3.3) $$\begin{eqnarray}\text{B}\simeq \text{M}_{m\text{sp}\prime }(\text{D}\text{sp}\prime ).\end{eqnarray}$$

The intersection $\mathfrak{b}=\mathfrak{a}\,\cap \,\text{B}$ is a hereditary order in $\text{B}$ .

Recall [Reference SécherreSéc04, Reference Mínguez and SécherreMS14b] that associated with $[\mathfrak{a},\unicode[STIX]{x1D6FD}]$ are compact open subgroups

$$\begin{eqnarray}\text{H}^{1}(\mathfrak{a},\unicode[STIX]{x1D6FD})\subseteq \text{J}^{1}(\mathfrak{a},\unicode[STIX]{x1D6FD})\subseteq \text{J}(\mathfrak{a},\unicode[STIX]{x1D6FD})\end{eqnarray}$$

of $\text{G}$ , together with a non-empty finite set $\mathscr{C}(\mathfrak{a},\unicode[STIX]{x1D6FD})$ depending on the choice of $\unicode[STIX]{x1D713}$ made in § 2. These groups are normal in $\text{J}(\mathfrak{a},\unicode[STIX]{x1D6FD})$ , and the elements of $\mathscr{C}(\mathfrak{a},\unicode[STIX]{x1D6FD})$ are $\text{R}$ -characters of $\text{H}^{1}(\mathfrak{a},\unicode[STIX]{x1D6FD})$ , called simple characters. Besides, $\text{H}^{1}(\mathfrak{a},\unicode[STIX]{x1D6FD})$ and $\text{J}^{1}(\mathfrak{a},\unicode[STIX]{x1D6FD})$ are pro- $p$ -groups, and $\text{J}(\mathfrak{a},\unicode[STIX]{x1D6FD})$ is equal to $\mathfrak{b}^{\times }\text{J}^{1}(\mathfrak{a},\unicode[STIX]{x1D6FD})$ .

Attached to a simple character $\unicode[STIX]{x1D703}\in \mathscr{C}(\mathfrak{a},\unicode[STIX]{x1D6FD})$ is an invariant called its endo-class. We do not recall the precise definition of this invariant, which can be found in [Reference Bushnell and HenniartBH96, Reference Broussous, Sécherre and StevensBSS12]. We only need a few properties of endo-classes, which we will recall when they are needed. Endo-classes form a set $\mathscr{E}(\text{F})$ which depends only on  $\text{F}$ .

The endo-class of a simple character in $\text{G}$ has degree dividing $n$ . Conversely, any endo-class of degree dividing $n$ occurs as the endo-class of some simple character in $\text{G}$ .

A $\unicode[STIX]{x1D6FD}$ -extension of a simple character $\unicode[STIX]{x1D703}\in \mathscr{C}(\mathfrak{a},\unicode[STIX]{x1D6FD})$ is an irreducible representation of $\text{J}(\mathfrak{a},\unicode[STIX]{x1D6FD})$ with coefficients in $\text{R}$ whose restriction to $\text{J}^{1}(\mathfrak{a},\unicode[STIX]{x1D6FD})$ is irreducible, whose restriction to $\text{H}^{1}(\mathfrak{a},\unicode[STIX]{x1D6FD})$ contains $\unicode[STIX]{x1D703}$ and which is intertwined by any element of $\text{B}^{\times }$ (see [Reference SécherreSéc05a, Reference Mínguez and SécherreMS14b]).

Assume now that $\mathfrak{b}$ is a maximal order in $\text{B}$ , in which case we say that the simple stratum $[\mathfrak{a},\unicode[STIX]{x1D6FD}]$ , the simple characters in $\mathscr{C}(\mathfrak{a},\unicode[STIX]{x1D6FD})$ and their $\unicode[STIX]{x1D6FD}$ -extensions are maximal. Let us fix an isomorphism (3.3) such that the image of $\mathfrak{b}$ is the maximal order made of all matrices with integer entries. There is a natural group isomorphism

$$\begin{eqnarray}\text{J}(\mathfrak{a},\unicode[STIX]{x1D6FD})/\text{J}^{1}(\mathfrak{a},\unicode[STIX]{x1D6FD})\simeq \text{GL}_{m\text{sp}\prime }(\boldsymbol{d}),\end{eqnarray}$$

where $\boldsymbol{d}$ is the residue field of $\text{D}\text{sp}\prime$ . We write $\mathscr{G}$ for the group on the right-hand side. Let us fix a $\unicode[STIX]{x1D6FD}$ -extension $\unicode[STIX]{x1D705}$ of some simple character $\unicode[STIX]{x1D703}\in \mathscr{C}(\mathfrak{a},\unicode[STIX]{x1D6FD})$ . We write $\text{J}=\text{J}(\mathfrak{a},\unicode[STIX]{x1D6FD})$ and $\text{J}^{1}=\text{J}^{1}(\mathfrak{a},\unicode[STIX]{x1D6FD})$ .

We fix a finite extension $\boldsymbol{k}$ of $\boldsymbol{d}$ of degree $m\text{sp}\prime$ . We write $\unicode[STIX]{x1D6F4}$ for the Galois group of this extension and $\text{X}$ for the group of $\text{R}$ -characters of $\boldsymbol{k}^{\times }$ . Given $\unicode[STIX]{x1D6FC}\in \text{X}$ , there is a unique subfield $\boldsymbol{d}\subseteq \boldsymbol{d}[\unicode[STIX]{x1D6FC}]\subseteq \boldsymbol{k}$ such that the $\unicode[STIX]{x1D6F4}$ -stabilizer of $\unicode[STIX]{x1D6FC}$ is $\text{Gal}(\boldsymbol{k}/\boldsymbol{d}[\unicode[STIX]{x1D6FC}])$ , and then a character $\unicode[STIX]{x1D6FC}_{0}$ of $\boldsymbol{d}[\unicode[STIX]{x1D6FC}]^{\times }$ such that $\unicode[STIX]{x1D6FC}$  is equal to $\unicode[STIX]{x1D6FC}_{0}$ composed with the norm of $\boldsymbol{k}$ over $\boldsymbol{d}[\unicode[STIX]{x1D6FC}]$ . If we write $u$ for the degree of $\boldsymbol{d}[\unicode[STIX]{x1D6FC}]$ over $\boldsymbol{d}$ , then $\unicode[STIX]{x1D6FC}_{0}$ defines a supercuspidal irreducible $\text{R}$ -representation $\unicode[STIX]{x1D70E}_{0}$ of $\text{GL}_{u}(\boldsymbol{d})$ ; see [Reference GreenGre96] for the case where $\text{R}$ has characteristic $0$ , and [Reference DipperDip85] or [Reference Mínguez and SécherreMS15] otherwise.

