Generic cuspidal representations of $U(2,1)$

TL;DR

This paper classifies generic cuspidal representations of the unitary group U(2,1) over non-Archimedean local fields with odd residue characteristic, using explicit constructions by Stevens.

Contribution

It provides a complete classification of generic cuspidal representations of U(2,1) over fields with odd residue characteristic, extending prior work with explicit methods.

Findings

Classification of all generic cuspidal representations of U(2,1)

Explicit construction methods for these representations

Extension of Stevens' techniques to unitary groups

Abstract

Let $F$ be any non-Archimedean local field with a Galois involution $\sigma$ and $F_0$ be the fixed field for the action of $\sigma$. When the residue characteristic of $F_0$ is odd, using the explicit construction of cuspidal representations of classical groups by Stevens, we classify generic cuspidal representations of $U(2,1)(F/F_0)$.

Equations

$$h(xv,yw)=x\sigma (y)\sigma (h(w,v))\text{for all}v,w\in V\text{and}x,y\in F.$$

$$l(\pi (u)v)=\Psi (u)l(v),\text{for all}u\in U,v\in V.$$

$$F[\beta ]=F[{\beta }_1]\oplus F{\beta }_2]\oplus \cdots \oplus F[{\beta }_k]$$

$$\{X\in {\mathrm{End}}_F(V):{\sigma }_h(X)=-X\}.$$

$$\{\mathrm{diag}(t,1,t^{-1}):t\in F_0^{\times }\}\text{and}\{\mathrm{diag}(z,z^{\prime },\sigma (z{)}^{-1}):z,z^{\prime }\in F^{\times },z^{\prime }\sigma (z^{\prime })=1\}$$

$$\left\{u(c,d):=\left(\begin{array}{ccc}1& c& d\\ 0& 1& -\sigma (c)\\ 0& 0& 1\end{array}\right):c,d\in F,c\sigma (c)+d+\sigma (d)=0\right\},$$

$$\left\{\bar{u}(c,d):=\left(\begin{array}{ccc}1& 0& 0\\ c& 1& 0\\ d& -\sigma (c)& 1\end{array}\right):c,d\in F,c\sigma (c)+d+\sigma (d)=0\right\}$$

$$U_{\mathrm{der}}(r):=\{u(0,y):y\in \delta {\mathfrak{p}}_{F_0}^r\}.$$

$$l(\pi (u)w)=\Psi (u)l(w)\text{for all}u\in U,w\in W.$$

$${\dim }_{\mathbb{C}}{\mathrm{Hom}}_U(\pi ,\Psi )\le 1,$$

$${\psi }_{\beta }(X)={\psi }_F(\text{tr}(\beta ({\mathrm{id}}_V-X)))\text{for all}X\in A.$$

$$\beta V_1\subset V_2.$$

$${\mathfrak{X}}_{\beta }(F_0)=\{B\in \mathfrak{B}(F_0):{\psi }_{\beta }\text{is a character of}{\mathrm{R}}_u(B)(F_0)\}.$$

$${\nu }_{F_0}(\delta h(g{e}_1,\beta g{e}_1))\le -r.$$

$${\nu }_{F_0}(\delta h(g{e}_{-1},\beta g{e}_{-1}))\le -r.$$

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

$${\psi }_{\beta }(gu(0,\delta d)g^{-1})={\psi }_0(d({\mathrm{tr}}_{F/F_0}(\delta \mathrm{tr}(\beta g{X}_{\text{der}}{g}^{-1}))).$$

$${\mathrm{tr}}_{F/F_0}(\delta h(g{e}_1,\beta g{e}_1))=2\delta h(g{e}_1,\beta g{e}_1).$$

$${\Lambda }^{\#}(n)=\Lambda (-n{)}^{\#},\text{for all}n\in \mathbb{Z}.$$

$$(a\Lambda +b)(n)=\Lambda (\lceil (n-b)/a\rceil ),\text{for all}n\in \mathbb{Z}.$$

$${\tilde{\mathfrak{a}}}_n(\Lambda )=\left\{T\in {\mathrm{End}}_F(V):T\Lambda (i)\subset \Lambda (i+n)\forall i\in \mathbb{Z}\right\}.$$

$${\psi }_{\beta }:{\mathrm{id}}_V+X\mapsto {\psi }_F(\mathrm{tr}\beta X),\text{for}X\in {\tilde{a}}_n(\Lambda ).$$

$$F[{\beta }_1]\oplus F[{\beta }_2]\oplus \cdots \oplus F[{\beta }_k].$$

$${\kappa }_{\theta }:\frac{J^1(\Lambda ,\beta )}{H^1(\Lambda ,\beta )}\times \frac{J^1(\Lambda ,\beta )}{H^1(\Lambda ,\beta )}\to {\mathbb{C}}^{\times }.$$

$$P_0({\Lambda }_{\beta })/P_1({\Lambda }_{\beta })\simeq J^0(\Lambda ,\beta )/J^1(\Lambda ,\beta ).$$

$${\mathrm{ind}}_{J^0(\Lambda ,\beta )}^G(\kappa \otimes \tau )$$

$$P_m({\Lambda }_{\beta })P_k(\Lambda )\cap N=P_k(\Lambda )\cap N.$$

$${\mathrm{res}}_{H^1(\Lambda ,\beta )\cap U}\theta ={\mathrm{res}}_{H^1(\Lambda ,\beta )\cap U}{\psi }_{\beta },$$

$$\pi \simeq {\mathrm{ind}}_{J^0(\Lambda ,\beta )}^G\kappa .$$

$${\Theta }_{\beta }(jh)={\psi }_{\beta }(j)\theta (h),\text{for all}j\in J^0(\Lambda ,\beta )\cap U,h\in H^1(\Lambda ,\beta ).$$

$$F_0$$

Full text

1. Introduction

Let $F$ be a non-Archimedean local field and let $\sigma$ be a non-trivial Galois involution with fixed field $F_{0}$. Let $V$ be a three dimensional $F$-vector space, and let $h:V\times V\rightarrow F$ be a non-degenerate hermitian form with

[TABLE]

Let $\sigma_{h}$ be the adjoint anti-involution induced by $h$ on $\operatorname{End}_{F}(V)$. Let $\operatorname{{\bf G}}$ be the unitary group associated to $(V,h)$, and let $G$ be the group of $F_{0}$-rational points of $\operatorname{{\bf G}}$. In this article we obtain a classification of generic cuspidal representations of $G$ using the underlying skew semisimple strata, in the construction of cuspidal representations of $G$.

Let $\operatorname{{\bf H}}$ be a quasi-split reductive algebraic group defined over $F_{0}$. Let $\operatorname{{\bf B}}$ be a Borel subgroup of $\operatorname{{\bf H}}$, defined over $F_{0}$, and let $\operatorname{{\bf U}}$ be the unipotent radical of $\operatorname{{\bf B}}$. Let $\operatorname{{\bf T}}$ be a maximal $F_{0}$-split torus of $\operatorname{{\bf H}}$ contained in $\operatorname{{\bf B}}$. For any algebraic group $\operatorname{{\bf R}}$, defined over $F_{0}$, we denote by $R$ the group of $F_{0}$-rational points of $\operatorname{{\bf R}}$. A character $\psi$ of $U$ is said to be non-degenerate if $\psi$ is non-trivial on each simple root group of $T$ in $U$. A representation $\pi$ of $H$ is called generic if $\operatorname{Hom}_{U}(\pi,\Psi)\neq 0$, for some non-degenerate character $\Psi$ of $U$.

A cuspidal representation $(\pi,V)$ of $H$ is a generic representation if and only if there exists a non-zero linear functional $l:V\rightarrow\mathbb{C}$ and a character $\Psi$ of $U$ such that

[TABLE]

If such a linear functional exists, for $(\pi,V)$, then the character $\Psi$ necessarily satisfies the non-degenerate condition on the character $\Psi$. The functional $l$ is called a Whittaker linear functional. Moreover, when $H$ equals $G$, genericity of an irreducible smooth representation does not depend on the choice of the pair $(U,\Psi)$. A Whittaker linear functional on an irreducible smooth representation, if it exists, is unique up to scalars (see [19] and [16]).

Whittaker functionals are first used by Jacquet and Langlands to define the local $L,\epsilon$-factors for ${\rm GL_{2}}(F_{0})$ (see [11, Theorem 2.18]). These methods and their generalisations have played a fundamental role in the Langlands program, and especially in the theory of automorphic $L$-functions (see [9] for a survey). The local Langlands correspondence gives a natural partition of irreducible smooth representations of $H$ into finite sets called the $L$-packets. It is conjectured by Shahidi that there exists a unique generic representation inside an $L$-packet consisting of irreducible tempered representations (see [18, Conjecture 9.4]). When the characteristic of $F$ is zero, the local Langlands correspondence for the group $G$ is established by Rogawski in the book [17], and Shahidi’s conjecture, for the group $G$, is proved in the paper [8]. We hope that the results of this article are useful in understanding an explicit version of the local Langlands correspondence for $G$, relating the $L$-packets of cuspidal representations to their inducing data.

Every cuspidal representation of ${\rm GL}_{n}(F_{0})$ is generic (see [1, Chapter 3, 5.18]), but this is no longer true for classical groups. The classification of generic positive depth cuspidal representations from the inducing data is expected to depend on some subtle arithmetic aspects of the inducing data. When the characteristic of $F_{0}$ is zero, $F/F_{0}$ is unramified, and the cardinality of the residue field of $F_{0}$ is odd, Murnaghan classified the generic cuspidal representations of $G$ in the article [14, Theorem 7.13]. The methods used in [14] are based on character formulas for cuspidal representations–the Murnaghan–Kirillov formula–and using a local character expansion to relate with Shalika germs. DeBacker and Reeder also studied genericity of very cuspidal representations, arising from an unramified torus, of an unramified $p$-adic group (see [7]). Blondel and Stevens, using different techniques from Murnaghan, have classified generic cuspidal representations of ${\rm Sp}_{4}(F_{0})$, for a non-Archimedean local field $F_{0}$ with odd residue characteristic (see [4]). The methods of this article are inspired by the work of Blondel and Stevens on ${\rm Sp}_{4}(F_{0})$.

The explicit construction of cuspidal representations of $G$, when $F/F_{0}$ is an unramified extension of $p$-adic fields, goes back to the work of Moy and Jabon in the articles [13] and [10] respectively. Later, Blasco, in the article [2], gave an explicit construction of cuspidal representations in the line of Bushnell–Kutzko’s work on the admissible dual of ${\rm GL}_{n}$. In this article, we use the generalisation of Bushnell–Kutzko construction of cuspidal representations to classical groups by Stevens, culminating in the paper [21].

We now describe the results of this article using the language of strata from the theory of types (see Section 3 and references in loc.cit). Let $\mathfrak{x}=[\Lambda,n,0,\beta]$ be any skew semisimple stratum in $\operatorname{End}_{F}(V)$, in particular, $\Lambda$ is a lattice sequence, $n$ is a non-negative integer, $\beta\in\operatorname{End}_{F}(V)$ with $\sigma_{h}(\beta)=-\beta$, and the $G$-stabilizer of $\beta$–for the adjoint action of $G$ on its Lie algebra–is isomorphic to a product of unitary groups. Let $\Pi_{\mathfrak{x}}$ be the set of all cuspidal representations containing a type, in the sense of Bushnell–Kutzko, constructed from the stratum $\mathfrak{x}$. Let $\psi$ be a fixed non-trivial additive character of $F$, and let $\psi_{\beta}$ be the function sending $X\in\operatorname{End}_{F}(V)$ to $\psi(\operatorname{tr}(\beta(\operatorname{id}_{V}-X)))$. Let $\mathfrak{X}_{\beta}(F_{0})$ be the set of all $F_{0}$-rational Borel subgroups $\operatorname{{\bf B}}$ of $\operatorname{{\bf G}}$ such that $\psi_{\beta}$ defines a character on the group of $F_{0}$-rational points of the unipotent radical of $\operatorname{{\bf B}}$. The main result of this article is the following theorem:

Theorem 1.0.1**.**

Let $F$ be a non-Archimedean local field with odd residue characteristic. Let $\mathfrak{x}=[\Lambda,n,0,\beta]$ be any skew semisimple stratum with $n>0$. The cuspidal representations in the set $\Pi_{\mathfrak{x}}$ are either all generic or all non-generic. If $\mathfrak{X}_{\beta}(F_{0})$ is empty, then every cuspidal representation in the set $\Pi_{\mathfrak{x}}$ is non-generic. Except when $\beta$ has a non-degenerate two dimensional eigenspace, a cuspidal representation in the set $\Pi_{\mathfrak{x}}$ is generic if and only if $\mathfrak{X}_{\beta}(F_{0})$ is non-empty. If $\beta$ has a non-degenerate two dimensional eigenspace, $V_{2}$, then the set $\mathfrak{X}_{\beta}(F_{0})$ is non-empty if and only if $(V_{2},h)$ is isotropic. However, every cuspidal representation in the set $\Pi_{\mathfrak{x}}$ is non-generic.

The set $\mathfrak{X}_{\beta}(F_{0})$ is the set of $F_{0}$-rational points of a closed subvariety $\mathfrak{X}_{\beta}$ of the variety of Borel subgroups of $\operatorname{{\bf G}}$. We determine necessary and sufficient conditions on $\beta$ for the non-emptiness of $\mathfrak{X}_{\beta}(F_{0})$. Hence, we obtain a more explicit form of Theorem 1.0.1, and for this, we refer to Theorem 9.0.1. The genericity of depth-zero cuspidal representations $G$ is well understood (see [6, Section 6]). However, we recall these results for giving a complete analysis of genericity of cuspidal representations of $G$, especially, for those results not stated in the literature, for instance, when $F/F_{0}$ is ramified.

In general, the proofs use the explicit construction of cuspidal representations for classical groups by Stevens in the articles [20] and [21] and Mackey-decomposition to understand the restriction of a cuspidal representation to a maximal unipotent subgroup of $G$. Blondel and Stevens, in the article [4], related the set $\mathfrak{X}_{\beta}(F_{0})$ with the problem of genericity of cuspidal representations of ${\rm Sp}_{4}(F_{0})$. We use the approach in [4] to classify generic cuspidal representations of $G$; however, there are significant differences from the case of ${\rm Sp}_{4}(F_{0})$. In the case of ${\rm Sp}_{4}(F_{0})$, the variety $\mathfrak{X}_{\beta}$ is a $\mathbb{P}^{1}$-fibre space over a quadratic hypersurface–in a $3$-dimensional projective space over $F_{0}$. Hence, the problem of finding rational points on $\mathfrak{X}_{\beta}$ is reduced to that of a quadratic hypersurface. For the unitary group in $3$-variables, the variety $\mathfrak{X}_{\beta}$ is the intersection of two quadratic hypersurfaces in a $5$-dimensional projective space over $F_{0}$. This poses some arithmetic difficulties in understanding the $F_{0}$-rational points on $\mathfrak{X}_{\beta}$. The other significant difference between ${\rm Sp}_{4}(F_{0})$ and $G$ is that the isomorphism class of a non-degenerate subspace of $(V,h)$ is not determined by its dimension. For instance, a $2$-dimensional non-degenerate subspace of $(V,h)$ can be either isotropic or anisotropic. The problem of genericity depends on these differences.

We briefly sketch the contents of each section. The algebra $F[\beta]$ is a direct sum of fields, say

[TABLE]

with $\beta=\sum_{i=1}^{k}\beta_{i}$ and $\sigma_{h}(\beta_{i})=-\beta_{i}$, for $1\leq i\leq k$. This decomposition of $F[\beta]$ corresponds to a maximal orthogonal decomposition of $V=\perp_{i=1}^{k}V_{i}$, for the property that $F[\beta]$ acts on $V_{i}$ via its projection onto $F[\beta_{i}]$, for $1\leq i\leq k$. The above decomposition is unique and is determined by $\beta$. In Section 2, we set up some preliminaries to prove non-genericity results. In Section 3, we review some useful results from the construction of cuspidal representations of $G$. In Section 4, we consider the case where $F[\beta]$ is a field. When the characteristic of $F_{0}$ is zero, the $L$-packet containing a cuspidal representation from the set $\Pi_{\mathfrak{x}}$ has cardinality $1$. Hence, any representation in the set $\Pi_{\mathfrak{x}}$ is expected to be generic. We first prove that $\mathfrak{X}_{\beta}(F_{0})$ is non-empty, and using this, we will show that every representation in the set $\Pi_{\mathfrak{x}}$ is generic.

In Section 5, we consider the case where $F[\beta]$ is a $3$-dimensional algebra, $F[\beta]=F[\beta_{2}]\oplus F[\beta_{1}]$, and $\beta=\beta_{2}+\beta_{1}$ such that $[F[\beta_{2}]:F]=2$. We will completely determine when $\mathfrak{X}_{\beta}(F_{0})$ is non-empty and this depends only on the valuation of $\beta_{i}$, in the field $F[\beta_{i}]$, and on the isomorphism class of the hermitian space $(V_{2},h)$. Then we will use these results to show that a representation in the set $\Pi_{\mathfrak{x}}$ is generic if and only if the set $\mathfrak{X}_{\beta}(F_{0})$ is non-empty. In this case, we sometimes have to find a nice integral model of $\mathfrak{X}_{\beta}$ and lift points from its special fibre.

In Sections 6 and 7, we treat the cases where $F[\beta]$ is a direct sum of two copies of $F$ and three copies of $F$, respectively. The strategy is similar to that of the previous sections. But, in Section 6, we will see examples when $\mathfrak{X}_{\beta}(F_{0})$ is non-empty, and yet every representation in the set $\Pi_{\mathfrak{x}}$ is non-generic. We note that $\beta$ is not a regular semisimple element in this case.

In Section 7, we have $\beta=\beta_{1}+\beta_{2}+\beta_{2}$, with $\beta_{i}\in F$ and $\sigma(\beta_{i})=-\beta_{i}$. When $F/F_{0}$ is unramified, the non-emptiness of the set $\mathfrak{X}_{\beta}(F_{0})$ depends only on the valuations of $\beta_{i}$. However, when $F/F_{0}$ is ramified the information on the valuations of $\beta_{i}$, for $1\leq i\leq 3$, is not enough to determine whether the set $\mathfrak{X}_{\beta}(F_{0})$ is empty or not. Although it is fairly easy to determine the conditions on $\beta$ for the non-emptiness of $\mathfrak{X}_{\beta}(F_{0})$, these conditions do not involve the natural invariants of the stratum $\mathfrak{x}$; hence, we did not make them explicit. Nonetheless, we will show that a representation in the set $\Pi_{\mathfrak{x}}$ is generic if and only if $\mathfrak{X}_{\beta}(F_{0})$ is non-empty.

**Acknowledgements ** I want to thank Maarten Solleveld for many useful discussions and clarifications during the course of this work. The author is supported by the NWO Vidi grant “A Hecke algebra approach to the local Langlands correspondence” (nr. 639.032.528). I want to thank Shaun Stevens for answering some questions on his paper, for his interest, and other clarifications. I want to thank Peter Badea for suggesting some useful references. I want to thank Kam-Fai Geo Tam for helpful discussions.

2. Preliminaries

All representations in this article are defined over $\mathbb{C}$-vector spaces. Let $G,H$ be two groups with $H\subset G$, and let $\rho$ be a representation of $H$. We denote by $\rho^{g}$ the representation of $g^{-1}Hg$ sending $h\in g^{-1}Hg$ to $\rho(ghg^{-1})$. The group $g^{-1}Hg$ is denoted by $H^{g}$.

For any real number $x$, we denote by $\lfloor x\rfloor$ the greatest integer less than or equal to $x$. Let $\lceil x\rceil$ be the smallest integer greater than or equal to $x$. Let $x+$ be the smallest integer strictly bigger than $x$ and $x-$ be the greatest integer strictly smaller than $x$.

Let $(W,h)$ be a pair consisting of an $F$-vector space and a non-degenerate hermitian form $h$ on $W$. Let $(W^{\prime},h)$ be a non-degenerate subspace of $(W,h)$. Then let ${\bf 1}_{W^{\prime}}$ be the projection of $W$ onto $W^{\prime}$ with kernel $(W^{\prime})^{\perp}$.

2.1.

For a non-Archimedean local field $K$, let $\mathfrak{o}_{K}$ be the ring of integers of $K$, let $\mathfrak{p}_{K}$ be the maximal ideal of $\mathfrak{o}_{K}$, let $k_{K}$ be the residue field $\mathfrak{o}_{K}/\mathfrak{p}_{K}$, and $q_{K}$ denotes the cardinality of the residue field $k_{K}$. Let $\nu_{K}$ be the normalised valuation of $K$. From now we assume that $q_{K}$ is odd.

