Dedicated to G. Lusztig on the occasion of his 60th birthday 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 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 G = G(F) and its central extension Ĝ . For instance our spherical Hecke algebra corresponds to the subgroup G(A) ⊂ G(F) where A ⊂ F is the subring OK((t)) where OK ⊂ K is the ring of integers. It turns out that for generic level (cf. [3]) 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).