Quantising proper actions on Spin c -manifolds
aa r X i v : . [ m a t h . DG ] J u l Quantising proper actions on Spin c -manifolds Peter Hochs ∗ and Varghese Mathai † October 30, 2018
Abstract
Paradan and Vergne generalised the quantisation commutes with reduction prin-ciple of Guillemin and Sternberg from symplectic to Spin c -manifolds. We extend theirresult to noncompact groups and manifolds. This leads to a result for cocompact ac-tions, and a result for non-cocompact actions for reduction at zero. The result forcocompact actions is stated in terms of K -theory of group C ∗ -algebras, and the resultfor non-cocompact actions is an equality of numerical indices. In the non-cocompactcase, the result generalises to Spin c -Dirac operators twisted by vector bundles. Thisyields an index formula for Braverman’s analytic index of such operators, in terms ofcharacteristic classes on reduced spaces. Contents
I Preliminaries 6 c -structures . . . . . . . . . . . . . . . . . . 9 Spin c -structures on reduced spaces 10 c -regular values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Spin c -slices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 Reduction and slices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 ∗ University of Adelaide, [email protected] † University of Adelaide, [email protected]
I Cocompact actions 17 µ ∇ . . . . . . . . . . . . . . . . . . . . . . . . 20 Spin c -structures on reduced spaces and fibred products 22 c -reduction at regular values . . . . . . . . . . . . . . . . . . . . . . . . . 225.2 Induced connections and momentum maps . . . . . . . . . . . . . . . . . . . 245.3 Spin c -reduction for fibred products . . . . . . . . . . . . . . . . . . . . . . . . 255.4 Spin c -structures on N ξ and M ξ . . . . . . . . . . . . . . . . . . . . . . . . . . 275.5 Quantisation commutes with induction . . . . . . . . . . . . . . . . . . . . . 28 III Non-cocompact actions 29 L -index . . . . . . . . . . . . . . . . . . . . . . . 306.2 Invariant quantisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.3 ρ -shifts and asymptotic results . . . . . . . . . . . . . . . . . . . . . . . . . . 336.4 An index formula for twisted Spin c -Dirac operators . . . . . . . . . . . . . . 34 A . . . . . . . . . . . . . . . . . . . . . . . . . . 39 D p,t . . . . . . . . . . . . . . . . . . . . . . . 428.2 Choosing the metric on M × g ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . 438.3 Proofs of the localisation estimates . . . . . . . . . . . . . . . . . . . . . . . . 458.4 Proofs of Theorems 6.6 and 6.8 . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Spin c -Dirac operators 47
10 Applications and examples 52 c -manifolds . . . . . . . . . . . . 5210.2 Generating examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5210.3 Characteristic classes and formal degrees . . . . . . . . . . . . . . . . . . . . 5410.4 Consequences of the index formula for twisted Dirac operators . . . . . . . 552 Introduction
Recently, Paradan and Vergne [28] generalised the quantisation commutes with reduction principle [13, 24, 25, 26, 33, 36] from the symplectic setting to the Spin c -setting. In thispaper, we extend their result to noncompact groups and manifolds. Whereas Paradanand Vergne use topological methods, we generalise Tian and Zhang’s analytic approach[15, 33] to possibly non-cocompact actions on Spin c -manifolds. This approach generalisesto Spin c -Dirac operators twisted by vector bundles, and implies an index formula forBraverman’s analytic index [7] of such operators. For cocompact actions, we generaliseand apply the KK -theoretic quantisation commutes with induction methods of [17, 18]. Ap-plications of our results include a proof of a Spin c -version of Landsman’s conjecture [20],and various topological properties of the index of twisted Spin c -Dirac operators for pos-sibly non-cocompact actions. The compact case
Cannas da Silva, Karshon and Tolman noted in [8] that Spin c -quantisation is the mostgeneral, and possibly natural, notion of geometric quantisation. This version of quanti-sation has a much greater scope for applications than geometric quantisation in the sym-plectic setting. It was shown in Theorem 3 of [8] that Spin c -quantisation commutes withreduction for circle actions on compact Spin c -manifolds, under a certain assumption onthe fixed points of the action. Paradan and Vergne’s result generalises this to actions byarbitrary compact, connected Lie groups, without the additional assumption made in [8].Paradan and Vergne considered a compact, connected Lie group K acting on a com-pact, connected, even-dimensional manifold M , equipped with a K -equivariant Spin c -structure. For a Spin c -Dirac operator D on M , they defined the Spin c -quantisation of theaction as Q Spin c K ( M ) := K -index ( D ) , which lies in the representation ring of K , and computed the multiplicities m π in Q Spin c K ( M ) = M π ∈ ^ K m π π. These multiplicities are expressed in terms of indices of Spin c -Dirac operators on reducedspaces M ξ := µ − (cid:0) Ad ∗ ( K ) ξ (cid:1) /K, where ξ ∈ k ∗ , and the Spin c -momentum map µ : M → k ∗ is a generalisation of the momen-tum map in symplectic geometry. The cocompact case
