2.1

Given any non-Archimedean locally compact field ${ F}$ , we write for its ring of integers, $\mathfrak {p}_{ F}$ for the maximal ideal of , $\boldsymbol {k}_{ F}$ for its residue field and $q_{ F}$ for the cardinality of $\boldsymbol {k}_{ F}$ .

We also write $\operatorname {val}_{ F}$ for the valuation of ${ F}$ taking any uniformizer to $1$ , and $|\cdot |_{ F}$ for the absolute value of ${ F}$ taking any uniformizer to the inverse of $q_{ F}$ .

Given any finite extension ${ L}$ of ${ K}$ , we write $\mathrm {N}_{{ L}/{ K}}$ and $\operatorname {tr}_{{ L}/{ K}}$ for the norm and trace maps.

2.2

Given a locally compact, totally disconnected topological group ${ G}$ and an algebraically closed field ${ R}$ of characteristic different from p, we consider smooth representations of ${ G}$ on ${ R}$ -vector spaces. We will abbreviate smooth ${ R}$ -representation to ${ R}$ -representation, or even representation if the coefficient field ${ R}$ is clear from the context.

An ${ R}$ -character (or character) of ${ G}$ is a group homomorphism from ${ G}$ to ${ R}^\times $ with open kernel.

Let $\pi $ be a representation of ${ G}$ . We write $\pi ^\vee $ for its contragredient. Given a character $\chi $ of ${ G}$ , we write $\pi \chi $ for the representation $g\mapsto \chi (g)\pi (g)$ of ${ G}$ .

Let $\pi $ be a representation of a closed subgroup ${{H}}$ of ${ G}$ . Given any element $g\in { G}$ , we write $\pi ^g$ for the representation $x\mapsto \pi (gxg^{-1})$ of ${{H}}^g=g^{-1}{{H}} g$ . Given any continuous involution $\sigma $ of ${ G}$ , we write $\pi ^\sigma $ for the representation $\pi \circ \sigma $ of $\sigma ({{H}})$ . Given any character $\mu $ of ${{H}}\cap { G}^\sigma $ , we say that $\pi $ is $\mu $ -distinguished if the space $\operatorname {Hom}_{{{H}}\cap { G}^\sigma }(\pi ,\chi )$ is nonzero. If $\mu $ is the trivial character, we will abbreviate $\mu $ -distinguished to ${{H}}\cap { G}^\sigma $ -distinguished, or just distinguished.

2.3

Let us fix a separable quadratic extension ${{ F}}/{ F}_0$ of non-Archimedean locally compact fields of residual characteristic p, and let $\sigma $ denote its nontrivial automorphism. Let R be an algebraically closed field of characteristic different from p. Let

(2.1) $$ \begin{align} \varkappa = \varkappa_{{{ F}}/{ F}_0} : { F}_0^\times \rightarrow \{-1,1\} = \mathbb{Z}^\times \end{align} $$

denote the $\mathbb {Z}$ -valued character of ${ F}_0^\times $ with kernel $\mathrm {N}_{{{ F}}/{ F}_0}({{ F}}^\times )$ . When needed, we will consider $\varkappa $ as a character with values in ${ R}$ . We abbreviate $q=q_{{ F}}$ and $q_0=q_{{ F}_0}$ . We fix a square root

(2.2) $$ \begin{align} q_0^{1/2}\in { R} \end{align} $$

of $q_0$ in ${ R}$ and define

(2.3) $$ \begin{align} q^{1/2} = \left\{ \begin{array}{ll} q_0^{1/2} & \text{if }{ F}/{ F}_0\text{ is ramified}, \\ q_0 & \text{if }{ F}/{ F}_0\text{ is unramified}, \end{array}\right. \end{align} $$

which we will use to normalize parabolic induction and restriction functors (see below).

2.4

Given a positive integer $n{\geqslant }1$ , the automorphism $\sigma $ acts on the group $\operatorname {GL}_n(F)$ componentwise, thus defines a continuous involution of $\operatorname {GL}_n(F)$ , still denoted $\sigma $ . Its fixed points form the subgroup $\operatorname {GL}_n(F_0)$ .

We denote by $\nu $ the unramified character ‘absolute value of the determinant’ of $\operatorname {GL}_n(F)$ and by $\nu ^{1/2}$ the unramified character taking any element whose determinant has valuation $1$ to $q^{-1/2}$ . We thus have $(\nu ^{1/2})^2=\nu $ . Similarly, we define the characters $\nu ^{\phantom {1}}_0$ and $\nu _0^{1/2}$ of $\operatorname {GL}_n(F_0)$ .

Given positive integers $n_1,\dots ,n_r$ such that $n_1+\dots +n_r=n$ and, for each $i=1,\ldots ,r$ , given an ${ R}$ -representation $\pi _i$ of $\operatorname {GL}_{n_i}(F)$ , we write

(2.4) $$ \begin{align} \pi_1\times \cdots \times \pi_r \end{align} $$

for the representation of $\operatorname {GL}_{n}(F)$ obtained by normalized parabolic induction from $\pi _1\otimes \cdots \otimes \pi _r$ along the parabolic subgroup generated by upper triangular matrices and the standard Levi subgroup $\operatorname {GL}_{n_1}(F)\times \dots \times \operatorname {GL}_{n_r}(F)$ .

