Applications of the full Kostant-Toda lattice and hyper-functions to unitary representations of the Heisenberg groups (Q6183445)

The full Kostant-Toda lattice is an integrable system constructed with a Lax operator \(L\in \Lambda + \bar{\mathfrak{b}}\), for the shift matrix \(\Lambda=\sum_{i=1}^{n-1} E_{i,i+1} \in \mathfrak{gl}_n(\mathbb{R})\) and \(\bar{\mathfrak{b}}\) the Borel subalgebra of lower triangular matrices. The definition of the system is related to a tower structure on the flag manifold \(G/B\) for \(G=\mathrm{GL}_n(\mathbb{R})\), whose base space is isomorphic to the \((2n-3)\)-dimensional Heisenberg group \[ U=\left\{ \left( \begin{array}{ccc} 1&{}^t\mathbf{0}&0 \\ \mathbf{a}&\mathrm{Id}_{n-2}& \mathbf{0} \\ c& {}^t\mathbf{b} &1 \end{array} \right) \Big| \mathbf{a},\mathbf{b}\in \mathbb{R}^{n-2} \right\} \subset \mathrm{GL}_n(\mathbb{R})\,. \] Let \(R\) be the centre of \(U\) and put \(X=U/R\). Thanks to the Kostant-Toda flow, we get a symplectic action of \(U\) on \(X\), and a corresponding polarization. As the author puts it, the paper `experiments' a decomposition into irreducible unitary representations of \(U\) on \(L^2\)-sections of an appropriate bundle on \(X\), see Section 5. The paper is clearly intended for experts.