Let $F$ be a non-Archimedean local field with a Galois involution $\sigma$. Let $F_{0}$ be the fixed field of $\sigma$. Let $\varpi$ be a uniformizer of $F$ such that $\sigma(\varpi)=(-1)^{e(F|F_{0})-1}\varpi$. When $F/F_{0}$ is ramified, we set $\varpi_{0}$ to be the element ${\rm Nr}_{F/F_{0}}(\varpi)$, and when $F/F_{0}$ is unramified, we set $\varpi_{0}=\varpi$. The element $\varpi_{0}$ is a uniformizer of $F_{0}$. Let $\psi_{0}$ be a fixed additive character of $F_{0}$ with conductor $\mathfrak{p}_{F_{0}}$. The character $\psi_{0}\circ{\rm tr}_{F/F_{0}}$ is denoted by $\psi_{F}$. Let $F=F_{0}[\delta]$, where $\sigma(\delta)=-\delta$ and $\nu_{F}(\delta)=e(F|F_{0})-1$. Let $\nu_{F/F_{0}}$ be the valuation of $F$ extending the normalised valuation of $F_{0}$.

For any $F_{0}$-scheme $\operatorname{{\bf X}}$, we denote by $X$ the set of $F_{0}$-rational points of $\operatorname{{\bf X}}$. If $\operatorname{{\bf H}}$ is any linear algebraic group over $F_{0}$, then the group $H$ is considered as a topological group whose topology is induced from the non-Archimedean metric on $F$.

2.2.

Let $V$ be a three dimensional $F$-vector space and let $h$ be a non-degenerate hermitian form on $V$, as defined in (1.1). Let $\sigma_{h}$ be the adjoint anti-involution on $\operatorname{End}_{F}(V)$ induced by the hermitian form $h$. The hermitian space $(V,h)$ is isotropic, and we assume that the determinant of $(V,h)$ is the trivial class in $F_{0}^{\times}/{\rm Nr}_{F/F_{0}}(F^{\times})$. Let $\operatorname{{\bf G}}$ be the unitary $F_{0}$-group scheme associated with the pair $(V,h)$. We identify the Lie algebra, $\mathfrak{g}$, of $\operatorname{{\bf G}}$ with

[TABLE]

From now the algebra $\operatorname{End}_{F}(V)$ is denoted by $A$.

2.3.

A basis, $(e_{1},e_{0},e_{-1})$, of $(V,h)$ is called a Witt-basis if $h(e_{1},e_{1})=h(e_{-1},e_{-1})=0$, $h(e_{1},e_{-1})=1$, and $e_{0}\in\langle e_{1},e_{-1}\rangle^{\perp}$ with $h(e_{0},e_{0})=1$. A basis, $(e_{1},e_{-1})$, for a two dimensional hermitian space $(V^{\prime},h^{\prime})$, is called a Witt-basis if $h(e_{1},e_{1})=h(e_{-1},e_{-1})=0$, and $h(e_{1},e_{-1})=1$. Let $\operatorname{{\bf B}}$ be any $F_{0}$-rational Borel subgroup of $\operatorname{{\bf G}}$, and let $\operatorname{{\bf U}}$ be the unipotent radical of $\operatorname{{\bf B}}$. Let $\operatorname{{\bf T}}$ be a maximal $F_{0}$-split torus of $\operatorname{{\bf G}}$ contained in $\operatorname{{\bf B}}$. Let $\bar{\operatorname{{\bf U}}}$ be the unipotent radical of the opposite Borel subgroup, $\bar{\operatorname{{\bf B}}}$, of $\operatorname{{\bf B}}$ with respect to $\operatorname{{\bf T}}$. Let $\operatorname{{\bf Z}}$ and $\operatorname{{\bf N}}$ be the centraliser and the normaliser of $\operatorname{{\bf T}}$ respectively. We denote by $W_{G}$ the Weyl group $\operatorname{{\bf N}}/\operatorname{{\bf Z}}$.

There exists a Witt-basis $(e_{1},e_{0},e_{-1})$ of $V$–giving an embedding of $G$ in ${\rm GL}_{3}(F)$–such that $B$ stabilises the line $\langle e_{1}\rangle$. The groups $T$ and $Z$ are identified with the groups

[TABLE]

respectively. The groups $U$ and $\bar{U}$ are identified with the groups

[TABLE]

respectively. The derived groups of $U$ and $\bar{U}$, denoted by $U_{\text{der}}$ and $\bar{U}_{\text{der}}$ respectively, and they are identified with the groups $\{u(0,d):d\in F,d+\sigma(d)=0\}$ and $\{\bar{u}(0,d):d\in F,d+\sigma(d)=0\}$ respectively. Let $\{U_{\rm der}(r):r\in\mathbb{Z}\}$ be a filtration of compact subgroups of $U_{\rm der}$ defined as follows:

[TABLE]

Similarly, we set $\bar{U}_{\rm der}(r)$ to be the group $\{\bar{u}(0,y):y\in\delta\mathfrak{p}_{F_{0}}^{r}\}$, for $r\in\mathbb{Z}$.

2.4.

Let $\operatorname{{\bf U}}$ be the unipotent radical of an $F_{0}$-rational Borel subgroup $\operatorname{{\bf B}}$ of $\operatorname{{\bf G}}$. An irreducible smooth representation $(\pi,W)$ of $G$ is called a generic representation if and only if there exists a non-zero linear functional $l:W\rightarrow\mathbb{C}$ and a non-trivial character $\Psi$ of $U$ such that

[TABLE]

The group $Z$ acts transitively on the set of non-trivial characters of $U$, and hence the genericity of an irreducible smooth representation $(\pi,W)$ of $G$ does not depend on a choice of the pair $(U,\Psi)$. The linear functional $l:W\rightarrow\mathbb{C}$ is called a Whittaker linear functional. We have

[TABLE]

for any irreducible smooth representation $\pi$ of $G$, and a non-trivial character $\Psi$ of $U$ (see [19] and [16]).

2.5.

Let $\beta$ be an element in the algebra $A$. Let $\psi_{\beta}$ be the function on $A$ given by

[TABLE]

Let $V_{1}\subset V_{2}\subset V$ be a complete flag of $F$-vector spaces, and let $P$ be the Borel subgroup of ${\rm GL}_{F}(V)$ fixing this flag. Let $Y$ be the unipotent radical of $P$. If $V_{2}=V_{1}^{\perp}$, then $Y\cap G$ is the unipotent radical of the Borel subgroup $P\cap G$ of $G$. The function $\psi_{\beta}$ is a character of $Y$ if and only if

[TABLE]

Let $\mathfrak{B}$ be the variety of Borel subgroups of $\operatorname{{\bf G}}$. Let $\mathfrak{X}_{\beta}(F_{0})$ be the following subset of $\mathfrak{B}(F_{0})$:

[TABLE]

Here, ${\rm R}_{u}(\operatorname{{\bf B}})$ is the unipotent radical of a Borel subgroup $\operatorname{{\bf B}}$ of $\operatorname{{\bf G}}$. Note that the set $\mathfrak{X}_{\beta}(F_{0})$ is the set of $F_{0}$-rational points of a closed sub-variety of $\mathfrak{B}$, to be denoted by $\mathfrak{X}_{\beta}$.

2.6.

The following lemma is frequently used in proving certain cuspidal representations are non-generic. Let $(e_{1},e_{0},e_{-1})$ be a Witt-basis for $(V,h)$ and let $B$ be the Borel subgroup of $G$ fixing the line $\langle e_{1}\rangle$. Let $U$ be the unipotent radical of $B$. Using the basis $(e_{1},e_{0},e_{-1})$, we identify $G$ as a subgroup of ${\rm GL}_{3}(F)$.

Lemma 2.6.1**.**

Let $g$ be an element of $G$, and let $r$ be an integer. The character $\psi_{\beta}^{g}$ of $U_{\rm der}(r)$ is non-trivial if and only if

[TABLE]

Similarly, the character $\psi_{\beta}^{g}$ of the group $\bar{U}_{\rm der}(r)$, is non-trivial if and only if

[TABLE]

Proof.

We prove the lemma for $U_{\rm der}(r)$, and the other case is similar. Let $X_{\rm der}$ be the $3\times 3$ matrix

[TABLE]

We have the following equality:

[TABLE]

Note that $\operatorname{tr}(\beta gX_{\text{der}}g^{-1})$ is equal to $\operatorname{tr}(g^{-1}\beta gX_{\text{der}})$, and $\operatorname{tr}(g^{-1}\beta gX_{\text{der}})$ is equal to $h(ge_{1},\beta ge_{1})$. Since $\sigma_{h}(\beta)=-\beta$, we get that

[TABLE]

Hence, the character $\psi_{\beta}^{g}$ is trivial on $U_{\text{der}}(r)$ if and only if the character $d^{\prime}\mapsto\psi_{0}(d^{\prime}\delta h(ge_{1},\beta ge_{1}))$, for $d^{\prime}\in\mathfrak{p}_{F_{0}}^{r}$, is trivial. Since the conductor of $\psi_{0}$ is equal to $\mathfrak{p}_{F_{0}}$, we get the required inequality. ∎

3. Strata and cuspidal representations

In this section, we recall the construction of cuspidal representations of $G$, via Bushnell–Kutzko’s theory of types. We refer to the articles [20], [21] and [12] for more details.

3.1.

An $\mathfrak{o}_{F}$-lattice sequence, $\Lambda$, on $V$ is a function from $\mathbb{Z}$ to the set of $\mathfrak{o}_{F}$-lattices in $V$ satisfying the following conditions:

1. (1)

$\Lambda(n+1)\subseteq\Lambda(n)$, for all $n\in\mathbb{Z}$, 2. (2)

there exists a positive integer $e(\Lambda)$ such that $\Lambda(n+e(\Lambda))=\mathfrak{p}_{F}\Lambda(n)$, for any $n\in\mathbb{Z}$.

Given any lattice $\mathcal{L}\subset V$, let $\mathcal{L}^{\#}$ be the lattice $\{v\in V:\ h(v,\mathcal{L})\subset\mathfrak{p}_{F}\}$. For any lattice sequence $\Lambda$, let $\Lambda^{\#}$ be the lattice sequence defined as:

[TABLE]

A lattice sequence $\Lambda$ is said to be self-dual if there exists an integer $d$ such that $\Lambda^{\#}(n)=\Lambda(n+d)$, for all $n\in\mathbb{Z}$. Since we only use $\mathfrak{o}_{F}$-lattice sequences, we call them directly as lattice sequences.

Let $W$ be a subspace of the vector space $V$, and let $\Lambda$ be a lattice sequence on $V$. We denote by $\Lambda\cap W$ the lattice sequence on $W$ sending $n$ to $\Lambda(n)\cap W$.

Given any lattice sequence $\Lambda$ and integers $a,b\in\mathbb{Z}$, the lattice sequence $a\Lambda+b$ is defined by setting

[TABLE]

The set of lattice sequences $\{a\Lambda+b:\ a,b\in\mathbb{Z}\}$ is called the affine class of $\Lambda$. For any self-dual lattice-sequence $\Lambda$, we can find a lattice sequence $\Lambda^{\prime}$ in the affine class of $\Lambda$ such that $e(\Lambda^{\prime})$ is an even integer, and $(\Lambda^{\prime})^{\#}=\Lambda^{\prime}-1$. Henceforth, we assume that all self-dual lattice sequences satisfy these conditions.

3.2.

Given any lattice sequence $\Lambda$, and an integer $n$, let $\tilde{\mathfrak{a}}_{n}(\Lambda)$ be the following sublattice of $\operatorname{End}_{F}(V)$:

[TABLE]

The decreasing sequence of lattices $\{\tilde{\mathfrak{a}}_{n}(\Lambda):n\geq 0\}$ has trivial intersection. Given any element $T\in\operatorname{End}_{F}(V)$, we denote by $\nu_{\Lambda}(T)$ the unique integer $k$ such that $T\in\tilde{\mathfrak{a}}_{k}(\Lambda)$ and $T\not\in\tilde{\mathfrak{a}}_{k+1}(\Lambda)$. Let $\tilde{P}_{0}(\Lambda)$ be the units in the ring ${\tilde{\mathfrak{a}}}_{0}(\Lambda)$. For any positive integer $n$, let $\tilde{P}_{n}(\Lambda)$ be the compact open subgroup $\operatorname{id}_{V}+{\tilde{\mathfrak{a}}}_{n}(\Lambda)$ of ${\rm GL}_{F}(V)$. For any self-dual lattice sequence $\Lambda$, the lattices $\tilde{\mathfrak{a}}_{n}(\Lambda)$ are stable under $\sigma_{h}$. For any non-negative integer $n$, let $P_{n}(\Lambda)$ be the compact open subgroup $\tilde{P}(\Lambda)\cap G$ of $G$.

The group $P_{0}(\Lambda)/P_{1}(\Lambda)$ is the set of $k_{F_{0}}$-rational points of a (not necessarily connected) reductive algebraic group over $k_{F_{0}}$, and let $P^{0}(\Lambda)$ be the inverse image of the $k_{F_{0}}$-rational points of its connected component. The compact subgroup $P^{0}(\Lambda)$ is called the parahoric subgroup associated to $\Lambda$. If $F/F_{0}$ is unramified, then $P^{0}(\Lambda)$ is equal to $P_{0}(\Lambda)$ and has index $2$ in $P_{0}(\Lambda)$ otherwise.

A stratum in $\operatorname{End}_{F}(V)$ is the data, $[\Lambda,n,r,\beta]$, consisting of a lattice sequence $\Lambda$ on $V$, integers $n\geq r\geq 0$, and an element $\beta\in\operatorname{End}_{F}(V)$ such that $\beta\in\tilde{a}_{-n}(\Lambda)$. Two strata $[\Lambda,n,r,\beta_{1}]$ and $[\Lambda,n,r,\beta_{2}]$ are said to be equivalent if $\beta_{2}-\beta_{1}\in\tilde{a}_{-r}(\Lambda)$. A stratum $[\Lambda,n,r,\beta]$ is called a zero stratum if $n=r$ and $\beta=0$. For $n\geq r\geq n/2>0$, the set of equivalence classes of strata are in bijection with the characters of the group $\tilde{P}_{r+1}(\Lambda)/\tilde{P}_{n+1}(\Lambda)$. The character corresponding to the equivalence class containing the stratum $[\Lambda,n,r,\beta]$ is given by

[TABLE]

A stratum is called skew if the lattice sequence $\Lambda$ is self-dual and $\beta\in\mathfrak{g}$. We have the same notion of equivalence on skew strata. For $n\geq r\geq n/2>0$, an equivalence class of skew strata corresponds to a character on the group $P_{r+1}(\Lambda)/P_{n+1}(\Lambda)$, given by $\operatorname{res}_{P_{r+1}(\Lambda)}\psi_{\beta}$.

3.3.

Recall that a stratum $[\Lambda,n,r,\beta]$ is called a simple stratum if it satisfies the following conditions:

1. (1)

We have $n\geq r\geq 0$, 2. (2)

The valuation of $\beta$ with respect to $\Lambda$, denoted by $\nu_{\Lambda}(\beta)$, is equal to $-n$. 3. (3)

The algebra $F[\beta]$ is a field and it normalises the lattice sequence $\Lambda$. 4. (4)

We have $r<-k_{0}(\Lambda,\beta)$, where $k_{0}(\Lambda,\beta)$ is the critical constant defined in [20, Section 1.2.2].

A stratum $[\Lambda,n,r,\beta]$ is called a semisimple stratum if it is either a zero stratum or if it satisfies the following conditions:

1. (1)

We have $n\geq r\geq 0$ and $\nu_{\Lambda}(\beta)=-n$. 2. (2)

There exists a decomposition $V=\oplus_{i=1}^{k}V_{i}$ for which $\Lambda(k)=\oplus_{i=1}^{k}(\Lambda(k)\cap V_{i})$, for all $k\in\mathbb{Z}$. 3. (3)

Let ${\bf 1}_{i}$ be the projection of $V$ onto $V_{i}$ with kernel $\oplus_{j\neq i}V_{j}$. We have $\beta=\sum_{i=1}^{k}\beta_{i}$, where $\beta_{i}={\bf 1}_{i}\beta{\bf 1}_{i}$, for $1\leq i\leq k$. 4. (4)

The stratum $[\Lambda_{i},q_{i},r,\beta_{i}]$–with $q_{i}=r$ if $\beta_{i}=0$ and $q_{i}=-\nu_{\Lambda_{i}}(\beta_{i})$ otherwise–is either a zero stratum or a simple stratum, and this data must satisfy the following crucial condition: 5. (5)

the stratum $[\Lambda_{i}+\Lambda_{j},q,r,\beta_{i}+\beta_{j}]$, with $q=\max\{q_{i},q_{j}\}$, is non-equivalent to a zero stratum or a simple stratum, for $1\leq i,j\leq k$ and $i\neq j$.

The decomposition $V=\oplus_{i=1}^{k}V_{i}$ is uniquely determined by the element $\beta$, called the underlying splitting of the semisimple stratum $[\Lambda,n,r,\beta]$. A semisimple stratum $[\Lambda,n,r,\beta]$ is called a skew semisimple stratum if the decomposition $V=\oplus_{i=1}^{k}V_{i}$ is an orthogonal decomposition with respect to the form $h$ on $V$, and $\sigma_{h}(\beta_{i})=-\beta_{i}$, for $1\leq i\leq k$. Observe that the algebra $F[\beta]$ is isomorphic to the algebra

[TABLE]

We use the notation $\mathfrak{x}$ for a general skew semisimple stratum $[\Lambda,n,0,\beta]$.

Let $C_{\beta}(A)$ be the centraliser of $F[\beta]$ in $\operatorname{End}_{F}(V)$. The group $G\cap C_{\beta}(A)$ is denoted by $G_{\beta}$. Let $n$ be any integer and let $\tilde{\mathfrak{b}}_{n}(\Lambda)$ and $\mathfrak{b}_{n}(\Lambda)$ be the groups $\tilde{\mathfrak{a}}_{n}(\Lambda)\cap C_{\beta}(A)$ and ${\mathfrak{a}}_{n}(\Lambda)\cap C_{\beta}(A)$ respectively. For any non-negative integer $n$, let $\tilde{P}_{n}(\Lambda_{\beta})$ and ${P}_{n}(\Lambda_{\beta})$ be the groups $\tilde{P}_{n}(\Lambda)\cap C_{\beta}(A)^{\times}$ and ${P}_{n}(\Lambda)\cap G_{\beta}$ respectively.

3.4.

We recall the construction of a cuspidal representation starting from a skew semisimple stratum $\mathfrak{x}=[\Lambda,n,0,\beta]$. Stevens, generalising the Bushnell–Kutzko’s construction, associates some special compact subgroups: $J^{0}(\Lambda,\beta)$ and $H^{0}(\Lambda,\beta)$ of $G$. Then a particular class of representations of $J^{0}(\Lambda,\beta)$ are compactly induced to the group $G$ to obtain cuspidal representations. We will not go into the details of the construction of these compact subgroups here. It will suffice to briefly recall the general scheme of this construction. However, we describe these compact subgroups, in more detail, as required in the later part of the article.

Let $J^{i}(\Lambda,\beta)$ be the compact open subgroup $J^{0}(\Lambda,\beta)\cap P_{i}(\Lambda)$, for any non-negative integer $i$. For any skew-semisimple stratum $\mathfrak{x}=[\Lambda,n,0,\beta]$, the construction of cuspidal representations of $G$ begins with a specific set of characters of the group $H^{1}(\Lambda,\beta)$ called skew semisimple characters, denoted by $\mathcal{C}(\Lambda,0,\beta)$, (see [20, Section 3.6] and the set $\mathcal{C}(\Lambda,0,\beta)$ is denoted by $\mathcal{C}_{-}(\Lambda,0,\beta)$ there). The group $P_{(n/2)+}(\Lambda)$ is contained in $H^{1}(\Lambda,\beta)$, and we have $\operatorname{res}_{P_{(n/2)+}(\Lambda)}\theta=\psi_{\beta}$, for any $\theta\in\mathcal{C}(\Lambda,0,\beta)$. For any character $\theta\in\mathcal{C}(\Lambda,0,\beta)$, the map sending $g_{1},g_{2}\in J^{1}(\Lambda,\beta)$ to $\theta([g_{1},g_{2}])$ induces a perfect alternating pairing:

[TABLE]

Using the theory of Heisenberg lifting, for any character $\theta\in\mathcal{C}(\Lambda,0,\beta)$, there exists a unique representation $\eta_{\theta}$ of $J^{1}(\Lambda,\beta)$ such that $\operatorname{res}_{H^{1}(\Lambda,\beta)}\eta_{\theta}$ is equal to a power of $\theta$. There are a particular set of extensions of the representation $\eta_{\theta}$ to the group $J^{0}(\Lambda,\beta)$ called $\beta$-extensions; these representations are denoted by $\kappa$ (see [21, Section 4]).

