Quantizations on nilpotent Lie groups and algebras having flat coadjoint orbits (Q2318050)

Let \(G\) be a simply connected and connected nilpotent Lie group and \(\mathfrak{g}\) its Lie algebra. The paper under review discusses various quantizations available and prove certain relations between them under the assumption of existence of flat co-adjoint orbits. The flatness condition is equivalent to the existence of an irreducible representation which is square integrable modulo the center. Main result is the equivalence of the Pedersen quantization and the global quantization on \(G\) which sends scalar valued functions on \(\mathfrak{g}^\ast\) to operator valued sections on \(\widehat{G}.\) For an irreducible representation \(\xi\) of \(G,\) let \(\Omega_\xi\) be the corresponding co-adjoint orbit in \(\mathfrak{g}^\ast\) given by the Kirillov theory. Then, \(\Omega_\xi\) is a homogeneous space and carries a canonical invariant measure \(\gamma_\xi.\) For \(\Psi \in \mathcal{S}(\Omega_\xi)\) define \[\widehat{\Psi} (X) = \int_{\Omega_\xi}~e^{-i\langle X, Y \rangle} \Psi(Y)\,d\gamma_\xi(Y), \quad X \in \mathfrak{g}.\] Then, \(\widehat{\Psi}\) is a \(C^\infty\) function on \(\mathfrak{g}\) and its restriction to the predual \(\omega_\xi\) is a linear topological isomorphism from \(\mathcal{S}(\Omega_\xi)\) to \(\mathcal{S}(\omega),\) denoted by \(F_\xi.\) If \(\psi \in \mathcal{S}(\omega_\xi)\) set \[\text{Dep}_\xi(\psi) = \int_{\omega_\xi}~\psi(X)~\xi(\exp X)~d\lambda_\xi(X), \] where \(\lambda_\xi\) is the Lebesgue measure (suitably normalized) on \(\omega.\) Then, for \(\Psi \in \mathcal{S}(\Omega_\xi)\) the Pedersen quantization is defined to be \[ \text{Ped}_\xi (\Psi) = \mathbf{Dep}_\xi (F_\xi (\Psi)). \] Identifying \(G\) with \(\mathfrak{g}\) via the exponential map and using the Fourier transform from \(\mathfrak{g}\) to \(\mathfrak{g}^\ast\) one gets a map \[\left ( \mathscr{F}^{-1}_{G, \mathfrak{g}^\ast}~w \right )(x) = \int_{\mathfrak{g}^\ast}~e^{i \langle \log x, X \rangle}~w(X)~dX.\] Let \(\mathscr{F}_{G, \widehat{G}}\) be the unitary group Fourier transform defined by \[\left ( \mathscr{F}_{G, \widehat{G}} u \right)(\xi) = \int_G~u(x)~\xi(x)^\ast~dx\] where \(dx\) is the Haar measure on \(G\) and \(\xi \in \widehat{G}.\) Finally define \(\mathscr{W}\) to be \[ \mathscr{W} = \mathscr{F}_{G, \widehat{G}} \circ \mathscr{F}_{G, \mathfrak{g}^\ast}^{-1}: \mathcal{S}(\mathfrak{g}^\ast) \to \mathscr{S}(\widehat{G})\] where \(\mathscr{S}(\widehat{G})\) is the image of \(\mathcal{S}(G)\) under the map \(\mathscr{F}_{G, \widehat{G}}.\) The main result is then the following: Theorem: Let \(G\) be an admissible group (that is, \(G\) admits a flat co-adjoint orbit). (i) For \(B \in \mathcal{S}(\mathfrak{g}^\ast)\) and \(\xi \in \widehat{G},\) set \[B_\xi = B|_{\Omega_\xi}~\text{and}~b(\xi) = \text{Ped}_\xi(B_\xi).\] Then \(b(\xi) \in \mathbb B(H_\xi)^\infty\) (the space of smooth vectors for \(\xi\)) and one has \[b = \mathscr{W}(B) \in \mathscr{S}(\widehat{G}).\] (ii) Conversely, let \[b \equiv \{b(\xi)|~\xi \in \widehat{G} \} \in \mathscr{S}(\widehat{G}).\] For every \(\xi \in \widehat{G}\) and every \(X \in \Omega_\xi,\) one has \[ \left [ \mathscr{W}^{-1}(b) \right ](X) = \int_{\omega_\xi}~e^{i \langle X, Y \rangle}~\operatorname{Tr}_\xi \left [ b(\xi) \xi(\exp Y)^\ast \right ]~d\lambda_\xi(Y).\]