  We apply Treumann's "Smith theory for sheaves" in the context of the
Iwahori--Whittaker model of the Satake category. We deduce two results in the
representation theory of reductive algebraic groups over fields of positive
characteristic: (a) a geometric proof of the linkage principle; (b) a character
formula for tilting modules in terms of the $\ell$-canonical basis, valid in
all blocks and in all characteristics.
