Geometric Quantization for Proper Moment Maps

We establish a geometric quantization formula for Hamiltonian actions of a compact Lie group acting on a non-compact symplectic manifold such that the associated moment map is proper. In particular, we resolve the conjecture of Vergne in this non-compact setting. 0. Introduction The famous geometric quantization conjecture of Guillemin and Sternberg [9] states that for a compact pre-quantizable symplectic manifold admitting a Hamiltonian action of a compact Lie group, the principle of “quantization commutes with reduction” holds. This conjecture was first proved independently by Meinrenken [14] and Vergne [23] for the case where the Lie group is abelian, and then by Meinrenken [15] in the general case. The singular reduction case was proved by Meinrenken-Sjamaar in [16]. There are also an analytic approach to the original conjecture developed by Tian and Zhang [20] as well as a proof developed by Paradan [17] by making use of the theory of transversally elliptic operators, see also [24] for an excellent survey. It is natural to consider the generalizations of the above results to actions on noncompact spaces. One of the aspects of this issue has been considered by Weitsman in [26], where the properness of the associated moment map is assumed. In her ICM2006 Plenary lecture [25], Vergne made a quantization conjecture (under the assumption that the zero point set of the vector field generated by the moment map is compact), which generalizes the original Guillemin-Sternberg conjecture to this non-compact setting. A special case of this conjecture had indeed been verified already by Paradan in [18] where he proved a quantization formula valid for the case where a maximal compact subgroup of a non-compact real semi-simple Lie group acts on the co-adjoint orbits of the real semi-simple Lie group itself. The purpose of this paper is to establish a general quantization formula in this framework of a compact group acting on a non-compact space with proper moment map. As we will see, our result could be viewed as an extended version of the conjecture of Vergne, in the sense that we do not make any extra assumptions beside the properness of the moment map. To be more precise, let (M,ω) be a non-compact symplectic manifold with symplectic form ω. We assume that (M,ω) is prequantizable, that is, there exists a complex line bundle L (called a prequantized line bundle) carrying a Hermitian metric h and a Date: December 22, 2008. 1 2 XIAONAN MA AND WEIPING ZHANG Hermitian connection ∇L such √ −1 2π ( ∇ )2 = ω. (0.1) We also assume that there exists an almost complex structure J on TM such that g(u, v) = ω(u, Jv), u, v ∈ TM (0.2) defines a Riemannian metric on TM . Let G be a compact connected Lie group with Lie algebra denoted by g. We assume that G acts on the left on M and this action can be lifted to an action on L. Moreover, we assume that G preserves g , J , h and ∇L. For any K ∈ g, let K be the vector field generated by K over M . Let μ : M → g∗ be defined by the Kostant formula [10] 2π √ −1μ(K) := μ(K) = ∇KM − LK , K ∈ g. (0.3) Then μ is the corresponding moment map, i.e. for any K ∈ g, dμ(K) = iKMω. (0.4) We call the G action with a moment map μ : M → g∗ verifying (0.4) a Hamiltonian action. From now on, we assume that the following fundamental assumption holds. Fundamental Assumption. The moment map μ : M → g∗ is proper, in the sense that the inverse image of a compact subset is compact. Let T be a maximal torus of G, CG ⊂ g∗ be a Weyl chamber associated to T , Λ∗ ⊂ g∗ be the weight lattice, and Ĝ = Λ∩CG be the set of dominate weights. Then the ring of characters R(G) of G has a Z-basis V G γ , γ ∈ Ĝ : V G γ is the irreducible G-representation with highest weight γ. Take any γ ∈ Ĝ. If γ is a regular value of the moment map μ, then one can construct the Marsden-Weinstein symplectic reduction (Mγ , ωγ), where Mγ = μ −1(G · γ)/G is a compact (as μ is proper) orbifold. Moreover, the line bundle L (resp. the almost complex structure J) induces a prequantized line bundle Lγ (resp. an almost complex structure Jγ) over (Mγ , ωγ). One can then construct the associated Spin -Dirac operator (twisted by Lγ), D Lγ + : Ω (Mγ , Lγ) → Ω(Mγ , Lγ) (cf. Section 1.6, (1.5)) on Mγ whose index Q(Lγ) := dimKer ( D Lγ + ) − dimCoker ( D Lγ + ) ∈ Z, (0.5) is well-defined. If γ ∈ Ĝ is not a regular value of μ, then by proceeding as in [16] (cf. [17, §7.4]), one still gets a well-defined quantization number Q(Lγ) extending the above definition. On the other hand, let g∗ be equipped with an AdG-invariant metric. Set H = |μ|2. Then since μ is proper, for any c > 0, Uc := H−1([0, c]) = {x ∈ M : H(x) 6 c} is a compact subset of M . 1See also Section 2.1 for a standard perturbative definition. GEOMETRIC QUANTIZATION FOR PROPER MOMENT MAPS 3 Recall that by Sard’s theorem, the set of critical values of the function H : M → R is of measure zero. Let XH = −J(dH)∗ be the Hamiltonian vector field associated to H. For any regular value c > 0 of H, one knows XH is nowhere zero on ∂Uc = H−1(c). Thus, according to Atiyah [1, §1, §3] and Paradan [17, §3] (cf. also Vergne [23]), the triple (Uc, X H, L) defines a transversally elliptic symbol (corresponding to the SpinDirac operator (twisted by L) on M) associated to the G-action on Uc. And according to Atiyah [1, §1], it admits a well-defined transversal index whose character is a distribution on G. For any γ ∈ Ĝ, let Q(L)c ∈ Z denote the γ-component of this transversal index. Theorem 0.1. For any γ ∈ Ĝ, there exists cγ > 0 such that Q(L)c ∈ Z does not depend on c > cγ, with c a regular value of H. And Q(L) c ∈ Z does not depend on c > 0, with c a regular value of H. According to Theorem 0.1, for any γ ∈ Ĝ, we have a well-defined integer Q(L)c not depending on the regular value c ≫ 0. From now on we denote it by Q(L) . We can now state our main result as follows. Theorem 0.2. For any γ ∈ Ĝ, the following identity holds, Q(L) = Q(Lγ). (0.6) Remark 0.3. If the zero set of XH is compact, then Theorem 0.1 was already known in [17] and [25], while Theorem 0.2 was conjectured by Vergne in [25, §4.3]. Thus Theorem 0.2 can be thought of as an extended version of the Vergne conjecture. If we set QG(L) −∞ = ⊕ γ∈ b G Q(L) V G γ , (0.7) then by Theorem 0.2, QG(L) −∞ equals to the formal geometric quantization in the sense of [26, Definition 4.1] and [19, Definition 1.2]. In particular, it verifies the functorial quantization property described as follows. It is clear that if M is compact, then Theorem 0.1 holds tautologically and Theorem 0.2 is the Guillemin-Sternberg conjecture proved in [16]. Let (N,ω) be such a pair with N being compact, F the notation for the prequantized line bundle over N , etc. Combining Theorem 0.2 with the result [26, Theorem 1] (cf. also [19, Theorem 1.5]) one gets the following functorial quantization result. Let L⊗F be the prequantized line bundle over M ×N obtained by the tensor product of the natural liftings of L and F to M ×N . Theorem 0.4. For the induced action of G on (M×N,ω⊕ωN) and L⊗F , the following identity holds,