Perverse 𝔽 p -sheaves on the affine Grassmannian

Abstract

Abstract For a reductive group over an algebraically closed field of characteristic p > 0 {p>0} we construct the abelian category of perverse 𝔽 p {\mathbb{F}_{p}} -sheaves on the affine Grassmannian that are equivariant with respect to the action of the positive loop group. We show this is a symmetric monoidal category, and then we apply a Tannakian formalism to show this category is equivalent to the category of representations of a certain affine monoid scheme. We also show that our work provides a geometrization of the inverse of the mod p Satake isomorphism. Along the way we prove that affine Schubert varieties are globally F-regular and we apply Frobenius splitting techniques to the theory of perverse 𝔽 p {\mathbb{F}_{p}} -sheaves.