Moyal product and representations of solvable Lie groups (Q1908108)

Let \(G = \text{exp}({\mathfrak g})\) be a simply connected connected solvable Lie group with Lie algebra \(\mathfrak g\). Let \({\mathcal O} = \text{Ad}^*(G)l\) be a coadjoint orbit of \(G\). Such an orbit is not simply connected in general. Pukanszky has given an explicit description of its universal covering \({\mathcal O}_0 = G/G(l)_0\), where \(G(l)_0\) denotes the connected component of the stabilizer \(G(l)\) of \(l\) in \(G\). Let \(\widehat {G} (l) \subset G(l)\) be the reduced stabilizer of \(l\). Let \(\widehat{\mathcal O} = G/\widehat{G}(l)\). Choosing a complex positive \({\mathfrak n} = [{\mathfrak g}, {\mathfrak g}]\) admissible polarization \(\mathfrak h\) at \(l\), we obtain the unitary representation \(\pi(l) = \text{Hol} (l, \chi_0, {\mathfrak h})\) of \(G\) associated with \(l\) and the character \(\chi_0\) of \(G(l)_0\) whose differential is \(\text{il}_{|{\mathfrak g}(l)}\). Denoting by \(\Pi\) the character group of the free abelian discrete group \(\widehat{G}(l)/G(l)_0\), we have that \(\pi(l) = \int_{\Pi} \pi^\eta d\eta\) where \(\pi^\eta\) is the canonical representation of \(G\) associated with \(l\) and \(\eta\). The authors use special Darboux coordinates on \({\mathcal O}_0\) to define a Liouville measure and a Moyal \(*\)-product on \({\mathcal O}_0\). With this \(*\)-product there is associated a unitary representation of \(G \times G\) on \(L^2({\mathcal O}_0)\) which is shown to be unitarily equivalent to the representation \(\pi(l) \otimes \pi(l)^*\) on \(HS({\mathcal H}_{\pi(l)})\) (Theorem 3). Denoting by \(\mathcal K\) the space of all decomposable operators \[ A = \int_{\Pi} A^\eta d\eta,\quad A^\eta \in HS({\mathcal H}_\eta),\quad |A|^2 = \int_{\Pi} |A^\eta|^2 d\eta < \infty \] the authors construct a unitary transform \(\widehat{T} : L^2(\widehat{\mathcal O}) \to {\mathcal K}\) and a \(*\)-product on \(L^2(\widehat{\mathcal O})\) such that \(\widehat{T}(u * v) = \widehat {T} (u) \circ\widehat T(v)\) for \(u\), \(v\) in a dense subspace of \(L^2(\widehat {\mathcal O})\) (Theorem 4). In the last section the authors consider type I groups \(G\). In that case \(\widehat{G} (l) = G(l)\) for all \(l \in {\mathfrak g}^*\). The authors define an adapted Fourier transform for \(G\) which is based on their \(*\)-products on the orbits \(\mathcal O\).