Hecke modules based on involutions in extended Weyl groups

TL;DR

This paper constructs modules over the extended Hecke algebra associated with involutions in the extended Weyl group, generalizing previous work and deepening the understanding of algebraic structures related to semisimple groups.

Contribution

It introduces a new class of modules over the extended Hecke algebra indexed by involutions, extending prior constructions to the extended Weyl group context.

Findings

Construction of H-modules indexed by involutions in the extended Weyl group

Generalization of previous involution-based modules to the extended Weyl group setting

Provides new algebraic tools for studying semisimple groups and their Weyl groups

Abstract

Let X be the group of weights of a maximal torus of a simply connected semisimple group over C and let W be the Weyl group. The semidirect product W(Q\otimes X/X) is called the extended Weyl group. There is a natural C(v)-algebra H called the extended Hecke algebra with basis indexed by the extended Weyl group which contains the usual Hecke algebra as a subalgebra. We construct an H-module with basis indexed by the involutions in the extended Weyl group. This generalizes a construction of the author and D.Vogan.