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

Abstract

Given a quaternionic form G of a p-adic classical group (p odd) we classify all cuspidal irreducible complex representations of G. It is a straight forward generalization of the classification in the padic classical group case. We prove two theorems: At first: Every irreducible cuspidal representation of G is induced from a cuspidal type, i.e. from a certain irreducible representation of a compact open subgroup of G, constructed from a β-extension and a cuspidal representation of a finite group. Secondly we show that two intertwining cuspidal types of G are up to equivalence conjugate under some element of G.

Representations of $p$-adic groups over coefficient rings

“…For R algebraically closed, the conjecture has been proved for the level 0 37 representations of any G or when G has rank 1 (Martin Weissman [201]), G is an inner form of GL(n, F ) (Minguez-Sécherre [138]), or of SL(n, F ) (Peyi Cui [36], [37]), G is a classical group and p = 2 (Shaun Stevens [185], Stevens-Kurinczuk-Daniel Skodlerak [131]) or a quaternionic form of G (Skodlerak [179]), G splits on a moderately ramified extension of F and p does not divide the order of the absolute Weyl group (Jessica Fintzen [66]), Algebraically closed is not necessary and there is an explicit list X of pairs (K, W ) of G where K is an open subgroup of G compact modulo the center and 34 supposing that a uniformizer of F acts trivially 35 ρP is half the sum of the roots of AM in LieP . The formula can be simplified !…”

Representations of a reductive $p$-adic group in characteristic distinct from $p$

“…Quaternionic form. Finally the case of a quaternionic form G of a classical group for odd p is obtained by Skodlerak [47] [48] for C = C. That case is a mix of the previous two and Skodlerak constucts a set of C-types satisfying irreducibility, exhaustion and unicity [48] Thm.1.1. The procedure to define semisimple characters is the same as for classical groups but starting with Aut D V where V is a right vector space of dimension m over a central quaternion division F -algebra D equipped with an anti-involution d → d (it is necessarily of the first kind), and a non-degenerate ǫ-hermitian form h on V with ǫ ∈ {1, −1}.…”

“…Indeed we show that if C a and C ′a are two algebraically closed fields with the same characteristic c = p and G admits a set of C a -types satisfying unicity, exhaustion, and Aut(C a )-stability, then G also admits a list of C ′a -types satisfying the same properties. For example, in the case of a quaternionic unitary group G, the set of cuspidal complex types constructed by Skodlerack in [48] gives rise to a set of cuspidal C-types satisfying exhaustion, unicity, intertwining, and Aut(C)-stability, for any characteristic 0 field C. In another case, only unicity for C a -types is lacking to apply Theorem 0.1: when G splits over a tamely ramified extension of F and p does not divide the order of the absolute Weyl group of G, Fintzen [24] shows that the constructions of Yu [56] give a list of types satisfying exhaustion, for any algebraically closed field C a of characteristic c = p. We prove here Aut(C a )-stability for those types, but we have not verified if the arguments of Hakim-Murnaghan [28], analysing the fibers of the map from data à la Yu to cuspidal representations, give unicity; that is a topic of the current Ph. D. thesis of R. Deseine.…”

International Congress of Mathematicians

“…G is a classical group (Stevens [187], Stevens-Kurinczuk-Skodlerak [131]) or a quaternionic form of G (Skodlerak [181]), if p ¤ 2.…”