On Non-unitary Representations of the Heisenberg Group

Let $G $ be a connected, simply connected nilpotent Lie group and $\mathfrak{g} $ be it $s $ Lie algebra. Irreducible unitary representations can be described by the orbit method due to A.Kirillov$([K]) $. Let $f $ be an element of the dual space $\mathfrak{g}’ $ of $\mathfrak{g} $. Let $G_{f} $ be the isotropy subgroup of $f $ in $G $ and $\mathfrak{g}_{f} $ b $e $ its Lie algebra. The polarization at $f