Parameters for Twisted Representations

TL;DR

This paper develops an explicit algorithm for computing Kazhdan-Lusztig-Vogan polynomials associated with Hermitian forms on real reductive groups, focusing on extensions of representations to extended groups, and implements it in computational software.

Contribution

It provides a complete algorithm for describing extensions of representations to extended groups, enabling the computation of new Kazhdan-Lusztig-Vogan polynomials.

Findings

Algorithm successfully computes the polynomials.

Implementation is integrated into Atlas of Lie Groups software.

Facilitates further research on Hermitian forms and representations.

Abstract

The study of Hermitian forms on a real reductive group $G$ gives rise, in the unequal rank case, to a new class of Kazhdan-Lusztig-Vogan polynomials. These are associated with an outer automorphism $\delta$ of $G$, and are related to representations of the extended group $<G,\delta>$. These polynomials were defined geometrically by Lusztig and Vogan in "Quasisplit Hecke Algebras and Symmetric Spaces", Duke Math. J. 163 (2014), 983--1034. In order to use their results to compute the polynomials, one needs to describe explicitly the extension of representations to the extended group. This paper analyzes these extensions, and thereby gives a complete algorithm for computing the polynomials. This algorithm is being implemented in the Atlas of Lie Groups and Representations software.