Harmonic analysis on algebraic groups over two-dimensional local fields of equal characteristic

Let K (K = K2 whose residue field is K1 whose residue field is K0, see the notation in section 1 of Part I) be a two-dimensional local field of equal characteristic. Thus K2 is isomorphic to the Laurent series field K1((t2)) over K1. It is convenient to think of elements of K2 as (formal) loops over K1. Even in the case where char (K1) = 0, it is still convenient to think of elements of K1 as (generalized) loops over K0 so that K2 consists of double loops. Denote the residue map OK2 → K1 by p2 and the residue map OK1 → K0 by p1. Then the ring of integers OK of K as of a two-dimensional local field (see subsection 1.1 of Part I) coincides with p−1 2 (OK1 ). Let G be a split simple simply connected algebraic group over Z (e.g. G = SL2 ). Let T ⊂ B ⊂ G be a fixed maximal torus and Borel subgroup of G; put N = [B,B], and let W be the Weyl group of G. All of them are viewed as group schemes. Let L = Hom(Gm, T ) be the coweight lattice of G; the Weyl group acts on L. Recall that I(K1) = p −1 1 (B(Fq)) is called an Iwahori subgroup of G(K1) and T (OK1 )N (K1) can be seen as the “connected component of unity” in B(K1). The latter name is explained naturally if we think of elements of B(K1) as being loops with values in B.