Some examples of Hecke algebras for two-dimensional local fields

Abstract

Let $\mathbf{K}$ be a local non-archimedian field, $\mathbf{F} = \mathbf{K}((t))$ and let $\mathbf{G}$ be a split semi-simple group. The purpose of this paper is to study certain analogs of spherical and Iwahori Hecke algebras for representations of the group $\mathbb{G} = G(\mathbf{F})$ and its central extension $\hat{\mathbb{G}}$. For instance our spherical Hecke algebra corresponds to the subgroup $G(\mathcal{A}) \subset G(\mathbf{F})$ where $\mathcal{A} \subset \mathbf{F}$ is the subring $\mathcal{O}_{\mathbf{K}}((t))$ where $\mathcal{O}_{\mathbf{K}} \subset \mathbf{K}$ is the ring of integers. It turns out that for generic level (cf. [4]) the spherical Hecke algebra is trivial; however, on the critical level it is quite large. On the other hand we expect that the size of the corresponding Iwahori-Hecke algebra does not depend on a choice of a level (details will be considered in another publication).