Remark 3.4. More precisely, if $\text{R}$ has characteristic $0$ , fix an embedding of $\boldsymbol{d}[\unicode[STIX]{x1D6FC}]$ in $\text{M}_{u}(\boldsymbol{d})$ . Then $\unicode[STIX]{x1D70E}_{0}$ is the unique (up to isomorphism) irreducible representation of $\text{GL}_{u}(\boldsymbol{d})$ such that

$$\begin{eqnarray}\text{tr}\,\unicode[STIX]{x1D70E}_{0}(g)=(-1)^{u-1}\cdot \mathop{\sum }_{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6FC}_{0}^{\unicode[STIX]{x1D6FE}}(g)\end{eqnarray}$$

for all $g\in \boldsymbol{d}[\unicode[STIX]{x1D6FC}]^{\times }$ of degree $u$ over $\boldsymbol{d}$ , where $\unicode[STIX]{x1D6FE}$ runs over $\text{Gal}(\boldsymbol{d}[\unicode[STIX]{x1D6FC}]/\boldsymbol{d})$ .

The character $\unicode[STIX]{x1D6FC}\in \text{X}$ thus defines a supercuspidal $\text{R}$ -representation

$$\begin{eqnarray}\boldsymbol{\unicode[STIX]{x1D70E}}(\unicode[STIX]{x1D6FC})=\unicode[STIX]{x1D70E}_{0}\otimes \cdots \otimes \unicode[STIX]{x1D70E}_{0}\end{eqnarray}$$

of the Levi subgroup $\text{GL}_{u}(\boldsymbol{d})\times \cdots \times \text{GL}_{u}(\boldsymbol{d})$ in $\mathscr{G}$ . Moreover, the fibres of the map $\unicode[STIX]{x1D6FC}\mapsto \boldsymbol{\unicode[STIX]{x1D70E}}(\unicode[STIX]{x1D6FC})$ are the $\unicode[STIX]{x1D6F4}$ -orbits of $\text{X}$ . Write $r$ for the integer defined by $ru=m\text{sp}\prime$ . The maximal order $\mathfrak{b}$ contains a unique principal order $\mathfrak{b}_{r}$ of period $r$ whose image under (3.3) consists of matrices with entries in the ring of integers of $\text{D}\text{sp}\prime$ whose reduction modulo its maximal ideal is upper triangular by blocks of size $r$ . We write $\mathfrak{a}_{r}$ for the unique order normalized by $\text{F}[\unicode[STIX]{x1D6FD}]^{\times }$ such that $\mathfrak{a}_{r}\,\cap \,\text{B}=\mathfrak{b}_{r}$ , and $\unicode[STIX]{x1D705}_{r}$ for the transfer of  $\unicode[STIX]{x1D705}$ with respect to the simple stratum $[\mathfrak{a}_{r},\unicode[STIX]{x1D6FD}]$ in the sense of [Reference Mínguez and SécherreMS14b, Proposition 2.3]. Considering $\boldsymbol{\unicode[STIX]{x1D70E}}(\unicode[STIX]{x1D6FC})$ as a representation of the group $\text{J}_{r}=\text{J}(\mathfrak{a}_{r},\unicode[STIX]{x1D6FD})$ that is trivial on $\text{J}^{1}(\mathfrak{a}_{r},\unicode[STIX]{x1D6FD})$ , we define

$$\begin{eqnarray}\boldsymbol{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D6FC})=\unicode[STIX]{x1D705}_{r}\otimes \boldsymbol{\unicode[STIX]{x1D70E}}(\unicode[STIX]{x1D6FC}),\end{eqnarray}$$

which is a simple supertype in $\text{G}$ defined on $\text{J}_{r}$ in the sense of [Reference Sécherre and StevensSS16]. Write $\unicode[STIX]{x1D6E4}$ for the Galois group of $\boldsymbol{k}$ over $\boldsymbol{e}$ , where $\boldsymbol{e}$ denotes the residue field of $\text{F}[\unicode[STIX]{x1D6FD}]$ .

Write $\boldsymbol{\unicode[STIX]{x1D6E9}}$ for the endo-class of the simple character $\unicode[STIX]{x1D703}\in \mathscr{C}(\mathfrak{a},\unicode[STIX]{x1D6FD})$ , and $\mathscr{T}(\text{G},\boldsymbol{\unicode[STIX]{x1D6E9}},\text{R})$ for the set of isomorphism classes of simple $\text{R}$ -supertypes in $\text{G}$ with endo-class $\boldsymbol{\unicode[STIX]{x1D6E9}}$ , that is, simple $\text{R}$ -supertypes whose associated simple character has endo-class $\boldsymbol{\unicode[STIX]{x1D6E9}}$ .

Recall [Reference Sécherre and StevensSS16, Definition 6.1] that two simple $\text{R}$ -supertypes in $\text{G}$ are said to be equivalent if the representations of $\text{G}$ obtained from them by compact induction are isomorphic.

Proposition 3.5. The map

(3.4) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}\mapsto \boldsymbol{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D6FC})\end{eqnarray}$$

induces a surjection from $\text{X}$ onto the set of equivalence classes of $\mathscr{T}(\text{G},\boldsymbol{\unicode[STIX]{x1D6E9}},\text{R})$ . The fibres of this map are the $\unicode[STIX]{x1D6E4}$ -orbits of $\text{X}$ .

Proposition 3.6. The bijection

(3.5) $$\begin{eqnarray}\{\unicode[STIX]{x1D6E4}\text{-orbits of }\text{X}\}\leftrightarrow \{\text{equivalence classes of }\mathscr{T}(\text{G},\boldsymbol{\unicode[STIX]{x1D6E9}},\text{R})\}\end{eqnarray}$$

depends only on the choice of $\unicode[STIX]{x1D705}$ , not on that of the isomorphism (3.3).

Remark 3.7. Suppose that $\boldsymbol{k}_{1}$ is another extension of $\boldsymbol{d}$ of degree $m\text{sp}\prime$ . Write $\text{X}_{1}$ for the group of $\text{R}$ -characters of its invertible elements and $\unicode[STIX]{x1D6E4}_{1}$ for the Galois group $\text{Gal}(\boldsymbol{k}_{1}/\boldsymbol{e})$ . Let $\mathbf{t}$ denote the bijection (3.5) and write $\mathbf{t}_{1}$ for its analogue obtained by replacing $\boldsymbol{k}$ with $\boldsymbol{k}_{1}$ . Choosing an isomorphism of $\boldsymbol{e}$ -algebras $\boldsymbol{k}\rightarrow \boldsymbol{k}_{1}$ induces a bijection

$$\begin{eqnarray}\mathbf{b}:\text{X}_{1}/\unicode[STIX]{x1D6E4}_{1}\rightarrow \text{X}/\unicode[STIX]{x1D6E4}\end{eqnarray}$$

which does not depend on this choice, and one has $\mathbf{t}_{1}=\mathbf{t}\circ \mathbf{b}$ .