We first generalise this result to cocompact actions by a Lie group G on a manifold M , i.e.actions for which M/G is compact. This is achieved by applying the quantisation com-3utes with induction machinery of [17, 18] to it, together with a Spin c -slice theorem. Inthe cocompact case, one can define Spin c -quantisation using the analytic assembly map,denoted by G -index, from the Baum–Connes conjecture [2]: Q Spin c G ( M ) := G -index ( D ) ∈ K ∗ ( C ∗ G ) , where K ∗ ( C ∗ G ) is the K -theory of the maximal group C ∗ -algebra of G . This notion ofquantisation was introduced by Landsman [20] in the symplectic setting. He conjecturedthat quantisation commutes with reduction at zero in that case.To obtain a statement for reduction at nonzero values of the momentum map, weapply the natural map r ∗ : K ∗ ( C ∗ G ) → K ∗ ( C ∗ r G ) , where C ∗ r G is the reduced C ∗ -algebra of G . The group K ∗ ( C ∗ r G ) has natural generators [ λ ] ,which have representation theoretic meaning in many cases. The first main result in thispaper, Theorem 4.7, yields an expression for the multiplicities m λ in(1.1) r ∗ (cid:0) Q Spin c G ( M ) (cid:1) = X λ m λ [ λ ] . The non-cocompact case
In the symplectic setting, the invariant part of geometric quantisation was defined in [15]for possibly non-cocompact actions. Braverman [7] then combined techniques from [6]and [15] to extend this definition to general Dirac operators, and proved important prop-erties of the resulting index. We generalise the main result from [15] from symplectic toSpin c -manifolds. In addition, we obtain a generalisation to Spin c -Dirac operators twistedby arbitrary vector bundles E → M . This allows us to express Braverman’s index of suchoperators in terms of topological data on M .To be more precise, let D Ep be the Spin c -Dirac operator on M , twisted by E via a con-nection on E , for the Spin c -structure whose determinant line bundle is the p ’th tensorpower of the determinant line bundle of a fixed Spin c -structure. Then in Theorem 6.12,we obtain the index formula(1.2) index G D Ep = Z M ch ( E ) e p2 c ( L ) ^ A ( M ) , for p ∈ N large enough, where index G denotes Braverman’s index [7]. Here E := ( E | ( µ ∇ ) − ( ) ) /G ,and L := ( L | ( µ ∇ ) − ( ) ) /G . This equality holds if M is smooth, and a generalisation of theKirwan vector field has a cocompact set of zeros. This implies that M is compact. If M and G are both compact, analogous results were obtained in [30, 34].If M/G is noncompact, it is not clear a priori how to define a topological couterpartto Braverman’s index. Gromov and Lawson [11] face a similar problem in their study ofDirac operators on noncompact manifolds. They define a relative topological index, rep-resenting the difference of the indices of two operators satisfying their criteria, although4hese indices are not defined for each operator separately. They prove that the relativetopological index equals the difference of the analytical indices of the operators in ques-tion (Theorem 4.18 in [11]). Localisation to ( µ ∇ ) − ( ) allows us to give the topologicalexpression (1.2) for the index of a single twisted Spin c -Dirac operator, i.e. an ‘absolute’rather than a relative index formula. Applications and examples If M/G is compact, Theorem 6.8 implies that the main result of [23], which to a largeextent solves Landsman’s conjecture mentioned above, generalises to the Spin c -setting.We give a way to construct examples where our results apply, from cases where the groupacting is compact. The main result (1.1) in the cocompact case has a purely geometricconsequence, not involving K -theory and C ∗ -algebras. A special case of this consequenceis an expression for the formal degree of a discrete series representation in terms of an ^ A -type genus of the corresponding coadjoint orbit. Finally, the index formula (1.2) allowsus to draw conclusions about topological properties of the index of twisted Spin c -Diracoperators. These include an excision property, and a twisted version of Hirzebruch’ssignature theorem in the noncompact case. Outline of this paper
