Geometric L-packets of Howe-unramified toral supercuspidal representations

Abstract

We show that L-packets of toral supercuspidal representations arising from unramified maximal tori of p-adic groups are realized by Deligne-Lusztig varieties for parahoric subgroups. We prove this by exhibiting a direct comparison between the cohomology of these varieties and algebraic constructions of supercuspidal representations. Our approach is to establish that toral irreducible representations are uniquely determined by the values of their characters on a domain of sufficiently regular elements. This is an analogue of Harish-Chandra's characterization of real discrete series representations by their character on regular elements of compact maximal tori, a characterization which Langlands relied on in his construction of L-packets of these representations. In parallel to the real case, we characterize the members of Kaletha's toral L-packets by their character on sufficiently regular elements of elliptic maximal tori.

Representations of reductive groups over local fields

“…There are two approaches to this problem. One, taken by Chan-Oi in [CO21], is to extend the validity of this formula to some more general elements of S(F ) ∩ G(F ) reg and prove that the resulting values are enough to characterize the representation; so far this has been successful under additional assumptions on (S, θ), including the assumption that S is unramified. One can hope that such methods can be generalized to yield the validity of Theorem 1.6 for all elements of S(F ) whose topologically semi-simple modulo center part is regular.…”

“…Nonetheless, a more complete solution is desirable. It is conceivable that the ideas of [CO21] might lead to a stronger version of this formula. Another approach is to allow arbitrary regular semi-simple elements of G(F ), in the vein of Theorem 1.7.…”

On a decomposition of $p$-adic Coxeter orbits

“…1.1] in this case. Our decomposition result, along with the Mackey-type formula shown in [DI20] (as well as techniques from [Lus76,CO21]), can be applied -similarly as in [CI19]-to study the cohomology of X c (b), Ẋc (b) and smooth G(k)-representations therein. Beside our main result, we prove some related results on rational conjugacy classes of tori and a version of Steinberg's cross section, see §1.4 below for a summary.…”

Characterization of supercuspidal representations and very regular elements

“…For L-packets of toral supercuspidals corresponding to unramified tori, the necessity of the quadratic twist ε j of (j, θ j ) was established by DeBacker-Spice [DS18]. In [CO21], the authors of the present paper proved that under a mild assumption on the size of the residue field of F , for these (j, θ j ), the parahoric Deligne-Lusztig induction of [CI21b] gives rise to an irreducible supercuspidal representation π geom (j(S),θj ) and that π geom (j(S),θj ) ∼ = π Yu (j(S),θj εj ) . This comparison was established by considering very regular elements, though the exact methods were different and for the most part somewhat simpler.…”