3.4

Recall [Reference Mínguez and SécherreMS14b] that any supercuspidal $\text{R}$ -representation $\unicode[STIX]{x1D70C}$ of $\text{G}$ contains a maximal simple character, uniquely determined up to $\text{G}$ -conjugacy. We define the endo-class of $\unicode[STIX]{x1D70C}$ to be the endo-class of any simple character contained in $\unicode[STIX]{x1D70C}$ . If we write $\boldsymbol{\unicode[STIX]{x1D6E9}}$ for this endo-class, then $\unicode[STIX]{x1D70C}$ contains a simple $\text{R}$ -supertype $\boldsymbol{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D6FC})\in \mathscr{T}(\text{G},\boldsymbol{\unicode[STIX]{x1D6E9}},\text{R})$ for some $\unicode[STIX]{x1D6FC}\in \text{X}$ with trivial $\unicode[STIX]{x1D6F4}$ -stabilizer.

3.5

We call an inertial class of supercuspidal pairs of $\text{G}$ simple if it contains a pair of the form

(3.6) $$\begin{eqnarray}(\text{GL}_{m/r}(\text{D})^{r},\unicode[STIX]{x1D70C}\otimes \cdots \otimes \unicode[STIX]{x1D70C})\end{eqnarray}$$

for some integer $r$ dividing $m$ and some supercuspidal $\text{R}$ -representation $\unicode[STIX]{x1D70C}$ of $\text{GL}_{m/r}(\text{D})$ , and we define the endo-class of such an inertial class to be the endo-class of $\unicode[STIX]{x1D70C}$ , that is, the endo-class of any simple character contained in $\unicode[STIX]{x1D70C}$ . By [Reference Sécherre and StevensSS16, § 8], there is a bijective correspondence between simple inertial classes of supercuspidal pairs of $\text{G}$ and equivalence classes of simple supertypes of $\text{G}$ , that preserves endo-classes. More precisely, the inertial class of (3.6), denoted by $\unicode[STIX]{x1D6FA}$ , corresponds to the equivalence class of a simple supertype $(\text{J},\unicode[STIX]{x1D706})$ if and only if the irreducible representations of $\text{G}$ occurring as a subquotient of the compact induction of $\unicode[STIX]{x1D706}$ to $\text{G}$ are exactly those irreducible representations of $\text{G}$ occurring as a subquotient of the parabolic induction to $\text{G}$ of an element of  $\unicode[STIX]{x1D6FA}$ .

From the previous subsection, we have an endo-class $\boldsymbol{\unicode[STIX]{x1D6E9}}$ and a maximal $\unicode[STIX]{x1D6FD}$ -extension  $\unicode[STIX]{x1D705}$ . Combining the map (3.4) with the correspondence between simple inertial classes of supercuspidal pairs and equivalence classes of simple supertypes, we get the following result. Given $\unicode[STIX]{x1D6FC}\in \text{X}$ , we write $\boldsymbol{\unicode[STIX]{x1D6FA}}(\unicode[STIX]{x1D6FC})$ for the inertial class of supercuspidal pairs of $\text{G}$ that corresponds to $\boldsymbol{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D6FC})$ .

Proposition 3.9. The map

(3.7) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}\mapsto \boldsymbol{\unicode[STIX]{x1D6FA}}(\unicode[STIX]{x1D6FC})\end{eqnarray}$$

induces a surjection from $\text{X}$ to the set of simple inertial classes of supercuspidal pairs of $\text{G}$ with associated endo-class $\boldsymbol{\unicode[STIX]{x1D6E9}}$ . Its fibres are the $\unicode[STIX]{x1D6E4}$ -orbits of $\text{X}$ .

Let us recall the following important result [Reference Mínguez and SécherreMS14a, Théorème 8.16]: given an irreducible $\text{R}$ -representation $\unicode[STIX]{x1D70B}$ of $\text{G}$ , there are integers $m_{1},\ldots ,m_{r}\geqslant 1$ such that $m_{1}+\cdots +m_{r}=m$ and supercuspidal irreducible representations $\unicode[STIX]{x1D70C}_{1},\ldots ,\unicode[STIX]{x1D70C}_{r}$ of $\text{GL}_{m_{1}}(\text{D}),\ldots ,\text{GL}_{m_{r}}(\text{D})$ , respectively, such that $\unicode[STIX]{x1D70B}$ occurs as a subquotient of the induced representation $\unicode[STIX]{x1D70C}_{1}\times \cdots \times \unicode[STIX]{x1D70C}_{r}$ . Moreover, up to renumbering, the supercuspidal representations $\unicode[STIX]{x1D70C}_{1},\ldots ,\unicode[STIX]{x1D70C}_{r}$ are unique. The conjugacy class of the supercuspidal pair $(\text{GL}_{m_{1}}(\text{D})\times \cdots \times \text{GL}_{m_{r}}(\text{D}),\unicode[STIX]{x1D70C}_{1}\,\otimes \,\ldots \,\otimes \,\unicode[STIX]{x1D70C}_{r})$ is called the supercuspidal support of $\unicode[STIX]{x1D70B}$ .

Let us call an irreducible $\text{R}$ -representation of $\text{G}$ simple if the inertial class of its supercuspidal support is simple. For instance, any discrete series $\text{R}$ -representation of $\text{G}$ is simple. We define the endo-class of a simple irreducible representation to be that of its supercuspidal support.

4.1

Let $\widetilde{\unicode[STIX]{x1D70B}}$ be an irreducible $\ell$ -adic representation of $\text{G}$ . Fix a representative $(\text{M},\widetilde{\unicode[STIX]{x1D70C}})$ in the inertial class of its cuspidal support, with $\text{M}$ a standard Levi subgroup $\text{GL}_{m_{1}}(\text{D})\times \cdots \times \text{GL}_{m_{r}}(\text{D})$ and $\widetilde{\unicode[STIX]{x1D70C}}$ of the form $\widetilde{\unicode[STIX]{x1D70C}}_{1}\otimes \cdots \otimes \widetilde{\unicode[STIX]{x1D70C}}_{r}$ where $\widetilde{\unicode[STIX]{x1D70C}}_{i}$ is an $\ell$ -adic cuspidal irreducible representation of $\text{GL}_{m_{i}}(\text{D})$ for $i\in \{1,\ldots ,r\}$ , and with $m_{1}+\cdots +m_{r}=m$ . Since $\widetilde{\unicode[STIX]{x1D70C}}_{i}$ is determined up to an unramified twist, we may assume that it is integral (see § 3.2) and fix an irreducible subquotient $\unicode[STIX]{x1D70C}_{i}$ of its reduction mod  $\ell$ . By the classification of mod  $\ell$ irreducible cuspidal representations in terms of supercuspidal representations [Reference Mínguez and SécherreMS14a, Théorème 6.14], there are a unique integer $u_{i}\geqslant 1$ dividing $m_{i}$ and a supercuspidal irreducible representation $\unicode[STIX]{x1D70F}_{i}$ of degree $u_{i}$ such that the supercuspidal support of $\unicode[STIX]{x1D70C}_{i}$ is inertially equivalent to

$$\begin{eqnarray}(\text{GL}_{u_{i}}(\text{D})\times \cdots \times \text{GL}_{u_{i}}(\text{D}),\unicode[STIX]{x1D70F}_{i}\otimes \cdots \otimes \unicode[STIX]{x1D70F}_{i})\end{eqnarray}$$

where the factors are repeated $k_{i}$ times, with $m_{i}=k_{i}u_{i}$ .