In Section 2, we first briefly recall the definition of Spin c -Dirac operators. Then we statethe definition of Spin c -reduction as in [28], and define stabilisation and destabilisation ofSpin c -structures in terms of Plymen’s two out of three lemma. We give conditions forreduced spaces to have naturally defined Spin c -structures in Section 3. We also discuss aSpin c -slice theorem, and its relation to Spin c -reduction.Section 4 contains the statements of Paradan and Vergne’s result from [28], and ourmain result on cocompact actions, Theorem 4.7. This result is proved in Section 5.The main result for untwisted Spin c -Dirac operators for possibly non-cocompact ac-tions, Theorem 6.8, is stated in Section 6. It is proved in Sections 7 and 8. The indexformula for Spin c -Dirac operators twisted by vector bundles is also stated in Section 6,and is proved in Section 9.Finally, in Section 10, we mention some applications of the main results, and a way toconstruct examples where they apply. Acknowledgements
The authors are grateful to Paul- ´Emile Paradan and Mich`ele Vergne, for useful commentsand explanations, and for making a preliminary version of their paper [29] available tothem. They would also like to thank Gennadi Kasparov for a helpful remark.The first author was supported by Marie Curie fellowship PIOF-GA-2011-299300 from5he European Union. The second author thanks the Australian Research Council for sup-port via the ARC Discovery Project grant DP130103924.
Notation and conventions
We will denote the dimension of a manifold Y by d Y . If a group H acts on Y , we denotethe quotient map Y → Y/H by q , or by q H to emphasise which group is acting. For afinite-dimensional representation space V of H , we write V Y for the trivial vector bundle M × V → M , with the diagonal H -action. (So that, for proper, free actions, V Y /H → Y/H is the vector bundle associated to the principal fibre bundle Y → Y/H .) If E → Y is areal vector bundle of rank r , we will refer to a principal Spin c ( r ) -bundle P E → Y suchthat E ∼ = P E × Spin c ( r ) R r as a Spin c -structure on E , without making explicit mention of thisisomorphism. Part I
Preliminaries
Let G be a Lie group, acting properly on a manifold M . Suppose M is equipped witha G -equivariant Spin c -structure. Let L → M be the associated determinant line bundle,and let a G -invariant, Hermitian connection ∇ on L be given. To these data, one canassociate a Spin c -Dirac operator on M in the usual way, as well as a Spin c -momentummap , as introduced by Paradan and Vergne [28]. This momentum map can be used todefine reduced spaces, which play a central role in the results in this paper. We mentionPlymen’s two out of three lemma, which we will use to construct Spin c -structures onthese reduced spaces in Section 3. Let S → M be the spinor bundle associated to the Spin c -structure on M . The connection ∇ and the Levi–Civita connection on TM (associated to the Riemannian metric inducedby the Spin c -structure), together induce a connection ∇ S on S , as discussed for examplein Proposition D.11 in [21]. The construction of the connection ∇ S involves local decom-positions S| U ∼ = S Spin U ⊗ L | on open sets U ⊂ M , where S Spin U is the spinor bundle associated to a local Spin-structure,to which the Levi–Civita connection lifts. 6et c : TM → End ( S ) be the Clifford action. Identifying T ∗ M ∼ = TM via the Riemannian metric, one gets anaction c : T ∗ M ⊗ S → S . The Spin c -Dirac operator associated to the Spin c -structure on M and the connection ∇ on L is then defined as the composition D : Γ ∞ ( S ) ∇ S −− → Ω ( M ; S ) c − → Γ ∞ ( S ) . Write d M := dim ( M ) . If { e , . . . , e d M } is a local orthonormal frame for TM , then, locally, D = d M X j = c ( e j ) ∇ S e j . For certain arguments, we will also need the operator D p on the vector bundle S p := S ⊗ L p , defined in the same way by a connection on S p which is induced by the Levi–Civitaconnection and ∇ , via local decompositions(2.1) S p | U ∼ = S Spin U ⊗ L | p + . Note that S p is the spinor bundle of the Spin c -structure on M obtained by twisting theoriginal Spin c -structure by the line bundle L p (see e.g. (D.15) in [21]). A Spin c -momentum map is a generalisation of the momentum map in symplectic geom-etry. It was used by Paradan and Vergne in [28]. (See also Definition 7.5 in [3].)For X ∈ g , let X M be the induced vector field on M , and let L EX be the Lie derivative ofsections of any G -vector bundle E → M . Definition 2.1.