The group $P_{0}(\Lambda_{\beta})$ is contained in $J^{0}(\Lambda,\beta)$. The inclusion of $P_{0}(\Lambda_{\beta})$ in $J^{0}(\Lambda,\beta)$ induces an isomorphism

[TABLE]

The group $P_{0}(\Lambda_{\beta})/P_{1}(\Lambda_{\beta})$ is the $k_{F_{0}}$-rational points of a (non-necessarily connected) reductive group over $k_{F_{0}}$. Let $\tau$ be a cuspidal representation of $P_{0}(\Lambda_{\beta})/P_{1}(\Lambda_{\beta})$. If $P^{0}(\Lambda_{\beta})$ is a maximal parahoric subgroup of $C_{\beta}(A)\cap G$, then the induced representation

[TABLE]

is irreducible, and this construction exhausts all cuspidal representations of $G$. The pair $(J^{0}(\Lambda,\beta),\kappa\otimes\tau)$ is a Bushnell–Kutzko type for the Bernstein component containing the representation (3.1). Let $\Pi_{\mathfrak{x}}$ be the set of cuspidal representations of $G$ containing a Bushnell–Kutzko type of the form $(J^{0}(\Lambda,\beta),\kappa\otimes\tau)$, for some $\kappa$ and $\tau$ as above.

3.5.

For the convenience of the reader, we recall some frequently used results from [4]. Let us begin with the following lemma, which is useful in calculating the group $H^{1}(\Lambda,\beta)\cap U$.

Lemma 3.5.1** (Blondel–Stevens).**

Let $[\Lambda,n,0,\beta]$ be a semisimple stratum in $\operatorname{End}_{F}(V)$ such that $C_{\beta}(A)$ does not contain any nilpotent elements. Let $N$ be a maximal unipotent subgroup of $G$. For $k\geq m\geq 1$, we have

[TABLE]

We refer to [4, Section 6.3, Lemma 6.5] for a proof of the above lemma. The following result is proved in greater generality by Blondel and Stevens (see [4, Section 4, Corollary 4.2, Theorem 4.3]); however, in the present context we will use the following simple version.

Proposition 3.5.2** (Blondel–Stevens).**

Let $[\Lambda,n,0,\beta]$ be a skew semisimple stratum such that $\mathfrak{X}_{\beta}(F_{0})$ is non-empty. Assume that $J^{0}(\Lambda,\beta)/J^{1}(\Lambda,\beta)$ is anisotropic. Let $\operatorname{{\bf B}}\in\mathfrak{X}_{\beta}(F_{0})$, and let $\operatorname{{\bf U}}$ be the unipotent radical of $\operatorname{{\bf B}}$. If

[TABLE]

for all $\theta\in\mathcal{C}(\Lambda,0,\beta)$, then every cuspidal representation in the set $\Pi_{\mathfrak{x}}$ is generic.

Proof.

Since we use this result in a crucial way, we briefly sketch the proof (see [4, Section 4, Corollary 4.2, Theorem 4.3]). Let $\pi$ be a cuspidal representation in the set $\Pi_{\mathfrak{x}}$. Let $(J^{0}(\Lambda,\beta),\kappa)$ be a Bushnell–Kutzko type contained in the representation $\pi$. We have

[TABLE]

Let $\theta$ be a skew semisimple character contained in $\operatorname{res}_{H^{1}(\Lambda,\beta)}\kappa$.

Let $\tilde{H}^{1}$ be the group $(J^{0}(\Lambda,\beta)\cap U)H^{1}(\Lambda,\beta)$ and $\Theta_{\beta}$ be the character of $\tilde{H}^{1}$ defined by:

[TABLE]

The group $\tilde{H}^{1}\cap J^{1}(\Lambda,\beta)$ is equal to $(J^{1}(\Lambda,\beta)\cap U)H^{1}(\Lambda,\beta)$, and it is a totally isotropic subspace for the pairing $\kappa_{\theta}$ on $J^{1}(\Lambda,\beta)/H^{1}(\Lambda,\beta)$. The representation $\eta_{\theta}$ is the induced representation from an extension of the character $\Theta_{\beta}$ to the inverse image in $J^{1}(\Lambda,\beta)$ of a maximal isotropic subspace of $J^{1}(\Lambda,\beta)/H^{1}(\Lambda,\beta)$. Hence, $\operatorname{res}_{J^{1}\cap U}\eta_{\theta}$ contains the character $\operatorname{res}_{J^{1}(\Lambda,\beta)\cap U}\psi_{\beta}$. Since the group $J^{0}(\Lambda,\beta)\cap U$ is equal to $J^{1}(\Lambda,\beta)\cap U$, using Mackey-decomposition, we get that $\operatorname{Hom}_{U}(\pi,\psi_{\beta})\neq 0$. ∎

It is convenient to partition the set of skew-semisimple strata in $\operatorname{End}_{F}(V)$ into four disjoint classes. A skew semisimple stratum $\mathfrak{x}=[\Lambda,n,0,\beta]$ is of type (A) if $F[\beta]$ is a field. A skew semisimple stratum $\mathfrak{x}$ is of type (B) if $F[\beta]$ is a direct sum of two fields with one of them a quadratic extension of $F$. A skew semisimple stratum $\mathfrak{x}$ is of type (C) if the algebra $F[\beta]$ is a direct sum of two copies of $F$, and finally a skew semisimple stratum $\mathfrak{x}$ is of type (D) if the algebra $F[\beta]$ is a direct sum of three copies of $F$.

4. The simple case.

4.1.

A skew semisimple stratum $[\Lambda,n,0,\beta]$ is of type (A) if the algebra $F[\beta]$ is a field. When the characteristic of $F_{0}$ is zero, it was shown by Blasco that the cardinality of the $L$-packet, containing a cuspidal representation in the set $\Pi_{\mathfrak{x}}$, is equal to one (see [3]). When the characteristic of $F_{0}$ is zero, a tempered $L$-packet is known to contain a generic member (see [8]), and hence every cuspidal representation in the set $\Pi_{\mathfrak{x}}$ is generic. In this section, for any non-Archimedean local field $F_{0}$ with odd residue characteristic, we directly prove that every cuspidal representation in the set $\Pi_{\mathfrak{x}}$ is generic.

Lemma 4.1.1**.**

Let $[\Lambda,n,0,\beta]$ be a skew simple stratum of type (A), then the set $\mathfrak{X}_{\beta}(F_{0})$ is non-empty.

Proof.

The involution $\sigma_{h}$ stabilises the field $F[\beta]$, and let $D$ be the fixed field of $\sigma_{h}$ in $F[\beta]$. We have the following diagram of field extensions:

[TABLE]

Let $\lambda$ be a $\sigma_{h}$-equivariant non-zero $F$-linear form on $F[\beta]$. There exists a unique hermitian form $h_{1}:V\times V\rightarrow F[\beta]$ such that

[TABLE]

[TABLE]

The set $\mathfrak{X}_{\beta}(F_{0})$ is non-empty if and only there exists a non-zero vector, $v\in V$, such that

[TABLE]

Observe that $h(v,\beta v)=\lambda(h_{1}(v,\beta v))=\lambda(\beta h_{1}(v,v))$. Since $\beta$ does not stabilise a proper non-zero subspace of $F[\beta]$, the $F$-linear forms $\lambda$ and $\lambda\circ\beta$ are linearly independent. Let $W$ be the $F$-vector space $\ker(\lambda)\cap\ker(\lambda\circ\beta)$, we have $\dim_{F}W=1$. Since $\sigma_{h}(\beta)=-\beta$ and $\lambda$ is $\sigma_{h}$-equivariant, the $F$-vector space $W$ is stable under the action of $\sigma_{h}$. Let $W_{0}$ be the $F_{0}$-vector space $W\cap D$, and note that $W_{0}\otimes_{F_{0}}F=W$.

The form $h_{1}(x,y)$ is equal to $xa\sigma_{h}(y)$, for some $a\in D$. Let $N_{a}$ be the set $\{xa\sigma_{h}(x):x\in F[\beta]^{\times}\}$. Assume that the inclusion of $F_{0}^{\times}$ in $D^{\times}$ induces a surjection of $F_{0}^{\times}$ onto the quotient $D^{\times}/({\rm Nr}_{F[\beta]/D}(F[\beta]^{\times})$. Let $w$ be a non-zero vector in $W_{0}$. If $w\not\in N_{a}$, then there exists some element $x\in F_{0}^{\times}$ such that $xw_{1}\in N_{a}$. Hence, the set $N_{a}\cap W_{0}$ is non-empty. Since $h(w,w)=h(w,\beta w)=0$, for all $w\in N_{a}\cap W_{0}$, the set $\mathfrak{X}_{\beta}(F_{0})$ is non-empty.

Now, we will prove that the inclusion of $F_{0}^{\times}$ in $D^{\times}$ induces a surjection of $F_{0}^{\times}$ onto the quotient

[TABLE]

Recall the notations $\varpi$ and $\varpi_{0}$ for a choice uniformizers of $F$ and $F_{0}$, respectively, from Paragraph 2.1. Fix a valuation $\nu:F[\beta]^{\times}\rightarrow 1/e[F[\beta]:F]\mathbb{Z}$ such that $\nu(\varpi)=1$. Assume that $\varpi_{0}={\rm Nr}_{F[\beta]/D}(x)$, for some $x\in F[\beta]$. Then, we have $\nu(\varpi)=2\nu(x)$. If $F/F_{0}$ is unramified, then the equality $\nu(\varpi)=2\nu(x)$ is impossible and therefore, we get that $\varpi_{0}\not\in{\rm Nr}_{F[\beta]/D}(F[\beta]^{\times})$. Consider the case where $F$ is a ramified extension of $F_{0}$. Let $x$ be an element of $\mathfrak{o}_{F_{0}}^{\times}$ such that $\bar{x}\in k_{F_{0}}$ is not a square in $k_{F_{0}}$. The element $x$ does not belong to ${\rm Nr}_{F[\beta]/D}(F[\beta]^{\times})$; since the involution $\sigma_{h}$ must act trivially on the residue field of $k_{F[\beta]}$ (note that $[k_{F[\beta]}:k_{F_{0}}]|3$). ∎

Proposition 4.1.2**.**

Let $\mathfrak{x}=[\Lambda,n,0,\beta]$ be any skew simple stratum of type (A). Then every cuspidal representation in the set $\Pi_{\mathfrak{x}}$ is generic.

Proof.

Let $\pi$ be a cuspidal representation in the set $\Pi_{\mathfrak{x}}$. Since $\dim_{F}V=3$, we may twist the representation $\pi$ by a character, if necessary, and assume that $\mathfrak{x}$ is minimal. Let $(J^{0}(\Lambda,\beta),\kappa)$ be a Bushnell–Kutzko type contained in $\pi$. Then the representation $\pi$ is isomorphic to $\operatorname{ind}_{J^{0}(\Lambda,\beta)}^{G}\kappa$. Here, the group $J^{0}(\Lambda,\beta)$ is equal to $\mathfrak{o}^{\times}_{F[\beta]}P_{(n/2)+}(\Lambda)$, and $\kappa$ is a $\beta$-extension of the Heisenberg lift $\eta_{\theta}$ of a skew semisimple character $\theta$ of $H^{1}(\Lambda,\beta)$. Now, Lemma 4.1.1 implies that the set $\mathfrak{X}_{\beta}(F_{0})$ is non-empty. Let $\operatorname{{\bf U}}$ be the unipotent radical of a Borel subgroup in the set $\mathfrak{X}_{\beta}(F_{0})$. Using Lemma 3.5.1, we get that the group $J^{0}(\Lambda,\beta)\cap U$ is equal to $P_{(n/2)+}(\Lambda)\cap U$. Hence, we have

[TABLE]

Now using Proposition 3.5.2, we get that the representation $\pi$ is generic. ∎

5. The non simple type (B) strata

A skew semisimple stratum $\mathfrak{x}=[\Lambda,n,0,\beta]$ is of type (B) if the underlying splitting of $\mathfrak{x}$ is of the form $V=V_{1}\perp V_{2}$ with $\dim_{F}(V_{2})=2$ and the algebra $F[\beta_{2}]$ is a quadratic extension of $F$. Recall that $\beta_{i}$ is equal to ${\bf 1}_{V_{i}}\beta{\bf 1}_{V_{i}}$, for $i\in\{1,2\}$. From the definition of a skew semisimple stratum, we have $\beta=\beta_{1}+\beta_{2}$. If $F/F_{0}$ is unramified, from the equality $\sigma_{h}(\beta_{2})=-\beta_{2}$, we get that the extension $F[\beta_{2}]/F$ is ramified. Similarly, if $F/F_{0}$ is ramified, then the extension $F[\beta_{2}]/F$ is unramified. We recall the notation that $q_{i}=-\nu_{\Lambda_{i}}(\beta_{i})$, for $i\in\{1,2\}$. Genericity of a cuspidal representation, in the set $\Pi_{\mathfrak{x}}$, depends only on the isomorphism class of the hermitian space $(V_{2},h)$ and the integers $q_{1}$ and $q_{2}$. Since $\dim_{F}(V_{2})=2$, after twisting by a character, if necessary, we may assume that $\beta_{2}$ is minimal.

5.1. Lattice sequences

Since $G_{\beta}$ is anisotropic, the lattice sequence $\Lambda$ is uniquely determined by $\beta$. First, we need to fix a Witt-basis of $V$ which provides a splitting for $\Lambda$.

5.1.1. The Unramified case

Let $F$ be an unramified quadratic extension of $F_{0}$, and consider the case where $(V_{2},h)$ is isotropic. Since $F/F_{0}$ is unramified, the extension $F[\beta_{2}]/F$ is ramified. In this case, we have $e(\Lambda)=4$. There exists a Witt-basis, $(e_{1},e_{-1})$, for the space $V_{2}$, and a unit vector $e_{0}\in V_{1}$ (i.e., $h(e_{0},e_{0})=1$) such that the lattice sequence $\Lambda$ is given by:

[TABLE]

The group $P(\Lambda)$ is an Iwahori subgroup of $G$. The filtration $\{P_{m}(\Lambda):m>0\}$ is sometimes called a non-standard filtration on the Iwahori subgroup $P(\Lambda)$ (see [14, Page 612]).

Now, consider the case where $F/F_{0}$ is unramified and $(V_{2},h)$ is anisotropic. There exists an orthogonal basis $(v_{2},v_{3})$ of $V_{2}$ such that $h(v_{2},v_{2})=1$ and $h(v_{3},v_{3})=\varpi_{0}$. Let $v_{1}$ be any vector in $V_{1}$ such that $h(v_{1},v_{1})=\varpi_{0}$. The period of the lattice sequence, $e(\Lambda)$, is equal to $2$, and

[TABLE]

Lemma 5.1.1**.**

Let $(V_{2},h)$ be anisotropic, and let $\Lambda$ be the lattice sequence in (5.1). There exists a Witt-basis $(e_{1},e_{-1})$ for $(\langle v_{1},v_{3}\rangle,h)$ such that

[TABLE]

Proof.

Let $\epsilon\in F$ be an element with $\epsilon\sigma(\epsilon)=-1$; such an element exists because $F/F_{0}$ is unramified. The vectors $e_{1}=\epsilon v_{1}/2+v_{3}/2$ and $\varpi_{0}e_{-1}=-\epsilon v_{1}+v_{3}$ are isotropic vectors, and $h(e_{1},e_{-1})=1$. This implies that $(e_{1},e_{-1})$ is a Witt-basis for $\langle v_{1},v_{3}\rangle$, and since $\nu_{F}(\epsilon)=0$, the tuple $(e_{1},\varpi_{0}e_{-1})$ is a $\mathfrak{o}_{F}$ basis for the lattice $\mathfrak{o}_{F}v_{1}\oplus\mathfrak{o}_{F}v_{3}$. ∎

When $(V_{2},h)$ is anisotropic as above, we set $e_{0}$ to be the vector $v_{2}$. The Witt-basis $(e_{1},e_{0},e_{-1})$, where $(e_{1},e_{-1})$ is a Witt-basis for $(\langle v_{1},v_{3}\rangle,h)$ as in Lemma 5.1.1 provides a splitting for $\Lambda$. In the basis $(e_{1},e_{0},e_{-1})$, the lattice sequence $\Lambda$, with $e(\Lambda)=2$, is given by

[TABLE]

5.1.2. Ramified case

Let $F/F_{0}$ be a ramified quadratic extension. Then the extension $F[\beta_{2}]/F$ is an unramified quadratic extension. First consider the case where $(V_{2},h)$ is isotropic. The lattice sequence $\Lambda$ has period $2$. There exists a Witt-basis, $(e_{1},e_{0},e_{-1})$, for the hermitian space $(V,h)$ with $e_{1},e_{-1}\in V_{2}$ such that

[TABLE]

Now, consider the case where $(V_{2},h)$ is anisotropic. There exists an orthogonal basis $(v_{2},v_{3})$ of $V_{2}$ and a non-zero vector $v_{1}\in V_{1}$ such that: $h(v_{i},v_{i})=\lambda_{i}\in\mathfrak{o}_{F}^{\times}$, for $1\leq i\leq 3$, and the hermitian space $(\langle v_{1},v_{3}\rangle,h)$ is isotropic. The period of the lattice sequence $\Lambda$ is equal to $2$, and we have:

[TABLE]

Lemma 5.1.2**.**

Let $(V_{2},h)$ be anisotropic. Then there exists a Witt-basis $(e_{1},e_{-1})$ for the hermitian space $(\langle v_{1},v_{3}\rangle,h)$ such that

[TABLE]

Proof.

We fix an $\epsilon\in F$ such that $\epsilon\sigma(\epsilon)=-\lambda_{3}\lambda_{1}^{-1}$. The vectors $e_{1}=\epsilon v_{1}/2+v_{3}/2$ and $e_{-1}=(-\epsilon v_{1}+v_{3})\lambda_{3}^{-1}$ are isotropic and $h(e_{1},e_{-1})=1$. Moreover, the vectors $e_{1}$ and $e_{-1}$ are a basis for the $\mathfrak{o}_{F}$-lattice $\mathfrak{o}_{F}v_{1}\oplus\mathfrak{o}_{F}v_{3}$. ∎

Let $(e_{1},e_{-1})$ be a Witt-basis for the hermitian space $(\langle v_{1},v_{3}\rangle,h)$ as in Lemma 5.1.2. Let $e_{0}$ be the vector $v_{2}$. In the basis $(e_{1},e_{0},e_{-1})$, the period $2$ lattice sequence $\Lambda$ is given by

[TABLE]

5.2.

In each of the above cases, we fixed a Witt-basis $(e_{1},e_{0},e_{-1})$ which gives a splitting for the lattice sequence $\Lambda$. Let $\operatorname{{\bf B}}$ be the Borel subgroup of $\operatorname{{\bf G}}$ such that $B$ fixes the space $\langle e_{1}\rangle$. Let $\operatorname{{\bf T}}$ be the maximal $F_{0}$-split torus of $\operatorname{{\bf G}}$ such that $T$ stabilises the decomposition $V=\langle e_{1}\rangle\oplus\langle e_{0}\rangle\oplus\langle e_{-1}\rangle$. Let $\bar{\operatorname{{\bf B}}}$ be the opposite Borel subgroup of $\operatorname{{\bf B}}$ with respect to $\operatorname{{\bf T}}$. Let $\operatorname{{\bf U}}$ (resp. $\bar{\operatorname{{\bf U}}}$) be the unipotent radical of $\operatorname{{\bf B}}$ (resp. $\bar{\operatorname{{\bf B}}}$). We also recall the notations $u(c,d)$ and $\bar{u}(c,d)$ for elements in $U$ and $\bar{U}$ respectively. In the basis $(e_{1},e_{0},e_{-1})$, let $\mathcal{I}$ be the Iwahori subgroup:

[TABLE]

From the Iwasawa decomposition, we get that

[TABLE]

5.3.

The integers $q_{1}$ and $q_{2}$ have the following constraints: If $F/F_{0}$ is unramified and $(V_{2},h)$ is isotropic, then $q_{2}=4m_{2}+2$ and $q_{1}=4m_{1}$ for some $m_{1},m_{2}\in\mathbb{Z}$. If $F/F_{0}$ is unramified and $(V_{2},h)$ is anisotropic, then $q_{2}=2m_{2}+1$ and $q_{1}=2m_{1}$, for some $m_{1},m_{2}\in\mathbb{Z}$. Finally consider the case where $F/F_{0}$ is ramified. Since the image of the element

[TABLE]

(here, $g=(q_{2},e(\Lambda_{2}))=2$) in $k_{F[\beta_{2}]}$ must generate the degree $2$ field extension $k_{F[\beta_{2}]}$ over $k_{F}$, we get that $q_{2}=4m_{2}$ and $q_{1}=4m_{1}+2$, for some $m_{1},m_{2}\in\mathbb{Z}$. Hence, in all the above cases $q_{1}\neq q_{2}$.

5.4.

We will need the structure of the compact subgroups $J^{0}(\Lambda,\beta)$ and $H^{0}(\Lambda,\beta)$ associated to $\mathfrak{x}$. First consider the case $q_{2}<q_{1}$, and in this case the constant $k_{0}(\Lambda,\beta)$, defined in [20, equation 3.6], is equal to $q_{2}$. The stratum $[\Lambda,n,q_{2},\beta_{1}]$ is a skew semisimple stratum equivalent to the stratum $[\Lambda,n,q_{2},\beta]$. We then have:

[TABLE]

If $q_{2}>q_{1}$, then we have

[TABLE]

5.5.

