Toeplitz Quantization and Asymptotic Expansions: Geometric Construction

TL;DR

This paper introduces a geometric method for constructing invariant differential operators on symmetric domains, providing an asymptotic expansion framework analogous to star-products in quantization of Kähler manifolds.

Contribution

It develops the concept of star-restriction as a real analogue of star-products and offers a geometric construction for invariant differential operators on symmetric domains.

Findings

Defined star-restriction for symmetric domains

Constructed invariant differential operators geometrically

Established asymptotic expansion framework

Abstract

For a real symmetric domain $G_{\mathbb R}/K_{\mathbb R}$, with complexification $G_{\mathbb C}/K_{\mathbb C}$, we introduce the concept of "star-restriction" (a real analogue of the "star-products" for quantization of K\"ahler manifolds) and give a geometric construction of the $G_{\mathbb R}$-invariant differential operators yielding its asymptotic expansion.