The multiplicity function on exponential and completely solvable homogeneous spaces (Q809193)

Let G be a simply connected completely solvable Lie group, H a connected closed subgroup, p: \({\mathfrak g}^*\to {\mathfrak h}^*\) the canonical projection. For any irreducible unitary representations \(\pi\) and \(\gamma\) of G and H respectively, one denotes by \({\mathcal O}_{\pi}\subset {\mathfrak g}^*\) and \({\mathcal O}_{\gamma}\subset {\mathfrak h}^*\) the corresponding co-adjoint orbits and by \(n^{\gamma}_{\pi}=\#({\mathcal O}_{\pi}\cap p^{-1}({\mathcal O}_{\gamma}))/H\) the multiplicity of \(\pi\) in \(Ind_ H^ G\gamma\). The author proves that \(n^{\gamma}_{\pi}\), as a function of \(\pi\), is always uniformly infinite or almost everywhere finite. It is finite if and only if \(\dim (G\phi)=2\dim (H\phi)-\dim O_{\gamma}\) for generic \(\phi \in p^{- 1}({\mathcal O}_{\gamma})\). In this case \(n^{\gamma}_{\pi}\) is bounded.