When $\mathfrak{x}$ is a skew semisimple stratum of type (B), we prove a necessary and sufficient condition for the non-emptiness of the set $\mathfrak{X}_{\beta}(F_{0})$. The field $F[\beta_{2}]$ is stable under the action of $\sigma_{h}$ and let $D$ be the fixed field of the automorphism $\sigma_{h}$. Let $\lambda$ be the $F$-linear $\sigma_{h}$-equivariant linear functional $\lambda:F[\beta_{2}]\rightarrow F$ such that $\lambda(\beta_{2})=\beta_{1}$ and $\lambda(1)=1$. There exists a unique hermitian form $h^{\prime}:V_{2}\times V_{2}\rightarrow F[\beta_{2}]$ such that

[TABLE]

and

[TABLE]

Since $F_{0}[\delta]$ is a quadratic extension of $F_{0}$, the kernel of $\lambda$ is equal to

[TABLE]

The set $\mathfrak{X}_{\beta}(F_{0})$ is non-empty if and only if there exists $v_{1}\in V_{1}$, $v_{2}\in V_{2}$ with $v_{1}+v_{2}\neq 0$ such that

[TABLE]

For any two vectors $v_{1}$ and $v_{2}$ as above we must have $v_{1}\neq 0$ and $v_{2}\neq 0$. Note that $(\beta_{1}-\beta_{2})h^{\prime}(v_{2},v_{2})$ is contained in the kernel of $\lambda$, and this implies that $h^{\prime}(v_{2},v_{2})\in F_{0}^{\times}$. If $d_{1}$ and $d_{2}$ are the determinants of $(V_{2},h^{\prime})$ and $(V_{1},h)$, then we have

[TABLE]

Conversely, if the condition (5.6) holds, then we can always find a simultaneous solution to the equations (5.4) and (5.5), and hence, the set $\mathfrak{X}_{\beta}(F_{0})$ is non-empty.

5.6. The case where $(V_{2},h)$ is isotropic

In this part, we treat the case where $(V_{2},h)$ is isotropic, and let us begin with the case where $q_{2}>q_{1}$.

Lemma 5.6.1**.**

Let $F/F_{0}$ be an unramified extension and let $\mathfrak{x}$ be a stratum of type (B). If $(V_{2},h)$ is isotropic and $q_{2}>q_{1}$, then every cuspidal representation in the set $\Pi_{\mathfrak{x}}$ is non-generic. Moreover, we have $\mathfrak{X}_{\beta}(F_{0})=\emptyset$.

Proof.

Let $\pi$ be a representation in the set $\Pi_{\mathfrak{x}}$. There exists a $\beta$-extension $\kappa$ of $J^{0}(\Lambda,\beta)$ such that $\pi=\operatorname{ind}_{J^{0}(\Lambda,\beta)}^{G}\kappa$. If $\pi$ is genetic, then there exists a non-trivial character $\Psi$ of $U$, $p\in\mathcal{I}$, and $w\in W_{G}$ such that

[TABLE]

Let $\overline{U^{w}}$ be the unipotent radical of the opposite Borel subgroup, $\overline{B^{w}}$, of $B^{w}$ containing $T$. Using the Iwahori decomposition of $\mathcal{I}$, with respect to $(\overline{B^{w}},T)$, we have $p=p^{+}u^{-}$ with $p^{+}\in B^{w}\cap\mathcal{I}$ and $u^{-}\in\overline{U^{w}}\cap\mathcal{I}$. Now the equation (5.7) implies that

[TABLE]

for some character $\Psi^{\prime}$ of $U^{w}$. The group $P_{(q_{2}/2)+}(\Lambda)$ is normalised by the element $u^{-}=\dot{u}(x,y)$, and hence, $P_{(q_{2}/2)+}(\Lambda)^{u^{-}}\cap U^{w}$ is equal to $P_{(q_{2}/2)+}(\Lambda)\cap U^{w}$.

We set $q_{2}=8k+2r$, for some integer $k$ and $r\in\{1,3\}$. For the following calculations it is convenient to refer to the appendix A.3 for an explicit description of the filtration $\{a_{m}(\Lambda):m\in\mathbb{Z}\}$. We have $(q_{2}/2)+=4k+r+1$ and the intersection $P_{(q_{2}/2)+}(\Lambda)\cap U^{w}$ is given by:

[TABLE]

We define $e_{w}$ and $e_{-w}$ by setting $e_{w}=we_{1}$ and $e_{-w}=we_{-1}$, $w\in W_{G}$. Then, we have

[TABLE]

Now, the valuation of the term $h(e_{w},\beta_{2}e_{w})$ is strictly less than the valuation of all other terms in the right-hand side of (5.9). Hence, using Iwasawa decomposition, we get that $h(ge_{1},\beta ge_{1}))\neq 0$, for all $g\in G$. This shows that the set $\mathfrak{X}_{\beta}(F_{0})$ is empty. We also get the valuation of $\delta h(u^{-}e_{w},\beta u^{-}e_{w})$ is equal to

[TABLE]

From the equations (5.8) and (5.10), we get that

[TABLE]

where $s$ is given by the equality

[TABLE]

Now, Lemma 2.6.1 implies that the character $\psi_{\beta}^{u^{-}}$ is non-trivial on the group $P_{(q_{2}/2)+}(\Lambda)\cap U^{w}$ and we get a contradiction to (5.7). ∎

Lemma 5.6.2**.**

Let $F/F_{0}$ be a ramified extension and let $\mathfrak{x}$ be a stratum of type (B) such that $(V_{2},h)$ is isotropic. If $q_{2}>q_{1}$, then the set $\mathfrak{X}_{\beta}(F_{0})$ is empty, and every cuspidal representation in the set $\Pi_{\mathfrak{x}}$ is non-generic.

Proof.

First, we note that the group $P^{0}(\Lambda)$ is a special parahoric subgroup of $G$. Let $\pi=\operatorname{ind}_{J^{0}(\Lambda,\beta)}^{G}\kappa$ be a generic representation in the set $\Pi_{\mathfrak{x}}$. Using the Iwasawa decomposition $G=P(\Lambda)TU$, and the Mackey decomposition for the representation $\operatorname{res}_{U}\pi$, we get that

[TABLE]

for some $g\in P(\Lambda)$, and a non-trivial character $\Psi$ of $U$. We refer to the appendix A.2 for an explicit description of the filtration $\{a_{m}(\Lambda):m\in\mathbb{Z}\}$.

We set $q_{2}=4k_{2}$ and $q_{1}=4k_{1}+2$, for some integers $k_{1}$ and $k_{2}$. The group $P(\Lambda)$ normalises $P_{(q_{2}/2)+}(\Lambda)$ and hence, $P_{(q_{2}/2)+}(\Lambda)^{g}\cap U_{\rm der}$ is equal to $P_{(q_{2}/2)+}(\Lambda)\cap U_{\rm der}$. We have

[TABLE]

Since, $\nu_{\Lambda}(e_{1})=1$, we have $ge_{1}=ae_{1}+\varpi be_{0}\oplus\varpi ce_{-1}$, for some $a,b,c\in\mathfrak{o}_{F}$. We now try to estimate the valuation of $h(ge_{1},\beta ge_{1})$. Observe that $h(ge_{1},\beta ge_{1})$ is equal to

[TABLE]

Recall that $F[\beta_{2}]$ is the unramified quadratic extension of $F$ and the element $\tilde{\beta}_{2}=\varpi^{q_{2}/2}\beta_{2}$ belongs to $a_{0}(\Lambda_{2})$. Since $\nu_{\Lambda}(e_{1})=1$, the constants $a$ and $c$ both cannot belong to $\mathfrak{p}_{F}$. From the observation that $\nu_{F}({\rm Nr}_{F[\beta_{2}]/F}(\beta_{2}))=0$, we have

[TABLE]

We note that

[TABLE]

Since $q_{2}>q_{1}$, we have $-k_{2}<-k_{1}-1/2$, we deduce that $h(ge_{1},\beta ge_{1})\neq 0$ and $\nu_{F/F_{0}}(\delta h(ge_{1},\beta ge_{1}))$ is equal to $-k_{2}+1$. From the equation (5.12), we get that

[TABLE]

where $P_{(q_{2}/2)+}(\Lambda)\cap U_{\rm der}=U_{\rm der}(s)$. Now, using Lemma 2.6.1, we arrive at a contradiction to (5.11). Hence, any representation $\pi$ in the set $\Pi_{\mathfrak{x}}$ is non-generic. Finally, using the Iwasawa decomposition $G=P(\Lambda)B$, we get that $h(ge_{1},\beta ge_{1})\neq 0$, for $g\in G$. Hence, the set $\mathfrak{X}_{\beta}(F_{0})$ is empty. ∎

Lemma 5.6.3**.**

Let $F/F_{0}$ be an unramified extension and let $\mathfrak{x}$ be a stratum of type (B) such that $(V_{2},h)$ is isotropic. If $q_{1}>q_{2}$, then the set $\mathfrak{X}_{\beta}(F_{0})$ is non-empty.

Proof.

We continue using the notations introduced in 5.5. Since $(V_{2},h)$ is isotropic, we get that $\ker(\lambda)$ has non-trivial intersection with the set $\{h^{\prime}(v,v):v\in V_{2},v\neq 0\}$. Since $F[\beta_{2}]$ is a ramified extension of $F$ and $F/F_{0}$ is an unramified extension, we get that $F_{0}^{\times}$ is contained in ${\rm Nr}_{F[\beta_{2}]/D}(F[\beta_{2}]^{\times})$. This implies that the determinant of the form $(V_{2},h^{\prime})$ is the class in $D^{\times}/{\rm Nr}_{F[\beta_{2}]/D}(F[\beta_{2}]^{\times})$ containing $\delta(\beta_{2}-\beta_{1})$. Since, $q_{1}>q_{2}$ we have

[TABLE]

From the observation that $d_{1}$, the determinant of $(V_{1},h)$, belongs to $F_{0}$, we see that

[TABLE]

Now, using the criterion (5.6), we get that the set $\mathfrak{X}_{\beta}(F_{0})$ is non-empty. ∎

Lemma 5.6.4**.**

Let $F/F_{0}$ be a ramified extension, and let $\mathfrak{x}$ be a stratum of type (B) such that $(V_{2},h)$ is isotropic. If $q_{1}\geq q_{2}$, then the set $\mathfrak{X}_{\beta}(F_{0})$ is non-empty.

Proof.

We continue using notations introduced in 5.5. Since the space $(V_{2},h)$ is isotropic, the set $\{h^{\prime}(v,v):v\in V_{2},v\neq 0\}$ has a nontrivial intersection with $\ker(\lambda)$. Hence, the determinant class of $(V,h^{\prime})$ is equal to the coset in $D^{\times}/{\rm Nr}_{F[\beta_{2}]/D}(F[\beta_{2}]^{\times})$ containing the element $\delta\beta_{1}(1-\beta_{2}\beta_{1}^{-1})$. We note that $(1-\beta_{2}\beta_{1}^{-1})$ belongs to $1+\mathfrak{p}_{F[\beta_{2}]}$ and $\beta_{1}\in\delta F_{0}$. Since $k_{D}$ is a quadratic extension of $k_{F_{0}}$, we get that $F_{0}^{\times}$ is contained in ${\rm Nr}_{F[\beta_{2}]/D}(F[\beta_{2}]^{\times})$. Hence, we have $\delta\beta_{1}\in F_{0}$ and we get that $\delta(\beta_{1}-\beta_{2})$ belongs to ${\rm Nr}_{F[\beta_{2}]/D}(F[\beta_{2}]^{\times})$. Now, using the criterion (5.6) we get that the set $\mathfrak{X}_{\beta}(F_{0})$ is non-empty. ∎

Lemma 5.6.5**.**

Let $F/F_{0}$ be a quadratic extension and let $\mathfrak{x}$ be a stratum of the type (B) such that $(V_{2},h)$ is isotropic. If $q_{1}>q_{2}$, then any representation in the set $\Pi_{\mathfrak{x}}$ is generic.

Proof.

Let $\operatorname{{\bf U}}$ be the unipotent radical of a Borel subgroup of in the set $\mathfrak{X}_{\beta}(F_{0})$. Let $\theta$ be any skew semisimple character in the set $\mathcal{C}(\Lambda,0,\beta)$. We will first check that

[TABLE]

The group $H^{1}(\Lambda,\beta)$ is equal to

[TABLE]

We define $H^{\prime}$ to be the subgroup $P_{1}(\Lambda_{\beta})P_{(q_{2}/2)+}(\Lambda)$. Now, using Lemma 3.5.1, we get that $H^{\prime}\cap U$ is equal to $P_{(q_{2}/2)+}(\Lambda)\cap U$. Since $H^{1}(\Lambda,\beta)\cap U$ is equal to $H^{1}(\Lambda,\beta)\cap(H^{\prime}\cap U)$, we get that $H^{1}(\Lambda,\beta)\cap U$ is equal to $P_{(q_{2}/2)+}(\Lambda_{\beta_{1}})P_{(q_{1}/2)+}(\Lambda)\cap U$. Let $g_{1}g_{2}\in U$ for some $g_{1}\in P_{(q_{2}/2)+}(\Lambda_{F[\beta_{1}]})$ and $g_{2}\in P_{(q_{1}/2)+}(\Lambda)$. We recall the notations defined in 5.5. Let $v=v_{1}+v_{2}$ be a non-trivial vector fixed by $U$. If $v_{1}=0$, then we have $\lambda(h^{\prime}(v_{2},v_{2}))=0$ and $\lambda(\beta_{2}h^{\prime}(v_{2},v_{2}))=0$. This implies that $\beta_{2}$ stabilises the kernel of $\lambda$ and this absurd. Hence, we get that $v_{1}\neq 0$. We have

[TABLE]

Now, comparing both sides we get that $x\in F^{\times}\cap P_{(q_{1}/2)+}(\Lambda)$. This implies that ${\bf 1}_{V_{1}}g_{1}{\bf 1}_{V_{1}}\in F^{\times}\cap P_{(q_{1}/2)+}(\Lambda)$. Since, the determinant of $g_{2}g_{1}$ is equal to $1$, we get that determinant of ${\bf 1}_{V_{2}}g_{1}{\bf 1}_{V_{2}}$ belongs to $F^{\times}\cap P_{(q_{1}/2)+}(\Lambda)$. From the definition of simple character (see [5, Definition 3.2.3(a)]), we get (5.13). The Lemma is now a consequence of Proposition 3.5.2. ∎

5.7. The case where $(V_{2},h)$ is

anisotropic

As in the case where $(V_{2},h)$ is isotropic, the genericity of a cuspidal representation in the set $\Pi_{\mathfrak{x}}$ depends only on the integers $q_{1}$ and $q_{2}$. However, the condition for genericity becomes the opposite to the case where $(V_{2},h)$ is isotropic, i.e., the inequality $q_{2}>q_{1}$ is necessary and sufficient condition for genericity of a cuspidal representation in $\Pi_{\mathfrak{x}}$.

Lemma 5.7.1**.**

Let $F/F_{0}$ be an unramified extension, and let $\mathfrak{x}$ be a stratum of type ${\bf(B)}$ such that $(V_{2},h)$ is anisotropic. If $q_{1}>q_{2}$, then the set $\mathfrak{X}_{\beta}(F_{0})$ is empty, and every cuspidal representation contained in the set $\Pi_{\mathfrak{x}}$ is non-generic.

Proof.

Let $\pi=\operatorname{ind}_{J^{0}(\Lambda,\beta)}^{G}\kappa$ be a generic representation in the set $\Pi_{\mathfrak{x}}$. The group $P(\Lambda)$ is a special parahoric subgroup of $G$. Using the Iwasawa decomposition $G=P(\Lambda)B$, we have

[TABLE]

for some $g\in P(\Lambda)$ and a character $\Psi$ of $U$.

Let $q_{1}=4m_{1}+2r$ and $q_{2}=2m_{2}+1$, for some integers $m_{1}$, $m_{2}$ and $r\in\{0,1\}$. Note that $P_{(q_{1}/2)+}(\Lambda)\cap U_{\rm der}$ is equal to $U_{\rm der}(m_{1})$ (see 5.1.1 and (A.2) for an explicit description of the lattice sequence $\Lambda$ and the induced filtrations on $\operatorname{End}_{F}(V)$). We have $\nu_{\Lambda}(e_{1})=1$, and we get that $ge_{1}=av_{1}+\varpi_{0}bv_{2}+cv_{3}$, for some $a,b,c\in\mathfrak{o}_{F}$. Since $e_{1}$ is isotropic, we get that

[TABLE]

Since $\nu_{\Lambda}(e_{1})=1$, the above equality implies that $a,c\in\mathfrak{o}_{F}^{\times}$. Note that $\nu_{F[\beta_{2}]}(\beta_{2}\beta_{1}^{-1})>0$, and in the basis $(v_{2},v_{3})$ for $V_{2}$, the element $\beta_{2}\beta_{1}^{-1}\in\operatorname{End}_{F}(V)$ belongs to the following lattice of $\operatorname{End}_{F}(V_{2})$:

[TABLE]

Hence, we simultaneously get that $h(ge_{1},\beta ge_{1})\neq 0$ and

[TABLE]

Now, using Lemma 2.6.1, we get that the character $\psi_{\beta}^{g}$ is non-trivial on the group $P_{(q_{1}/2)+}(\Lambda)\cap U_{\rm der}$. Hence, we get a contradiction for the assumption (5.14). This shows that the representation $\pi$ in the set $\Pi_{\mathfrak{x}}$ is non-generic. Using the Iwasawa decomposition $G=P(\Lambda)B$, we get that $h(ge_{1},\beta ge_{1})\neq 0$, for all $g\in G$. Hence, the set $\mathfrak{X}_{\beta}(F_{0})$ is empty. ∎

Lemma 5.7.2**.**

Let $F/F_{0}$ be a ramified extension, and let $\mathfrak{x}$ be a stratum of type (B) such that $(V_{2},h)$ is anisotropic. If $q_{1}>q_{2}$, then the set $\mathfrak{X}_{\beta}(F_{0})$ is empty, and every representation in the set $\Pi_{\mathfrak{x}}$ is non-generic.

Proof.

Let $\pi=\operatorname{ind}_{J^{0}(\Lambda,\beta)}^{G}\kappa$ be a generic representation in the set $\Pi_{\mathfrak{x}}$. The group $P^{0}(\Lambda)$ is a special parahoric subgroup of $G$, and using the Iwasawa decomposition $G=P(\Lambda)B$, we get that

[TABLE]

for some $g\in P(\Lambda)$, and a non-trivial character $\Psi$ of $U$.

Let $q_{1}=4m_{1}+2$ and $q_{2}=4m_{2}$, for some integers $m_{1}$ and $m_{2}$. Observe that $P_{(q_{1}/2)+}(\Lambda)^{g}\cap U_{\rm der}$ is equal to $U_{\rm der}(\lceil m_{1}/2\rceil)$. Since $\nu_{\Lambda}(e_{1})=0$, we get that $ge_{1}=av_{1}+bv_{2}+cv_{3}$ for some $a,b,c\in\mathfrak{o}_{F}$. As the vector $e_{1}$ is isotropic we get that

[TABLE]

Now, the space $(V_{2},h)$ is anisotropic and this implies that $a\in\mathfrak{o}_{F}^{\times}.$ Now, we have

[TABLE]

Since, $\nu_{F[\beta_{2}]}(\beta_{2}\beta_{1}^{-1})>0$, we simultaneously get that $h(ge_{1},\beta ge_{1})\neq 0$, and $\nu_{F}(\delta h(ge_{1},\beta ge_{1}))$ is equal to $\nu_{F}(\delta\beta_{1})$. From the inequality $\nu_{F}(\delta\beta_{1})=-m_{1}\leq-\lceil m_{1}/2\rceil$, we get that $\psi_{\beta}^{g}$ is a non-trivial character on $P_{(q_{1}/2)+}(\Lambda)^{g}\cap U_{\rm der}$. This is a contradiction to the assumption (5.15). Using the Iwasawa decomposition $G=P(\Lambda)B$, we get that $h(ge_{1},\beta ge_{1})\neq 0$, for all $g\in G$. This shows that the set $\mathfrak{X}_{\beta}(F_{0})$ is empty. ∎

In the case where $(V_{2},h)$ is anisotropic, we could not use the criterion in 5.5. However, the following observation motivates the fact that $\mathfrak{X}_{\beta}(F_{0})$ is non-empty in the case where $q_{2}>q_{1}$. We suppose that $F/F_{0}$ is unramified. In the basis $(v_{1},v_{2},v_{3})$ consider a skew element $\beta$ of the form

[TABLE]

Let $e_{1}$ be an isotropic vector in $\langle v_{1},v_{3}\rangle$. Let $\operatorname{{\bf B}}$ be a Borel subgroup of $\operatorname{{\bf G}}$ such that $B$ fixes the space $\langle e_{1}\rangle$. The Borel subgroup $\operatorname{{\bf B}}$ belongs to the set $\mathfrak{X}_{\beta}(F_{0})$. Now, the class of $\beta$ in the quotient $a_{-n}(\Lambda)/a_{1-n}(\Lambda)$, is represented by a matrix as in (5.16). This suggests that we may lift a point from the special fibre of an integral model of $\mathfrak{X}_{\beta}$, and we do this in the following lemma.