The Spin c -momentum map associated to the connection ∇ is the map µ ∇ : M → g ∗ defined by (2.2) ∇ X = ∇ X M − L LX ∈ End ( L ) = C ∞ ( M ) , for any X ∈ g . Here µ ∇ X denotes the pairing of µ ∇ with X . In [28], a factor − i/2 is used instead of . Our convention is consistent with [15, 33]. c -momentum map is a special case of the notion of an abstractmoment map , as for example in Definition 3.1 of [12]. This is an equivariant map Φ : M → g ∗ such that for all X ∈ g , the pairing Φ X of Φ with X is locally constant on the set Crit ( X M ) ofzeros of the vector field X M . A Spin c -momentum map is an abstract moment map in thissense. This was already noted in the introduction to [26], and follows from the followingwell-known fact. Lemma 2.2.
For any G-equivariant line bundle L → M and a G -invariant connection ∇ on L ,and any X ∈ g , one has ∇ X = R ∇ (− , X M ) , with R ∇ the curvature of ∇ .Proof. Let u be any vector field on M . Then for all X ∈ g and s ∈ Γ ∞ ( L ) , one has(2.3) ∇ u ( µ ∇ X s ) = ( µ ∇ X ) s + ∇ X ∇ u s. This is also equal to(2.4) ∇ u (cid:0) ∇ X M − L LX (cid:1) s. Now ∇ u ∇ X M = ∇ X M ∇ u + ∇ [ u,X M ] + R ∇ ( u, X M ) . Also, by G -invariance of ∇ , ∇ u L LX s = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = ∇ u exp ( tX ) s = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = exp ( tX ) ∇ exp (− tX ) ∗ u s = L LX ∇ u s − ∇ [ X M ,u ] s. We conclude that (2.4) equals ∇ X M ∇ u s + ∇ [ u,X M ] s + R ∇ ( u, X M ) s − L LX ∇ u s + ∇ [ X M ,u ] s = ∇ X ∇ u s + R ∇ ( u, X M ) s. Since this expression equals (2.3), we find that ( µ ∇ X ) = R ∇ ( u, X M ) . Analogously to symplectic reduction [22], one can define reduced spaces in the Spin c -setting. 8 efinition 2.3. For any ξ ∈ g ∗ , the space M ξ := ( µ ∇ ) − ( ξ ) /G ξ = ( µ ∇ ) − (cid:0) Ad ∗ ( G ) ξ (cid:1) /G is the reduced space at ξ .As in the symplectic case, the stabiliser G ξ acts infinitesimally freely on µ − ( ξ ) , if ξ isa regular value of µ ∇ . Since M ξ ∼ = ( µ ∇ ) − ( ξ ) /G ξ , this implies that the reduced space M ξ is an orbifold if ξ is a regular value of µ ∇ , and the action is proper. Lemma 2.4.
In the setting of Lemma 2.2, let ξ ∈ g ∗ be a regular value of µ ∇ . Then for all m ∈ µ − ( ξ ) , the infinitesimal stabiliser g m is zero.Proof. In the situation of the lemma, let X ∈ g m . Then for all v ∈ T m M , we saw in Lemma2.2 that h T m µ ∇ ( v ) , X i = v ( µ ∇ X )( m ) = ∇ m ( v, X Mm ) = since X Mm = . Because T m µ ∇ is surjective, it follows that X = .(See Lemma 5.4 in [12] for a version of this lemma where G is a torus and µ ∇ is replacedby any abstract momentum map.) Spin c -structures To study Spin c -structures on reduced spaces, we will use the notions of stabilisation and destabilisation of Spin c -structures. These will also be used to obtain a Spin c -slice theoremin Subsection 3.2.Stabilisation and destabilisation are based on Plymen’s two out of three lemma . Lemma 2.5.
