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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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