Resumen de An inverse Satake isomorphism in characteristic p

Rachel Ollivier

• Let FF be a local field with finite residue field of characteristic pp and kk an algebraic closure of the residue field. Let GG be the group of FF -points of a FF -split connected reductive group. In the apartment corresponding to a maximal split torus TT , we choose a hyperspecial vertex and denote by KK the corresponding maximal compact subgroup of GG . Given an irreducible smooth kk -representation ρρ of KK , we construct an isomorphism from the affine semigroup kk -algebra k[X+∗(T)]k[X∗+(T)] of the dominant cocharacters of TT onto the spherical kk -algebra H(G,ρ)H(G,ρ) . In the case when the derived subgroup of GG is simply connected, we prove furthermore that our isomorphism is the inverse to the mod pp Satake isomorphism constructed by Herzig (Compos Math 147(1):263–283, 2011).