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Let $K$ be a compact Lie group acting in a Hamiltonian way on a symplectic manifold $(M,\Omega )$ which is pre-quantized by a Kostant-Souriau line bundle. We suppose here that the moment map $\Phi$ is proper so that the reduced space $M_{\mu }:={\Phi }^{-1}(K·\mu )/K$ is compact for all $\mu$. Then, we can define the “formal geometric quantization” of $M$ as $${𝒬}_K^{-\infty }\left(M\right):=\sum _{\mu \in \widehat{K}}𝒬\left(M_{\mu }\right)V_{\mu }^K.$$ The aim of this article is to study the functorial properties of the assignment $(M,K)\to {𝒬}_K^{-\infty }\left(M\right)$.

In this paper we show that the multiplicities of holomorphic discrete series representations relative to reductive subgroups satisfy the credo “quantization commutes with reduction”.