The Steinberg quotient of a tilting character

Let G be a simple algebraic group over an algebraically closed field of prime characteristic. If M is a finite dimensional G-module that is projective over the Frobenius kernel of G, then its character is divisible by the character of the Steinberg module. In this paper we study such quotients, showing that if M is an indecomposable tilting module, then the multiplicities of the orbit sums appearing in its “Steinberg quotient” are well behaved.