An irreducible R-representation of $\operatorname {GL}_n(F)$ is said to be cuspidal (respectively, supercuspidal) if it does not occur as a subrepresentation (respectively, a subquotient) of any representation of the form (2.4) with $r{\geqslant }2$ . Any supercuspidal representation of $\operatorname {GL}_n(F)$ is cuspidal. When R has characteristic $0$ , any cuspidal representation of $\operatorname {GL}_n(F)$ is supercuspidal. When R has characteristic $\ell>0$ , the group $\operatorname {GL}_n(F)$ may have cuspidal nonsupercuspidal representations (see §3.4).

Given a representation $\pi $ of $\operatorname {GL}_n(F)$ and a character $\chi $ of $F^\times $ , we will write $\pi \chi $ for $\pi (\chi \circ \mathrm {det})$ .

2.5

Let us fix an algebraic closure ${\overline {\mathbb {Q}}_\ell }$ of the field of $\ell $ -adic numbers. Let ${\overline {\mathbb {Z}}_\ell }$ denote its ring of integers, and ${\overline {\mathbb {F}}_{\ell }}$ denote the residue field of ${\overline {\mathbb {Z}}_\ell }$ .

We call an irreducible representation $\pi $ of a locally compact, totally disconnected group ${ G}$ on a ${\overline {\mathbb {Q}}_\ell }$ -vector space V integral if it stabilizes a ${\overline {\mathbb {Z}}_\ell }$ -lattice L in V. In this case, we obtain a smooth ${\overline {\mathbb {F}}_{\ell }}$ -representation $L\otimes {\overline {\mathbb {F}}_{\ell }}$ of ${ G}$ whose isomorphism class may depend on the choice of L.

If ${ G}$ is either the group of rational points of a connected reductive linear algebraic F-group or a finite group (see [Reference Vignéras42, Theorem 1] and the Brauer–Nesbitt principle), the smooth ${\overline {\mathbb {F}}_{\ell }}$ -representation $L\otimes {\overline {\mathbb {F}}_{\ell }}$ has finite length, and its semisimplification is independent of the choice of L. This semisimplification is called the reduction modulo $\ell $ of $\pi $ and is denoted by ${{\mathbf {\textsf {r}}}}_\ell (\pi )$ .

Given an irreducible ${\overline {\mathbb {F}}_{\ell }}$ -representation $\rho $ of G, we call an irreducible integral ${\overline {\mathbb {Q}}_\ell }$ -representation with reduction modulo $\ell $ equal to $\rho $ a ${\overline {\mathbb {Q}}_\ell }$ -lift of $\rho $ .

3.1

Fundamental results of Flicker and Prasad [Reference Flicker16, Reference Prasad32, Reference Prasad33] on irreducible complex representations of $\operatorname {GL}_n({ F})$ distinguished by $\operatorname {GL}_n({ F}_0)$ have been extended to irreducible ${ R}$ -representations in [Reference Sécherre35] Theorem 4.1.

We will say that a representation $\pi $ of $\operatorname {GL}_n(F)$ is $\sigma $ -self-dual if $\pi ^\vee $ is isomorphic to $\pi ^\sigma $ .

3.2

For supercuspidal representations, we have the following dichotomy and disjunction theorem ([Reference Kable23] Theorem 4, [Reference Anandavardhanan, Kable and Tandon2] Corollary 1.6 if $\ell =0$ , [Reference Sécherre35] Theorem 10.8 if $p\neq 2$ and [Reference Cui, Lanard and Lu12] Theorem 3.14 if $\ell \neq 0,2$ ).

3.3

In this paragraph, $\ell $ is a prime number different from p and we will consider representations with coefficients in ${\overline {\mathbb {Q}}_\ell }$ or ${\overline {\mathbb {F}}_{\ell }}$ . The following theorem is [Reference Kurinczuk and Matringe25] Theorem 3.4.

It follows that any $\sigma $ -self-dual cuspidal ${\overline {\mathbb {F}}_{\ell }}$ -representation of $\operatorname {GL}_n({ F})$ having a distinguished lift to ${\overline {\mathbb {Q}}_\ell }$ is distinguished. For supercuspidal representations, one has the following converse (see [Reference Sécherre35] Theorem 10.11 if $p\neq 2$ , and [Reference Cui, Lanard and Lu12] Theorem 3.4):

We also have the following distinguished lift theorem, making Theorem 3.4 more precise.

3.4

From now on, we consider the case of cuspidal nonsupercuspidal R-representations, thus $\ell $ is a prime number different from p. Let us recall how they are classified in terms of their supercuspidal support.

Recall that a representation $\pi $ of $\operatorname {GL}_n(F)$ on an ${ R}$ -vector space V is generic if V carries a nonzero ${ R}$ -linear form $\mathit {\Theta }$ such that $\mathit {\Theta } (\pi (u)v) = {\theta }(u)v$ for all $v\in V$ and all unipotent upper triangular matrices u, where ${\theta }(u) = \psi (u_{1,2}+\dots +u_{n-1,n})$ and $\psi $ is a nontrivial R-character of F.

