Strict quantization of coadjoint orbits

Abstract

For every semisimple coadjoint orbit $\hat{\mathcal{O}}$ of a complex connected semisimple Lie group $\hat{G}$, we obtain a family of $\hat{G}$-invariant products ${\hat{\ast }}_{\hslash }$ on the space of holomorphic functions on $\hat{\mathcal{O}}$. For every semisimple coadjoint orbit $\mathcal{O}$ of a real connected semisimple Lie group $G$, we obtain a family of $G$-invariant products ${\ast }_{\hslash }$ on a space $\mathcal{A}(\mathcal{O})$ of certain analytic functions on $\mathcal{O}$ by restriction. $\mathcal{A}(\mathcal{O})$, endowed with one of the products ${\ast }_{\hslash }$, is a $G$-Fréchet algebra, and the formal expansion of the products around $\hslash =0$ determines a formal deformation quantization of $\mathcal{O}$, which is of Wick type if $G$ is compact. Our construction relies on an explicit computation of the canonical element of the Shapovalov pairing between generalized Verma modules and complex analytic results on the extension of holomorphic functions.