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 $\\hat{}\_\\hbar$ 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 $\\hbar$ 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 $\*\\hbar$, is a $G$-Fréchet algebra, and the formal expansion of the products around $\\hbar=0$ 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.