Let $k{\geqslant }1$ be a positive integer, and $\rho $ be a supercuspidal ${ R}$ -representation of $\operatorname {GL}_k({{ F}})$ . According to [Reference Mínguez and Sécherre29] 8.1, for any $r{\geqslant }1$ , the induced representation

(3.1) $$ \begin{align} \rho\nu^{-(r-1)/2}\times\cdots\times\rho\nu^{(r-1)/2} \end{align} $$

contains a unique generic irreducible subquotient, denoted $\operatorname {St}_{r}(\rho )$ .

Let $e(\rho )$ be the smallest integer $i{\geqslant }1$ such that $\rho \nu ^i$ is isomorphic to $\rho $ and $t(\rho )$ be the torsion number of $\rho $ , that is, the number of unramified characters $\chi $ of ${{ F}}^\times $ such that $\rho \chi $ is isomorphic to $\rho $ . By [Reference Mínguez and Sécherre31] Lemme 3.6, these integers are related by the identity

(3.2) $$ \begin{align} e(\rho) = \text{order of }q^{t(\rho)}\text{ mod }\ell. \end{align} $$

By [Reference Mínguez and Sécherre29] Théorème 6.14, one has the following classification.

Note that, conversely, by the same references, if $\rho $ is a supercuspidal representation of $\operatorname {GL}_k(F)$ and $r=e(\rho )\ell ^v$ for some $v{\geqslant }0$ , the representation $\operatorname {St}_r(\rho )$ is cuspidal.

It will be convenient to set $r(\pi )=1$ for any supercuspidal ${ R}$ -representation $\pi $ of $\operatorname {GL}_n({{ F}})$ .

3.5

We now classify $\sigma $ -self-dual cuspidal representations.

Proof. The representation $\operatorname {St}_r(\rho )$ is the unique generic irreducible subquotient of (3.1). The representation $\operatorname {St}_r(\rho )^{\vee \sigma }$ is thus generic, irreducible and it is a subquotient of

$$ \begin{align*} \rho^{\vee\sigma}\nu^{(r-1)/2}\times\cdots\times\rho^{\vee\sigma}\nu^{-(r-1)/2}. \end{align*} $$

Equivalently (see [Reference Mínguez and Sécherre29] Proposition 2.6), it is a subquotient of $\rho ^{\vee \sigma }\nu ^{-(r-1)/2}\times \cdots \times \rho ^{\vee \sigma }\nu ^{(r-1)/2}$ . By uniqueness of the generic irreducible subquotient, we thus have

$$ \begin{align*} \operatorname{St}_r(\rho)^{\vee\sigma} \simeq \operatorname{St}_r(\rho^{\vee\sigma}). \end{align*} $$

The representation $\operatorname {St}_r(\rho )$ is thus $\sigma $ -self-dual if and only if $\operatorname {St}_r(\rho ^{\vee \sigma })$ is isomorphic to $\operatorname {St}_r(\rho )$ . The result then follows from Proposition 3.6.

3.6

We will need the finite field analogue of 3.4 (see [Reference Vignéras41] III.2.5 or [Reference Cabanes10] Theorem 19.3).

3.7

As in the previous paragraph, $\boldsymbol {k}$ is a finite field of characteristic p. Let us recall how to parametrize cuspidal representations of $\operatorname {GL}_n(\boldsymbol {k})$ by regular characters ([Reference Green19], [Reference Dipper14] Theorem 3.5 and [Reference Dipper and James15, Reference James21]).

Let $\overline {\boldsymbol {k}}$ be an algebraic closure of $\boldsymbol {k}$ . For any integer $s{\geqslant }1$ , let $\boldsymbol {k}_{s}$ be the extension of $\boldsymbol {k}$ of degree s contained in $\overline {\boldsymbol {k}}$ . Let $\Delta $ denote the group $\operatorname {Gal}(\boldsymbol {k}_n/\boldsymbol {k})$ . A character of $\boldsymbol {k}_n^\times $ is $\Delta $ -regular if it is fixed by no nontrivial element of $\Delta $ .

By reduction mod $\ell $ , we get the following classification.

3.8

Finally, we will need the following distinction criterion for cuspidal ${\overline {\mathbb {Q}}_\ell }$ -representations (see [Reference Hakim and Murnaghan20] Proposition 6.1 and [Reference Coniglio-Guilloton11] Lemme 3.4.10) of $\operatorname {GL}_n(\boldsymbol {k})$ when p is odd.

Remark 3.14. Suppose that $\mathrm {W}_\xi $ is $\operatorname {GL}_u(\boldsymbol {k})\times \operatorname {GL}_u(\boldsymbol {k})$ -distinguished. By [Reference Sécherre35] Lemma 2.6, the cuspidal ${\overline {\mathbb {F}}_{\ell }}$ -representation $\overline {\mathrm {W}}_\xi $ is $\operatorname {GL}_u(\boldsymbol {k})\times \operatorname {GL}_u(\boldsymbol {k})$ -distinguished as well. More precisely, if we fix a nonzero $\operatorname {GL}_u(\boldsymbol {k})\times \operatorname {GL}_u(\boldsymbol {k})$ -invariant ${\overline {\mathbb {Q}}_\ell }$ -linear form $\mathit {\Lambda }$ on $\mathrm {W}_\xi $ together with a $\operatorname {GL}_n(\boldsymbol {k})$ -stable ${\overline {\mathbb {Z}}_\ell }$ -lattice ${L}\subseteq \mathrm {W}_\xi $ , then the associated ${\overline {\mathbb {F}}_{\ell }}$ -linear form

