On Donkin’s tilting module conjecture I: lowering the prime

Abstract

In this paper the authors provide a complete answer to Donkin’s Tilting Module Conjecture for all rank 2 2 semisimple algebraic groups and SL 4 ⁡ ( k ) \operatorname {SL}_{4}(k) where k k is an algebraically closed field of characteristic p > 0 p>0 . In the process, new techniques are introduced involving the existence of ( p , r ) (p,r) -filtrations, Lusztig’s character formula, and the G r G_{r} T-radical series for baby Verma modules.