Semisimple characters for inner forms II: Quaternionic forms of 𝑝-adic classical groups (𝑝 odd)

Abstract

In this article we consider the set G G of rational points of a quaternionic form of a symplectic or an orthogonal group defined over a non-Archimedean local field of odd residue characteristic. We construct all full self-dual semisimple characters for G G and we classify their intertwining classes using endo-parameters. We compute the set of intertwiners between self-dual semisimple characters, and prove an intertwining and conjugacy theorem. Finally we count all G G -intertwining classes of full self-dual semisimple characters which lift to the same G ~ \tilde {G} -intertwining class of a full semisimple character for the ambient general linear group G ~ \tilde {G} for G G .

Intertwining Semisimple Characters for -Adic Classical Groups

“…Semisimple characters (or, more precisely, their endo-classes) will give a decomposition of the category of smooth -modular representations of classical groups, and each subcategory should be equivalent to the subcategory of depth zero representations of some other (endoscopic) group, for which other techniques are available. Current work of the first author (see [Sko17] for the start of this) aims at generalizing the results proved here to proper inner forms of classical groups, where additional problems arise, analogous to those in the case of inner forms of general linear groups [BSS12]. One would then expect that a Jacquet–Langlands correspondence between inner forms would respect the decompositions of the categories by endo-class, as for general linear groups [SS16], and that this would be a major step in making such a correspondence explicit.…”

Cuspidal irreducible complex or l-modular representations of quaternionic forms of p-adic classical groups for odd p

“…This article is a continuation of [27] and [28] which we call I and II. We mainly follow their notation, but there is a major change, see the remark below, and there are slight changes to adapt the notation to [33].…”

“…This work is the third part in a series of three papers, the first two being [28] and [27]. Let F be a non-Archimedean local field with odd residue characteristic p. The construction and classification of cuspidal irreducible complex representation of the set of rational points G(F) of a reductive group G defined over F has already been successfully studied for general linear groups ( [8] Bushnell-Kutzko, [23], [2], [24] Broussous-Secherre-Stevens) and for p-adic classical groups ( [33] Stevens, [17] Kurinczuk-Skodlerack-Stevens).…”

“…We fix a skew-field D of index 2 over F together with an anti-involution (¯) on D and an ǫ-hermitian form h ∶ V × V → D on a finite dimensional D-vector space V. Let G be the group of isometries of h. At first we describe the construction of the cuspidal types (imitating the Bushnell-Kutzko-Stevens framework): A cuspidal type is a certain irreducible representation λ of a certain compact open subgroup J of G. The arithmetic core of λ is given by a skew-semisimple stratum ∆ = [Λ, n, 0, β]. It provides the following data (see [27] for more information):…”

“…(v) For the intertwining implies conjugacy part of Theorem 1.1 we use [27] and follow [18], see Section 10.…”