Seminar

Let $G$ be a simple, simply connected algebraic group over an algebraically closed field of prime characteristic. 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 a Steinberg module with a simple module with restricted highest weight admits a good filtration. In this talk, I will survey results in this area and present new results where we verify the aforementioned good filtration statement (i.e., Steinberg tensored with restricted simple module) when (i) $p\geq 2h-4$ ($h$ is the Coxeter number), (ii) for all rank two groups, (iii) for $p\geq 3$ when the simple module corresponding to a fundamental weight and (iv) for a number of cases when the rank is less than or equal to five.