DECOMPOSITION OF DOUBLE COSETS IN p-ADIC GROUPS

Let G be a split group over a locally compact field F with non-trivial discrete valuation. Employing the structure theory of such groups and the theory of Coxeter groups, we obtain a general formula for the decomposition of double cosets P1σP2 of subgroups P1, P2 ⊂ G(F) containing an Iwahori subgroup into left cosets of P2. When P1 and P2 are the same hyperspe-cial subgroup, we use this result to derive a formula of Iwahori for the degrees of elements of the spherical Hecke algebra. 1. Introduction. Let G be a semisimple algebraic group which is split over a locally compact field F with non-trivial discrete valuation and let I be an Iwahori subgroup of G(F). In [6], Iwahori and Matsumoto show that the double cosets in I\G(F)/I are indexed by the elements of the extended affine Weyl group W