Parity sheaves and Smith theory

Abstract

Let p be a prime number and let X be a complex algebraic variety with an action of ℤ / p ⁢ ℤ {\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 ℤ / p ⁢ ℤ b ⁢ ( X , ℤ p ) {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 ℤ / p ⁢ ℤ {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.

Modular representations and reflection subgroups

“…Instead I have tried to take a more speculative perspective, and to dream a little. The main inspiration is recent work of Treumann [Tre19], Hazi [Haz17], Abe [Abe19] and Leslie-Lonergan [LL18]. As I was finishing these notes I became aware of the work of McDonnell [McD19], which comes to similar conclusions to these notes.…”

“…However when we are in characteristic p (and in particular to get the equivalences ( 14) and (15)) we need Smith theory. Here we follow work of Treumann [Tre19] and Leslie-Lonergan [LL18].…”

“…Parity sheaves and Smith theory. The ideas of this section (and even its title) are taken from [LL18]. Their crucial observation is that the theory of parity shaves is well-adapted to Smith theory.…”

“…At the centre of Smith theory, is the group µ p " Z{pZ. This group appears at first sight to be poorly adapted to the theory of parity sheaves because H m µp ppt, kq " k for all m ě 0, for any field k of characteristic p. With a little work (see [LL18,§2]) this calculation implies that Hom Perfppt,T k q pk pt , k pt rmsq " k for all m P Z. Thus parity vanishing fails rather dramatically in Perfppt, T k q, even on a point!…”

“…Remark 4.20. The work of Leslie-Lonergan [LL18] builds on the expectation that the theory of perverse sheaves is "compatible with Smith theory". This expectation was first suggested by Treumann; see the final two sections of [Tre19].…”