Some Non-Trivial Kazhdan-Lusztig Coefficients of an Affine Weyl Group of Type $\tilde A_n$

TL;DR

This paper demonstrates that certain Kazhdan-Lusztig polynomial coefficients in an affine Weyl group of type A_n are equal to n+2, impacting the understanding of extension groups in finite groups of Lie type.

Contribution

It establishes a specific value for leading coefficients of Kazhdan-Lusztig polynomials in affine A_n groups, revealing new algebraic properties.

Findings

Leading coefficient (y,w) equals n+2 for certain pairs in A_n

Implications for the dimension of first extension groups in finite Lie type groups

Enhanced understanding of Kazhdan-Lusztig polynomial structure 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 A_n $ is $n+2$. This fact has some consequences on the dimension of first extension groups of finite groups of Lie type with irreducible coefficients.