Let
E, F → M be oriented vector bundles with metrics, over a manifold M . Then Spin c -structures on two of the three vector bundles E , F and E ⊕ F determine a unique Spin c -structure on the third. The determinant line bundles L E , L F and L E ⊕ F of the respective Spin c -structures are related by L E ⊕ F = L E ⊗ L F . Proof.
See Section 3.1 of [32]. The uniqueness part of the statement refers to the construc-tions given there.
Remark 2.6.
Suppose a group G acts on the vector bundles E and F in Lemma 2.5, andthe two Spin c -structures initially given in the lemma are G -equivariant. Then the Spin c -structure on the third bundle, as constructed in Section 3.1 of [32], is also G -equivariant.Here one uses the fact that the actions by G on the spinor bundles associated to the Spin c -structures on E , F and E ⊕ F are compatible, since they are induced by the actions by G on E and F . 9 efinition 2.7. In the setting of Lemma 2.5, suppose E and F have Spin c -structures. Let P E be the Spin c -structure on E . Then the resulting Spin c -structure on E ⊕ F is the stabilisation Stab F ( P E ) → M. If F and E ⊕ F have Spin c -structures, and P E ⊕ F is the Spin c -structure on E ⊕ F , then theresulting Spin c -structure on E is the destabilisation Destab F ( P E ⊕ F ) → M. The terms stabilisation and destabilisation are motivated by the case where F is a trivialvector bundle. See also Section 3.2 in [32], Lemma 2.4 in [8] and Section D.3.2 in [12].We will use the following properties of stabilisation and destabilisation of Spin c -structures. Lemma 2.8.
Let
E, F → M be vector bundles with Spin c -structures over a manifold M . Then Stab E ◦ Destab E = id ; (2.5) Destab E ◦ Stab E = id ; (2.6) Stab E ◦ Stab F = Stab E ⊕ F ; (2.7) Destab E ◦ Destab F = Destab E ⊕ F . (2.8) (Here id means leaving Spin c -structures on the relevant bundles unchanged.)Proof. The relations (2.5) and (2.6) follow from the uniqueness part of Lemma 2.5. Theexplicit constructions in Section 3.1 of [32] imply that (2.7) and (2.8) hold. Spin c -structures on reduced spaces Consider the setting of Subsection 2.2. One can define quantisation of smooth or orbifoldreduced spaces using Spin c -structures induced by the Spin c -structure on M . If G is atorus, these are described in Proposition D.60 of [12]. In general, we will see that theSpin c -structure on M induces one on reduced spaces at Spin c -regular values of the Spin c -momentum map µ ∇ . In Proposition 3.5, we give a relation between Spin c -regular valuesand usual regular values. We then discuss how Abels’ slice theorem for proper actionscan be used in the Spin c -context, and how it is related to Spin c -reduction. The proofs ofthe main statements in this section will be given Section 5. Spin c -regular values For ξ ∈ g ∗ , we will denote the quotient map ( µ ∇ ) − ( ξ ) → M ξ by q . Definition 3.1.
A value ξ ∈ µ ∇ ( M ) of µ ∇ is a Spin c -regular value if • ( µ ∇ ) − ( ξ ) is smooth; 10 G ξ acts locally freely on ( µ ∇ ) − ( ξ ) ; and • there is a G ξ -invariant splitting TM | ( µ ∇ ) − ( ξ ) = q ∗ TM ξ ⊕ N ξ , for a vector bundle N ξ → ( µ ∇ ) − ( ξ ) with a G ξ -equivariant Spin-structure. Remark 3.2.
