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Geometric quantization, reduction and decomposition of group representations. (English) Zbl 1149.53323

Summary: We consider a Hamiltonian action of a connected group \(G\) on a symplectic manifold \((P, \omega )\) with an equivariant momentum map \(J : P \rightarrow {\mathfrak{g}^{*}}\) and its quantization in terms of a Kähler polarization which gives rise to a unitary representation \({\mathcal{U}}\) of \(G\) on a Hilbert space \({\mathcal{H}}\). If \(O\) is a co-adjoint orbit of \(G\) quantizable with respect to a Kähler polarization, we describe geometric quantization of algebraic reduction of \(J ^{ - 1}(O)\). We show that the space of normalizable states of quantization of algebraic reduction of \(J ^{ - 1}(O)\) gives rise to a projection operator onto a closed subspace of \({\mathcal{H}}\) on which \({\mathcal{U}}\) is unitarily equivalent to a multiple of the irreducible unitary representation of \(G\) corresponding to \(O\). This is a generalization of the results of Guillemin and Sternberg obtained under the assumptions that \(G\) and \(P\) are compact and that the action of \(G\) on \(P\) is free. None of these assumptions are needed here.

MSC:

53D50 Geometric quantization
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