Computing twisted KLV polynomials

TL;DR

This paper presents an explicit algorithm for computing twisted Kazhdan-Lusztig-Vogan polynomials, essential for analyzing Hermitian forms on representations of real reductive groups, and implements it in the Atlas software.

Contribution

It provides a detailed algorithmic approach to compute twisted KLV polynomials, advancing computational methods in representation theory.

Findings

Algorithm successfully implemented in Atlas software

Enables explicit computation of twisted KLV polynomials

Facilitates analysis of Hermitian forms in real reductive groups

Abstract

In order to compute Hermitian forms on representations of real reductive groups, in the unequal rank case, it is necessary to compute twisted Kazhdan-Lusztig-Vogan polynomials. These were defined by Lusztig and Vogan (Quasisplit Hecke algebras and Symmetric Spaces, Duke, 2014) and discussed further by Adams and Vogan (Parameters for twisted representations, 2015). These notes contain the details necessary to go from what is in those papers to an explicit algorithm. This algorithm has been implemented in the Atlas of Lie Groups and Representations software.