Perverse ${\mathbb{F}}_p$-sheaves on the affine Grassmannian

Abstract

For a reductive group over an algebraically closed field of characteristic $p>0$ we construct the abelian category of perverse ${\mathbb{F}}_p$-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 ${\mathbb{F}}_p$-sheaves.

Final version

Cited by in corpus (3)

• A note on Integral Satake isomorphisms
• Central elements in affine mod $p$ Hecke algebras via perverse ${\mathbb{F}}_p$-sheaves
• Tame twistings and $\Theta$-data