A bar operator for involutions in a Coxeter group

TL;DR

This paper introduces a new, elementary algebraic definition of a bar operator for involutions in Coxeter groups, extending previous geometric approaches to a broader algebraic context.

Contribution

It provides a completely elementary, geometry-free definition of the Hecke algebra action and bar operator applicable to any Coxeter group.

Findings

New elementary definition of the Hecke algebra action

Extension of bar involution to arbitrary Coxeter groups

Simplification of previous geometric constructions

Abstract

In [LV] the authors defined a Hecke algebra action and a bar involution on a vector space spanned by the involutions in a Weyl group. In this paper we give a new definition of the Hecke algebra action and the bar operator which, unlike the one in [LV], is completely elementary (does not use geometry) and in particular it makes sense for an arbitrary Coxeter group.