Definition 4.1. Let $\widetilde{\unicode[STIX]{x1D70B}}$ be an irreducible $\ell$ -adic representation of $\text{G}$ as above. Let us write

$$\begin{eqnarray}\text{L}=\text{GL}_{u_{1}}(\text{D})^{k_{1}}\times \cdots \times \text{GL}_{u_{r}}(\text{D})^{k_{r}},\quad \unicode[STIX]{x1D70F}=\underbrace{\unicode[STIX]{x1D70F}_{1}\otimes \cdots \otimes \unicode[STIX]{x1D70F}_{1}}_{k_{1}\text{ times}}\otimes \cdots \otimes \underbrace{\unicode[STIX]{x1D70F}_{r}\otimes \cdots \otimes \unicode[STIX]{x1D70F}_{r}}_{k_{r}\text{ times}}.\end{eqnarray}$$

The inertial class in $\text{G}$ of the supercuspidal pair $(\text{L},\unicode[STIX]{x1D70F})$ , denoted by $\text{}\text{i}_{\ell }(\widetilde{\unicode[STIX]{x1D70B}})$ , is uniquely determined by the irreducible representation $\widetilde{\unicode[STIX]{x1D70B}}$ . It is called the mod- $\ell$ inertial supercuspidal support of $\widetilde{\unicode[STIX]{x1D70B}}$ .

Let $\widetilde{\unicode[STIX]{x1D70B}}$ be an irreducible $\ell$ -adic representation of $\text{G}$ as above. By definition, $\text{}\text{i}_{\ell }(\widetilde{\unicode[STIX]{x1D70B}})$ depends only on the inertial class of the supercuspidal support of $\widetilde{\unicode[STIX]{x1D70B}}$ . Assume that $\widetilde{\unicode[STIX]{x1D70B}}$ is integral.

Proof. The representation $\widetilde{\unicode[STIX]{x1D70B}}$ is a subquotient of $\widetilde{\unicode[STIX]{x1D70C}}_{1}\times \cdots \times \widetilde{\unicode[STIX]{x1D70C}}_{r}$ . Since $\widetilde{\unicode[STIX]{x1D70B}}$ is integral, all the $\widetilde{\unicode[STIX]{x1D70C}}_{i}$ are integral and, by Proposition 3.2, for each $i$ there is an integer $a_{i}\geqslant 1$ such that

$$\begin{eqnarray}\text{}\text{r}_{\ell }(\widetilde{\unicode[STIX]{x1D70C}}_{i})=\unicode[STIX]{x1D70C}_{i}+\unicode[STIX]{x1D70C}_{i}\unicode[STIX]{x1D708}+\cdots +\unicode[STIX]{x1D70C}_{i}\unicode[STIX]{x1D708}^{a_{i}-1},\end{eqnarray}$$

where $\unicode[STIX]{x1D708}$ denotes the unramified mod- $\ell$ character ‘absolute value of the reduced norm’. Thus any irreducible subquotient of $\text{}\text{r}_{\ell }(\widetilde{\unicode[STIX]{x1D70B}})$ occurs as a subquotient of $\unicode[STIX]{x1D70C}_{1}\unicode[STIX]{x1D708}^{i_{1}}\times \cdots \times \unicode[STIX]{x1D70C}_{r}\unicode[STIX]{x1D708}^{i_{r}}$ for some integers $i_{1},\ldots ,i_{r}\in \mathbf{N}$ . The result now follows by looking at the supercuspidal support of each $\unicode[STIX]{x1D70C}_{i}$ .◻

4.2

Let $\widetilde{\unicode[STIX]{x1D70B}}$ be a simple irreducible $\ell$ -adic representation of $\text{G}$ . There are an integer $r\geqslant 1$ dividing  $m$ and a cuspidal irreducible representation $\widetilde{\unicode[STIX]{x1D70C}}$ of $\text{G}_{m/r}$ such that the inertial class of its cuspidal support contains

(4.1) $$\begin{eqnarray}(\text{GL}_{m/r}(\text{D})^{r},\widetilde{\unicode[STIX]{x1D70C}}\otimes \cdots \otimes \widetilde{\unicode[STIX]{x1D70C}}).\end{eqnarray}$$

We may assume that $\widetilde{\unicode[STIX]{x1D70C}}$ is integral. We fix an irreducible subquotient $\unicode[STIX]{x1D70C}$ of its reduction modulo $\ell$ . As in § 4.1, there are a unique integer $u\geqslant 1$ dividing $m/r$ and a supercuspidal irreducible representation $\unicode[STIX]{x1D70F}$ of degree $u$ such that the supercuspidal support of $\unicode[STIX]{x1D70C}$ is inertially equivalent to $(\text{GL}_{u}(\text{D})\times \cdots \times \text{GL}_{u}(\text{D}),\unicode[STIX]{x1D70F}\otimes \cdots \otimes \unicode[STIX]{x1D70F})$ , with $m=kur$ . Therefore, the mod- $\ell$ inertial supercuspidal support $\text{}\text{i}_{\ell }(\widetilde{\unicode[STIX]{x1D70B}})$ of the $\ell$ -adic simple irreducible representation $\widetilde{\unicode[STIX]{x1D70B}}$ is the inertial class of the pair

$$\begin{eqnarray}(\text{GL}_{u}(\text{D})^{kr},\unicode[STIX]{x1D70F}\otimes \cdots \otimes \unicode[STIX]{x1D70F}).\end{eqnarray}$$

In particular, it is simple.

Recall that, according to [Reference Mínguez and SécherreMS14a, Théorème 6.11], any supercuspidal irreducible mod- $\ell$ representation can be lifted to an $\ell$ -adic irreducible representation. The following lemma is an immediate consequence of the definition of the mod- $\ell$ inertial supercuspidal support.

Lemma 4.5. Let $\widetilde{\unicode[STIX]{x1D70F}}$ be an $\ell$ -adic lift of $\unicode[STIX]{x1D70F}$ . Any simple irreducible $\ell$ -adic representation whose cuspidal support is inertially equivalent to

$$\begin{eqnarray}(\text{GL}_{u}(\text{D})^{kr},\widetilde{\unicode[STIX]{x1D70F}}\otimes \cdots \otimes \widetilde{\unicode[STIX]{x1D70F}})\end{eqnarray}$$

is in the same $\ell$ -block as $\widetilde{\unicode[STIX]{x1D70B}}$ . In particular, the $\ell$ -adic discrete series representation $\text{L}(\widetilde{\unicode[STIX]{x1D70F}},kr)$ is in the same $\ell$ -block as $\widetilde{\unicode[STIX]{x1D70B}}$ .

