A generalised $\tau$-invariant for the unequal parameter case

TL;DR

This paper introduces a broad framework for defining a generalized tau-invariant applicable to Hecke algebras with unequal parameters, extending Vogan's original concept for primitive ideals and left cells in Weyl groups.

Contribution

It develops a new, general method for constructing tau-invariants that works for Hecke algebras with unequal parameters, broadening the scope of previous invariants.

Findings

Framework successfully applies to unequal parameter cases

Extends the utility of tau-invariants in representation theory

Provides tools for analyzing left cells in finite Weyl groups

Abstract

In 1979, Vogan proposed a generalised $\tau$-invariant for characterising primitive ideals in enveloping algebras. Via a known dictionary this translates to an invariant of left cells of finite Weyl groups. 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.