Generic Representation Theory of the Heisenberg Group

In this article we extend a result for representations of the additive group G_a given in 4 Suslin , A. , Friedlander , E. M. , Bendel , C. P. ( 1997 ). Infinitesimal 1-parameter subgroups and cohomology . Journal of the AMS 10 ( 3 ): 693 – 728 .Web of Science ®] , Google Scholar]] to the Heisenberg group H₁. Namely, if p is greater than 2d, then all d-dimensional characteristic p representations for H₁ can be factored into commuting products of representations, with each factor arising from a representation of the Lie algebra of H₁, and conversely any commuting collection of Lie algebra representations gives rise to a representation of H₁ in this fashion. In this sense, for a fixed dimension and large enough p, all representations for H₁ look generically like representations for direct powers of it over a field of characteristic zero.

The following originally appeared as Chapter 13 of the author's dissertation 1 Crumley , M. ( 2010 ). Ultraproducts of Tannakian Categories and Generic Representation Theory of Unipotent Algebraic Groups. Ph.D. dissertation, The University of Toledo, Department of Mathematics . Google Scholar]].