4.3

Recall that we have fixed in § 2 a smooth character $\unicode[STIX]{x1D713}_{\ell }^{}:\text{F}\rightarrow \overline{\mathbf{Q}}_{\ell }^{\times }$ that is trivial on  $\mathfrak{p}$ but not on $\mathscr{O}$ . Since $\text{F}$ is the union of the $\mathfrak{p}^{-i}$ for $i\geqslant 1$ and $p$ is invertible in $\overline{\mathbf{Z}}_{\ell }$ , it has values in $\overline{\mathbf{Z}}_{\ell }^{\times }$ . For any simple stratum $[\mathfrak{a},\unicode[STIX]{x1D6FD}]$ in $\text{M}_{m}(\text{D})$ , the set of simple $\ell$ -adic characters associated with $[\mathfrak{a},\unicode[STIX]{x1D6FD}]$ will be defined with respect to $\unicode[STIX]{x1D713}_{\ell }$ (see § 3.3), whereas the set of $\ell$ -modular simple characters associated with $[\mathfrak{a},\unicode[STIX]{x1D6FD}]$ will be defined with respect to the reduction mod $\ell$ of $\unicode[STIX]{x1D713}_{\ell }^{}$ . Reduction mod  $\ell$ thus induces a bijection between $\ell$ -adic and $\ell$ -modular simple characters associated with $[\mathfrak{a},\unicode[STIX]{x1D6FD}]$ . It also induces a bijection between endo-classes of $\ell$ -adic and $\ell$ -modular simple characters. Thus we will speak of endo-classes of simple characters, without referring to the coefficient field.

Let $\boldsymbol{\unicode[STIX]{x1D6E9}}$ be the endo-class of §§ 3.3–3.5. Fix a $\unicode[STIX]{x1D6FD}$ -extension $\widetilde{\unicode[STIX]{x1D705}}$ of a maximal $\ell$ -adic simple character in $\text{G}$ of endo-class $\boldsymbol{\unicode[STIX]{x1D6E9}}$ , and write $\text{X}_{\ell }$ for the group of $\ell$ -adic characters of $\boldsymbol{k}^{\times }$ . The map (3.4) gives us a bijection $\widetilde{\boldsymbol{\unicode[STIX]{x1D706}}}_{\ell }$ from $\text{X}_{\ell }/\unicode[STIX]{x1D6E4}$ onto the set of equivalence classes of $\mathscr{T}(\text{G},\boldsymbol{\unicode[STIX]{x1D6E9}},\overline{\mathbf{Q}}_{\ell })$ . Also write $\text{Y}_{\ell }$ for the group of $\ell$ -modular characters of $\boldsymbol{k}^{\times }$ , and $\unicode[STIX]{x1D705}$ for the reduction mod $\ell$ of $\widetilde{\unicode[STIX]{x1D705}}$ . This gives us a bijection $\boldsymbol{\unicode[STIX]{x1D706}}_{\ell }$ from $\text{Y}_{\ell }/\unicode[STIX]{x1D6E4}$ onto the set of equivalence classes of $\mathscr{T}(\text{G},\boldsymbol{\unicode[STIX]{x1D6E9}},\overline{\mathbf{F}}_{\ell })$ . These two bijections are compatible in the following sense.

We keep in mind the following straightforward but important fact.

The converse does not hold in general, but we have the following result. Given $\unicode[STIX]{x1D6FC}\in \text{X}$ , write $[\unicode[STIX]{x1D6FC}]$ for its $\unicode[STIX]{x1D6E4}$ -orbit and $\unicode[STIX]{x1D719}$ for its reduction mod $\ell$ . The orbit $[\unicode[STIX]{x1D719}]$ depends only on $[\unicode[STIX]{x1D6FC}]$ and is called the reduction mod $\ell$ of $[\unicode[STIX]{x1D6FC}]$ .

5.1

We fix a prime number $\ell$ different from $p$ and an isomorphism of fields $\unicode[STIX]{x1D704}_{\ell }:\mathbf{C}\simeq \overline{\mathbf{Q}}_{\ell }$ . If $\unicode[STIX]{x1D70B}$ is a complex representation of $\text{G}$ , write $\unicode[STIX]{x1D704}_{\ell }^{\ast }\unicode[STIX]{x1D70B}$ for the $\ell$ -adic representation of $\text{G}$ obtained by extending scalars from $\mathbf{C}$ to $\overline{\mathbf{Q}}_{\ell }$ along $\unicode[STIX]{x1D704}_{\ell }$ .

5.2

Recall that we have fixed in § 2 a smooth character $\unicode[STIX]{x1D713}:\text{F}\rightarrow \mathbf{C}^{\times }$ that is trivial on $\mathfrak{p}$ but not on $\mathscr{O}$ . For any simple stratum $[\mathfrak{a},\unicode[STIX]{x1D6FD}]$ , the set of simple complex characters associated with $[\mathfrak{a},\unicode[STIX]{x1D6FD}]$ will be defined with respect to this choice (see §§ 3.3 and 4.3). We may and will assume that the character $\unicode[STIX]{x1D704}_{\ell }\circ \unicode[STIX]{x1D713}$ is the character $\unicode[STIX]{x1D713}_{\ell }$ of § 4.3. This gives us a bijection between endo-classes of complex and $\ell$ -adic simple characters of $\text{G}$ . Again, we will speak of endo-classes of simple characters, without referring to the coefficient field.

Let $\unicode[STIX]{x1D705}$ be a $\unicode[STIX]{x1D6FD}$ -extension of some maximal complex simple character in $\text{G}$ having endo-class  $\boldsymbol{\unicode[STIX]{x1D6E9}}$ . Write $\text{X}$ for the group of complex characters of $\boldsymbol{k}^{\times }$ .

Lemma 5.3. Let $\unicode[STIX]{x1D70B}$ be a simple irreducible complex representation of $\text{G}$ with endo-class  $\boldsymbol{\unicode[STIX]{x1D6E9}}$ . Then

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}\in \text{X}(\unicode[STIX]{x1D705},\unicode[STIX]{x1D70B})\quad \Leftrightarrow \quad \unicode[STIX]{x1D704}_{\ell }^{}\circ \unicode[STIX]{x1D6FC}\in \text{X}_{\ell }^{}(\unicode[STIX]{x1D704}_{\ell }^{\ast }\unicode[STIX]{x1D705},\unicode[STIX]{x1D704}_{\ell }^{\ast }\unicode[STIX]{x1D70B}).\end{eqnarray}$$

Given $\unicode[STIX]{x1D6FC}\in \text{X}$ , the orbit $[\unicode[STIX]{x1D6FC}_{\ell }]$ depends only on $[\unicode[STIX]{x1D6FC}]$ . It is called the $\ell$ -regular part of $[\unicode[STIX]{x1D6FC}]$ , denoted by  $[\unicode[STIX]{x1D6FC}]_{\ell }$ .

5.3

Recall that $q$ is the cardinality of the residue field of $\text{F}$ . For each prime number $\ell$ dividing

(5.1) $$\begin{eqnarray}(q^{n}-1)(q^{n-1}-1)\ldots (q-1)\end{eqnarray}$$

we fix an isomorphism of fields $\unicode[STIX]{x1D704}_{\ell }:\mathbf{C}\simeq \overline{\mathbf{Q}}_{\ell }$ .

