Kazhdan-Lusztig parameters and extended quotients

TL;DR

This paper reveals that Kazhdan-Lusztig parameters have a geometric structure as extended quotients, unifying them with Springer parameters within a framework that interpolates between different parameter regimes.

Contribution

It demonstrates the geometric nature of Kazhdan-Lusztig parameters as extended quotients and unifies them with Springer parameters through a new interpolating framework.

Findings

Kazhdan-Lusztig parameters form an extended quotient structure.

Unified framework for Kazhdan-Lusztig and Springer parameters.

Interpolation between Springer and Kazhdan-Lusztig parameters via a complex parameter s.

Abstract

The Kazhdan-Lusztig parameters are important parameters in the representation theory of $p$-adic groups and affine Hecke algebras. We show that the Kazhdan-Lusztig parameters have a definite geometric structure, namely that of the extended quotient $T//W$ of a complex torus $T$ by a finite Weyl group $W$. More generally, we show that the corresponding parameters, in the principal series of a reductive $p$-adic group with connected centre, admit such a geometric structure. This confirms, in a special case, our recently formulated geometric conjecture. In the course of this study, we provide a unified framework for Kazhdan-Lusztig parameters on the one hand, and Springer parameters on the other hand. Our framework contains a complex parameter $s$, and allows us to interpolate between $s = 1$ and $s = \sqrt q$. When $s = 1$, we recover the parameters which occur in the Springer correspondence; when $s = \sqrt q$, we recover the Kazhdan-Lusztig parameters.