The third point in Definition 3.1 appears to have a choice of the bundle N ξ in it, but these are all isomorphic; the condition is really that the quotient bundle TM | ( µ ∇ ) − ( ξ ) /q ∗ TM ξ has a G ξ -equivariant Spin-structure.Note that a Spin-structure is equivalent to a Spin c -structure with a trivial determinantline bundle. In the equivariant setting, an equivariant Spin-structure is equivalent toa Spin c -structure with an equivariantly trivial determinant line bundle. Indeed, if thedeterminant line bundle of a Spin c -structure is equivariantly trivial, then its spinor bundleequals the spinor bundle of the underlying Spin-structure as equivariant vector bundles. Lemma 3.3. If ξ is a Spin c -regular value of µ ∇ , then the Spin c -structure on M induces anorbifold Spin c -structure on M ξ , with determinant line bundle L ξ := ( L | ( µ ∇ ) − ( ξ ) ) /G ξ → M ξ Proof.
We generalise the proof of Proposition D.60 in [12] to cases where G may not be atorus.We apply the equivariant version of Lemma 2.5 (see Remark 2.6) to the vector bundles q ∗ TM ξ and N ξ . This yields a G ξ -equivariant Spin c -structure on q ∗ TM ξ , with determinantline bundle L | ( µ ∇ ) − ( ξ ) . On the quotient M ξ , this induces an orbifold Spin c -structure, withdeterminant line bundle L ξ . Remark 3.4. If G ξ acts freely on ( µ ∇ ) − ( ξ ) , then one can also use the Spin c -structure(3.1) ( P M | ( µ ∇ ) − ( ξ ) ) /G ξ → M ξ , on ( TM | ( µ ∇ ) − ( ξ ) ) /G ξ , where P M → M is the given Spin c -structure on M . The determinantline bundle of (3.1) is L ξ . By the assumption on N ξ , Lemma 2.5 yields a Spin c -structure on TM ξ , with the same determinant line bundle.If G ξ only acts locally freely on ( µ ∇ ) − ( ξ ) , then one would need an orbifold version ofLemma 2.5 to use this argument.In the language of Definition 2.7, the Spin c -structure P M ξ on M ξ induced by the Spin c -structure P M on M equals(3.2) P M ξ = Destab N ξ (cid:0) P M | ( µ ∇ ) − ( ξ ) (cid:1) /G ξ . In Definition 3.1, it was not assumed that ξ is a regular value of µ ∇ in the usual sense,since this will not necessarily be the case in the situation considered in Subsection 3.2. If ξ is a regular value, then the first two conditions of Definition 3.1 hold by Lemma 2.4. Onecan use the following fact to check the third condition.11 roposition 3.5. Suppose that ξ is a regular value of µ ∇ , and that • G and G ξ are unimodular; • there is an Ad ( G ξ ) -invariant, nondegenerate bilinear form on g ; • there is an Ad ( G ξ ) -invariant subspace V ⊂ g such that g = g ξ ⊕ V ; and • there is an Ad ( G ξ ) -invariant complex structure on V .Then ξ is a Spin c -regular value of µ ∇ . Example 3.6. If g ξ = g , then the last two conditions in Proposition 3.5 are vacuous. There-fore, • if G is Abelian , any regular value of µ ∇ is a Spin c -regular value; • if is a regular value of µ ∇ , and G is semisimple, then is a Spin c -regular value. Example 3.7. If G is unimodular, and G ξ is compact (i.e. ξ is strongly elliptic), then one canuse an Ad ( G ξ ) -invariant inner product on g . Together with the standard symplectic formon V := g ⊥ ξ ∼ = g / g ξ ∼ = T ξ ( G · ξ ) , this induces an Ad ( G ξ ) -invariant complex structure on V (see e.g. Example D.12 in [12]).For semisimple Lie groups, strongly elliptic elements and coadjoint orbits correspondto discrete series representations, under an integrality condition. (See also [27].) Remark 3.8.
