Smith theory and geometric Hecke algebras

TL;DR

This paper extends Smith theory to constructible sheaves and functions, connecting Z/p-localization on loop groups with geometric Satake correspondence and special algebraic group homomorphisms in small characteristic.

Contribution

It introduces a novel Z/p-localization framework for sheaves and functions, linking Smith theory with geometric Satake and algebraic group homomorphisms in new settings.

Findings

Smith theory on loop groups relates to geometric Satake correspondence.

Z/p-localization connects to special algebraic group homomorphisms.

Results apply to fields of small characteristic.

Abstract

In 1960 Borel proved a "localization" result relating the rational cohomology of a topological space X to the rational cohomology of the fixed points for a torus action on X. This result and its generalizations have many applications in Lie theory. In 1934, P. Smith proved a similar localization result relating the mod p cohomology of X to the mod p cohomology of the fixed points for a Z/p-action on X. In this paper we study Z/p-localization ("Smith theory") for constructible sheaves and functions. We show that Smith theory on loop groups is related via the geometric Satake correspondence to some special homomorphisms that exist between algebraic groups defined over a field of small characteristic.