An inverse Satake isomorphism in characteristic p

Let F be a local field with finite residue field of characteristic p and k an algebraic closure of the residue field. Let G be the group of F-points of a F-split connected reductive group. In the apartment corresponding to a chosen maximal split torus of T, we fix a hyperspecial vertex and denote by K the corresponding maximal compact subgroup of G. Given an irreducible smooth k-representation $\rho$ of K, we construct an isomorphism from the affine semigroup k-algebra of the dominant cocharacters of T onto the Hecke algebra $H(G, \rho)$. In the case when the derived subgroup of G is simply connected, we prove furthermore that our isomorphism is the inverse to the Satake isomorphism constructed by Herzig.