If the bilinear form in the second point of Proposition 3.5 is positive definiteon g ξ , then one can take V = g ⊥ ξ , and the third condition in Proposition 3.5 holds.If, on the other hand, the bilinear form is positive definite on V , then one has an in-duced Ad ( G ξ ) -invariant complex structure on V (as in Example 3.7), so the fourth condi-tion in Proposition 3.5 holds.We will prove Proposition 3.5 in Subsection 5.1. Spin c -slices Let G be an almost connected Lie group, and let K < G be a maximal compact subgroup.Let M be any smooth manifold, on which G acts properly. Then Abels showed (see p. 2 of[1]) that there is a K -invariant submanifold (or slice ) N ⊂ M such that the map [ g, n ] → g · n is a G -equivariant diffeomorphism G × K N ∼ = M. G × N by the K -action given by k · ( g, n ) = ( gk − , kn ) , for k ∈ K , g ∈ G and n ∈ N .Fix an Ad ( K ) -invariant inner product on g , and let p ⊂ g be the orthogonal com-plement to k . After replacing G by a double cover if necessary, we may assume thatAd : K → SO (p) lifts to(3.3) f Ad : K → Spin (p) . Indeed, consider the diagram e K f Ad / / π K (cid:15) (cid:15) Spin (p) π 2 : (cid:15) (cid:15) K Ad / / SO (p) , where e K := { ( k, a ) ∈ K × Spin (p); Ad ( k ) = π ( a ) } ; π K ( k, a ) := k ; f Ad ( k, a ) := a, for k ∈ K and a ∈ Spin (p) . Then for all k ∈ K , π − ( k ) ∼ = π − ( Ad ( k )) ∼ = Z , so π K is a double covering map. In what follows, we will assume the lift (3.3) exists.It was shown in Section 3.2 of [17] that a K -equivariant Spin c -structure P N on N inducesa G -equivariant Spin c -structure P M on M . In terms of stabilisation of Spin c -structures(Definition 2.7), one has(3.4) P M = G × K Stab p N ( P N ) . Here p N → N is the trivial vector bundle N × p → N , equipped with the K -action k ( n, X ) = ( kn, Ad ( k ) X ) , for k ∈ K , n ∈ N and X ∈ p . It has the K -equivariant Spin-structure(3.5) N × Spin (p) → N, with the diagonal K -action defined via the lift (3.3) of the adjoint action. To show that (3.4)defines a Spin c -structure on M , one uses the isomorphism(3.6) TM = G × K ( TN ⊕ p N ) c -structure P M → M on M , consider the K -equivariant Spin c -structure(3.7) P N := Destab p N ( P M | N ) → N on N . Here we again use (3.6). Lemma 3.9.
The constructions (3.4) and (3.7) are inverse to one another.Proof.
Starting with a K -equivariant Spin c -structure P N → N on N , we see that (2.6) im-plies that Destab p N (cid:0)(cid:0) G × K Stab p N ( P N ) (cid:1) | N (cid:1) = Destab p N (cid:0) Stab p N ( P N ) (cid:1) = P N . On the other hand, suppose P M → M is a G -equivariant Spin c -structure on M . Then wehave by (2.5), G × K Stab p N (cid:0) Destab p N ( P M | N ) (cid:1) = G × K ( P M | N ) , which is ismorphic to P M via the map [ g, f ] → g · f , for g ∈ G and f ∈ P M | N .Combining Abels’ theorem and Lemma 3.9, we obtain the following Spin c -slice theo-rem. Proposition 3.10.
For any G-equivariant
Spin c -structure P M on a proper G -manifold M , thereis a K -invariant submanifold N ⊂ M and a K -equivariant Spin c -structure P N → N such that M ∼ = G × K N , and P M = G × K Stab p N ( P N ) . Consider the situation of Subsection 3.2, and fix N and P N as in Proposition 3.10. Torelate Spin c -reductions of the actions by G on M and by K on N , we will use a relationbetween Spin c -momentum maps for these two actions. Let L M → M and L N → N be thedeterminant line bundles of P M and P N , respectively. Let ∇ M be a G -invariant Hermitianconnection on L M , let j : N ֒ → M be the inclusion map, and consider the connection ∇ N := j ∗ ∇ M on L N . Let µ ∇ M : M → g ∗ and µ ∇ N : N → k ∗ be the Spin c -momentum mapsassociated to these connections. Let Res gk : g ∗ → k ∗ be the restriction map. Lemma 3.11.
One has1. L N = L M | N ;2. L M = G × K L N ;3. µ ∇ N = Res gk ◦ µ ∇ M | N ; . if µ ∇ M ( n ) ∈ k ∗ for all n ∈ N , then (3.8) µ ∇ M ([ g, n ]) = Ad ∗ ( g ) µ ∇ N ( n ) , for all g ∈ G and n ∈ N . In the fourth point of this lemma, and in the rest of this paper, we embed k ∗ into g ∗ asthe annihilator of p . Proof.