$$ \begin{align*} \overline{\mathit{\Lambda}} : {L}\otimes{\overline{\mathbb{F}}_{\ell}} \to \mathit{\Lambda}({L})\otimes{\overline{\mathbb{F}}_{\ell}} \end{align*} $$

is nonzero and $\operatorname {GL}_u(\boldsymbol {k})\times \operatorname {GL}_u(\boldsymbol {k})$ -invariant. Moreover, if s acts on $\mathit {\Lambda }$ by a sign $c\in \{-1,1\}\subseteq \overline {\mathbb {Q}}{}^\times _\ell $ , then s acts on $\overline {\mathit {\Lambda }}$ by the image of c in $\overline {\mathbb {F}}{}^\times _\ell $ .

4.1

Let us recall the definitions and main results of [Reference Bushnell and Kutzko9, Reference Bushnell and Henniart8, Reference Mínguez and Sécherre30, Reference Anandavardhanan, Kurinczuk, Matringe, Sécherre and Stevens3] which we will need.

Let $[\mathfrak {a},{\beta }]$ be a simple stratum in the algebra $\boldsymbol {\mathsf {M}}_{n}({ F})$ of $n\times n$ matrices with entries in ${ F}$ . Recall that $\mathfrak {a}$ is a hereditary -order of $\boldsymbol {\mathsf {M}}_{n}({ F})$ and ${\beta }$ is an element of $\boldsymbol {\mathsf {M}}_{n}({ F})$ such that

• ○ the ${ F}$ -algebra ${ E}={ F}[{\beta }]$ is a field, and

• ○ the multiplicative group ${ E}^\times $ normalizes $\mathfrak {a}$

(plus an extra technical condition on ${\beta }$ which is not necessary to recall here: See [Reference Bushnell and Kutzko9] 1.5.5).

Let be the normalizer of $\mathfrak {a}$ in ${ G}$ and $\mathfrak {p}_{\mathfrak {a}}$ be its Jacobson radical, and set $\mathrm {U}^1_{\mathfrak {a}} = 1 + \mathfrak {p}_{\mathfrak {a}}$ . Let ${ B}$ be the centralizer of ${ E}$ in $\boldsymbol {\mathsf {M}}_{n}({ F})$ . The intersection $\mathfrak {b}=\mathfrak {a}\cap { B}$ is a hereditary order in ${ B}$ .

Associated with $[\mathfrak {a},{\beta }]$ in [Reference Bushnell and Kutzko9] Chapter 3, there are compact mod centre open subgroups

and a nonempty finite set of characters of ${{H}}^1(\mathfrak {a},{\beta })$ called simple characters, depending on the choice of the character (4.1). We write $\mathbf {J}=\mathbf {J}(\mathfrak {a},{\beta })$ , $\mathbf {J}^0=\mathbf {J}^0(\mathfrak {a},{\beta })$ , $\mathbf {J}^1=\mathbf {J}^1(\mathfrak {a},{\beta })$ and ${{H}}^1={{H}}^1(\mathfrak {a},{\beta })$ for simplicity.

We will only be interested in the case where $\mathfrak {b}$ is a maximal order in ${ B}$ , in which case the simple stratum $[\mathfrak {a},{\beta }]$ and the simple characters in are said to be maximal. For the following result, see [Reference Bushnell and Henniart8] 2.1, 3.2 and [Reference Bushnell and Kutzko9] 5.1.1.

The representation $\eta $ of (4.b) is called the Heisenberg representation associated with ${\theta }$ . If $\boldsymbol {\kappa }$ is a representation of $\mathbf {J}$ extending $\eta $ , any other extension of $\eta $ to $\mathbf {J}$ has the form $\boldsymbol {\kappa }\boldsymbol {\xi }$ for a unique character $\boldsymbol {\xi }$ of $\mathbf {J}$ trivial on $\mathbf {J}^1$ .

A character ${\theta }$ of an open pro-p-subgroup ${{H}}$ of ${ G}$ will be called a maximal simple character if there is a maximal simple stratum $[\mathfrak {a},{\beta }]$ in $\boldsymbol {\mathsf {M}}_n({ F})$ such that ${{H}}={{H}}^1(\mathfrak {a},{\beta })$ and . Given a maximal simple character ${\theta }$ of ${ G}$ , we will write ${{H}}^1_{\theta }$ for the group on which ${\theta }$ is defined, $\mathbf {J}^{}_{\theta }$ for its ${ G}$ -normalizer, $\mathbf {J}^0_{\theta }$ for its unique maximal compact subgroup, $\mathbf {J}^1_{\theta }$ for its unique maximal normal pro-p-subgroup and ${ T}$ for the maximal tamely ramified extension of ${ F}$ in ${ E}={ F}[{\beta }]$ . The following result shows how the latter depends on the choice of $[\mathfrak {a},{\beta }]$ (see [Reference Bushnell and Henniart8] 2.1, 2.5 and 2.6).

