On the quantization of spherical nilpotent orbits (Q2855938)

Let \( G \) be the real symplectic group \(\mathrm{Sp}(2n, \mathbb R) \) and denote by \( \mathfrak{g}_0 \) its Lie algebra and by \( \mathfrak{g} \) the complexification. The philosophy of the orbit method suggests that to each coadjoint \(G\)-orbit in \( \mathfrak{g}_0^{\ast} \) there is a unitary representation of \( G \). However, the coadjoint orbits which contribute to the unitary representations should be ``integrable'', i.e., there should be an integrable half-volume-form bundle on it. The pairs of orbits and the half-volume-form bundle are called admissible datum.NEWLINENEWLINEThe author considers spherical nilpotent orbits \( \mathcal O^G \) and an admissible datum \( L_{\chi} \) on it. Since the orbits are not simply connected, there are finitely many choices of \( L_{\chi} \). For \( G = Sp(2n, \mathbb R) \), there are at most four choices for each orbit.NEWLINENEWLINELet \( K \) be a maximal compact subgroup of \( G \) and \( K_{\mathbb C} \) its complexification. Then by the Kostant-Sekiguchi correspondence we can associate \( \mathcal O^G \) with a nilpotent \( K_{\mathbb C} \) orbit \( \mathcal O^K \). For an admissible datum \( (\mathcal O^K, L_{\chi}) \), the author considers the space of algebraic global sections \( \Gamma(\mathcal O^K, L_{\chi}) \) and determines the \( K \)-types on it. If \( \mathcal O^K \) has the boundary of codimension \( \geq 2 \), it coincides with the global sections over the closure \(\overline{\mathcal O^K} \), since the spherical nilpotent orbits are normal. The author compares it with the theta lifted representation \( \pi \) from the characters of \( O(p, q) \) in the stable range, and concludes that their \( K \)-characters match. So \( \Gamma(\mathcal O^K, L_{\chi}) \) inherits the structure of irreducible unitary \( (\mathfrak{g}, K_{\mathbb C}) \) modules which is predicted by Vogan.