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 [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] constructed by I. Fesenko.