It follows that the ${ G}$ -conjugacy class of the simple character ${\theta }$ uniquely determines the integer m in equation (4.3), as well as the extension ${ T}$ up to ${ G}$ -conjugacy (or equivalently up to ${ F}$ -isomorphism). However, the fields ${ E}$ , ${ E}'$ need not be isomorphic (see [Reference Bushnell and Henniart8] Example 2.1).

4.2

In this paragraph only, we let n vary among all positive integers, and consider the set

where the union is taken over all maximal simple strata of $\boldsymbol {\mathsf {M}}_n({ F})$ , for any $n{\geqslant }1$ . It is endowed with an equivalence relation called endo-equivalence ([Reference Bushnell and Henniart6, Reference Bushnell and Henniart7]). An equivalence class for this equivalence relation is called an endo-class (see [Reference Anandavardhanan, Kurinczuk, Matringe, Sécherre and Stevens3] 3.2).

Given a maximal simple character with endo-class $\boldsymbol {\Theta }$ , the degree $[{ E}:{ F}]$ and the ${ F}$ -isomorphism class of its tame parameter field ${ T}$ only depend on $\boldsymbol {\Theta }$ . They are called the degree and tame parameter field of $\boldsymbol {\Theta }$ , respectively.

For a given n, any two maximal simple characters of $\operatorname {GL}_n({ F})$ are endo-equivalent if and only if they are $\operatorname {GL}_n({ F})$ -conjugate.

4.3

We go back to the situation of Paragraph 4.1, assuming further that the character $\psi $ of equation (4.1) is $\sigma $ -invariant, which is possible since $p\neq 2$ . As in [Reference Sécherre35], we will say that:

• ○ a simple stratum $[\mathfrak {a},{\beta }]$ in $\boldsymbol {\mathsf {M}}_n({ F})$ is $\sigma $ -self-dual if $\mathfrak {a}$ is $\sigma $ -stable and $\sigma ({\beta })=-{\beta }$ ,

• ○ a simple character ${\theta }$ is $\sigma $ -self-dual if the group ${{H}}^1_{\theta }$ is $\sigma $ -stable and ${\theta }^{-1}\circ \sigma ={\theta }$ ,

• ○ an endo-class $\boldsymbol {\Theta }$ of (maximal) simple characters is $\sigma $ -self-dual if for some (or equivalently for any) ${\theta }\in \boldsymbol {\Theta }$ , the character ${\theta }^{-1}\circ \sigma $ is in $\boldsymbol {\Theta }$ .

If ${\theta }$ is a $\sigma $ -self-dual maximal simple character, we will write ${ T}_0$ for the maximal tamely ramified extension of ${ F}_0$ in ${ E}_0$ , that is, ${ T}_0={ T}\cap { E}_0$ . By [Reference Anandavardhanan, Kurinczuk, Matringe, Sécherre and Stevens3] Lemma 4.10, the canonical homomorphism

(4.6) $$ \begin{align} { T}_0\otimes_{{ F}_0}{ F} \to { T} \end{align} $$

is an isomorphism. Also, ${ T}/{ T}_0$ and ${ E}/{ E}_0$ have the same ramification index. By [Reference Anandavardhanan, Kurinczuk, Matringe, Sécherre and Stevens3] Lemma 4.29, the ${ F}_0$ -isomorphism class of ${ T}_0$ is uniquely determined by the endo-class $\boldsymbol {\Theta }$ of ${\theta }$ . And it follows from the isomorphism (4.6) that the ${ F}_0$ -isomorphism class of ${ T}_0$ determines the ${ F}$ -isomorphism class of ${ T}$ .

The following result is given by [Reference Sécherre35] Proposition 6.12, Lemma 6.20 and [Reference Sécherre36] Lemme 3.28. (The latter reference in [Reference Sécherre36] is for representations with coefficients in ${ R}=\mathbb {C}$ , but its proof is still valid in the $\ell $ -modular case.)

Conversely, let $\boldsymbol {\Theta }$ be a $\sigma $ -self-dual endo-class of degree dividing n. By [Reference Anandavardhanan, Kurinczuk, Matringe, Sécherre and Stevens3] Section 4, it contains a $\sigma $ -self-dual maximal simple character ${\theta }$ in ${ G}$ , and we have the following classification.

Remark 4.8. When ${ T}/{ T}_0$ is ramified, we define (as in [Reference Anandavardhanan, Kurinczuk, Matringe, Sécherre and Stevens3, Reference Sécherre35]) the index of a $\sigma $ -self-dual maximal simple character ${\theta }$ to be the integer $i\in \{0,\dots ,\lfloor m/2\rfloor \}$ associated with its ${ G}^\sigma $ -conjugacy class. By [Reference Anandavardhanan, Kurinczuk, Matringe, Sécherre and Stevens3] Remark 4.28 or [Reference Sécherre35] 5.D, if ${\theta }$ has index $0$ and if

$$ \begin{align*} y_i = \mathrm{diag}(\varpi,\dots,\varpi,1,\dots,1) \in\operatorname{GL}_{m}({ E})\simeq{ B}^\times, \quad \varpi\text{ a uniformizer of }{ E}\text{ occurring }i\text{ times,} \end{align*} $$

for some $i\in \{0,\dots ,\lfloor m/2\rfloor \}$ , then ${\theta }^{y_i}$ is a $\sigma $ -self-dual maximal simple character of index i.