Two linked simple complex representations of $\text{G}$ have the same endo-class (see Remark 4.7). The converse is given by the following proposition.

Proof. Assume that $\unicode[STIX]{x1D70B}$ and $\unicode[STIX]{x1D70B}\text{sp}\prime$ are simple irreducible complex representations with the same endo-class $\boldsymbol{\unicode[STIX]{x1D6E9}}$ . Let $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FC}\text{sp}\prime$ be characters in $\text{X}(\unicode[STIX]{x1D70B})$ and $\text{X}(\unicode[STIX]{x1D70B}\text{sp}\prime )$ , respectively, and write $\unicode[STIX]{x1D709}=\unicode[STIX]{x1D6FC}\text{sp}\prime \unicode[STIX]{x1D6FC}^{-1}$ . Let $\ell _{1},\ldots ,\ell _{r}$ be the prime numbers dividing (5.1). The character $\unicode[STIX]{x1D709}$ decomposes uniquely as

$$\begin{eqnarray}\unicode[STIX]{x1D709}=\unicode[STIX]{x1D709}_{1}\ldots \unicode[STIX]{x1D709}_{r},\end{eqnarray}$$

where the order of $\unicode[STIX]{x1D709}_{i}$ is a power of $\ell _{i}$ , for $i\in \{1,\ldots ,r\}$ . Write $\unicode[STIX]{x1D6FC}_{0}=\unicode[STIX]{x1D6FC}$ and define inductively

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{i}=\unicode[STIX]{x1D6FC}_{i-1}\cdot \unicode[STIX]{x1D709}_{i}\end{eqnarray}$$

for all $i\in \{1,\ldots ,r\}$ . Let $\unicode[STIX]{x1D70B}_{i}$ be a simple irreducible complex representation of endo-class $\boldsymbol{\unicode[STIX]{x1D6E9}}$ and parametrizing class $[\unicode[STIX]{x1D6FC}_{i}]$ . The result follows from Proposition 5.5.◻

5.4

Let $\unicode[STIX]{x1D70B}$ be an irreducible complex representation of $\text{G}$ . Fix a representative ( $\text{M},\unicode[STIX]{x1D70C}$ ) in its cuspidal support, with $\text{M}=\text{GL}_{m_{1}}(\text{D})\times \cdots \times \text{GL}_{m_{r}}(\text{D})$ and $\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}_{1}\otimes \cdots \otimes \unicode[STIX]{x1D70C}_{r}$ , where $\unicode[STIX]{x1D70C}_{i}$ is a cuspidal irreducible representation of $\text{GL}_{m_{i}}(\text{D})$ , for $i\in \{1,\ldots ,r\}$ , and $m_{1}+\cdots +m_{r}=m$ . Write $\boldsymbol{\unicode[STIX]{x1D6E9}}_{i}$ for the endo-class of $\unicode[STIX]{x1D70C}_{i}$ and $g_{i}$ for the degree of $\boldsymbol{\unicode[STIX]{x1D6E9}}_{i}$ . We define the semi-simple endo-class of $\unicode[STIX]{x1D70B}$ to be the formal sum

(5.2) $$\begin{eqnarray}\boldsymbol{\unicode[STIX]{x1D6E9}}(\unicode[STIX]{x1D70B})=\mathop{\sum }_{i=1}^{r}\frac{m_{i}d}{g_{i}}\cdot \boldsymbol{\unicode[STIX]{x1D6E9}}_{i}\end{eqnarray}$$

in the free abelian semigroup generated by all $\text{F}$ -endo-classes. It depends only on the inertial class of the cuspidal support of $\unicode[STIX]{x1D70B}$ .

Note that if $\unicode[STIX]{x1D70B}$ is a simple irreducible representation with endo-class $\boldsymbol{\unicode[STIX]{x1D6E9}}$ , then its semi-simple endo-class is $\boldsymbol{\unicode[STIX]{x1D6E9}}(\unicode[STIX]{x1D70B})=(n/g)\cdot \boldsymbol{\unicode[STIX]{x1D6E9}}$ where $g$ is the degree of $\boldsymbol{\unicode[STIX]{x1D6E9}}$ .

The following theorem, which is our first main result, generalizes Proposition 5.8.

Proof. Any two linked irreducible complex representations automatically have the same semi-simple endo-class. We therefore start with two irreducible complex representations $\unicode[STIX]{x1D70B}$ and $\unicode[STIX]{x1D70B}\text{sp}\prime$ with the same semi-simple endo-class. By [Reference Mínguez and SécherreMS14b, Théorème 4.16], the representation $\unicode[STIX]{x1D70B}$ can be written as

$$\begin{eqnarray}\unicode[STIX]{x1D70B}=\unicode[STIX]{x1D70B}_{1}\times \unicode[STIX]{x1D70B}_{2}\times \cdots \times \unicode[STIX]{x1D70B}_{k}\end{eqnarray}$$

where $\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2},\ldots ,\unicode[STIX]{x1D70B}_{k}$ are simple irreducible representations whose inertial cuspidal supports are pairwise distinct, and this decomposition is unique up to renumbering. We have the following straightforward lemma.

Lemma 5.11. Let $\unicode[STIX]{x1D6FF}$ be an irreducible complex representation of $\text{GL}_{m-k}(\text{D})$ for some integer $k\in \{1,\ldots ,m-1\}$ . Let $\unicode[STIX]{x1D70E}$ and $\unicode[STIX]{x1D70E}\text{sp}\prime$ be two irreducible complex representations of $\text{GL}_{k}(\text{D})$ , and let $\unicode[STIX]{x1D70B}$ and $\unicode[STIX]{x1D70B}\text{sp}\prime$ be irreducible subquotients of $\unicode[STIX]{x1D70E}\times \unicode[STIX]{x1D6FF}$ and $\unicode[STIX]{x1D70E}\text{sp}\prime \times \unicode[STIX]{x1D6FF}$ , respectively. If $\unicode[STIX]{x1D70E}$ and $\unicode[STIX]{x1D70E}\text{sp}\prime$ are linked, then $\unicode[STIX]{x1D70B}$ and  $\unicode[STIX]{x1D70B}\text{sp}\prime$ are linked.

For each $i\in \{1,\ldots ,k\}$ , thanks to Lemma 5.11 and Proposition 5.8 we may and will assume that $\unicode[STIX]{x1D70B}_{i}$ is a discrete series representation of the form $\text{L}(\unicode[STIX]{x1D70C}_{i},r_{i})$ for some cuspidal representation $\unicode[STIX]{x1D70C}_{i}$  of $\text{GL}_{m_{i}}(\text{D})$ with the same endo-class as $\unicode[STIX]{x1D70B}_{i}$ and for some integer $r_{i}$ , such that $m_{1}r_{1}+\cdots +m_{k}r_{k}=m$ . We may even assume that $\unicode[STIX]{x1D70C}_{i}$ has minimal degree among all cuspidal irreducible representations of $\text{GL}_{a}(\text{D})$ , $a\geqslant 1$ , with the same endo-class as $\unicode[STIX]{x1D70B}_{i}$ . This amounts to saying that $m_{i}$ is equal to $g_{i}/(g_{i},d)$ , where $g_{i}$ is the degree of the endo-class of $\unicode[STIX]{x1D70B}_{i}$ .

