Abstract
We show that some types for supercuspidal representations of tamely ramified p-adic groups that appear in Jiu-Kang Yu’s work are geometrizable. To do so, we define a function-sheaf dictionary for one-dimensional characters of arbitrary smooth group schemes over finite fields. In previous work we considered the case of commutative smooth group schemes and found that the standard definition of character sheaves produced a dictionary with a nontrivial kernel. In this paper we give a modification of the category of character sheaves that remedies this defect, and is also extensible to non-commutative groups. We then use these commutative character sheaves to geometrize the linear characters that appear in the types introduced by Jiu-Kang Yu, assuming that the character vanishes on a certain derived subgroup. To define geometric types, we combine commutative character sheaves with Gurevich and Hadani’s geometrization of the Weil representation and Lusztig’s character sheaves.
Highlights
The combined work of Ju-Lee Kim in [Kim07] and Jessica Fintzen in [Fin[18], Fin19] establishes that all irreducible supercuspidal representations of a tamely ramified p-adic group G can be built from “data” introduced by Jiu-Kang Yu [Yu01, § 15], as long as p does not divide the order of the Weyl group of G
In the sense of Bushnell & Kutzko [BK98], of a supercuspidal representation built from Yu data can be constructed directly from the datum, it is convenient to consider an intermediate object, introduced in [Yu01, Remark 15.4], which we call a Yu type datum
Up to some linear characters, all the ingredient representations are on groups of the form H(O), where H is a smooth group scheme over [a Henselian discrete valuation ring with finite residue field κ] O, and the representations are inflated from H(κ)
Summary
The combined work of Ju-Lee Kim in [Kim07] and Jessica Fintzen in [Fin[18], Fin19] establishes that all irreducible supercuspidal representations of a tamely ramified p-adic group G can be built from “data” introduced by Jiu-Kang Yu [Yu01, § 15], as long as p does not divide the order of the Weyl group of G. In order to provide further justification for referring to objects in CCS(G) as commutative character sheaves, suppose for the moment that G is a connected, reductive algebraic group over k. Armed with the function-sheaf dictionary for smooth group schemes over finite fields, we return to the task of geometrizing Yu type data. The proof of Theorem 5.2 requires: Yu’s work on smooth integral models [Yu15]; the geometrization of the character of the Heisenberg–Weil representation over finite fields by Gurevich & Hadani [GH07]; Lusztig’s character sheaves on reductive groups over finite fields; and the function-sheaf dictionary for characters of smooth group schemes over finite fields, at our disposal in Theorem 3.12.
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