Connection between the Plancherel formula and the Howe dual pair \((Sp(2n,\mathbb R),O(k))\) (Q1566426)

The author explicitly describes the Plancherel formula for a weighted Bergman space \(H_k^2 (D)\) under the action of \(\widetilde{Sp}(2n,{\mathbb R})\times \widetilde{Sp}(2n,{\mathbb R})\). Here \(D\) is a bounded symmetric domain associated with the symplectic group. The author begins by realizing the \(k\)-th tensor power \(\sigma _k\) of the harmonic representation for the symplectic group on Fock space \(F_k^n\). One may use the dual pair \((Sp(2n,{\mathbb R}),O(k))\) to decompose \(\sigma _k\) into irreducibles under the symplectic group, obtaining \[ \sigma_k=\bigoplus_{\lambda\in\Sigma}\text{ dim}(V_\lambda)\sigma _k (\lambda), \] where \((\lambda, V_\lambda)\in \hat{O}(k)\), and \(\Sigma\) is as described by Kashiwara and Vergne. Next, a necessary and sufficient condition for \(\sigma _k (\lambda)\) to be in the discrete series is obtained. This condition is simply expressed in terms of a highest weight vector for \(\sigma _k (\lambda)\). Finally, the author exhibits a unitary isomorphism from \(H_k^2(D)\) to the space of \(O(k)\)-invariant functions on \(F^{2n}_k\). This, together with the decomposition mentioned above, induces a decomposition of \(H^2_k(D)\) under action of \(\widetilde{Sp}(2n,{\mathbb R})\times \widetilde{Sp}(2n,{\mathbb R})\). The Plancherel formula is then obtained through the identification of \(H^2_k(D)\) with \(H^2\)-functions on two-fold covering of the interior of an associated Olshanski-Stanton domain.