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

Abstract

In this paper we compute the leading coefficients $\hat{I}¼(u,w)$ 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 $\hat{I}¼(u,w)$ explicitly. With part of these values $\hat{I}¼(u,w)$, 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.

31 pages