Lemma 5.7.3**.**

Let $F/F_{0}$ be a quadratic extension and let $\mathfrak{x}$ be a stratum of type (B) such that $(V_{2},h)$ is anisotropic. If $q_{2}>q_{1}$, then the set $\mathfrak{X}_{\beta}(F_{0})$ is non-empty.

Proof.

We will lift a point from the special fibre of a smooth model for $\mathfrak{X}_{\beta}$. Let $(v_{1},v_{2},v_{3})$ be a basis of $V$ as defined in 5.1.1, when $F/F_{0}$ is unramified, and let $(v_{1},v_{2},v_{3})$ be a basis of $V$ as defined in 5.1.2, when $F/F_{0}$ is ramified. Let $\beta=(\beta_{ij})$ be the matrix representation of $\beta$ in the basis $(v_{1},v_{2},v_{3})$. We have $\beta_{11}=\beta_{1}$.

First consider the case where $F/F_{0}$ is unramified. A Borel subgroup $\operatorname{{\bf B}}$, corresponding to the line $\langle xv_{1}+yv_{2}+zv_{3}\rangle$, belongs to $\mathfrak{X}_{\beta}$ if and only if $(x,y,z)$ satisfy the following equations:

[TABLE]

and

[TABLE]

Changing $y$ to $\varpi y^{\prime}$ and rescaling the second equation by $\varpi^{\alpha-2}$ with $\alpha=-\nu_{F}(\beta_{32})$, we get the following set of equations:

[TABLE]

and

[TABLE]

Note that the coefficients of (5.18) are integral and the two equations (5.17) and (5.18) define a flat closed subscheme $\mathcal{X}_{\beta}$ of $\mathbb{P}^{5}_{\mathfrak{o}_{F_{0}}}$ such that the generic fibre is $\mathfrak{X}_{\beta}$. The special fibre is given by the set of equations

[TABLE]

and

[TABLE]

where $C_{1}=\overline{\varpi^{\alpha-1}\beta_{23}}$, $C_{2}=\overline{\varpi^{\alpha-1}\beta_{1}}$, and $C_{3}=\overline{\varpi^{\alpha-1}\beta_{33}}$. Note that $C_{1}\neq 0$, and therefore, the special fibre $\overline{\mathcal{X}}_{\beta}$ is smooth. Since $\mathcal{X}_{\beta}$ is flat we get that $\mathcal{X}_{\beta}$ is smooth over $\mathfrak{o}_{F}$. The special fibre has a rational point and hence by Hensel’s lemma the set $\mathfrak{X}_{\beta}(F_{0})$ is non-empty.

Now, consider the case where $F/F_{0}$ is a ramified extension. A Borel subgroup $\operatorname{{\bf B}}$ of $\operatorname{{\bf G}}$ fixing the isotropic subspace $\langle av_{1}+bv_{2}+cv_{3}\rangle$, belongs to $\mathfrak{X}_{\beta}$ if and only if:

[TABLE]

and

[TABLE]

In the present case $F[\beta_{2}]$ is an unramified extension of $F$. After rescaling by a power of $\varpi$, if necessary, we may assume that

[TABLE]

and $\beta_{1}\in\mathfrak{o}_{F}$. Because $\beta_{1}$ is skew, we get that $\beta_{1}\in\mathfrak{p}_{F}$. By a change of variable $b$ to $\varpi b^{\prime}$, the above set of equations become:

[TABLE]

and

[TABLE]

Since $\beta$ is skew, we get that $\sigma(\beta_{22})+\beta_{22}=0$, $\sigma(\beta_{33})+\beta_{33}=0$, and $\lambda_{1}\beta_{23}=-\lambda_{3}\sigma(\beta_{32})$. Hence, the above two equations have integral coefficients. The above two equations define a flat projective sub-variety $\mathcal{X}_{\beta}$ in $\mathbb{P}^{5}_{\mathfrak{o}_{F_{0}}}$ with generic fibre $\mathfrak{X}_{\beta}$. The special fibre is given by

[TABLE]

and

[TABLE]

where $C_{1}=\overline{\varpi^{-1}\lambda_{1}\beta_{1}}$, $C_{2}=\overline{\varpi^{-1}\lambda_{3}\beta_{33}}$, and $C_{3}=\overline{\lambda_{2}\beta_{23}}$. Clearly $C_{3}\neq 0$ as the element $\beta_{2}$ is minimal. The special fibre is smooth and hence, $\mathcal{X}_{\beta}$ is a smooth model for $\mathfrak{X}_{\beta}$. Note that the special fibre has a rational point. Using Hensel’s lemma, we get that the set $\mathfrak{X}_{\beta}(F_{0})$ is non-empty. ∎

Lemma 5.7.4**.**

Let $F/F_{0}$ be a quadratic extension and let $\mathfrak{x}$ be a stratum of type (B) such that $(V_{2},h)$ is anisotropic. If $q_{2}>q_{1}$, then every representation in the set $\Pi_{\mathfrak{x}}$ is generic.

Proof.

Using Lemma 5.7.3, we get that the set $\mathfrak{X}_{\beta}(F_{0})$ is non-empty. Let $\operatorname{{\bf U}}$ be the unipotent radical of a Borel subgroup in the set $\mathfrak{X}_{\beta}(F_{0})$. Note that the group $H^{1}(\Lambda,\beta)$ is equal to $P_{1}(\Lambda_{\beta})P_{(q_{2}/2)+}(\Lambda)$ and it follows from Lemma 3.5.1 that $H^{1}(\Lambda,\beta)\cap U$ is equal to $P_{(q_{2}/2)+}(\Lambda)\cap U$. Hence, we get that

[TABLE]

where $\theta$ is any skew semisimple character of $H^{1}(\Lambda,\beta)$. Now, genericity is a consequence of Proposition 3.5.2. ∎

6. Non simple type (C) strata

We say that a skew semisimple stratum $[\Lambda,n,0,\beta]$, denoted by $\mathfrak{r}$, is of type (C) if the underlying splitting of $\mathfrak{x}$ is given by $V=V_{1}\perp V_{2}$ with $\dim_{F}V_{i}=i$, and $[F[\beta_{i}]:F]=1$. Here, $\beta_{i}$ is equal to ${\bf 1}_{V_{i}}\beta{\bf 1}_{V_{i}}$, for $i\in\{1,2\}$. From the definition of skew semisimple stratum, we have $\sigma(\beta_{i})=-\beta_{i}$, for $i\in\{1,2\}$, and $\beta=\beta_{1}+\beta_{2}$. We will show that every representation contained in the set $\Pi_{\mathfrak{x}}$ is non-generic.

6.1. Lattice sequences

We will describe the lattice sequences up to $G_{\beta}$-conjugacy. Note that $\Lambda$ is a lattice sequence on $V$ such that $P^{0}(\Lambda_{\beta})$ is a maximal parahoric subgroup of $G_{\beta}$. We will fix a Witt-basis of $(V,h)$ which gives a splitting for these lattice sequences.

6.1.1. The unramified case

Consider the case where $F/F_{0}$ is unramified and $(V_{2},h)$ is isotropic. We fix a Witt-basis $(e_{1},e_{0},e_{-1})$ for $(V,h)$ such that $e_{1},e_{-1}\in V_{2}$. The lattice sequence $\Lambda$, upto $G_{\beta}$-conjugation, is given by one of the following two lattice sequences:

[TABLE]

or

[TABLE]

If $F/F_{0}$ is an unramified extension and $(V_{2},h)$ is anisotropic, we fix vectors $v_{1}\in V_{1}$, $v_{2},v_{3}\in V_{2}$ such that $(v_{1},v_{2},v_{3})$ is an orthogonal basis for $V$ and

[TABLE]

The lattice sequence $\Lambda$, upto $G_{\beta}$-conjugation, is given by the following lattice sequence with

[TABLE]

Using Lemma 5.1.1, there exists a Witt-basis $(e_{1},e_{0},e_{-1})$ of $V$ with $e_{1},e_{-1}\in\langle v_{1},v_{3}\rangle$ such that $(e_{1},e_{0},e_{-1})$ provides a splitting for the lattice sequence $\Lambda$, and we have

[TABLE]

6.1.2. The ramified case

In this part, we assume that the extension $F/F_{0}$ is ramified. If $(V_{2},h)$ is isotropic, we fix a Witt-basis $(e_{1},e_{-1})$ for $(V_{2},h)$ and an unit vector $e_{0}\in V_{1}$ such that $(e_{1},e_{0},e_{-1})$ is a Witt-basis for $(V,h)$. The lattice sequence $\Lambda$, upto $G_{\beta}$-conjugation, is given by the following lattice sequence:

[TABLE]

Now assume that $(V_{2},h)$ is anisotropic, and fix an orthogonal basis $(v_{1},v_{2},v_{3})$ of $(V,h)$ such that $v_{1}\in V_{1}$ and $v_{2},v_{3}\in V_{2}$, $h(v_{i},v_{i})\in\mathfrak{o}_{F}^{\times}$, and the hermitian space $(\langle v_{1},v_{3}\rangle,h)$ is isotropic. We denote by $\lambda_{i}$ the constant $h(v_{i},v_{i})$, for $1\leq i\leq 3$. Up to $G_{\beta}$-conjugacy, the lattice sequence $\Lambda$ is given by the following lattice sequence:

[TABLE]

There exists a Witt-basis $(e_{1},e_{0},e_{-1})$ for the space $(V,h)$ with $e_{1},e_{-1}\in\langle v_{1},v_{3}\rangle$ such that

[TABLE]

The groups $J^{0}(\Lambda,\beta)$ and $H^{1}(\Lambda,\beta)$ are given by

[TABLE]

and

[TABLE]

respectively.

6.2. The case where

$(V_{2},h)$ is anisotropic

In this subsection, we assume that the hermitian space $(V_{2},h)$ is anisotropic. To show that any representation $\pi\in\Pi_{\mathfrak{x}}$ is non-generic, we will prove that the character $\psi_{\beta}^{g}$ is non-trivial on $U_{\rm der}\cap P_{(n/2)+}(\Lambda)$, for all $g\in P(\Lambda)$.

Lemma 6.2.1**.**

Let $F/F_{0}$ be any quadratic extension and let $\mathfrak{x}$ be a strata of type (C) such that $(V_{2},h)$ is anisotropic. Every cuspidal representation in the set $\Pi_{\mathfrak{x}}$ is non-generic.

Proof.

We recall, from Subsection 6.1 the two $F$-basis $(v_{1},v_{2},v_{3})$ and $(e_{1},e_{0},e_{-1})$ for the vector space $V$, when $(V_{2},h)$ is anisotropic. Note that $\Lambda$ is uniquely determined by $\beta$, and recall the description from Subsection 6.1. Let $\operatorname{{\bf U}}$ be the unipotent radical of a Borel subgroup $\operatorname{{\bf B}}$ of $\operatorname{{\bf G}}$ such that $U$ fixes the vector $e_{1}$. Since $(e_{1},e_{0},e_{-1})$ provides a splitting for the lattice $\Lambda$ and $P^{0}(\Lambda)$ is a special parahoric subgroup, we get that $G=P(\Lambda)B$. Let $\pi$ be a cuspidal representation in the set $\Pi_{\mathfrak{x}}$. Then we have $\pi\simeq\operatorname{ind}_{J^{0}(\Lambda,\beta)}^{G}\kappa$, where $(J^{0}(\Lambda,\beta),\kappa)$ is a Bushnell–Kutzko type contained in $\pi$. Now, assume that $\pi$ is generic. Then there exists a $g\in P(\Lambda)$ and a non-trivial character $\Psi$ of $U$ such that

[TABLE]

First consider the case where $F/F_{0}$ is a ramified extension. Let $ge_{1}=av_{1}+bv_{2}+cv_{3}$, for some $a,b,c\in\mathfrak{o}_{F}$. Since $e_{1}$ is isotropic, we get that

[TABLE]

Since $-\lambda_{2}\lambda_{3}^{-1}\not\in{\rm Nr}_{F/F_{0}}(F^{\times})$, the above equality implies that $a\in\mathfrak{o}_{F}^{\times}$. Let $\nu_{\Lambda}(\beta)=-n=-(4m+2r)$, for some integer $m$ and $r\in\{0,1\}$. Since $\sigma(\beta)=-\beta$, we get that $r=1$. Now, $(n/2)+=2m+2$ and we have $P_{(n/2)+}(\Lambda)\cap U_{\rm der}$ is equal to $U_{\rm der}([(m+1)/2])$. We also have

[TABLE]

As $\nu_{F/F_{0}}(h(ge_{1},ge_{1})\leq-[(m+1)/2]$, for any $m\geq 1$, we get that $\psi_{\beta}^{g}$ is a non-trivial character on the group $P_{(n/2)+}(\Lambda)\cap U_{\rm der}$. This is a contradiction to the equation (6.1). Hence, any representation in the set $\Pi_{\mathfrak{x}}$ is non-generic.

Consider the case where $F/F_{0}$ is unramified. Since the isotropic vector $e_{1}$ belongs to the lattice $\Lambda(1)$, we get that $ge_{1}=av_{1}+bv_{2}+cv_{3}$, for some $a,c\in\mathfrak{p}_{F}$ and $b\in\mathfrak{p}_{F}$, with

[TABLE]

By a change of variable: $b^{\prime}=\varpi b$, we have

[TABLE]

Since $\nu_{\Lambda}(e_{1})=1$, the above equality implies that $a,c\in\mathfrak{o}_{F}^{\times}$. Now, we have

[TABLE]

We note that $2\nu_{F}(\beta_{1})=\nu_{\Lambda_{1}}(\beta_{1})$ and $\nu_{F}(\beta_{2})=\nu_{\Lambda_{2}}(\beta_{2})$. We also have

[TABLE]

Assume that $-n=\nu_{\Lambda_{1}}(\beta_{1})\leq\nu_{\Lambda_{2}}(\beta_{2})$. In this case, $n=4m+2r$, where $m$ is an integer and $r\in\{0,1\}$. Now, we have $(n/2)+=2m+r+1$ and the group $P_{(n/2)+}(\Lambda)\cap U_{\rm der}$ is equal to $U_{\rm der}(m)$. We may have two possibilities either $\nu_{\Lambda_{2}}(\beta_{2})\leq\nu_{F}(\beta_{1})=-(2m+r)$ or $\nu_{\Lambda_{2}}(\beta_{2})\geq\nu_{F}(\beta_{1})=2m+r$. In the first case, we have

[TABLE]

In the second case, we have

[TABLE]

Hence, the character $\psi_{\beta^{g}}$ on $P_{(n/2)+}(\Lambda)$ is non-trivial on $P_{(n/2)+}(\Lambda)\cap U_{\rm der}$ and we obtain a contradiction to the equation (6.1).

Assume that $-n=\nu_{\Lambda_{2}}(\beta_{2})\leq\nu_{\Lambda_{1}}(\beta_{1})$ and set $n=4m+r$, for some integer $m$ and $0\leq r\leq 3$. In this case, the group $P_{(n/2)+}(\Lambda)\cap U_{\rm der}$ is equal to $U_{\rm der}(m)$. Since $\nu_{\Lambda_{1}}(\beta)\leq\nu_{F}(\beta_{1})$, we get that

[TABLE]

The above inequality implies that the character $\psi_{\beta}^{g}$ is non-trivial on the group $P_{(n/2)+}(\Lambda)\cap U_{\rm der}$ and hence we get a contradiction to the equation (6.1). In every case, we get that the representation $\pi$ is non-generic. ∎

6.3. $(V_{2},h)$ is isotropic

In this part, we assume that $(V_{2},h)$ is isotropic. Note that the set $\mathfrak{X}_{\beta}(F_{0})$ is non-empty. However, it turns out that every cuspidal representation in the set $\Pi_{\mathfrak{x}}$ is non-generic; the essential reason for this is that any cuspidal representation of $P_{0}(\Lambda_{\beta})/P_{1}(\Lambda_{\beta})$ is generic. The group $P_{0}(\Lambda_{\beta})/P_{1}(\Lambda_{\beta})$ is equal to $U(1,1)(k_{F}/k_{F_{0}})\times U(1)(k_{F}/k_{F_{0}})$, when $F/F_{0}$ is unramified and is equal to ${\rm SL}_{2}(k_{F})\times\{\pm 1\}$, when $F/F_{0}$ is ramified.

Let $\operatorname{{\bf B}}$ be the Borel subgroup of $\operatorname{{\bf G}}$ such that $B$ fixes the subspace $\langle e_{1}\rangle$. Let $\operatorname{{\bf U}}$ be the unipotent radical of $\operatorname{{\bf B}}$. Although $P^{0}(\Lambda)$ is a special parahoric subgroup of $G$, it is convenient to use the decomposition

[TABLE]

for Mackey-decompositions. Here, $\mathcal{I}$ is the Iwahori subgroup contained in the subgroups $P(\Lambda)$, where $\Lambda$ varies over the two (representatives for $G_{\beta}$ conjugacy classes) lattice sequences in 6.1.1 and 6.1.2 when $F/F_{0}$ is unramified and ramified respectively.

6.3.1. Shallow elements

To understand Mackey decompositions, we need to control the conjugation by elements in $\mathcal{I}$. For the present purposes, it is easy to understand the conjugation action of an element which is contained in a sufficiently small compact subgroup of $\mathcal{I}$. So, we introduce a measure of shallowness, relative to the group $P_{(n/2)+}(\Lambda)$, on the elements $u(x,y)$ and $\bar{u}(x,y)$ in $\mathcal{I}\cap U$ and $\mathcal{I}\cap U^{w}$ respectively (here, $w$ is the non-trivial element in $W_{G}$).

We begin with defining an integer $d(\mathfrak{x},w,x)$. The main purpose of the definition of $d(\mathfrak{x},w,x)$ becomes apparent in Lemma 6.3.1. Let $n=4m+2r$, for some positive integer $m$ and $r\in\{0,1\}$. Consider the case where $F/F_{0}$ is unramified. For any $x,y\in F$ such that $x\sigma(x)+y+\sigma(y)=0$, $w\in W_{G}$ and $\Lambda$ a lattice sequence defined in 6.1.1 or in 6.1.2, we set $d(\mathfrak{x},w,x)$ to be

[TABLE]

If $F/F_{0}$ is ramified, then the lattice sequences $\Lambda$, defined in 6.1.2, up to $G_{\beta}$ conjugacy, is the unique lattice sequence such that $P^{0}(\Lambda_{F[\beta]})$ is a maximal parahoric subgroup in $G_{\beta}$. Note that $\Lambda(0)=\mathfrak{o}_{F}e_{1}\oplus\mathfrak{p}_{F}e_{-1}$. We define the integer $d(\mathfrak{x},w,x)$ as follows:

[TABLE]

Note that $d(\mathfrak{x},w,x)$ is a constant for $\nu_{F}(x)>>0$, and we denote this constant by $d(\mathfrak{x},w)$. For example, when $F/F_{0}$ is unramified, $\Lambda(0)=\mathfrak{o}_{F}e_{1}\oplus\mathfrak{p}_{E}e_{-1}$, and $w\neq\operatorname{id}$ we have $d(\mathfrak{x},w)=2$. If $F/F_{0}$ is ramified and $w\neq\operatorname{id}$, we have $d(\mathfrak{x},w)=1$.

6.3.2.

With these preliminaries we are ready to prove that any representation in the set $\Pi_{\mathfrak{x}}$ is non-generic. Let $\overline{\operatorname{{\bf U}}^{w}}$ be the unipotent radical of the opposite Borel subgroup of $\operatorname{{\bf B}}^{w}$ with respect to the torus $\operatorname{{\bf T}}$, for all $w\in W_{G}$. Recall that $\operatorname{{\bf T}}$ is the maximal $F_{0}$-split torus of $\operatorname{{\bf G}}$ such that $T$ stabilises the decomposition

[TABLE]

Lemma 6.3.1**.**

Let $F/F_{0}$ be any quadratic extension and let $\mathfrak{x}$ be a skew semisimple strata of type (C) such that $(V_{2},h)$ is isotropic. Let $u^{-}=\bar{u}(x,y)$ be an element of $\mathcal{I}\cap\overline{U^{w}}$, then we have

[TABLE]

Proof.

