Some examples of Hecke algebras for two-dimensional local fields

Let K be a local non-archimedian field, F = K((t)) and let G be & 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 double-struck G sign = G(F) and its central extension G. For instance our spherical Hecke algebra, corresponds to the subgroup G(A) ⊂ G(F) where A ⊂ F is the subring script O sign_K((t)) where script O sign _K ⊂ 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).