Iwahori-hecke Algebras of Sl2 over 2-dimensional Local Fields

Hecke algebras were first studied because of their role in the representation theory of p-adic groups, or algebraic groups over 1-dimensional local fields. There are two important classes of Hecke algebras. One is spherical Hecke algebras attached to maximal compact open subgroups, and the other is Iwahori-Hecke algebras attached to Iwahori subgroups. A spherical Hecke algebra is isomorphic to the center of the corresponding Iwahori-Hecke algebra. Recently, the representation theory of algebraic groups over 2-dimensional local fields was initiated by the works of Kapranov, Kazhdan and Gaitsgory [7, 2, 3, 4]. In their development, Cherednik’s double affine Hecke algebras appear as an analogue of Iwahori-Hecke algebras. But a double affine Hecke algebra has no center in generic case. It makes one miss the other important class of Hecke algebras—an analogue of spherical Hecke algebras. On the other hand, Kim and Lee introduced in [8] an analogue of spherical Hecke algebras of SL2 over 2dimensional local fields. The main difference in their approach is the use of the rank 2 integral structure of 2-dimensional local fields. They also constructed a Satake isomorphism using Fesenko’s R((X))-valued measure defined in [1]. In particular, the algebra is proved to be commutative. A similar result is expected in the case of GLn and, eventually, in the case of reductive algebraic groups.