Parity Sheaves and Smith Theory

TL;DR

This paper develops a categorified version of Smith theory using parity complexes in a 2-periodic localization of equivariant derived categories, with applications to affine Grassmannians and geometric Satake.

Contribution

It introduces a new functor from parity sheaves on a variety to those on its fixed points, generalizing Smith theory in a geometric and categorical framework.

Findings

Defines a functor from parity sheaves on X to fixed points X^{Z/pZ}

Provides a geometric construction of Frobenius-contraction functor

Applies to affine Grassmannians and relates to geometric Satake

Abstract

Let $p$ be a prime number and let $X$ be a complex algebraic variety with an action of $\mathbb{Z}/p\mathbb{Z}$. We develop the theory of parity complexes in a certain $2$-periodic localization of the equivariant constructible derived category $D^b_{\mathbb{Z}/p\mathbb{Z}}(X,\mathbb{Z}_p)$. Under certain assumptions, we use this to define a functor from the category of parity sheaves on $X$ to the category of parity sheaves on the fixed-point locus $X^{\mathbb{Z}/p\mathbb{Z}}$. This may be thought of as a categorification of Smith theory. When $X$ is the affine Grassmannian associated to some complex reductive group, our functor gives a geometric construction of the Frobenius-contraction functor recently defined by M. Gros and M. Kaneda via the geometric Satake equivalence.