Tame cuspidal representations in non-defining characteristics

Abstract

Let k be a non-archimedean local field of residual characteristic p = 2. Let G be a (connected) reductive group that splits over a tamely ramified field extension of k. We revisit Yu's construction of smooth complex representations of G(k) from a slightly different perspective and provide a proof that the resulting representations are supercuspidal.Moreover, we show that an analogous construction yields smooth, irreducible, cuspidal representations over an arbitrary algebraically closed field R of characteristic different from p and that this construction provides all smooth, irreducible, cuspidal R-representations if p does not divide the order of the Weyl group of G.

Regular Bernstein blocks

“…Yu's construction [21] produces a supercuspidal representation π out of the datum Σ. Recently, Fintzen [11] has shown that if the residue characteristic p does not divide the order of the Weyl group, this construction yields all supercuspidal representations of G(F ). 4.2.…”

Local Langlands correspondences

“…Although the proof contained no error, it was based on a statement that had been published with a misprint, and was false. A correct proof of Yu's theorem was given more recently by Fintzen [Fi19], in connection with her proof of the fundamental theorem [Fi21] that we have already mentioned several times.…”

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

“…When C is algebraically closed, Z(0)-exhaustion (and unicity) is in [39] and [40] when C = C and the arguments of [40] carry over to C; exhaustivity was implicit in [53], and is established by Fintzen at the end of [24] (the hypothesis on G of [24] plays not role for level 0 representations).…”

“…(G n ) x,rn/2 (G n+1 ) x,0+ inflating the representation ρ of (G n+1 ) [x] . In ( [24], §2.4), J is denoted by K and λ 0 is still denoted by ρ. To describe the representation κ of K we introduce more notations following ([24] 2.5).…”

“…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.…”