Tits groups of Iwahori-Weyl groups and presentations of Hecke algebras

Abstract

Let G be a connected reductive group over a non-archimedean local field F and I be an Iwahori subgroup of G(F ). Let In is the n-th Moy-Prasad filtration subgroup of I. The purpose of this paper is two-fold: to give some nice presentations of the Hecke algebra of connected, reductive groups with In-level structure; and to introduce the Tits group of the Iwahori-Weyl group of groups G that split over an unramified extension of F .The first main result of this paper is a presentation of the Hecke algebra H(G(F ), In), generalizing the previous work of Iwahori-Matsumoto on the affine Hecke algebras. For split GLn, Howe gave a refined presentation of the Hecke algebra H(G(F ), In). To generalize such a refined presentation to other groups requires the existence of some nice lifting of the Iwahori-Weyl group W to G(F ). The study of a certain nice lifting of W is the second main motivation of this paper, which we discuss below.In 1966, Tits introduced a certain subgroup of G(k), which is an extension of W by an elementary abelian 2-group. This group is called the Tits group and provides a nice lifting of the elements in the finite Weyl group. The "Tits group" T for the Iwahori-Weyl group W is a certain subgroup of G(F ), which is an extension of the Iwahori-Weyl group W by an elementary abelian 2-group. The second main result of this paper is a construction of Tits group T for W when G splits over an unramified extension of F . As a consequence, we generalize Howe's presentation to such groups. We also show that when G is ramified over F , such a group T of W may not exist.are the characteristic functions ½ gIn for g ∈ I I n and ½ Inm(w)In , where w runs over elements 2010 Mathematics Subject Classification. 22E70, 20C08.

Affine Deligne-Lusztig varieties associated with generic Newton points

“…This normalization of y i differs from[GH21+] but is consistent with the normalization used in Lusztig's theory of total positivity.…”