Some multiplication formulas in an affine Hecke algebra

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 $C_{rt}{S}_{\hat{I}}»$, which imply some relations between the Kazhdan-Lusztig coefficients $\hat{I}¼(y,w)$ and representations of some algebraic groups. Here $C_{rt}$ is one element in the Kazhdan-Lusztig basis of $\mathcal{H}$ and $S_{\hat{I}}»$ is an element in the center of $\mathcal{H}$.

11 pages