Invariant bilinear forms for Heisenberg group (Q1903388)

In the paper under review non-unitary representations of the Heisenberg group of \((2n + 1)\)-dimensions are studied. If \(G\) is a connected, simply connected nilpotent Lie group, \(\mathfrak g\) its Lie algebra and \(f\) an element of the dual space \({\mathfrak g}'\) of \(\mathfrak g\), then there are fixed a real standard polarization \(\mathfrak h\) at \(f \in {\mathfrak g}'\) and a complex linear form \(\Lambda \in ({\mathfrak g}')_\mathbb{C}\) on \(\mathfrak g\) such that \(\mathfrak h\) is isotropic with respect to the bilinear form \(\varphi_\Lambda (x,y) = \Lambda (x,y)\), \(x, y \in {\mathfrak g}_\mathbb{C}\). A representation \(\tau_\Lambda\) of \(D = \text{exp} ({\mathfrak g} \cap {\mathfrak h})\) is defined by \(\tau_\Lambda (\text{exp }X) = \text{exp} (\sqrt{-1}\Lambda(X))\), \(X \in {\mathfrak g} \cap {\mathfrak h}\). There is also defined a non-unitary representation \(U^{\Lambda,{\mathfrak h}}\) of \(G\) induced from \(\tau_\Lambda\). The author considers the case where the representation space of \(U^{\Lambda,{\mathfrak h}}\) is the space \({\mathcal D} (R^n) = {\mathcal D}^{\mathfrak h}_\Lambda\) of \(C^\infty\)-functions on \(R^n\) with compact support \((f \neq 0\) on the center of \(\mathfrak g\), \(\mathfrak h\) is Abelian). There are proved ten theorems. The first of them gives a necessary and sufficient condition for the existence of an invariant bilinear form on \({\mathcal D}^{\mathfrak h}_{\Lambda_1} \times {\mathcal D}^{\mathfrak h}_{\Lambda_2}\).