Generic Representation Theory of the Additive and Heisenberg Groups

Abstract

In this paper we give an intimate connection between the characteristic zero representation theories of the Additive and Heisenberg groups, and their characteristic p >0 theories when p is much larger than the dimension a representation. In particular, if p >> dimension, then all characteristic p representations for these groups can be factored into commuting products of representations, with each factor arising from a representation of the Lie algebra of the group, one for each of the representation's Frobenius layers. In this sense, for a fixed dimension and large enough p, all representations for these groups look generically like representations for direct powers of themselves over a field of characteristic zero.