Characterization of supercuspidal representations and very regular elements

Abstract

We prove that regular supercuspidal representations of p-adic groups are uniquely determined by their character values on very regular elements-a special class of regular semisimple elements on which character formulae are very simple-provided that this locus is sufficiently large. As a consequence, we resolve a question of Kaletha by giving a description of Kaletha's L-packets of regular supercuspidal representations which mirrors Langlands' construction for real groups following Harish-Chandra's characterization theorem for discrete series representations. Our techniques additionally characterize supercuspidal representations in general, giving p-adic analogues of results of Lusztig on reductive groups over finite fields. In particular, we establish an easy, non-cohomological characterization of unipotent supercuspidal representations when the residue field of the base field is sufficiently large.

Local Jacquet-Langlands correspondence for regular supercuspidal representations

“…Remark 4.2. In fact, as explained below, this theorem follows from general results in [FKS21], [CO23] if we make suitable assumptions on n, p, q and so on. Our proof works without any assumptions (except for p = 2, which is necessary for the right-hand side to make sense), especially thanks to the work [BH11].…”

“…Res E/F G m for some tamely ramified extension E/F of degree n. Under the assumption on S below, [CO23, Theorem 9.10] describes LJLC cl (π) in terms of a tame elliptic regular pair of G and thus one can check that it agrees with LJLC Kal (π) to deduce Theorem 4.1 (note that the modified parametrization of [FKS21], which we mention in Remark 3.1, is adopted in [CO23]). The basic assumption in [CO23] reads in our setting that p does not divide n!. In addition to this, the assumption of [CO23, Theorem 9.10] requires that either of the following should be satisfied (here we write e = e(E/F ), f = f (E/F ) and φ is Euler's totient function):…”

“…Let us briefly comment on this assumption and also on the approach. In [CO23], it is proved among other things that a regular supercuspidal representation can be characterized by the restriction of the character to "elliptic very regular elements", where the character values are easier to compute, provided that there exist sufficiently many such elements ([CO23, Theorem 9.7]). As a sample application of the theorem, LJLC cl (π) is determined in [CO23, Theorem 9.10] by studying the restriction of the characters.…”

“…The above assumption ensures that "Henniart inequality" holds, which quantifies the sufficiency of elliptic very regular elements. Our proof of Theorem 4.1 may be close in spirit to that of [CO23], as the former relies on Bushnell-Henniart [BH11], where LJLC cl (π) is investigated mainly through the evaluation of the character at certain special elements.…”

“…We must remark that Theorem 0.1 follows from more general results in Fintzen-Kaletha-Spice [FKS21] or Chan-Oi [CO23], if we assume that the characteristic ch F of F is zero and p is sufficiently large, or if we make a suitable assumption on a field extension E/F of degree n associated to π. Theorem 0.1 holds without any such assumptions, essentially because none of the results we use, especially [BH11], makes any assumptions (see Remark 4.2 for more details).…”