Moreover, if $\unicode[STIX]{x1D70C}_{i}$ and $\unicode[STIX]{x1D70C}_{j}$ have the same endo-class for some $i,j\in \{1,\ldots ,k\}$ , then they have the same degree, and so they are linked. We may thus assume that $\unicode[STIX]{x1D70C}_{1},\ldots ,\unicode[STIX]{x1D70C}_{k}$ have distinct endo-classes, denoted by $\boldsymbol{\unicode[STIX]{x1D6E9}}_{1},\ldots ,\boldsymbol{\unicode[STIX]{x1D6E9}}_{k}$ , respectively.

Similarly, we may assume that the representation $\unicode[STIX]{x1D70B}\text{sp}\prime$ decomposes as a product $\unicode[STIX]{x1D70B}\text{sp}\prime _{1}\times \unicode[STIX]{x1D70B}\text{sp}\prime _{2}\times \cdots \times \unicode[STIX]{x1D70B}\text{sp}\prime _{t}$ , where $\unicode[STIX]{x1D70B}\text{sp}\prime _{j}$ is a discrete series representation of the form $\text{L}(\unicode[STIX]{x1D70C}_{j}^{\prime },s_{j}^{})$ for some cuspidal representation $\unicode[STIX]{x1D70C}\text{sp}\prime _{j}$ of $\text{GL}_{m_{j}^{\prime }}(\text{D})$ and some integer $s_{j}^{}\geqslant 1$ , and we may assume that the endo-classes $\boldsymbol{\unicode[STIX]{x1D6E9}}\text{sp}\prime _{1},\ldots ,\boldsymbol{\unicode[STIX]{x1D6E9}}\text{sp}\prime _{t}$ of $\unicode[STIX]{x1D70C}_{1}^{\prime },\ldots ,\unicode[STIX]{x1D70C}_{t}^{\prime }$ are distinct. It follows that $k=t$ and, up to renumbering, we may assume that we have $\boldsymbol{\unicode[STIX]{x1D6E9}}_{i}^{\prime }=\boldsymbol{\unicode[STIX]{x1D6E9}}_{i}^{}$ for each $i\in \{1,\ldots ,k\}$ . It then follows that $\unicode[STIX]{x1D70C}\text{sp}\prime _{i}$ and $\unicode[STIX]{x1D70C}_{i}^{}$ have the same degree by minimality of  $m_{i}$ .

Since $\unicode[STIX]{x1D70B}$ and $\unicode[STIX]{x1D70B}\text{sp}\prime$ have the same semi-simple endo-class, we have $s_{i}=r_{i}$ for all $i$ , so $\unicode[STIX]{x1D70B}_{i}^{}$ and  $\unicode[STIX]{x1D70B}\text{sp}\prime _{i}$ have the same degree. Proposition 5.8 then implies that $\unicode[STIX]{x1D70B}_{i}^{}$ and  $\unicode[STIX]{x1D70B}\text{sp}\prime _{i}$ are linked. Theorem 5.10 now follows from Lemma 5.11 again.◻

6.1

We fix an isomorphism of fields $\unicode[STIX]{x1D704}_{\ell }:\mathbf{C}\simeq \overline{\mathbf{Q}}_{\ell }$ and write (as in [Reference Mínguez and SécherreMS17])

(6.2) $$\begin{eqnarray}\widetilde{\boldsymbol{\unicode[STIX]{x1D70B}}}_{\ell }:\mathscr{D}(\text{G},\overline{\mathbf{Q}}_{\ell })\rightarrow \mathscr{D}(\text{H},\overline{\mathbf{Q}}_{\ell })\end{eqnarray}$$

for the $\ell$ -adic local Jacquet–Langlands correspondence between $\ell$ -adic discrete series representations of $\text{G}$ and $\text{H}$ . The correspondence (6.2) does not depend on the choice of $\unicode[STIX]{x1D704}_{\ell }$ (see [Reference Mínguez and SécherreMS17, Remarque 10.1]). According to [Reference BadulescuBad07, § 3.1], there is a unique surjective group homomorphism

$$\begin{eqnarray}\widetilde{\text{}\text{J}}_{\ell }:\mathscr{R}(\text{H},\overline{\mathbf{Q}}_{\ell })\rightarrow \mathscr{R}(\text{G},\overline{\mathbf{Q}}_{\ell })\end{eqnarray}$$

where $\mathscr{R}(\text{G},\overline{\mathbf{Q}}_{\ell })$ is the Grothendieck group of finite-length $\ell$ -adic representations of $\text{G}$ , with the following property: given positive integers $n_{1},\ldots ,n_{r}$ such that $n_{1}+\cdots +n_{r}=n$ and an $\ell$ -adic discrete series representation $\widetilde{\unicode[STIX]{x1D70E}}_{i}$ of $\text{GL}_{n_{i}}(\text{F})$ for each $i$ , the image of the product $\widetilde{\unicode[STIX]{x1D70E}}_{1}\times \cdots \times \widetilde{\unicode[STIX]{x1D70E}}_{r}$ by $\widetilde{\text{}\text{J}}_{\ell }$ is $0$ if $n_{i}$ is not divisible by $d$ for at least one $i$ , and is $\widetilde{\unicode[STIX]{x1D70B}}_{1}\times \cdots \times \widetilde{\unicode[STIX]{x1D70B}}_{r}$ otherwise, where $n_{i}=m_{i}d$ and $\widetilde{\unicode[STIX]{x1D70B}}_{i}$ is the $\ell$ -adic discrete series representation of $\text{GL}_{m_{i}}(\text{D})$ whose Jacquet–Langlands transfer is $\widetilde{\unicode[STIX]{x1D70E}}_{i}$ for each  $i$ .

By [Reference Mínguez and SécherreMS17, Théorème 12.4], there exists a unique surjective group homomorphism of Grothendieck groups $\text{}\text{J}_{\ell }:\mathscr{R}(\text{H},\overline{\mathbf{F}}_{\ell })\rightarrow \mathscr{R}(\text{G},\overline{\mathbf{F}}_{\ell })$ such that the diagram

is commutative, where $\mathscr{R}(\text{G},\overline{\mathbf{Q}}_{\ell })^{\text{e}}$ is the subgroup of $\mathscr{R}(\text{G},\overline{\mathbf{Q}}_{\ell })$ generated by integral irreducible representations and $\mathscr{R}(\text{G},\overline{\mathbf{F}}_{\ell })$ is the Grothendieck group of $\ell$ -modular representations of  $\text{G}$ .

