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An analytic proof of the geometric quantization conjecture of Guillemin-Sternberg. (English) Zbl 0944.53047
This paper gives a new proof of the conjecture of V. Guillemin and S. Sternberg [Invent. Math. 67, 515-538 (1982; Zbl 0503.58018)] which loosely says that ‘geometric quantization commutes with symplectic reduction’. Let \(M\) be a compact symplectic manifold and let \(L\) be a Hermitian line bundle with a Hermitian connection whose curvature is the symplectic form on \(M\). Given a compatible almost complex structure on the tangent bundle of \(M\), one can construct a \(\text{Spin}^c\)-Dirac operator \(D^L\) on \(\Omega^{0,*}(M,L)\). Then the geometric quantization \(Q(M,L)\) can be defined as the virtual vector space which is the intersection of the kernel of \(D^L\) with \(\Omega^{0,\text{even}}\) minus its intersection with \(\Omega^{0,\text{odd}}\). Now, suppose that a compact connected Lie group \(G\) with Lie algebra \({\mathfrak g}\) acts symplectically on \(M\) with moment map \(\mu:M \to {\mathfrak g}^*\). If \(G\) acts freely on \(\mu^{-1}(0)\) then \(M_G = \mu^{-1}(0)/G\) is a symplectic manifold, known as the Marsden-Weinstein or symplectic reduction of \(M\) by \(G\) at 0. If the action of \(G\) on \(M\) lifts to a suitable action on \(L\), then \(L\) and its connection induce a Hermitian line bundle \(L_G\) with connection on \(M_G\), the almost complex structure on \(M\) induces an almost complex structure on \(M_G\), and the geometric quantization \(Q(M_G,L_G)\) can be constructed. The conjecture of Guillemin and Sternberg (which they themselves proved in a special case when \(M\) is Kähler) is that \(\dim Q(M,L)^G = \dim Q(M_G,L_G)\) where \(Q(M,L)^G\) is the \(G\)-fixed part of \(Q(M,L)\) which is now a virtual representation of \(G\). By the well known shifting trick, this implies similar formulas for the multiplicities of the nontrivial irreducible representations of \(G\) in \(Q(M,L)\) in terms of reductions of \(M\) at nonzero coadjoint orbits of \(G\). The conjecture has now been proved by several different methods under various different hypotheses. The paper under review gives a direct analytic proof of the conjecture and of some other related results, using a deformation of the \(\text{Spin}^c\)-Dirac operator and a Bochner-type formula for the Laplacian of the deformed operator, together with a modification of the work of J.-M. Bismut and G. Lebeau in [Publ. Math. Inst. Hautes Étud. Sci. 74, 1-297 (1991; Zbl 0784.32010)].

MSC:
53D50 Geometric quantization
53D20 Momentum maps; symplectic reduction
53C27 Spin and Spin\({}^c\) geometry
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