We prove the lemma in the case where $w=\operatorname{id}$; the case where $w\neq\operatorname{id}$ is entirely similar. We have to show that $\{U_{{\rm der}}(d(\mathfrak{x},\operatorname{id},x))\}^{u^{-}}$ is contained in the group $H^{1}(\Lambda,\beta)$. First we note the following matrix identity

[TABLE]

Using the equality (6.4), for any element $u(0,a)\in U_{{\rm der}}(d(\mathfrak{x},\operatorname{id},x))$, we get that

[TABLE]

The lemma now follows because we have

[TABLE]

∎

Lemma 6.3.2**.**

With the same assumptions and notations as in Lemma 6.3.1, for any skew semisimple character $\theta\in\mathcal{C}(\Lambda,0,\beta)$ of the group $H^{1}(\Lambda,\beta)$, we have

[TABLE]

Proof.

Let $u^{+}=u(0,a)$ be an element in $U_{{\rm der}}(d(\mathfrak{x},w,x))$, and $u^{-}=\bar{u}(x,y)$ be any element as in Lemma 6.3.1. Assume that $u^{-}u^{+}(u^{-})^{-1}=g_{1}g_{2}$, where $g_{1}\in P_{1}(\Lambda_{F[\beta]})$ and $g_{2}\in P_{(n/2)+}(\Lambda)$. From the matrix identity 6.4 and from the definition of $d(\mathfrak{x},\operatorname{id},x)$, the constant $1+ax\sigma(x)$ belongs to $F^{\times}\cap P_{(n/2)+}(\Lambda)$. Hence, we get that ${\bf 1}_{V_{1}}g{\bf 1}_{V_{1}}\in F^{\times}\cap P_{(n/2)+}(\Lambda)$. This implies that the determinant of the element ${\bf 1}_{V_{2}}g_{2}{\bf 1}_{V_{2}}$ is contained in $F^{\times}\cap P_{(n/2)+}(\Lambda)$. Now, from the definition of a simple character, we get that $\theta(u^{-}u^{+}(u^{-})^{-1})$ is equal to $\psi_{\beta}(u^{-}u^{+}(u^{-})^{-1})$ and we get the lemma. ∎

Lemma 6.3.3**.**

With the same assumptions in the lemma 6.3.1, the restriction

[TABLE]

is equivalent to a direct sum of non-trivial characters.

Proof.

We essentially follow ideas from [15, Theorem 2.6] and [4, Theorem 4.3]. We prove this lemma in the case where $w=\operatorname{id}$, and the other case is similar. If $F/F_{0}$ is unramified, then $P_{0}(\Lambda)/P_{1}(\Lambda)$ is isomorphic to

[TABLE]

If $F/F_{0}$ is ramified, then $P_{0}(\Lambda)/P_{1}(\Lambda)$ is isomorphic to

[TABLE]

Let $\tilde{J}_{1}$ be the group $(J^{0}(\Lambda,\beta)\cap U)J^{1}(\Lambda,\beta)$ and observe that we have $\tilde{J}_{1}$ is equal to $(J^{0}(\Lambda,\beta)\cap U_{\text{der}})J^{1}(\Lambda,\beta)$. Note that the image of $J^{0}(\Lambda,\beta)\cap U$ in the quotient $P_{0}(\Lambda)/P_{1}(\Lambda)$ is its $p$-Sylow subgroup. Let $\tilde{H}_{1}$ be the group $(J^{0}(\Lambda,\beta)\cap U)H^{1}(\Lambda,\beta)$.

We observe that $\psi_{\beta}(u)=1$, for all $u\in U$. Let $u$ be an element of $H^{1}(\Lambda,\beta)\cap U$. Write $u=g_{2}g_{1}g_{1}^{\prime}$ where $g_{2}\in P_{(n/2)+}(\Lambda)$, $g_{2}\in P_{1}(\Lambda_{\beta_{1}})\cap U(V_{2},h)$, and $g_{1}^{\prime}\in P_{1}(\Lambda_{\beta_{1}})\cap U(V_{1},h)$. Since $g_{1}^{\prime}e_{1}=e_{1}$, we get that $g_{2}g_{1}e_{1}=e_{1}$. Hence, we get that $g_{2}g_{1}\in U$ and $g_{1}^{\prime}=\operatorname{id}$. Now, the determinant of $g_{2}^{-1}$ and $g_{1}$ are the same. With this observation, we get that

[TABLE]

Let $\theta$ be a skew semisimple character of $H^{1}(\Lambda,\beta)$ and $\eta$ be a Heisenberg lift of $\theta$ to the group $J^{1}(\Lambda,\beta)$. Using the equality (6.5), we define a character $\tilde{\theta}$ of the group $\tilde{H}_{1}$ by setting:

[TABLE]

Using [15, Lemma 2.5], we get that the representation $\operatorname{ind}_{\tilde{H}_{1}\cap J^{1}(\Lambda,\beta)}^{J^{1}(\Lambda,\beta)}\tilde{\theta}$ is isomorphic to $\eta$.

Note that the group $J^{1}(\Lambda,\beta)\cap U_{\text{der}}$ is equal to $H^{1}(\Lambda,\beta)\cap U_{\text{der}}$ and we get that the representation

[TABLE]

This implies that $\eta$ extends as a representation of $\tilde{J}_{1}$ such that $J^{1}(\Lambda,\beta)\cap U_{\text{der}}$ acts trivially on this extension; let us denote this extension by $\tilde{\eta}$. By Frobenius reciprocity we get a map

[TABLE]

The representation $\eta$ is irreducible. The dimension of the representation $\operatorname{ind}_{\tilde{H}_{1}}^{\tilde{J}_{1}}\tilde{\theta}$ is equal to $[J^{1}(\Lambda,\beta):H^{1}(\Lambda,\beta)]^{1/2}$. Hence, the map (6.6) is an isomorphism.

Let $[\lambda^{m},n_{1},0,\beta]$ be a skew semisimple stratum such that $\tilde{J}_{1}$ is equal to $(P_{1}(\Lambda^{m}_{\beta})\cap U_{\rm der})J^{1}(\Lambda,\beta)$. Let $\theta_{m}$ be the skew semisimple character of $H^{1}(\Lambda^{m},\beta)$ obtained as a transfer from the skew semisimple character $\theta$ of $H^{1}(\Lambda^{m},\beta)$. We note that the groups $\tilde{H}_{1}$ and $H^{1}(\Lambda^{m},\beta)$ have the Iwahori decomposition with respect to the pair $(B,T)$. We then get that

[TABLE]

This implies that

[TABLE]

From the uniqueness properties of $\beta$-extensions, we get that $\operatorname{res}_{\tilde{J}_{1}}\kappa\simeq\tilde{\eta}$. This shows that the representation

[TABLE]

is a direct sum of non-trivial characters. ∎

The following lemma is a numerical verification to be used in the subsequent Lemma 6.3.5.

Lemma 6.3.4**.**

Let $F/F_{0}$ be any quadratic extension and let $x,y\in\mathfrak{o}_{F}$ such that $x\sigma(x)+y+\sigma(y)=0$. If $d(\mathfrak{x},w,x)>d(\mathfrak{x},w)$, then we have

[TABLE]

Proof.

Assume that $F/F_{0}$ is unramified and $\Lambda(0)\cap V_{2}=\mathfrak{o}_{F}e_{1}\oplus\mathfrak{o}_{F}e_{-1}$. If $d(\mathfrak{x},w,x)>d(\mathfrak{x},w)$, then we get that $\nu_{F}(x)<m$, and this implies that

[TABLE]

Now, consider the case where $\Lambda(0)\cap V_{2}=\mathfrak{o}_{F}e_{1}\oplus\mathfrak{p}_{F}e_{-1}$ and assume that $d(\mathfrak{x},w,x)>d(\mathfrak{x},w)$. If $r=0$ and $w=\operatorname{id}$, then we get that $\nu_{F}(x)<m$ which implies that

[TABLE]

If $r=0$ and $w\neq\operatorname{id}$, then we get that $\nu_{F}(x)<m-1$ and hence

[TABLE]

If $r=1$ and $w=\operatorname{id}$, then we have $\nu_{F}(x)<m+1$ and hence we get that

[TABLE]

Finally, we consider the case where $r=1$ and $w\neq\operatorname{id}$; we then have $\nu_{F}(x)<m$. Hence, we get that

[TABLE]

Now, assume that $F/F_{0}$ is a ramified extension and $d(\mathfrak{x},w,x)>d(\mathfrak{x},w)$. Note that $\nu_{F/F_{0}}(\beta_{1}-\beta_{2})=-(2m+1)$, for some $m\in\mathbb{Z}$. If $w=\operatorname{id}$, then we have $\nu_{F/F_{0}}(x)<m/2$ and hence

[TABLE]

If $w\neq\operatorname{id}$, then we have $\nu_{F/F_{0}}(x)<(m-1)/2$ and we have

[TABLE]

From the above inequalities, in all exhaustive cases, gives the required inequality:

[TABLE]

∎

Lemma 6.3.5**.**

Let $F/F_{0}$ be an unramified extension and let $\mathfrak{x}$ be a strata of type (C) such that $(V_{2},h)$ is isotropic. Every cuspidal representation in the set $\Pi_{\mathfrak{x}}$ is non-generic.

Proof.

Let $\pi$ be a cuspidal representation in the set $\Pi_{\mathfrak{x}}$. Then we have $\pi\simeq\operatorname{ind}_{J^{0}(\Lambda,\beta)}^{G}(\kappa\otimes\tau)$, where $(J^{0}(\Lambda,\beta),\kappa\otimes\tau)$ is a Bushnell–Kutzko type contained in $\pi$. Assume that the representation $\pi$ is generic. Then there exists a $g\in\mathcal{I}$, an element $w\in W_{G}$, and a character $\Psi$ of $U^{w}$ such that

[TABLE]

We write $g=p^{+}u^{-}$ such that $p^{+}\in B^{w}\cap\mathcal{I}$ and $u^{-}\in\overline{U^{w}}\cap\mathcal{I}$, where $\overline{U^{w}}$ is the unipotent radical of the opposite Borel subgroup of $B^{w}$ with respect to the torus $T$. From the expression (6.7), we get that

[TABLE]

for some character $\Psi^{\prime}$ of $U^{w}$. We set $e_{w}=we_{1}$ and $e_{-w}=we_{-1}$, we then have

[TABLE]

Hence, $\nu_{F}(\delta h(u^{-}e_{w},\beta u^{-}e_{w}))$ is equal to $\nu_{F}(\delta(\beta_{1}-\beta_{2})x\sigma(x))$. If $d(\mathfrak{x},w,x)>d(\mathfrak{x},w)$, then Lemma 6.3.4 implies that

[TABLE]

Now, using Lemmas 6.3.2 and 2.6.1, we get that

[TABLE]

But, this is a contradiction to the assumption in (6.7). Hence, we obtain $d(\mathfrak{x},w,x)=d(\mathfrak{x},w)$ and this implies that $u^{-}\in H^{1}(\Lambda,\beta)$. We may as well assume that $u^{-}=\operatorname{id}$. The lemma now follows from Lemma 6.3.3. ∎

7. Non simple type (D) strata

7.1. Inducing data

A skew semisimple stratum $[\Lambda,n,0,\beta]$, denoted by $\mathfrak{x}$, is of type (D) if the underlying splitting is of the form $V=V_{1}\perp V_{2}\perp V_{3}$ with $\dim_{F}V_{i}=1$, for $1\leq i\leq 3$. We use the notation $W_{i}$ for the space $\oplus_{j\neq i}V_{j}$, for $1\leq i\leq 3$.

When $F/F_{0}$ is unramified, we fix a vector $v_{i}\in V_{i}$ such that $\nu_{F_{0}}(h(v_{i},v_{i}))\in\{0,1\}$, for $1\leq i\leq 3$. When $F/F_{0}$ is ramified, we fix a vector $v_{i}\in V_{i}$ such that $\nu_{F_{0}}(h(v_{i},v_{i}))=0$, for $1\leq i\leq 3$. We denote by $\lambda_{i}$ the inner product $h(v_{i},v_{i})$, for $1\leq i\leq 3$. We have $\beta=\beta_{1}+\beta_{2}+\beta_{3}$, where $\beta_{i}={\bf 1}_{V_{i}}\beta{\bf 1}_{V_{i}}$, for $1\leq i\leq 3$. The lattice sequence $\Lambda$ is uniquely determined by the element $\beta$ and we have $e(\Lambda)=2$. Let $\Lambda_{i}$ be the $\mathfrak{o}_{F}$-lattice sequence $\Lambda\cap V_{i}$, for $1\leq i\leq 3$.

Since, $\mathfrak{x}$ is a skew semisimple stratum of type (D), we have

[TABLE]

Without loss of generality, we assume that

[TABLE]

Moreover, we assume that

[TABLE]

We have $e(\Lambda_{i})=2$, for $1\leq i\leq 3$, and

[TABLE]

As the lattice sequence $\Lambda$ depends on various possibilities on $V_{i}$, we will describe these lattice sequences and a Witt-basis, which gives a splitting for $\Lambda$, as required in each individual case. However, the group $J^{0}(\Lambda,\beta)$ and $H^{1}(\Lambda,\beta)$ are given by

[TABLE]

and

[TABLE]

respectively.

7.2. Criterion for non-emptiness of the set

$\mathfrak{X}_{\beta}(F_{0})$

When $F/F_{0}$ is unramified, the non-emptiness of the set $\mathfrak{X}_{\beta}(F_{0})$ depends only on the integers $\{\nu_{F}(\beta_{i}),\nu_{F_{0}}(\lambda_{i}):1\leq i\leq 3\}$. This will be made precise in the following lemmas. However, when $F/F_{0}$ is ramified, one requires more information on $\{\beta_{1},\beta_{2},\beta_{3}\}$ to determine whether the set $\mathfrak{X}_{\beta}(F_{0})$ empty or not. In the case where $F/F_{0}$ is ramified we will not make these conditions explicit, but we will show that a cuspidal representation in $\Pi_{\mathfrak{x}}$ is generic if and only if $\mathfrak{X}_{\beta}(F_{0})$ is non-empty.

Lemma 7.2.1**.**

Let $F/F_{0}$ be an unramified extension and let $\mathfrak{x}$ be a stratum of type (D) such that $(W_{i},h)$ is isotropic, for $1\leq i\leq 3$. The set $\mathfrak{X}_{\beta}(F_{0})$ is non-empty if and only if $\nu_{F}(\beta_{1})-\nu_{F}(\beta_{2})$ is an even integer.

Proof.

Since the extension $F/F_{0}$ is unramified, we may assume that $\lambda_{i}=1$, for $1\leq i\leq 3$. The set $\mathfrak{X}_{\beta}(F_{0})$ is non-empty if and only if there exists a non-zero vector $v\in V$ such that

[TABLE]

Let $v=av_{1}+bv_{2}+cv_{3}$, for some $a,b,c\in F$. From the equation (7.3), we get that

[TABLE]

Using the assumption (7.2) on the skew semisimple stratum $\mathfrak{x}$ we get that $(1-\beta_{2}\beta_{1}^{-1})\in\mathfrak{o}_{F}^{\times}$ and $(1-\beta_{3}\beta_{2}^{-1})\in\mathfrak{o}_{F}^{\times}$. The set of equations (7.4) imply that

[TABLE]

Hence, we get that $\nu_{F}(\beta_{2})-\nu_{F}(\beta_{1})$ is an even integer. Conversely, if $\nu_{F}(\beta_{2})-\nu_{F}(\beta_{1})$ is even, we find can find a non-zero tuple $(a,b,c)\in F^{3}$ satisfying the equalities in equation (7.4); therefore, the set $\mathfrak{X}_{\beta}(F_{0})$ is non-empty. ∎

Lemma 7.2.2**.**

Let $F/F_{0}$ be an unramified extension and let $\mathfrak{x}$ be a stratum of type (D) such that $(W_{i},h)$ is anisotropic, for some $i$, $1\leq i\leq 3$. The set $\mathfrak{X}_{\beta}(F_{0})$ is non-empty if and only if $\nu_{F}(\lambda_{2})=\nu_{F}(\lambda_{3})=1$ and $\nu_{F}(\beta_{1})-\nu_{F}(\beta_{2})$ is an odd integer.

Proof.

Recall that the determinant of $(V,h)$ is the trivial class in $F_{0}^{\times}/{\rm Nr}_{F/F_{0}}(F^{\times})$. With the hypothesis on the spaces $W_{i}$, there exists an unique $i\in\{1,2,3\}$ such that $\nu_{F}(\lambda_{i})=0$. The set $\mathfrak{X}_{\beta}(F_{0})$ is non-empty if and only if the following equations

[TABLE]

have a non-trivial simultaneous solution. Recall that $(1-\beta_{2}\beta_{1}^{-1})\in\mathfrak{o}_{F}^{\times}$ and $(1-\beta_{3}\beta_{2}^{-1})\in\mathfrak{o}_{F}^{\times}$. Assume that there exists a non-trivial simultaneous solution to the equations in (7.5). Then, we get that

[TABLE]

This implies that $\nu_{F}(\lambda_{2})=\nu_{F}(\lambda_{3})$. From the assumption on the spaces $W_{i}$, for $1\leq i\leq 3$, we get that $\nu_{F}(\lambda_{2})=\nu_{F}(\lambda_{3})=1$ and $\nu_{F}(\lambda_{1})=0$. From the equation (7.5) we have

[TABLE]

Hence, $\mathfrak{X}_{\beta}(F_{0})$ is non-empty if and only if $(W_{1},h)$ is isotropic and $\nu_{F}(\beta_{2})-\nu_{F}(\beta_{1})$ is an odd integer. ∎

7.3. Estimating the valuation of

$h(gv,\beta gv)$

As observed in the previous sections, our approach to show non-genericity is by showing an appropriate inequality on the function sending $g\in P(\Lambda)$ to $\nu_{F}(h(gv,gv))$, where $v$ is a well chosen isotropic vector with respect to $P(\Lambda)$. Hence, we need some technical lemmas to understand the growth of this function.

Lemma 7.3.1**.**

Let $F/F_{0}$ be an unramified extension and let $\mathfrak{x}$ be a skew semisimple stratum of type (D) such that $\mathfrak{X}_{\beta}(F_{0})$ is the empty set. Let $v$ be an isotropic vector in $V$, and let $g\in G$. Assume that $gv=av_{1}+bv_{2}+cv_{3}$ for some $a,b,c\in F$. If $a\neq 0$ and $\beta_{2}\lambda_{2}b\sigma(b)+\beta_{3}\lambda_{3}c\sigma(c)\neq 0$, then we have

[TABLE]

Remark 7.3.2**.**

Since the set $\mathfrak{X}_{\beta}(F_{0})$ is the empty-set, we get that $h(v,\beta v)\neq 0$, for all isotropic vectors $v\in V$.

Proof of Lemma 7.3.1.

Before we begin the proof, it is useful to recall that the determinant of $(V,h)$ is the trivial class in $F_{0}^{\times}/{\rm Nr}_{F/F_{0}}(F^{\times})$. Since $v$ is an isotropic vector, we get that

[TABLE]

Observe that rescaling the constants $a,b,c$ does not effect the lemma. Hence, rescaling $a$, $b$ and $c$, if necessary, we assume that $a,b,c\in\mathfrak{o}_{F}$ and the $\mathfrak{o}_{F}$-ideal $(a,b,c)$ is equal to $\mathfrak{o}_{F}$.

First consider the case where $(W_{i},h)$ is isotropic, for $1\leq i\leq 3$. In this case we have $\nu_{F_{0}}(\lambda_{i})=0$, for $1\leq i\leq 3$ and $\nu_{F}(\beta_{1})-\nu_{F}(\beta_{2})$ is an odd integer. If $\nu_{F}(a)=0$, we get that

[TABLE]

Thus we prove the lemma in the case where $\nu_{F}(a)=0$. Consider the case where $a\in\mathfrak{p}_{F}$; we necessarily have $b,c\in\mathfrak{o}_{F}^{\times}$. We have

[TABLE]

Since $\mathfrak{x}$ is a skew semisimple stratum, we get that $1-\beta_{3}\beta_{2}^{-1}\in\mathfrak{o}_{F}^{\times}$. This implies that $\nu_{F}(\lambda_{2}b\sigma(b)+\lambda_{3}\beta_{3}\beta_{2}^{-1}c\sigma(c))=0$. Observe that $\nu_{F}(a\sigma(a))$ is an even integer. Therefore, we conclude that

[TABLE]

and this proves the lemma in the case where $(W_{i},h)$ is isotropic, for $1\leq i\leq 3$.

