Pseudo-Kaehler Quantization on Flag Manifolds

TL;DR

This paper introduces a unified framework for geometric, symbol, and deformation quantizations on flag manifolds with pseudo-Kaehler structures, connecting classical and quantum perspectives.

Contribution

It develops a comprehensive approach to quantization on flag manifolds, linking geometric and deformation methods, and relates to Berezin's quantization in specific cases.

Findings

Realizes the Hilbert space via Bott-Borel-Weil theorem

Establishes a deformation quantization with separation of variables

Connects to Berezin's quantization through covariant and contravariant symbols

Abstract

A unified approach to geometric, symbol and deformation quantizations on a generalized flag manifold endowed with an invariant pseudo-Kaehler structure is proposed. The Hilbert space of states is realized via the Bott-Borel-Weil theorem in the sheaf cohomology of the geometric quantization line bundle. The corresponding deformation quantization is a quantization with separation of variables. In particular cases we arrive at Berezin's quantization via covariant and contravariant symbols.