Proof. Let us write $\widetilde{\unicode[STIX]{x1D70E}}_{i}=\text{L}(\widetilde{\unicode[STIX]{x1D70C}}_{i},r_{i})$ and $k_{i}=k(\widetilde{\unicode[STIX]{x1D70C}}_{i})$ for $i=1,2$ . Then $k_{1}r_{1}=k_{2}r_{2}$ , which we denote by $v$ , and the mod- $\ell$ inertial supercuspidal support of $\widetilde{\unicode[STIX]{x1D70E}}_{1}$ and $\widetilde{\unicode[STIX]{x1D70E}}_{2}$ contains the supercuspidal pair

$$\begin{eqnarray}(\text{GL}_{u}(\text{F})\times \cdots \times \text{GL}_{u}(\text{F}),\unicode[STIX]{x1D70F}\otimes \cdots \otimes \unicode[STIX]{x1D70F})\end{eqnarray}$$

with $uv=m$ and for some mod- $\ell$ supercuspidal representation $\unicode[STIX]{x1D70F}$ of $\text{GL}_{u}(\text{D})$ . Fix an $\ell$ -adic lift $\widetilde{\unicode[STIX]{x1D70F}}$ of $\unicode[STIX]{x1D70F}$ and write $\widetilde{\unicode[STIX]{x1D70E}}=\text{L}(\widetilde{\unicode[STIX]{x1D70F}},v)$ . The representation $\widetilde{\unicode[STIX]{x1D70E}}$ is in the same $\ell$ -block as $\widetilde{\unicode[STIX]{x1D70E}}_{1}$ and $\widetilde{\unicode[STIX]{x1D70E}}_{2}$ , by Lemma 4.5. If we write $\widetilde{\unicode[STIX]{x1D70B}}$ for the $\ell$ -adic discrete series representation of $\text{G}$ whose transfer to $\text{H}$ is $\widetilde{\unicode[STIX]{x1D70E}}$ , then it is enough to prove that $\widetilde{\unicode[STIX]{x1D70B}}$ is in the same $\ell$ -block as  $\widetilde{\unicode[STIX]{x1D70B}}_{1}$ .

In the remainder of the proof, it will be more convenient to deal with Speh representations than with discrete series representations, as in [Reference Mínguez and SécherreMS17]. We therefore apply the Zelevinski involution to $\widetilde{\unicode[STIX]{x1D70B}},\widetilde{\unicode[STIX]{x1D70B}}_{1}$ and $\widetilde{\unicode[STIX]{x1D70E}},\widetilde{\unicode[STIX]{x1D70E}}_{1}$ and thus get $\ell$ -adic Speh representations.

Let us write $\widetilde{\unicode[STIX]{x1D70E}}^{\ast }$ for the Zelevinski dual of $\widetilde{\unicode[STIX]{x1D70E}}$ . Its reduction mod $\ell$ is the $\ell$ -modular super-Speh representation $\text{Z}(\unicode[STIX]{x1D70F},v)$ , by [Reference Mínguez and SécherreMS14a, Théorème 9.39]. If we write $\widetilde{\unicode[STIX]{x1D70B}}^{\ast }=\text{Z}(\widetilde{\unicode[STIX]{x1D6FC}},t)$ for the Zelevinski dual of $\widetilde{\unicode[STIX]{x1D70B}}$ , for some $t$ dividing $m$ and some cuspidal irreducible representation $\widetilde{\unicode[STIX]{x1D6FC}}$ of $\text{GL}_{m/t}(\text{D})$ , then its reduction mod $\ell$ contains the Speh representation $\text{Z}(\unicode[STIX]{x1D6FC},t)$ where $\unicode[STIX]{x1D6FC}$ is an irreducible component of the reduction mod $\ell$ of $\widetilde{\unicode[STIX]{x1D6FC}}$ (see for instance [Reference Mínguez and SécherreMS17, Proposition 1.10]). The cuspidal representation  $\unicode[STIX]{x1D6FC}$ need not be supercuspidal but, according to Proposition 3.1, it can be written as $\text{Sp}(\unicode[STIX]{x1D6FD},k)$ for $k=k(\unicode[STIX]{x1D6FC})$ and some supercuspidal irreducible representation  $\unicode[STIX]{x1D6FD}$ .

We now look at the reduction mod $\ell$ of the Zelevinski dual of $\widetilde{\unicode[STIX]{x1D70E}}_{1}$ . It is $\text{Z}(\unicode[STIX]{x1D70C}_{1},r_{1})$ where $\unicode[STIX]{x1D70C}_{1}$ , the reduction mod $\ell$ of $\widetilde{\unicode[STIX]{x1D70C}}_{1}$ , can be written as $\text{Sp}(\unicode[STIX]{x1D70F}\unicode[STIX]{x1D712},k_{1})$ for some unramified character $\unicode[STIX]{x1D712}$ . By twisting $\widetilde{\unicode[STIX]{x1D70B}}_{1}$ by an unramified character of $\text{G}$ , we may assume that $\unicode[STIX]{x1D712}$ is trivial. According to [Reference Mínguez and SécherreMS14a, Lemme 9.41], the representation $\text{Z}(\unicode[STIX]{x1D70C}_{1},k_{1})$ decomposes as a $\mathbf{Z}$ -linear combination of products of the form

$$\begin{eqnarray}\text{Z}(\unicode[STIX]{x1D70F}\unicode[STIX]{x1D708}^{i_{1}},v_{1})\times \cdots \times \text{Z}(\unicode[STIX]{x1D70F}\unicode[STIX]{x1D708}^{i_{r}},v_{r})\end{eqnarray}$$

with $v_{1}+\cdots +v_{r}=v$ and $i_{1},\ldots ,i_{r}\in \mathbf{Z}$ , where $\unicode[STIX]{x1D708}$ stands for the absolute value of the reduced norm, as usual. (For an explicit formula for this decomposition, see [Reference Mínguez and SécherreMS17, §§ 11 and 12].) Thanks to the commutative diagram above, the reduction modulo $\ell$ of the Zelevinski dual of $\widetilde{\unicode[STIX]{x1D70B}}_{1}$ will be made of products of the form

$$\begin{eqnarray}\text{Z}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D708}^{i_{1}},t_{1})\times \cdots \times \text{Z}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D708}^{i_{r}},t_{r})\end{eqnarray}$$

with $t_{1}+\cdots +t_{r}=t$ and $i_{1},\ldots ,i_{r}\in \mathbf{Z}$ , all of whose irreducible subquotients have supercuspidal support inertially equivalent to $(\text{GL}_{w}(\text{D})\times \cdots \times \text{GL}_{w}(\text{D}),\unicode[STIX]{x1D6FD}\otimes \cdots \otimes \unicode[STIX]{x1D6FD})$ with $wkt=m$ . The result follows from Corollary 4.4.◻

6.2

Proposition 6.1 implies that two complex discrete series representations $\unicode[STIX]{x1D70B}_{1}$ and $\unicode[STIX]{x1D70B}_{2}$ of $\text{G}$ are linked if their Jacquet–Langlands transfers are linked. We have the following refinement.