Parity Sheaves and Smith Theory

Let $p$ be a prime number and let $X$ be a complex algebraic variety with an action of $\mathbb{Z}\mathrm{/}p\mathbb{Z}$. We develop the theory of parity complexes in a certain $2$-periodic localization of the equivariant constructible derived category $D_{\mathbb{Z}\mathrm{/}p\mathbb{Z}}^b(X,{\mathbb{Z}}_p)$. 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 $X^{\mathbb{Z}\mathrm{/}p\mathbb{Z}}$. 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.