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Zbl 1091.53059
Paradan, Paul-Émile
Spin$^c$-quantization and the $K$-multiplicities of the discrete series. (English)
[J] Ann. Sci. Éc. Norm. Supér. (4) 36, No. 5, 805-845 (2003). ISSN 0012-9593

Summary: In the 1970s, W. Schmid has shown that the representations of the discrete series of a real semi-simple Lie group $G$ could be realized as the quantization of elliptic coadjoint orbits. We show that such orbits, equipped with the Hamiltonian action of a maximal compact subgroup $K \subset G$, are non-compact examples where the philosophy of Guillemin-Sternberg -- Quantization commutes with reduction -- applies. If $\cal H_{\cal O}$ is a representation of the discrete series of $G$ associated to a coadjoint orbit $\cal O$, we express the $K$-multiplicities of $\cal H_{\cal O}$ in terms of Spin$^c$-index on symplectic reductions of $\cal O$.

MSC 2000:: *53D50 Geometric quantization
22E45 Analytic repres.of Lie and linear algebraic groups over real fields
53D20 Momentum maps; symplectic reduction
58J20 Index theory and related fixed point theorems

Cited in: Zbl 1193.22012

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