Quantization commutes with reduction in the non-compact setting: the case of holomorphic discrete series

TL;DR

This paper proves that in the context of holomorphic discrete series representations, the process of quantization commutes with reduction when considering reductive subgroups, extending the principle to non-compact settings.

Contribution

It establishes the quantization commutes with reduction principle specifically for holomorphic discrete series in non-compact groups, a significant extension of existing theory.

Findings

Multiplicities of holomorphic discrete series satisfy the quantization commutes with reduction principle.

The result applies to non-compact reductive groups and their subgroups.

Provides a new understanding of representation multiplicities in this setting.

Abstract

In this paper we show that the multiplicities of holomorphic discrete series representations relatively to reductive subgroups satisfy the credo "Quantization commutes with reduction".