The based ring the lowest generalized two-sided cell of an extended affine Weyl group

Abstract

Let ${\mathbf{c}}_0$ be the lowest generalized two-sided cell of an extended affine Weyl group W. We determine the structure of the based ring of ${\mathbf{c}}_0$. For this we show that certain conjectures of Lusztig on generalized cells (called P1-P15) hold for ${\mathbf{c}}_0$. As an application, we use the structure of the based ring to study certain simple modules of Hecke algebras of $W$ with unequal parameters, namely those attached to ${\mathbf{c}}_0$. Also we give a set of prime ideals $\mathfrak{p}$ of the center $\mathcal{Z}$ of the generic affine Hecke algebra $\mathcal{H}$ such that the reduced affine Hecke algebra $k_{\mathfrak{p}}\mathcal{H}$ is simple over $k_{\mathfrak{p}}$, where $k_{\mathfrak{p}}=\mathrm{Frac}(\mathcal{Z}/\mathfrak{p})$ is the residue field of $\mathcal{Z}$ at $\mathfrak{p}$. In particular, we show that the algebra $\mathcal{H}{\otimes }_{\mathcal{Z}}\mathrm{Frac}(\mathcal{Z})$ is a split simple algebra over the field $\mathrm{Frac}(\mathcal{Z})$.

23pages, 1 figure; second version; An error (in last section of last version), pointed by an anonymous referee , is corrected

References in corpus (1)

• A factorization theorem for affine Kazhdan-Lusztig basis elements