The Spin-structure (3.5) on p N induces a Spin c -structure with equivariantly trivialdeterminant line bundle L p N → N . Since P N = Destab p N ( P M | N ) , Lemma 2.5 implies that L N = L N ⊗ L p N = L M | N . So the first claim holds, and the second claim follows from this: L M = G · L M | N = G × K L N .To prove the third claim, we use the first claim, and note that for all X ∈ k , ∇ N X = ∇ NX N − L L N X = (cid:16) ∇ MX M − L L M X (cid:17)(cid:12)(cid:12)(cid:12) Γ ∞ ( L N ) = ∇ M X | N . The fourth claim follows from the third.In the symplectic case, it was shown in Proposition 2.8 of [17] that one may take N =( µ ∇ M ) − (k ∗ ) . Then the condition in the fourth point of Lemma 3.11 holds, so one has (3.8).In the Spin c -setting, we use an arbitrary slice N . In Subsection 5.2, we show that a K -invariant connection ∇ N on L N induces a G -invariant connection ∇ M on L M such that thecondition in the fourth point of Lemma 3.11 is satisfied (see Lemma 5.3). From now on,we suppose that ∇ M was chosen in this way, so that (3.8) holds.In that case, a regular value of µ ∇ N is not necessarily a regular value of µ ∇ M . Indeed,any tangent vector to M at [ e, n ] , for n ∈ N , is of the form T ( e,n ) q ( X, v ) = X M [ e,n ] + v , for X ∈ g and v ∈ T n N . Using (3.8) one computes that T [ e,n ] µ ∇ M ( X M [ e,n ] + v ) = ad ∗ ( X ) (cid:0) µ ∇ N ( n ) (cid:1) + T n µ ∇ N ( v ) . If, for example, µ ∇ N ( n ) = , then T [ e,n ] µ ∇ M can only be surjective if g = k , even if is aregular value of µ ∇ N . However, all regular values of µ ∇ N are Spin c -regular values of µ ∇ M . Proposition 3.12. If ξ is a regular value of µ ∇ N , then it is a Spin c -regular value of µ ∇ M . Note that by the third point of Lemma 3.11, ξ is a regular value of µ ∇ N if and only if itis a regular value of Res gk ◦ µ ∇ M . Fix ξ ∈ k ∗ satisfying this condition, and let P M ξ → M ξ bethe Spin c -structure on M ξ as in Lemma 3.3.There is another way to define a Spin c -structure on M ξ , using the following fact.15 emma 3.13. For any η ∈ k ∗ , the inclusion map map N ֒ → M induces a homeomorphism N η ∼ = M η . Since ξ is a regular value of µ ∇ N , Proposition 3.5 implies that the Spin c -structure on N induces a Spin c -structure on N ξ , which equals M ξ . In the proof of Theorem 4.7, we willuse the fact that the two Spin c structures P M ξ and P N ξ are the same. Proposition 3.14.
The
Spin c -structures P M ξ and P N ξ on M ξ ∼ = N ξ are equal. Lemma 3.13 and Propositions 3.12 and 3.14 will be proved in Subsections 5.2–5.4.We end this section by mentioning a compatibility property of stabilising and desta-bilising Spin-structures with the fibred product construction that appears in the slice the-orem. This property will be used in the proof of Proposition 3.14. Suppose
H < G isany closed subgroup, acting on a manifold N , and let E → N be an H -vector bundle withan H -equivariant Spin c -structure P E → N . Then G × H P E → G × H N is a G -equivariantSpin c -structure for the G -vector bundle G × H E → G × H N (see Lemma 3.7 in [17]). In theproof of Proposition 3.14, we will use the fact that this construction is compatible withstabilisation and destabilisation. Lemma 3.15.
In the above setting, let F → N be another H -vector bundle.1. If P F → N is an H -equivariant Spin c -structure on P F , then G × H Stab E ( P F ) = Stab G × H E ( G × H P F ) .
2. If P E ⊕ F → N is an H -equivariant Spin c -structure on P E ⊕ F , then G × H Destab E ( P E ⊕ F ) = Destab G × H E ( G × H P E ