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Abstract Let H_n be the 2n+1 dimensional Heisenberg group and let chi be a non-trivial unitary character of its center Z. The celebrated Stone von-Neuman theorem says that there is a unique irreducible unitary representation of H_n with central character chi. This uniqueness result has many applications -- in particular it plays a crucial role in the construction of the oscillator representation for the metaplectic group.

We give an extension of this result to non-unitary and non-irreducible representations. Our main result is that there is an equivalence of categories between non-degenerate representations of Z,
suitably defined, and those of H_n. We give both algebraic and analytic versions of this equivalence. The algebraic version, which is easier, is closely related to Kashiwara's lemma from the theory of D-modules. The analytic version, which is much more delicate, is an equivalence in the setting of smooth nuclear Frechet representations of moderate growth. Finally, we show how to
extend the oscillator representation to the smooth setting, and we give an application to degenerate Whittaker models for representations of reductive groups.
This is joint work with Raul Gomez (UANL, Mexico) and Dmitry Gourevitch (Weizmann, Israel).

Archive paper arXiv:2201.12638
The web address of the paper is https://arxiv.org/abs/2201.12638