Smith-Treumann theory and the linkage principle

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 $\mathrm{\ell }$-canonical basis, valid in all blocks and in all characteristics.