Hecke algebras and involutions in Weyl groups

TL;DR

This paper introduces a new polynomial related to involutions in Weyl groups, which is significant for understanding unitary representations of complex reductive groups, and provides an algorithm for its computation.

Contribution

It defines a novel polynomial P^\sigma_{y,w} related to Weyl group involutions and presents an algorithm for its calculation, advancing representation theory.

Findings

New polynomial P^\sigma_{y,w} defined for Weyl group involutions

Algorithm developed for computing these polynomials

Potential applications in the theory of unitary representations

Abstract

For any two involutions y,w in a Weyl group (y\le w), let P_{y,w} be the polynomial defined in [KL]. In this paper we define a new polynomial P^\sigma_{y,w} whose i-th coefficient is a_i-b_i where the i-th coefficient of P_{y,w} is a_i+b_i (a_i,b_i are natural numbers). These new polynomials are of interest for the theory of unitary representations of complex reductive groups. We present an algorithm for computing these polynomials.