Regular supercuspidal representations

Abstract

We show that, in good residual characteristic, most supercuspidal representations of a tamely ramified reductive p p -adic group G G arise from pairs ( S , θ ) (S,\theta ) , where S S is a tame elliptic maximal torus of G G , and θ \theta is a character of S S satisfying a simple root-theoretic property. We then give a new expression for the roots of unity that appear in the Adler-DeBacker-Spice character formula for these supercuspidal representations and use it to show that this formula bears a striking resemblance to the character formula for discrete series representations of real reductive groups. Led by this, we explicitly construct the local Langlands correspondence for these supercuspidal representations and prove stability and endoscopic transfer in the case of toral representations. In large residual characteristic this gives a construction of the local Langlands correspondence for almost all supercuspidal representations of reductive p p -adic groups.

On the Unicity of Types for Toral Supercuspidal Representations

“…We restrict our attention here to those sequences for which the smallest twisted Levi subgroup is an anisotropic (also called elliptic) maximal torus of G. For the purposes of this paper we call these toral supercuspidal representations, though we caution the reader that some authors reserve "toral" to mean the more restrictive case that the twisted Levi sequence has length d = 1. Relating to work of T. Kaletha [9], F. Murnaghan calls our representations "positive regular" (as justified in [17]).…”

Multiplicity upon restriction to the derived subgroup

“…Suppose also that the restriction of θ to T(f) ∩ SU(2d)(f) remains regular so that the restriction of ρ to SU(2d)(f) remains irreducible. The torus T × T ⊂ U(2d) × U(2d) lifts to give an unramified torus T ⊂ GU(V), and the character θ ⊗ θχ can be inflated and extended to give a character Θ of T. The representation π of GU(V) that we have constructed in the theorem is a regular supercuspidal representation in the sense of Kaletha [5], but the irreducible components of its restriction to SU(V) are not since our character Θ of T, when restricted to T ∩ SU(V), is not regular because of the presence of the element g 0 ∈ GU(V). For depth-zero supercuspidal representations of quasi-split unitary groups, the parahoric that we have used is the only one that can lead to higher multiplicities.…”

“…In a future work, we will expand upon the example given in the Theorem, whose essence is the following. Given a supercuspidal representation of G 2 (k) whose restriction to G 1 (k) has regular components (in the sense of Kaletha [5]), then the components occur with multiplicity one. (Nevins [9] already verified this for many cases.)…”

Distinguished regular supercuspidal representations

“…The first part of this article is concerned with the properties of distinguished regular supercuspidal representations in terms of the inducing data tame regular elliptic pairs. We refer to [Kal,§2] or Section 2.2.1 for the basic notion and knowledge on regular supercuspidal representations.…”

“…We further assume that G is quasi-split. Kaletha [Kal,§5] defines the notion regular supercuspidal L-parameters ϕ for G. For each rigid inner twist (G ′ , ξ, z) of G, he also constructs L-packets Π ϕ (G ′ ) that consists of certain regular supercuspidal representations of G ′ (F ). We consider not merely the distinction problem for G, but also for all other rigid inner twists (G ′ , ξ, z) of G such that the fixed involution θ of G can be "transferred" to an involution θ ′ of G ′ .…”