INVERSE SATAKE ISOMORPHISM AND CHANGE OF WEIGHT

Let G be any connected reductive p-adic group. Let K subset of G be any special parahoric subgroup and V,V' be any two irreducible smooth (F-p) over bar [K]-modules. The main goal of this article is to compute the image of the Hecke bimodule End((Fp) over bar) (c-Ind(K)(G) V, c-Ind(K)(G) V') by the generalized Satake transform and to give an explicit formula for its inverse, using the pro-p Iwahori Hecke algebra of G. This immediately implies the "change of weight theorem" in the proof of the classification of mod p irreducible admissible representations of G in terms of supersingular ones. A simpler proof of the change of weight theorem, not using the pro-p Iwahori Hecke algebra or the Lusztig-Kato formula, is given when G is split and in the appendix when G is quasi-split, for almost all K.