Friday, July 20, 14:00, 8.339

Dan Nakano (University of Georgia, Athens), On Tensoring with the Steinberg Representation.

Abstract:
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 ≥ 2h-4 (h is the Coxeter number),
(ii) for all rank two groups,
(iii) for p ≥ 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.