  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.