Assume that $(W_{i},h)$ is anisotropic for some $1\leq i\leq 3$. Using Lemma 7.2.2, the set $\mathfrak{X}_{\beta}(F_{0})$ is empty in either of the following cases: case (I) where $\nu_{F}(\lambda_{2})\neq\nu_{F}(\lambda_{3})$, case (II) where $\nu_{F}(\lambda_{2})=\nu_{F}(\lambda_{3})$ and $\nu_{F}(\beta_{1})-\nu_{F}(\beta_{2})$ is an even integer. We first assume that $\nu_{F_{0}}(\lambda_{2})=0$, and we get that $\nu_{F_{0}}(\lambda_{1})=\nu_{F_{0}}(\lambda_{3})=1$. We may also assume that $\lambda_{2}=1$ and $\lambda_{1}=\lambda_{3}=\varpi$. Using equation (7.6), we get that $b\in\mathfrak{p}_{F}$, and set $b=\varpi b^{\prime}$, for some $b^{\prime}\in\mathfrak{o}_{F}$. Since we have

[TABLE]

we get that $a,c\in\mathfrak{o}_{F}^{\times}$. We now have

[TABLE]

and note that

[TABLE]

Hence, we get that $\nu_{F}(h(ge_{1},ge_{1}))=\nu_{F}(\beta_{1})+1$. Note that $\nu_{F}(\beta_{2}\varpi^{2}b^{\prime}\sigma(b^{\prime})+\beta_{3}\varpi c\sigma(c))$ is equal to $\nu_{F}(\beta_{2})+1$. Hence, the lemma follows in case (I), from the observation that

[TABLE]

The case where $\nu_{F}(\lambda_{2})=1$–in which case $\nu_{F}(\lambda_{1})=\nu_{F}(\lambda_{2})=1$ and $\nu_{F}(\lambda_{3})=0$–is entirely similar.

Assume that we are in case (II). In this case we may assume that $\lambda_{2}=\lambda_{3}=\varpi$ and $\lambda_{1}=1$. Then the equation (7.6) implies that $b,c\in\mathfrak{o}_{F}^{\times}$ and $a\in\mathfrak{p}_{F}$. We now have

[TABLE]

Since $\mathfrak{x}$ is a skew semisimple strata and $b,c\in\mathfrak{o}_{F}^{\times}$, we get that $\nu_{F}((b\sigma(b)+\beta_{3}\beta_{2}^{-1}c\sigma(c)))=0$. The lemma now follows from the observation that the integer $\nu_{F}(\lambda_{1}a\bar{a})$ is even and the integer $\nu_{F}(\beta_{2}\beta_{1}^{-1}\varpi_{F_{0}}(b\sigma(b)+\beta_{3}\beta_{2}^{-1}c\sigma(c)))$ is always odd. ∎

Lemma 7.3.3**.**

Let $F/F_{0}$ be a ramified extension and let $\mathfrak{x}$ be a stratum of type (D) such that $\mathfrak{X}_{\beta}(F_{0})$ is the empty set. Let $v$ be any isotropic vector in $V$ and let $g\in G$. Assume that $gv=av_{1}+bv_{2}+cv_{3}$ for some $a,b,c\in F$. If $a\neq 0$ and $\beta_{2}\lambda_{2}b\sigma(b)+\beta_{3}\lambda_{3}c\sigma(c)\neq 0$, then we have

[TABLE]

Proof.

Since $v$ is an isotropic vector, we get that

[TABLE]

Rescaling the constants $a$, $b$, and $c$, if necessary, we assume that $a,b,c\in\mathfrak{o}_{F}$ and the $\mathfrak{o}_{F}$-ideal $(a,b,c)$ is equal to $\mathfrak{o}_{F}$.

Since $\mathfrak{x}$ is a skew semisimple stratum, using the assumptions in (7.2), we get that $(1-\beta_{i}\beta_{j}^{-1})\in\mathfrak{o}_{F}^{\times}$, for $1\leq j<i\leq 3$. The set $\mathfrak{X}_{\beta}(F_{0})$ is empty in one of the two cases

1. (1)

The case where

[TABLE] 2. (2)

The case where

[TABLE]

and

[TABLE]

In Case $(1)$, unless $b,c\in\mathfrak{p}_{F}$, we have

[TABLE]

Hence, we get that

[TABLE]

Now consider Case $(2)$ and assume that

[TABLE]

if the condition (7.10) is false we get the lemma immediately. If $\nu_{F}(\beta_{1})=\nu_{F}(\beta_{2})$, then we have

[TABLE]

But, the above containment is a contradiction to the second condition in case $(2)$, the equation (7.8). Hence, we get that $\nu_{F}(\beta_{1})\neq\nu_{F}(\beta_{2})$. If $a\in\mathfrak{o}_{F}^{\times}$, the valuation of $h(gv,\beta gv)$ is equal to

[TABLE]

Note that the lemma follows from the observation that

[TABLE]

We consider the case where $a\in\mathfrak{p}_{F}$ and this implies that $b,c\in\mathfrak{o}_{F}^{\times}$. Thus we have $\lambda_{2}b\sigma(b)+\beta_{3}\beta_{2}^{-1}\lambda_{3}c\sigma(c)\in\mathfrak{o}_{F}^{\times}$. Suppose

[TABLE]

then we have

[TABLE]

Using the equality (7.7), we get that

[TABLE]

This is a contradiction to the condition (7.8). Hence, we obtain

[TABLE]

∎

7.4. Generic cuspidal representations of type

(D)

Lemma 7.4.1**.**

Let $F/F_{0}$ be any quadratic extension and let $\mathfrak{x}$ be a stratum of type (D) such that $\mathfrak{X}_{\beta}(F_{0})$ is non-empty. Every cuspidal representation contained in the set $\Pi_{\mathfrak{x}}$ is generic.

Proof.

Let $\theta$ be any semisimple character in $\mathcal{C}(\Lambda,0,\beta)$. Let $\operatorname{{\bf U}}$ be the unipotent radical of a Borel subgroup in the set $\mathfrak{X}_{\beta}(F_{0})$. We will first show that

[TABLE]

If $q_{1}=q_{2}$, then the group $H^{1}(\Lambda,\beta)$ is equal to $P_{1}(\Lambda_{\beta})P_{q_{1}/2}(\Lambda)$ and from Lemma 3.5.1, we get that $H^{1}(\Lambda,\beta)\cap U$ is equal to $P_{q_{1}/2+}(\Lambda)\cap U$. In the case where $q_{1}>q_{2}$, the group $H^{1}(\Lambda,\beta)$ is equal to

[TABLE]

Let $H^{\prime}$ be the group $P_{1}(\Lambda_{\beta})P_{(q_{2}/2)+}(\Lambda)$. From Lemma 3.5.1 we get that the group $H^{\prime}\cap U$ is equal to $P_{(q_{2}/2)+}(\Lambda)\cap U$ and hence, $H^{1}(\Lambda,\beta)\cap U$ is equal to $P_{(q_{2}/2)+}(\Lambda_{\beta_{1}})P_{(q_{1}/2)+}(\Lambda)\cap U$.

Let $v=av_{1}+bv_{2}+cv_{3}$ be an isotropic vector fixed by $U$. Since $\mathfrak{x}$ is a skew semisimple stratum of type (D), we get that $a\neq 0$. Assume that $g_{1}g_{2}\in U$, for some $g_{1}\in P_{(q_{2}/2)+}(\Lambda_{\beta_{1}})$ and $g_{2}\in P_{(q_{1}/2)+}(\Lambda)$. From the equality

[TABLE]

we get that

[TABLE]

Which implies that the determinant of ${\bf 1}_{W_{1}}g_{1}{\bf 1}_{W_{1}}$ belongs to $F^{\times}\cap P_{(q_{1}/2)+}(\Lambda)$. Hence, the simple character $\theta(g_{1}g_{2})$ is equal to $\psi_{\beta}(g_{1}g_{2})$. Now, the lemma follows from Proposition 3.5.2. ∎

7.5. Non-generic cuspidal

representations of type (D)

We will show that $\pi\in\Pi_{\mathfrak{x}}$ is non-generic if and only if the set $\mathfrak{X}_{\beta}(F_{0})$ is empty. We will devide the proof into several cases beginning with the easier case where $(W_{1},h)$ is anisotropic; in which case we will show that $\psi_{\beta}^{g}$ is non-trivial on $P_{(n/2)+}(\Lambda)\cap U_{\rm der}$, for all $g\in P(\Lambda)$. In the case where $(W_{1},h)$ is isotropic, the method of proof is more involved and we had to deal with conjugation of some shallow elements in the group $P(\Lambda)$.

7.5.1. The case where $(W_{1},h)$ is

anisotropic

Lemma 7.5.1**.**

Let $F/F_{0}$ be a quadratic extension and let $\mathfrak{x}$ be a skew semisimple stratum of type (D) such that $(W_{1},h)$ is anisotropic. If the set $\mathfrak{X}_{\beta}(F_{0})$ is empty, then every representation in the set $\Pi_{\mathfrak{x}}$ is non-generic.

Proof.

Let us begin with the case where $F/F_{0}$ is ramified. In this case, $e(\Lambda)=2$, and we have

[TABLE]

It is possible that neither of the spaces $(W_{2},h)$ and $(W_{3},h)$ are isotropic. However, there exists an orthogonal $\mathfrak{o}_{F}$-basis, $(\tilde{v}_{2},\tilde{v}_{3})$, for the lattice $\mathfrak{o}_{F}v_{2}\oplus\mathfrak{o}_{F}v_{3}$ such that $\langle v_{1},\tilde{v}_{2}\rangle$ is isotropic. Using arguments similar to Lemma 5.1.2, we can we choose a Witt-basis $(e_{1},e_{-1})$ of $\langle v_{1},\tilde{v}_{2}\rangle$ and an unit vector $e_{0}\in\langle v_{1},\tilde{v}_{2}\rangle^{\perp}$ such that the Witt-basis $(e_{1},e_{0},e_{-1})$ provides a splitting for the lattice sequence $\Lambda$. In the basis $(e_{1},e_{0},e_{-1})$, we have

[TABLE]

Let $\operatorname{{\bf U}}$ be the unipotent radical of the Borel subgroup, $\operatorname{{\bf B}}$, such that $\langle e_{1}\rangle$ is fixed by $B$. We set

[TABLE]

for some integer $m$. Assume that $\pi\in\Pi_{\mathfrak{x}}$ is a generic representation. Let $(J^{0}(\Lambda,\beta),\kappa)$ be a Bushnell–Kutzko type contained in $\pi$. There exists a $g\in P(\Lambda)$, and a non-trivial character $\Psi$ of $U$ such that

[TABLE]

Since $g$ normalises $P_{r}(\Lambda)$, for $r>0$, we get that the group $P_{(n/2)+}(\Lambda)\cap U_{\rm der}$ is equal to $U_{\rm der}(\lceil m/2\rceil)$. Let $ge_{1}=av_{1}+bv_{2}+cv_{3}$ for some $a,b,c\in\mathfrak{o}_{F}$. Then we have

[TABLE]

If $a\in\mathfrak{p}_{F}$, then $b,c\in\mathfrak{p}_{F}$ as $(W_{1},h)$ is anisotropic. Since $\nu_{\Lambda}(e_{1})=0$, we get that $a\in\mathfrak{o}_{F}^{\times}$. Now, we have

[TABLE]

If $\nu_{F}(\beta_{2})>\nu_{F}(\beta_{1})$, then we get that $\nu_{F}(h(ge_{1},\beta ge_{1}))$ is equal to $\nu_{F}(\beta_{1})$. Assume that $\nu_{F}(\beta_{1}=\nu_{F}(\beta_{2})$. Since, both the constants $b,c$ cannot be in $\mathfrak{p}_{F}$, without loss of generality, we assume that $b\in\mathfrak{o}_{F}^{\times}$. If

[TABLE]

then we get that

[TABLE]

therefore, we get a contradiction to the assumption that the set $\mathfrak{X}_{\beta}(F_{0})$ is empty. Hence, we have

[TABLE]

Thus we get that

[TABLE]

Now, Lemma 2.6.1 implies that the character $\psi_{\beta}^{g}$ is non-trivial on the group $P_{(n/2)+}(\Lambda)\cap U_{\rm der}$, and we get a contradiction to the assumption that $\pi$ is generic.

Consider the case where $F/F_{0}$ is unramified. In this case, we may assume that $\lambda_{1}=\varpi$ and $(\lambda_{2},\lambda_{3})\in\{(\varpi,1),(1,\varpi)\}$. So we define $\tilde{v}_{3}$ to be the vector in the set $\{v_{2},v_{3}\}$ with $h(\tilde{v}_{3},\tilde{v}_{3})=\varpi$, and the remaining vector in the set $\{v_{2},v_{3}\}$ is denoted by $\tilde{v}_{2}$. The notation $\tilde{\beta}_{i}$ will be used for ${\bf 1}_{\langle\tilde{v}_{i}\rangle}\beta{\bf 1}_{\langle\tilde{v}_{i}\rangle}$, for $i\in\{2,3\}$. The period $2$ lattice sequence $\Lambda$ is given by

[TABLE]

Let $e_{0}$ be the vector $\tilde{v}_{2}$. Since the space $\langle v_{1},\tilde{v}_{3}\rangle$ is isotropic, and there exists a Witt-basis $(e_{1},e_{-1})$ for the space $\langle v_{1},\tilde{v}_{3}\rangle$ such that

[TABLE]

Let $\operatorname{{\bf U}}$ be the unipotent radical of the Borel subgroup, $\operatorname{{\bf B}}$, such that $\langle e_{1}\rangle$ is fixed by $B$. Assume that $\pi\in\Pi_{\mathfrak{x}}$, containing a Bushnell–Kutzko type $(J^{0}(\Lambda,\beta),\kappa)$, is a generic representation. There exists a $g\in P(\Lambda)$ and a non-trivial character $\Psi$ of $U$ such that

[TABLE]

Note that $\nu_{\Lambda}(e_{1})=1$ and hence we get that $ge_{1}=av_{1}+\varpi b\tilde{v}_{2}+c\tilde{v}_{3}$, for some $a,b,c\in\mathfrak{o}_{F}$. Since $e_{1}$ is an isotropic vector, we get that

[TABLE]

From the above equality and the fact that $\nu_{\Lambda}(e_{1})=1$, we get that $a,c\in\mathfrak{o}_{F}^{\times}$. We set $n=4m+2r$, for some integer $m$ and $r\in\{0,1\}$. The valuation of $h(ge_{1},\beta ge_{1})$ is equal to

[TABLE]

We observe that $g$ normalises the group $P_{(n/2)+}(\Lambda)$; therefore, we get that $P_{(n/2)+}(\Lambda)^{g}\cap U_{\rm der}$ is equal to $U_{\rm der}(m)$. Since $\nu_{F}(h(ge_{1},\beta ge_{1}))\leq-m$, using Lemma 2.6.1, we get that the character $\psi_{\beta}^{g}$ is non-trivial on the group $P_{(n/2)+}(\Lambda)^{g}\cap U_{\rm der}$. Thus we obtain a contradiction to the assumption on the genericity of the representation $\pi\in\Pi_{\mathfrak{x}}$. ∎

7.5.2. The case where $(W_{1},h)$ is

isotropic

Lattice sequences:

We begin with the description of the lattice sequence $\Lambda$. Let $(e_{1},e_{-1})$ be a Witt-basis for the space $(W_{1},h)$, and $e_{0}$ be an unit vector in $V_{1}$. Let $\operatorname{{\bf U}}$ be the unipotent radical of the Borel subgroup, $\operatorname{{\bf B}}$, such that $\langle e_{1}\rangle$ is fixed by $B$. Let $\operatorname{{\bf T}}$ be the maximal torus of $\operatorname{{\bf G}}$, contained in $\operatorname{{\bf B}}$ such that $T$ fixes the decomposition

[TABLE]

If $F/F_{0}$ is unramified and $h(v_{2},v_{2})=h(v_{3},v_{3})=\varpi$, then $e(\Lambda)=2$ and

[TABLE]

If $F/F_{0}$ is unramified and $h(v_{2},v_{2})=h(v_{3},v_{3})=1$, then $e(\Lambda)=2$ and

[TABLE]

If $F/F_{0}$ is ramified, then $e(\Lambda)=2$ and

[TABLE]

Let $\mathcal{I}$ be the Iwahori subgroup defined, in its matrix form with respect to the basis $(e_{1},e_{0},e_{-1})$, by

[TABLE]

Recall that we use the notation $q_{i}$ for the integer $-\nu_{\Lambda_{i}}(\beta_{i})$, for $1\leq i\leq 3$. We set $q_{1}=4m_{1}+2r_{1}$ and $q_{2}=4m_{2}+2r_{2}$, for some integers $m_{1},m_{2}$ and $r_{1},r_{2}\in\{0,1\}$. Also recall our convention that $q_{1}\geq q_{2}$.

Shallow elements:

For the purpose of understanding linear functionals supported on $J^{0}(\Lambda,\beta)uU$, for $u\in\mathcal{I}$, we need a certain measure of depth of an element $u$ with respect to the group $P_{(n/2)+}(\Lambda)$. Let $x,y\in\mathfrak{o}_{F}$ be two elements such that $x\sigma(x)+y+\sigma(y)=0$, and let $w$ be an element in $W_{G}$. The function $d(\mathfrak{x},w,x)$, defined below, measures the depth of an element $u(x,y)^{w}$ with respect to $P_{(n/2)+}(\Lambda)$. When $F/F_{0}$ is an unramified extension, we set

[TABLE]

When $F/F_{0}$ is a ramified extension, the lattice sequences $\Lambda$ is uniquely determined (see subsection 7.5.2). We define the integer $d(\mathfrak{x},w,x)$ as follows:

[TABLE]

Lemma 7.5.2**.**

Let $\mathfrak{x}$ be a skew semisimple stratum of type (D). Let $w$ be an element of $W_{G}$. Let $u=\dot{u}(x,y)$ be an element of $\mathcal{I}\cap\overline{U^{w}}$, where $\overline{\operatorname{{\bf U}}^{w}}$ is the unipotent radical of the opposite Borel subgroup of $\operatorname{{\bf B}}^{w}$ with respect to $\operatorname{{\bf T}}$. We have

[TABLE]

Proof.

We will prove the above lemma when $w=\operatorname{id}$, and the proof is entirely similar in the case where $w\neq\operatorname{id}$. The group $H^{1}(\Lambda,\beta)$ is equal to

[TABLE]

Let $u=\bar{u}(x,y)$ be an element of $\mathcal{I}\cap\bar{B}$. Using the the matrix identity

[TABLE]

and the definition of $d(\mathfrak{x},\operatorname{id},x)$, we see that the group $\{U_{{\rm der}}(d(\mathfrak{x},\operatorname{id},x))\}^{u}$ is contained in the group $H^{1}(\Lambda,\beta)$. ∎

Lemma 7.5.3**.**

With the same notations as in Lemma 7.5.2, and for any skew semisimple character $\theta$ in $\mathcal{C}(\Lambda,0,\beta)$ we have

[TABLE]

Proof.

We prove this in the case where $\Lambda\cap W_{1}$ equal to $\mathfrak{o}_{F}e_{1}\oplus\mathfrak{p}_{F}e_{-1}$ and $w\neq\operatorname{id}$. The rest of the cases are similar and are simpler. Let $u^{+}$ be the element $u(x,y)\in\mathcal{I}\cap U$, and let $u=\bar{u}(0,a)$ be an element in the group $U^{w}_{\rm der}(d(\mathfrak{x},w,x))$. Using the identity (7.13), we get that the element $u^{+}u(u^{+})^{-1}$ is of the form $g_{1}g_{2}$, where $g_{1}\in P_{(q_{2}/2)+}(\Lambda_{F[\beta_{1}]})$ and $g_{2}\in P_{(n/2)+}(\Lambda)$. From the definition of the integer $d(\mathfrak{x},w,x)$, we get that $1+ax\sigma(x)\in F^{\times}\cap P_{(n/2)+}(\Lambda)$; therefore, we get that ${\bf 1}_{V_{1}}g_{1}{\bf 1}_{V_{1}}$ belongs to $F^{\times}\cap P_{(n/2)+}(\Lambda)$. Hence, the determinant of ${\bf 1}_{W_{1}}g_{1}{\bf 1}_{W_{1}}$ belongs to $F^{\times}\cap P_{(n/2)+}(\Lambda)$. The lemma follows from the defining property, [5, Definition 3.2.3(a)], of the character $\theta$. ∎

Lemma 7.5.4**.**

Let $F/F_{0}$ be a any quadratic extension and $\mathfrak{x}=[\Lambda,n,0,\beta]$ be a stratum of type (D) such that $\mathfrak{X}_{\beta}(F_{0})$ is the empty set. Every cuspidal representation contained in the set $\Pi_{\mathfrak{x}}$ is non-generic.

Proof.

Let $\pi$ be a cuspidal representation in the set $\Pi_{\mathfrak{x}}$. Let $(J^{0}(\Lambda,\beta),\kappa)$ be a Bushnell–Kutzko’s type contained in the representation $\pi$. Assume that the representation $\pi$ is generic. Then there exists a $w\in W_{G}$, an element $u=\dot{u}(x,y)\in\mathcal{I}\cap\overline{U^{w}}$, and a non-trivial character $\Psi$ of $U$ such that

[TABLE]

In particular, the above identity implies that

[TABLE]

where $\theta$ is the skew semisimple character contained in $\operatorname{res}_{H^{1}(\Lambda,\beta)}\kappa$.

