Some multiplication formulas in an affine Hecke algebra

TL;DR

This paper derives multiplication formulas in an affine Hecke algebra of type B_2, revealing relations between Kazhdan-Lusztig coefficients and algebraic group representations.

Contribution

It provides new explicit formulas for products involving specific basis elements in an affine Hecke algebra of type B_2, linking algebraic and representation-theoretic properties.

Findings

Formulas for products involving C_{rt} and S_{lambda} in the affine Hecke algebra.

Relations established between Kazhdan-Lusztig coefficients and algebraic group representations.

Abstract

In this paper we consider the Hecke algebra $\mathcal {H}$ associated to an extended affine Weyl group of type $\widetilde{B_2}$. We give some interesting formulas on $C_{rt}S_{\lambda}$, which imply some relations between the Kazhdan-Lusztig coefficients $\mu (y,w)$ and representations of some algebraic groups. Here $C_{rt}$ is one element in the Kazhdan-Lusztig basis of $\mathcal {H}$ and $S_{\lambda}$ is an element in the center of $\mathcal {H}$.