Monomial representations of certain exponential Lie groups (Q1122008)

Let G be a connected Lie group with Lie algebra \({\mathfrak g}\). We note by \(\Delta_ G\) the modular function of G. Let \(\pi\) be an irreducible unitary representation of G. We denote by \({\mathcal H}^{\infty}_{\pi}\) the space of \(C^{\infty}\) vectors for \(\pi\), provided with the usual topology, and by \({\mathcal H}_{\pi}^{-\infty}\) its antidual. Given a closed subgroup H and its unitary character \(\chi\), define the space of semi-invariant generalized vectors for (\(\pi\),H,\(\chi)\); \[ ({\mathcal H}_{\pi}^{-\infty})^{H,\chi \Delta^{1/2}_{H,G}}=\{a\in {\mathcal H}_{\pi}^{-\infty};\quad \pi (h)a=\chi (h)\Delta^{1/2}_{H,G}(h)a,\quad \forall h\in H\}, \] where \(\Delta_{H,G}=\Delta_ H/\Delta_ G\). On the other hand, one constructs a monomial representation \(\tau =ind^ G_ H\chi\) of G, which is decomposed into irreducible constituents \(\tau =\int^{\oplus}_{\hat G}m(\pi)\pi d\mu (\pi)\) with respect to some Borel measure \(\mu\) on the unitary dual \(\hat G\) of G. For multiplicities m(\(\pi)\), it is natural to ask a kind of reciprocity, whether the equalities \(m(\pi)=\dim ({\mathcal H}_{\pi}^{-\infty})^{H,\chi \Delta^{1/2}_{H,G}}\) hold \(\mu\)-almost everywhere. Now we assume G exponential and use the orbit method. Let \(\theta\) : \({\mathfrak g}^*\to \hat G\) be the Kirillov-Bernat mapping. Assume that H is connected and let \({\mathfrak h}\) denote its Lie algebra. Then there exists a linear form f on \({\mathfrak g}\) such that f([\({\mathfrak h},{\mathfrak h}])=0\) and that \(\chi =\chi_ f\), \(\chi_ f(h)=e^{\sqrt{-1}f(\log h)}\) for \(\forall h\in H=\exp {\mathfrak h}\). We denote by \({\mathfrak h}^{\perp}\) the annihilator of \({\mathfrak h}\) in \({\mathfrak g}^*\). In these situations, \(\mu\) is supported on \(\theta (f+{\mathfrak h}^{\perp})\) and m(\(\pi)\) is equal to the number of H-orbits contained in \((f+{\mathfrak h}^{\perp})\cap \theta^{- 1}(\pi)\). The author studies the space (\({\mathcal H}_{\pi}^{- \infty})^{H,\chi_ f\Delta^{1/2}_{H,G}}\) in the case where \({\mathfrak g}\) is a normal j-algebra and that \(\theta^{-1}(\pi)\) is open in \({\mathfrak g}^*\). She shows (\({\mathcal H}_{\pi}^{-\infty})^{H,\chi_ f\Delta^{1/2}_{H,G}}=\{0\}\) if \((f+{\mathfrak h}^{\perp})\cap \theta^{-1}(\pi)=\emptyset\). The main result of this paper is to determine (\({\mathcal H}_{\pi}^{-\infty})^{H,\chi_ f\Delta^{1/2}_{H,G}}\) for H satisfying 2dim H\(=\dim G\), which gives \(m(\pi)=\dim ({\mathcal H}_{\pi}^{-\infty})^{H,\chi_ f\Delta^{1/2}_{H,G}}\).