4.4

Let ${\theta }$ be a maximal simple character, and $[\mathfrak {a},{\beta }]$ be a simple stratum such that . As in 4.1, write $\mathbf {J}=\mathbf {J}_{\theta }^{}$ , $\mathbf {J}^0=\mathbf {J}^0_{\theta }$ , $\mathbf {J}^1=\mathbf {J}^1_{\theta }$ and ${{H}}^1={{H}}^1_{\theta }$ . Let $\eta $ be the Heisenberg representation of $\mathbf {J}^1$ associated with ${\theta }$ . In this paragraph, we give a slightly generalized version of [Reference Bushnell and Henniart8] 3.3.

Fix an -lattice in ${ V}={ F}^n$ whose endomorphism algebra is $\mathfrak {b}$ . (It is uniquely determined up to homothety as $\mathfrak {b}$ is maximal.) Fix a divisor $u{\geqslant }1$ of m and decompositions

(4.9)

in which ${ V}_*$ is an ${ E}$ -vector space of dimension $m/u$ and is an -lattice of rank $m/u$ in ${ V}_*$ . It defines a Levi subgroup ${ M}$ of ${ G}$ . Fix a pair $({ Q}_-,{ Q}_+)$ of ${ M}$ -opposite parabolic subgroups of ${ G}$ with Levi component ${ M}$ , and write ${ N}_-$ , ${ N}_+$ for their unipotent radicals. Define . It is a hereditary order, and $[\mathfrak {a}_*,{\beta }]$ is a maximal simple stratum in $\operatorname {End}_{ F}({ V}_*)$ . Write $\mathbf {J}^{}_*$ , $\mathbf {J}^{0}_*$ , $\mathbf {J}^{1}_*$ and ${{H}}^1_*$ for the subgroups associated with it. Compare with [Reference Bushnell and Henniart8] 3.3, which corresponds to the particular case where $u=m$ .

The next result follows from [Reference Bushnell and Henniart6] Example 10.9 (compare with Lemma 1 in [Reference Bushnell and Henniart8] 3.3).

Moreover, the map ${\theta }\mapsto {\theta }_*$ defined by Lemma 4.9(2) is a bijection from to : this is an instance of the transfer of [Reference Bushnell and Kutzko9] 3.6.

Let $\eta _*^{}$ denote the Heisenberg representation of $\mathbf {J}_*^1$ associated with ${\theta }_*^{}$ . Compare the next result with Lemma 2 in [Reference Bushnell and Henniart8] 3.3 and the discussion after it.

Proof. That $\mathbf {J}_+$ is a group follows from the fact that ${{H}}^1$ is normalized by $\mathbf {J}$ , thus by $\mathbf {J}\cap { Q}_+$ . The existence of $\boldsymbol {\kappa }_+$ follows from the containment

$$ \begin{align*} (\mathbf{J}^0\cap{ N}_+)\cdot({{H}}^1\cap{ N}_-) \subseteq ({{H}}^1\cap{ N}_-)\cdot(\mathbf{J}^1\cap{ M})\cdot(\mathbf{J}^0\cap{ N}_+) \end{align*} $$

(see the argument of [Reference Sécherre34] 2.3). Mackey’s formula implies that the restriction of $\widetilde {\boldsymbol {\kappa }}_+$ to $\mathbf {J}^1$ is

$$ \begin{align*} \operatorname{Ind}^{\mathbf{J}^1}_{\mathbf{J}^1\cap\mathbf{J}_+} (\boldsymbol{\kappa}_+). \end{align*} $$

The restriction of $\boldsymbol {\kappa }_+$ to $\mathbf {J}^1\cap \mathbf {J}_+=({{H}}^1\cap { N}_-)\cdot (\mathbf {J}^1\cap { Q}_+)$ is the unique representation $\eta _+$ which is trivial on ${{H}}^1\cap { N}_-$ , $\mathbf {J}^1\cap { N}_+$ and agrees with $\eta _*\otimes \dots \otimes \eta _*$ on $\mathbf {J}^1\cap { M}$ . The representation it induces to $\mathbf {J}^1$ is isomorphic to $\eta $ : Indeed, this representation contains ${\theta }$ by Lemma 4.9(2), and it has dimension

$$ \begin{align*} \operatorname{dim} (\eta_*\otimes\dots\otimes\eta_*) \cdot (\mathbf{J}^1\cap{ N}_-:{{H}}^1\cap{ N}_-) &= (\mathbf{J}^1\cap{ M}:{{H}}^1\cap{ M})^{1/2}\cdot (\mathbf{J}^1\cap{ N}_-:{{H}}^1\cap{ N}_-) \\ &= (\mathbf{J}^1:{{H}}^1)^{1/2} \end{align*} $$

which is the dimension of $\eta $ (see [Reference Mínguez and Sécherre30] 2.3).

It remains to prove (3). First, uniqueness follows from the fact that any two such extensions differ from a character of $\mathbf {J}$ trivial on $\mathbf {J}^1\mathbf {J}_+$ , and such a character is trivial since $p\neq 2$ . Existence follows from [Reference Bushnell and Kutzko9] 5.2.4 (see [Reference Mínguez and Sécherre30] 2.4 in the modular case).

