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We study the center of the pro-p Iwahori-Hecke ring H of a connected splitp-adic reductive group G. For k an algebraically closed field withcharacteristic p, we prove that the center of the k-algebra H_k:= H\otimes_Z kcontains an affine semigroup algebra which is naturally isomorphic to the Heckealgebra attached to any irreducible smooth k-representation of a givenhyperspecial maximal compact subgroup of G. This isomorphism is obtained usingthe inverse Satake isomorphism constructed in arXiv:1207.5557. We apply this toclassify the simple supersingular H_k-modules, study the supersingular block inthe category of finite length H_k-modules, and relate the latter tosupersingular representations of G.

Compatibility between Satake and Bernstein-type isomorphisms in characteristic p