On tensoring with the steinberg representation

Let G be a simple, simply connected algebraic group over an algebraically closed field of prime characteristic p > 0. Recent work of Kildetoft and Nakano and of Sobaje has shown close connections between two long-standing conjectures of Donkin: one on tilting modules and the lifting of projective modules for Frobenius kernels of G and another on the existence of certain filtrations of G -modules. A key question related to these conjectures is whether the tensor product of the r th Steinberg module with a simple module with p r th restricted highest weight admits a good filtration. In this paper we verify this statement (i) when p ≥ 2 h − 4 ( h is the Coxeter number), (ii) for all rank two groups, (iii) for p ≥ 3 when the simple module corresponds to a fundamental weight and (iv) for a number of cases when the rank is less than or equal to five.