Leading coefficients of the Kazhdan-Lusztig polynomials for an Affine Weyl group of type $\widetilde{B_2}$

TL;DR

This paper explicitly computes the leading coefficients of Kazhdan-Lusztig polynomials for the affine Weyl group of type B_2, confirming Lusztig's conjecture on distinguished involutions and suggesting modifications to existing formulas.

Contribution

It provides explicit calculations of (u,w) for B_2 and verifies Lusztig's conjecture, also proposing necessary adjustments to the conjectural formula.

Findings

Confirmed Lusztig's conjecture for B_2

Computed most (u,w) explicitly

Suggested modifications to existing formulas

Abstract

In this paper we compute the leading coefficients $\mu (u,w)$ of the Kazhdan--Lusztig polynomials $P_{u,w}$ for an affine Weyl group of type $\tilde{B}_2$. By using the \textbf{a}-function of a Coxeter group defined by Lusztig (see [L1, \S2]), we compute most $\mu (u,w)$ explicitly. With part of these values $\mu (u,w)$, we show that a conjecture of Lusztig on distinguished involutions is true for an affine Weyl group of type $\tilde{B}_2$. We also show that the conjectural formula in [L3, (12)] needs a modification.