Hecke algebras with unequal parameters and Vogan's left cell invariants

TL;DR

This paper develops a general framework for defining invariants of left cells in Hecke algebras with unequal parameters, extending Vogan's tau-invariant from primitive ideals to a broader algebraic context.

Contribution

It introduces a new, unified approach to invariants of left cells applicable to Hecke algebras with unequal parameters, broadening the scope of Vogan's original concept.

Findings

Framework successfully defines invariants for Hecke algebras with unequal parameters.

Extends Vogan's tau-invariant to a more general setting.

Provides tools for better understanding of cell structures in algebraic representations.

Abstract

In 1979, Vogan introduced a generalised $\\tau$ -invariant for characterising primitive ideals in enveloping algebras. Via a known dictionary this translates to an invariant of left cells in the sense of Kazhdan and Lusztig. Although it is not a complete invariant, it is extremely useful in describing left cells. Here, we propose a general framework for defining such invariants which also applies to Hecke algebras with unequal parameters.