Smith theory and geometric Hecke algebras

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, 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 for constructible sheaves and functions. We show that Z/p-localization 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.