Spherical Hecke algebras of SL2 over 2-dimensional local fields

In this paper, we study spherical Hecke algebras of SL2 over two dimensional local fields. In order to define the convolution product, we make explicit use of coset decompositions. We also consider spherical Hecke algebras of the torus of SL2 and construct the Satake isomorphism between two spherical Hecke algebras. In order to define the Satake isomorphism, we use the invariant measure on two-dimensional local fields with values in R((X)) constructed by I. Fesenko.