Multidimensional quantization and the degenerate principal series representations (Q1423547)

Let \(G\) be a connected Lie group, \(g\) its Lie algebra, \(g_C\) the complexification of \(g\). For an element \(Z\) of \(g_C\) we denote its conjugate by \(\overline Z\). If \(a\) is a sub-set of \(g_C\) we pose \(a:= \{\overline Z;Z\in a\}\). Let \(g^*\) be the dual space of the Lie algebra \(g\), \({\mathcal O}(G)\) the orbit space of \(G\), \(\Omega\in{\mathcal O}(G)\) a \(K\)-orbit, \(F\in\Omega\) a fixed point on \(\Omega\), which is admissible in the sense of \textit{M. Duflo} [Lect. Notes Math., vol. 1077, 101--165 (1984; Zbl 0546.22014)], \(G_F\) the stabilizator of the point \(F\), \(\widetilde \sigma_{\chi_F}\) an irreducible unitary representation of \(G_F\), the restriction of which on the connected component \((G_F)_0\) is a multiple of \(\chi_F\), where \[ \chi_F \bigl(\exp(.) \bigr):=\exp \left(\frac{i} {\hbar} \langle F,.\rangle\right) \] and \(\hbar:=\frac {h}{2\pi}\) is the normalized Planck constant. Making use of the notion of polarization introduced by the author in [C. R. Acad. Sci., Paris, Sér. A 291, 295--298 (1980; Zbl 0459.58009)] one has as the main result the following theorem. Theorem 1. The polarization \((p,\rho, \sigma_0)\) with \(\dim \sigma_0=1\), or equivalently, the corresponding to \(L\) tangent \(G\) distribution, is being Lagrangian, is maximal if and only if \[ \sigma_0= \chi_F,\quad \text{codim}_g h=\tfrac 12\dim \Omega_F \] and \(h\) is subordinate to the functional \(F\), \(\langle F,[h,h]\rangle=0\). In other words, \(h\) is a polarization in the sense of M. Duflo.