Applications of the full Kostant–Toda lattice and hyper-functions to unitary representations of the Heisenberg groups

We consider a new orbit method for unitary representations which determines the explicit values of the multiplicities of the irreducible components of unitary representations of the connected Lie groups. We provide the polarized symplectic affine space on which the Lie group acts. This polarization is obtained by the Hamiltonian flows of the full Kostant–Toda lattice. The Hamiltonian flows of the ordinary Toda lattice are not sufficient for constructing this polarization. In this paper we do an experiment on the case of the unitary representations of the Heisenberg groups. The irreducible representations of the Heisenberg group are obtained and classified by $\mathbb{R}$ by the Stone–von Nuemann theorem. The multiplicities are obtained by using spontaneous symmetry breaking and Sato hyper‑functions.

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Received 3 May 2022

Accepted 4 February 2023

Published 22 December 2023