The reader will pay attention to the fact that, although $\mathbf {J}^0\cap { M}$ is equal to $\mathbf {J}_*^0\times \cdots \times \mathbf {J}_*^0$ , the group $\mathbf {J}\cap { M}$ is not equal to $\mathbf {J}_*\times \cdots \times \mathbf {J}_*$ in general (unless $u=1$ ). It is generated by $\mathbf {J}^0\cap { M}$ and ${ E}^\times $ , the latter being diagonal in ${ M}$ .

Proof. The case where $u=m$ is given by [Reference Bushnell and Henniart8] 3.3, Corollary 1. For the general case, first note that, if $\boldsymbol {\kappa }$ is the image of $\boldsymbol {\kappa }_*$ by the map (4.10), and if $\boldsymbol {\xi }_*$ is a character of $\mathbf {J}_*$ trivial on $\mathbf {J}_*^0$ , then the image of $\boldsymbol {\kappa }_*\boldsymbol {\xi }_*$ by the map (4.10) is equal to $\boldsymbol {\kappa }\boldsymbol {\xi }^u$ , where $\boldsymbol {\xi }$ is the character of $\mathbf {J}$ trivial on $\mathbf {J}^0$ coinciding with $\boldsymbol {\xi }_*$ on any uniformizer of E. Now, start with a representation $\boldsymbol {\kappa }$ extending $\eta $ . Let $\widetilde {\boldsymbol {\kappa }}_+$ denote its restriction to $\mathbf {J}^1\mathbf {J}_+$ and $\boldsymbol {\kappa }_+$ denote the representation of $\mathbf {J}_+$ on the space of $\mathbf {J}^1\cap { N}_-$ -invariants of $\widetilde {\boldsymbol {\kappa }}_+$ . The restriction of $\boldsymbol {\kappa }_+$ to $\mathbf {J}^0\cap { M}$ has the form

$$ \begin{align*} \boldsymbol{\kappa}_*^0 \otimes\dots\otimes \boldsymbol{\kappa}_*^0 \end{align*} $$

for a uniquely determined representation $\boldsymbol {\kappa }_*^0$ of $\mathbf {J}_*^0$ extending $\eta _*$ . Since any two extensions of $\boldsymbol {\kappa }_*^0$ to $\mathbf {J}_*$ are twists of each other by a character of $\mathbf {J}_*=E^\times \mathbf {J}_*^0$ trivial on $\mathbf {J}_*^0$ , the result follows.

4.5

Suppose now that the simple character ${\theta }$ and the simple stratum $[\mathfrak {a},{\beta }]$ of 4.4 are $\sigma $ -self-dual. The groups $\mathbf {J}$ , $\mathbf {J}^0$ , $\mathbf {J}^1$ and ${{H}}^1$ are thus $\sigma $ -stable. Suppose also that the decompositions (4.9) are $\sigma $ -stable. (To obtain such decompositions, consider the vertex in the reduced building of $B^\times $ defined by the -lattice (see [Reference Bruhat and Tits5]) and choose a $\sigma $ -stable apartment containing this vertex, whose existence is granted by [Reference Delorme and Sécherre13] since $p\neq 2$ .)

The Levi subgroup ${ M}$ is thus $\sigma $ -stable, and we may assume that ${ Q}_-$ , ${ Q}_+$ are $\sigma $ -stable as well. Also, the simple stratum $[\mathfrak {a}_*,{\beta }]$ and the simple character ${\theta }_*$ given by Lemma 4.9 are $\sigma $ -self-dual. Let ${ G}_*$ denote the group $\operatorname {Aut}_{ F}({ V}_*)$ . It is isomorphic to $\operatorname {GL}_{n/u}({ F})$ .

We may also assume that our choice of basis induces an isomorphism of groups (4.4) between $\mathbf {J}^0/\mathbf {J}^1$ and $\operatorname {GL}_m(\boldsymbol {l})$ as in Proposition 4.5(4), transporting the action of $\sigma $ on $\mathbf {J}^0/\mathbf {J}^1$ to

• ○ the action of the nontrivial element of $\operatorname {Gal}(\boldsymbol {l}/\boldsymbol {l}_0)$ on $\operatorname {GL}_m(\boldsymbol {l})$ if ${ T}/{ T}_0$ is unramified,

• ○ the adjoint action of (4.5) on $\operatorname {GL}_m(\boldsymbol {l})$ for some $i\in \{0,\dots ,\lfloor m/2\rfloor \}$ , if ${ T}/{ T}_0$ is ramified.

Let $\boldsymbol {\kappa }_*$ be a representation of $\mathbf {J}_*$ extending $\eta _*$ , and let $\boldsymbol {\kappa }$ correspond to it by the map (4.10).

We will need the following lemma. Let $\varpi $ be a uniformizer of ${ E}$ such that

(4.11) $$ \begin{align} \sigma(\varpi)= \left\{ \begin{array}{rl} \varpi & \text{if }{ T}/{ T}_0\text{ is unramified}, \\ -\varpi & \text{if }{ T}/{ T}_0\text{ is ramified}. \end{array}\right. \end{align} $$

Note that the group $\mathbf {J}$ is generated by $\mathbf {J}^0$ and $\varpi $ .

