Unitary and Nonunitary Representations of the Heisenberg-Weyl Lie Algebra

Abstract

We examine unitary and nonunitary representations of the Heisenberg-Weyl Lie algebra $\mathfrak{hw}_n$, with particular emphasis on tensor products of unitary representations and on indecomposable nonunitary representations. In the unitary setting, the irreducible representations with nontrivial central character are the Schr\"odinger representations, as classified by the Stone-von Neumann theorem. Although tensor products of these representations are considered in the literature, we give a detailed Lie-algebraic analysis and construct explicit unitary intertwining operators, including the case where the central characters sum to zero. In the nonunitary setting, we consider a natural realization of $\mathfrak{hw}_n$ as a subalgebra of the real symplectic Lie algebra $\mathfrak{sp}_{2n+2}(\mathbb R)$ and prove that every finite-dimensional complex irreducible representation of $\mathfrak{sp}_{2n+2}(\mathbb{R})$ remains indecomposable upon restriction to $\mathfrak{hw}_n$. This yields a large natural family of finite-dimensional, nonunitary indecomposable representations of $\mathfrak{hw}_n$.