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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, 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.