Proof. If ${ T}/{ T}_0$ is unramified: See [Reference Sécherre35] Lemma 9.1. Suppose that ${ T}/{ T}_0$ is ramified, and assume that there is an $x\in \mathbf {J}\cap { G}^\sigma $ , $x\notin \langle \varpi ^2,\mathbf {J}^0\cap { G}^\sigma \rangle $ . We have $x=\varpi ^{v}y$ , where $v\in \mathbb {Z}$ is odd and $y\in \mathbf {J}^0$ satisfies $\sigma (y)=-y$ . Reducing mod $\mathbf {J}^1$ , we get an $\alpha \in \operatorname {GL}_m(\boldsymbol {l})$ such that $\sigma (\alpha )=-\alpha $ . Since the involution $\sigma $ acts on $\operatorname {GL}_m(\boldsymbol {l})$ by conjugacy by

$$ \begin{align*} {\delta} = \mathrm{diag}(-1,\dots,-1,1,\dots,1) \end{align*} $$

(where $-1$ occurs i times), this implies that ${\delta }$ and $-{\delta }$ are $\operatorname {GL}_m(\boldsymbol {l})$ -conjugate, thus $m=2i$ . Conversely, if $m=2i$ , then

(4.12)

is $\sigma $ -anti-invariant, and $\varpi '=\varpi w$ has the required property.

We now investigate the behavior of the map (4.10) with respect to distinction. The case where $u=m$ will be sufficient for our purpose (see Paragraph 4.6).

Proof. The representation $\boldsymbol {\kappa }_+$ is $\mathbf {J}_+\cap { G}^\sigma $ -distinguished, thus $\widetilde {\boldsymbol {\kappa }}_+$ is $\mathbf {J}^1\mathbf {J}_+\cap { G}^\sigma $ -distinguished. It follows that $\boldsymbol {\kappa }$ is $\mathbf {J}^1\mathbf {J}_+\cap { G}^\sigma $ -distinguished. Let $\chi $ be the character of $\mathbf {J}\cap { G}^\sigma $ associated with $\boldsymbol {\kappa }$ by Proposition 4.6. It is trivial on $\mathbf {J}^1\mathbf {J}_+\cap { G}^\sigma $ . Restricting to $\mathbf {J}^0\cap { G}^\sigma $ , it is a character of

$$ \begin{align*} (\mathbf{J}^0\cap{ G}^\sigma)/(\mathbf{J}^1\cap{ G}^\sigma) \simeq \operatorname{GL}_m(\boldsymbol{l})^\sigma. \end{align*} $$

Since $\mathbf {J}\cap { M}\subseteq \mathbf {J}_+$ , it is trivial on the image of $(\mathbf {J}\cap { M})\cap (\mathbf {J}^0\cap { G}^\sigma )$ in $\operatorname {GL}_m(\boldsymbol {l})^\sigma $ , which is made of the $\sigma $ -fixed points of the diagonal torus $\textsf {M}=\boldsymbol {l}^\times \times \cdots \times \boldsymbol {l}^\times $ .

If ${ T}/{ T}_0$ is unramified, we have $\operatorname {GL}_m(\boldsymbol {l})^\sigma =\operatorname {GL}_m(\boldsymbol {l}_0)$ and $\textsf {M}^\sigma =\boldsymbol {l}_0^\times \times \cdots \times \boldsymbol {l}_0^\times $ , thus $\chi $ is trivial on $\mathbf {J}^0\cap { G}^\sigma $ . If ${ T}/{ T}_0$ is ramified, we have $\operatorname {GL}_m(\boldsymbol {l})^\sigma =\operatorname {GL}_i(\boldsymbol {l})\times \operatorname {GL}_{m-i}(\boldsymbol {l})$ and $\textsf {M}^\sigma =\textsf {M}$ . Again, $\chi $ is trivial on $\mathbf {J}^0\cap { G}^\sigma $ .

By Lemma 4.14, it remains to consider the value of $\chi $ at $\varpi '$ . If ${ T}/{ T}_0$ is unramified, or if ${ T}/{ T}_0$ is ramified and $m\neq 2i$ , we have $\varpi ' \in \mathbf {J}^1\mathbf {J}_+\cap { G}^\sigma $ , thus $\chi $ is trivial.

Now, assume that ${ T}/{ T}_0$ is ramified and $m=2i$ . Let $\boldsymbol {\xi }$ be the quadratic character of $\mathbf {J}$ trivial on $\mathbf {J}^0$ defined by $\xi (\varpi ')=\chi (\varpi ')$ . Then $\boldsymbol {\kappa }\boldsymbol {\xi }$ is $\mathbf {J}\cap { G}^\sigma $ -distinguished.

We will prove in Paragraph 4.6 that the quadratic character $\boldsymbol {\xi }$ of Lemma 4.15(2) is always trivial: See Corollary 4.19.

4.6

As in Paragraph 4.5, the simple character ${\theta }$ and the simple stratum $[\mathfrak {a},{\beta }]$ are both maximal $\sigma $ -self-dual, and $\eta $ is the Heisenberg representation of $\mathbf {J}^1$ associated with ${\theta }$ . The next proposition, which says that $\eta $ has a canonical extension to $\mathbf {J}$ , is the core of our proof of Theorem 4.41.