On the Kazhdan--Lusztig order on cells and families

TL;DR

This paper explores the Kazhdan--Lusztig order on irreducible characters of finite Coxeter groups, providing a new characterization and computational method, and relating it to unipotent class closures in algebraic groups.

Contribution

It offers a novel characterization and effective computation of the Kazhdan--Lusztig order on families of irreducible characters, connecting it to Springer correspondence and unipotent class closures.

Findings

Effective characterization of the order $q_{LR}$ in the character ring.

Method to compute the order using standard operations in the character ring.

Interpretation of the order in terms of unipotent class closures for Weyl groups.

Abstract

We consider the set $\Irr(W)$ of (complex) irreducible characters of a finite Coxeter group $W$. The Kazhdan--Lusztig theory of cells gives rise to a partition of $\Irr(W)$ into "families" and to a natural partial order $\leq_{\cLR}$ on these families. Following an idea of Spaltenstein, we show that $\leq_{\cLR}$ can be characterised (and effectively computed) in terms of standard operations in the character ring of $W$. If, moreover, $W$ is the Weyl group of an algebraic group $G$, then $\leq_{\cLR}$ can be interpreted, via the Springer correspondence, in terms of the closure relation among the "special" unipotent classes of $G$.