Consider the case where $F/F_{0}$ is any quadratic extension and $\Lambda\cap W_{1}$ is equal to $\mathfrak{o}_{F}e_{1}\oplus\mathfrak{o}_{F}e_{-1}$. In this case, we have $\nu_{\Lambda}(e_{w})=0$; therefore, we get that $ue_{w}=bv_{2}+xv_{1}+cv_{3}$, for some $b,c\in\mathfrak{o}_{F}$. If $\nu_{F/F_{0}}(x)=0$, then we have

[TABLE]

If $\nu_{F/F_{0}}(x)>0$, then we have $b,c\in\mathfrak{o}_{F}^{\times}$. Since $\mathfrak{x}$ is skew semisimple stratum, we have $(1-\beta_{2}\beta_{3}^{-1})\in\mathfrak{o}_{F}^{\times}$. From the assumption that $\nu_{F/F_{0}}(x)>0$, we get that $\lambda_{2}b\sigma(b)+\lambda_{3}c\sigma(c)\in\mathfrak{p}_{F}$. Hence, we have

[TABLE]

Using Lemmas 7.3.1 and 7.3.3, we get that

[TABLE]

Consider the case where $F/F_{0}$ is unramified and $\Lambda\cap W_{1}$ is equal to $\mathfrak{o}_{F}e_{1}\oplus\mathfrak{p}_{F}e_{-1}$. In this case, we have $\nu_{\Lambda}(e_{\pm 1})=\pm 1$; therefore, we get that

[TABLE]

If $w=\operatorname{id}$, we observe that $b\sigma(b)+c\sigma(c)\in\mathfrak{p}_{F}$; which together with $\nu_{F}(e_{1})=1$ implies that $b,c\in\mathfrak{o}_{F}^{\times}$. Since, $\mathfrak{x}$ is skew semisimple we have get that $b\sigma(b)+\beta_{3}\beta_{2}^{-1}c\sigma(c)\in\mathfrak{o}_{F}^{\times}$. From Lemma 7.3.1, we get that

[TABLE]

If $w\neq\operatorname{id}$, then similar arguments as above imply that $\nu_{F}(b)=\nu_{F}(c)=-1$, and $\nu_{F}(b\sigma(b)+\beta_{3}\beta_{2}^{-1}c\sigma(c))=-2$. Hence, using lemma we get that

[TABLE]

Claim 1**.**

We claim that

[TABLE]

Assuming Claim 1 we complete the proof of the lemma. Using Lemma 2.6.1, we get that the character $\psi_{\beta}^{u}$ is non-trivial on the group $U_{\rm der}(d(\mathfrak{x},w,x))$. Thus, we get a contradiction to the equation (7.14), and hence, the cuspidal representation $\pi$ is non-generic.

Proof of Claim 1.

Case 1: First consider the case where $F/F_{0}$ is unramified and

[TABLE]

The integer $\nu_{F}(h(ue_{1},ue_{1}))$ is equal to $\min\{-(2m_{2}+r_{2}),-(2m_{1}+r_{1})+2\nu_{F}(x)\}$. Recall that $-d(\mathfrak{x},w,x)$ is equal to $\min\{-m_{2}-1,-m_{1}-1+\nu_{F}(x)\}$. Observe that $-(2m_{2}+r_{2})\leq-m_{2}-1$, unless $m_{2}+r_{2}=0$; since the stratum $\mathfrak{x}$ is a skew semisimple stratum of type (D), we have $m_{2}+r_{2}>0$ (see the assumption in the equation (7.1)). Assume that $-(2m_{2}+r_{2})$ is equal to $\min\{-(2m_{2}+r_{2}),-(2m_{1}+r_{1})+2\nu_{F}(x)\}$. Then we have

[TABLE]

Using Lemma 7.2.1, we get that $r_{1}-r_{2}$ is an odd integer. Hence, we get that $m_{2}-1+(r_{1}+r_{2})/2)\geq-1/2$. Therefore, we have

[TABLE]

Assume that $-(2m_{1}+r_{1})+2\nu_{F}(x)$ is equal to $\min\{-(2m_{2}+r_{2}),-(2m_{1}+r_{1})+2\nu_{F}(x)\}$. We have

[TABLE]

Since, $r_{1}+r_{2}$ is odd, we get that $m_{2}+(r_{1}+r_{2})/2-1\geq-1/2$. Hence, we the inequality $\nu_{F}(x)\leq m_{1}+r_{1}-1$ and we deduce that $-(2m_{1}+r_{1})+2\nu_{F}(x)\leq-(m_{1}+1)+\nu_{F}(x)$. Finally, using the inequality

[TABLE]

we complete the verification of our claim in the present case.

Case 2: Let us consider the case where $F/F_{0}$ is unramified and

[TABLE]

In this case, we have $r_{1}=r_{2}$. First, assume that $w=\operatorname{id}$. Then we have

[TABLE]

If $-(2m_{2}+r_{2})+1$ is equal to $\nu_{F}(h(ue_{1},ue_{1}))$, then we get that

[TABLE]

Since $m_{2}+(r_{2}-r_{1})/2-1/2\geq-1/2$, we get that

[TABLE]

If $-(2m_{1}+r_{1})+2\nu_{F}(x)$ is equal to $\nu_{F}(h(ue_{1},ue_{1}))$, then we that

[TABLE]

Since $\nu_{F}(x)$ is an integer, we get that $\nu_{F}(x)\leq m_{1}$. Therefore, we get that

[TABLE]

Hence, in the case where $w=\operatorname{id}$ we get that

[TABLE]

Let us continue with the case considered in the previous paragraph but with $w\neq\operatorname{id}$. We have

[TABLE]

If $-(2m_{2}+r_{2})-1$ is equal to $\nu_{F}(h(ue_{-1},ue_{-1}))$, then we get that

[TABLE]

From this we get that $\nu_{F}(x)\geq m_{1}$; therefore, we have

[TABLE]

If $-(2m_{1}+r_{1})+2\nu_{F}(x)$ is equal to $\nu_{F}(h(ue_{-1},ue_{-1}))$, then we get that $\nu_{F}(x)\leq m_{1}-m_{2}-1/2$. Since $\nu_{F}(x)$ is an integer, we get that $\nu_{F}(x)\leq m_{1}-1$. Hence, we complete the verification of the claim in the present case.

Case 3: Let us consider the case where $F/F_{0}$ is a ramified extension. In this case, we have

[TABLE]

We clearly have $-m_{2}\leq-\lceil m_{2}/2\rceil$, for $m_{2}\geq 0$. Now, assume that $-m_{2}$ is equal to $\nu_{F/F_{0}}(\delta h(ue_{w},ue_{w}))$. Then we have

[TABLE]

From the above inequality we get that $m_{2}\geq\lceil m_{1}/2-\nu_{F/F_{0}}(x)\rceil$. We assume that $-m_{1}+2\nu_{F/F_{0}}(x)$ is equal to $\nu_{F/F_{0}}(\delta h(ue_{w},ue_{w}))$. Then we get that $\nu_{F/F_{0}}(x)\leq m_{1}/2-m_{2}/2$. Hence, we have $\nu_{F/F_{0}}(x)\leq m_{1}/2$ and this is equivalent to the inequality

[TABLE]

With this we complete the verification of the claim in all cases. ∎

∎

8. The depth-zero case.

When $F/F_{0}$ is unramified, the classification of generic depth-zero cuspidal representations of $G$ can be deduced from the general work of DeBacker–Reeder in the article [6, Section 6]. From their results, generic depth-zero cuspidal representations are precisely the representations of the form

[TABLE]

where $P_{0}(\Lambda)$ is a parahoric subgroup such that $P_{0}(\Lambda)/P_{1}(\Lambda)$ is isomorphic to $U(2,1)(k_{F}/k_{F_{0}})$, and $\sigma$ is the inflation of a cuspidal generic representation of $P_{0}(\Lambda)/P_{1}(\Lambda)$. Now, we assume that $F/F_{0}$ is a ramified extension and consider a cuspidal representation of $G$, isomorphic to

[TABLE]

where $P^{0}(\Lambda)$ is a maximal parahoric subgroup of $G$, and $\sigma$ is the inflation of a cuspidal representation of $P_{0}(\Lambda)/P_{1}(\Lambda)$.

If $F/F_{0}$ is ramified, the groups $P_{0}(\Lambda)/P_{1}(\Lambda)$ is the $k_{F}$-rational points of a disconnected reductive group over $k_{F}$. An irreducible representation $\sigma$ of $P_{0}(\Lambda)/P_{1}(\Lambda)$ is called a cuspidal representation if $\operatorname{res}_{P^{0}(\Lambda)/P_{1}(\Lambda)}\sigma$ is a direct sum of cuspidal representations. An irreducible representation $\sigma$ of $P_{0}(\Lambda)/P_{1}(\Lambda)$ is called generic if and only if its restriction to a $p$-Sylow subgroup, say $H$, contains a non-trivial character of $H$.

Let $(e_{1},e_{0},e_{-1})$ be any Witt-basis for $(V,h)$ then upto $G$ conjugation there are two lattice sequences $\Lambda_{1}$ and $\Lambda_{2}$ such that $P^{0}(\Lambda_{i})$ is a maximal parahoric subgroup, for $i\in\{1,2\}$. We have $e(\Lambda_{i})=2$, for $i\in\{1,2\}$, and

[TABLE]

[TABLE]

Let $\operatorname{{\bf B}}$ be the Borel subgroup of $\operatorname{{\bf G}}$ such that $\langle e_{1}\rangle$ is fixed by $\operatorname{{\bf B}}$. Let $\operatorname{{\bf U}}$ be the unipotent radical of $\operatorname{{\bf B}}$. The groups $P^{0}(\Lambda_{1})$ and $P^{0}(\Lambda_{2})$ are special maximal compact subgroups of $G$, and we have the Iwasawa decomposition

[TABLE]

for $i\in\{1,2\}$. The representation of the form (8.1) is generic if and only if

[TABLE]

for some character $\Psi$ of $U$.

Lemma 8.0.1**.**

Let $F/F_{0}$ be a ramified quadratic extension. A depth-zero cuspidal representation $\pi$ of $G$ is generic if and only if

[TABLE]

where $\sigma$ is a generic cuspidal representation of $P_{0}(\Lambda_{1})/P_{1}(\Lambda_{1})$.

Proof.

Let $\pi$ be a depth zero cuspidal representation isomorphic to $\operatorname{ind}_{P_{0}(\Lambda_{1})}^{G}\sigma$. The image of $U\cap P_{0}(\Lambda_{1})$ in the quotient $P_{0}(\Lambda_{1})/P_{1}(\Lambda_{1})$ is the pro-$p$ Sylow subgroup of $P_{0}(\Lambda_{1})/P_{1}(\Lambda_{1})$. Note that $P_{0}(\Lambda_{1})\cap U$ is equal to

[TABLE]

and the group $P_{1}(\Lambda_{1})\cap U$ is equal to

[TABLE]

The quotient $(P_{0}(\Lambda_{1})\cap U)/(P_{1}(\Lambda_{1})\cap U)$ is isomorphic to $\{u(x,-x^{2}/2):x\in k_{F}\}$. Let $\Psi$ be any non-trivial character of $U$ such that $\operatorname{res}_{P_{0}(\Lambda)\cap U}\Psi\neq\operatorname{id}$ and $\operatorname{res}_{P_{1}(\Lambda)\cap U}\Psi=\operatorname{id}$. For such a character $\Psi$, the space

[TABLE]

if and only if $\sigma$ is the inflation of a generic cuspidal representation of $P_{0}(\Lambda_{1})/P_{1}(\Lambda_{1})$. Hence, $\pi$ is generic if and only if $\sigma$ is generic.

Let $\pi$ be a depth-zero cuspidal representation of the form $\pi\simeq\operatorname{ind}_{P_{0}(\Lambda_{2})}^{G}\sigma$. Assume that $\pi$ is generic, then we get that

[TABLE]

The image of the group $P_{0}(\Lambda_{2})\cap U_{\rm der}$ in the quotient $P_{0}(\Lambda_{2})/P_{1}(\Lambda_{2})\simeq{\rm SL}_{2}(k_{F})\times\{\pm 1\}$, is a $p$-Sylow subgroup, say $H$. Note that $\operatorname{res}_{H}\sigma$ is a direct sum of non-trivial characters of $H$. Thus we get a contradiction to the condition to the equation (8.2). Hence, the representation $\pi$ is non-generic. ∎

9. Main theorem

In this section, we combine the results obtained so far in the following theorem. Recall the following notation: if $W$ is a non-degenerate subspace of $V$, then ${\bf 1}_{W}$ is the projection onto $W$ with kernel $W^{\perp}$.

Theorem 9.0.1**.**

Let $F$ be a non-Archimedean local field with odd residue characteristic. Let $\mathfrak{x}=[\Lambda,n,0,\beta]$ be a skew semisimple stratum and let $\Pi_{\mathfrak{x}}$ be the set of cuspidal representations containing a Bushnell–Kutzko type of the form $(J^{0}(\Lambda,\beta),\lambda)$. The cuspidal representations in the set $\Pi_{\mathfrak{x}}$ are either all generic or all non-generic. Furthermore, the following holds.

1. (A)

Let $\mathfrak{x}$ is a skew simple stratum, i.e., the case where $F[\beta]$ is a degree $3$ field extension of $F$. Then the set $\mathfrak{X}_{\beta}(F_{0})$ is non-empty, and every representation contained in the set $\Pi_{\mathfrak{x}}$ is generic. 2. (B)

Let $\mathfrak{x}$ be a skew semisimple stratum with the underlying splitting $V=V_{1}\perp V_{2}$ such that $\dim_{F}V_{i}=i$, for $i\in\{1,2\}$. Assume that $\beta=\beta_{1}+\beta_{2}$, where $\beta_{i}$ is equal to ${\bf 1}_{V_{i}}\beta{\bf 1}_{V_{i}}$, $\sigma_{h}(\beta_{i})=-\beta_{i}$, $F[\beta_{2}]$ is a degree $2$ field extension of $F$. Let $q_{i}$ be the integer $\nu_{\Lambda_{i}}(\beta_{i})$, for $i\in\{1,2\}$. If $q_{1}>q_{2}$, then a cuspidal representation in the set $\Pi_{\mathfrak{x}}$ is generic if and only if $(V_{2},h)$ is isotropic. If $q_{2}>q_{1}$, then a cuspidal representation in the set $\Pi_{\mathfrak{x}}$ is generic if and only if $(V_{2},h)$ is anisotropic. In this case, a cuspidal representation in the set $\Pi_{\mathfrak{x}}$ is generic if and only if the set $\mathfrak{X}_{\beta}(F_{0})$ is non-empty. 3. (C)

Let $\mathfrak{x}$ be a skew semisimple stratum with the underlying splitting $V=V_{1}\perp V_{2}$ such that $\dim_{F}V_{i}=i$, for $i\in\{1,2\}$. We assume that $\beta=\beta_{1}+\beta_{2}$, where $\beta_{i}$ is equal to ${\bf 1}_{V_{i}}\beta{\bf 1}_{V_{i}}$, $\beta_{i}\in F$, and $\sigma(\beta_{i})=-\beta_{i}$, for $i\in\{1,2\}$. Every representation in the set $\Pi_{\mathfrak{x}}$ is non-generic. The set $\mathfrak{X}_{\beta}(F_{0})$ is non-empty if and only if $(V_{2},h)$ is isotropic. 4. (D)

Let $\mathfrak{x}$ be a skew semisimple stratum with the underlying splitting $V=V_{1}\perp V_{2}\perp V_{3}$. Then a representation in the set $\Pi_{\mathfrak{x}}$ is generic if and only if $\mathfrak{X}_{\beta}(F_{0})$ is non-empty.

In part $(2)$ of the above theorem, we have $q_{1}\neq q_{2}$ (see paragraph 5.3). In the case where the underlying splitting of a skew semisimple strata $\mathfrak{x}$ is equal to $V=V_{1}\perp V_{2}\perp V_{3}$, it is fairly easy to determine the necessary and sufficient conditions on $\beta$ for the non-emptiness of the set $\mathfrak{X}_{\beta}(F_{0})$. Assume that $\beta_{i}={\bf 1}_{V_{i}}\beta{\bf 1}_{V_{i}}$, for $1\leq i\leq 3$. When $F/F_{0}$ is unramified, the non-emptiness of $\mathfrak{X}_{\beta}(F_{0})$ depends only on the set of integers $\{\nu_{F}(\beta_{1}),\nu_{F}(\beta_{2}),\nu_{F}(\beta_{3})\}$ and the isomorphism classes of $(V_{i},h)$, for $1\leq i\leq 3$. We refer to Lemmas 7.2.1 and 7.2.2 for these results. However, in the case where $F/F_{0}$ is ramified, one requires more invariants on $\beta$ to determine whether $\mathfrak{X}_{\beta}(F_{0})$ is empty or not. Since these invariants are not the natural invariants attached to a stratum, we did not make them explicit. For details, we refer to the proof of Lemma 7.3.3.

Proof of Theorem 9.0.1.

We indicate the precise references to proofs of various parts enumerated in the theorem. The first part, case $(A)$, follows from Proposition 4.1.2. In case $(B)$, when $q_{1}>q_{2}$, the corresponding statements are proved in Lemmas 5.6.5, 5.7.1, and 5.7.2. In case $(B)$, when $q_{2}>q_{1}$, the corresponding statements are proved in Lemmas 5.6.1, 5.6.2, and 5.7.4. The statements in case $(C)$ are proved in Lemmas 6.2.1 and 6.3.5. The statements in case $(D)$ are proved in 7.4.1, 7.5.1 and 7.5.4. Now, the main statement of the theorem on genericity or non-genericity of all representations in $\Pi_{\mathfrak{x}}$ follows from statements in cases $(A),(B),(C)$, and $(D)$. ∎

Appendix A Appendix: Filtration of

$U_{\rm der}$ induced by lattice sequences

In this section we fix some representatives for $G$-conjugacy classes of self-dual lattice sequences on $V$ and describe $a_{n}(\Lambda)$, for $n\in\mathbb{Z}$. Then we use them to determine $U_{\rm der}\cap a_{n}(\Lambda)$, for $n\in\mathbb{Z}$. These calculations are used in showing certain representations are non-generic.

A.1. The unramified case:

We begin with the case where $F/F_{0}$ is unramified. Let $\Lambda_{1}$ be the lattice sequence of periodicity $2$ and

[TABLE]

The filtration $\{a_{n}(\Lambda)\ |\ n\in\mathbb{Z}\}$ of $\operatorname{End}_{F}(V)$ is given by

[TABLE]

for all $m\in\mathbb{Z}$. Let $\Lambda_{2}$ be a period $2$ lattice sequence given by

[TABLE]

The filtration $\{a_{n}(\Lambda_{2})\ |\ n\in\mathbb{Z}\}$ is given by:

[TABLE]

for all $m\in\mathbb{Z}$. Let $\Lambda_{3}$ be the lattice sequence of period $4$ given by

[TABLE]

The filtration $\{a_{n}(\Lambda_{3})\ |\ n\in\mathbb{Z}\}$ on $\mathfrak{g}$ is given by:

[TABLE]

Although, there is a lattice sequence, say $\Lambda_{4}$, with period $6$, we do not need to write it down explicitly. This corresponds to type (A) strata and in this case all representations are generic. The filtration $\{U_{\rm der}\cap a_{n}(\Lambda_{1})\ |\ n\in\mathbb{Z}\}$ is given by:

[TABLE]

for $m\in\mathbb{Z}$. The filtration $\{U_{\rm der}\cap a_{n}(\Lambda_{2})\ |\ n\in\mathbb{Z}\}$ is given by

[TABLE]

The filtration $\Lambda_{3}$ is given by

[TABLE]

A.2. The ramified case:

Now, assume that $F/F_{0}$ is a ramified extension and $\Lambda_{1}$ and $\Lambda_{2}$ be the lattice sequence of period $2$ given by

[TABLE]

and

[TABLE]

The filtration $\{a_{n}(\Lambda_{1})\ |\ n\in\mathbb{Z}\}$ is similar to the filtration in (A.1). The flitration $\{a_{n}(\Lambda_{2})\ |\ n\in\mathbb{Z}\}$, in this case, is similar to the filtration in (A.2). We will not require to write the filtrations $\{a_{n}(\Lambda^{\prime})\ |\ n\in\mathbb{Z}\}$ for which $P^{0}(\Lambda^{\prime})$ is an Iwahori subgroup of $G$. The filtration $\{U_{\rm der}\cap a_{n}(\Lambda_{1})\ |\ n\in\mathbb{Z}\}$ is given by

[TABLE]

for all $m\in\mathbb{Z}$. The filtration $\{U_{\rm der}\cap a_{n}(\Lambda_{2})\ |\ n\in\mathbb{Z}\}$ is given by

[TABLE]

for any $m\in\mathbb{Z}$.