Generic Representation Theory of the Heisenberg Group
  In this paper we extend a result for representations of the Additive group $G_a$ given in [3] to the Heisenberg group $H_1$. Namely, if $p$ is greater than 2d then all $d$-dimensional characteristic $p$ representations for $H_1$ can be factored into commuting products of representations, with each factor arising from a representation of the Lie algebra of $H_1$, one for each of the the representation's Frobenius layers. In this sense, for a fixed dimension and large enough $p$, all representations for $H_1$ look generically like representations for direct powers of it over a field of characteristic zero.   The reader may consult chapter 13 of [1] for a fuller account of what follows. 
Michael Crumley
math.RT