Non-trivial Kazhdan-Lusztig coefficients of affine Weyl groups

TL;DR

This paper calculates specific leading coefficients of Kazhdan-Lusztig polynomials for affine Weyl groups of types B, C, and D, revealing their exact values related to the rank of the groups.

Contribution

It provides explicit values for the leading coefficients of Kazhdan-Lusztig polynomials in affine Weyl groups of types B, C, and D, which were previously unknown.

Findings

Leading coefficient $(y,w)$ equals $n$ for type $ ilde B_n$

Leading coefficient $(y,w)$ equals $n+1$ for types $ ilde C_n$ and $ ilde D_n$

Results clarify the structure of Kazhdan-Lusztig polynomials in affine Weyl groups.

Abstract

In this paper we show that the leading coefficient $\mu(y,w)$ of some Kazhdan-Lusztig polynomials $P_{y,w}$ with $y,w$ in an affine Weyl group of type $\tilde B_n$ (resp. $\tilde C_n$ or $\tilde D_n$) is $n$ (resp. $n+1$).