Asymptotic lowest two-sided cell

TL;DR

This paper proves a conjecture about the stability of Kazhdan-Lusztig cells in affine Weyl groups under certain weight function conditions, specifically for the lowest two-sided cell.

Contribution

It confirms the semicontinuity conjecture for affine Weyl groups, showing the lowest two-sided cell remains stable under large weight perturbations.

Findings

Proves the semicontinuity conjecture for affine Weyl groups.

Establishes stability of the lowest two-sided cell under specific weight conditions.

Advances understanding of cell structures in Coxeter systems.

Abstract

To a Coxeter system $(W,S)$ (with $S$ finite) and a weight function $L : W \to \NM$ is associated a partition of $W$ into Kazhdan-Lusztig (left, right or two-sided) $L$-cells. Let $S^\circ = \{s \in S | L(s)=0\}$, $S^+=\{s \in S | L(s) > 0\}$ and let $C$ be a Kazhdan-Lusztig (left, right or two-sided) $L$-cell. According to the semicontinuity conjecture of the first author, there should exist a positive natural number $m$ such that, for any weight function $L' : W \to \NM$ such that $L(s^+)=L'(s^+) > m L'(s^\circ)$ for all $s^+ \in S^+$ and $s^\circ \in S^\circ$, $C$ is a union of Kazhdan-Lusztig (left, right or two-sided) $L'$-cells. The aim of this paper is to prove this conjecture whenever $(W,S)$ is an affine Weyl group and $C$ is contained in the lowest two-sided $L$-cell.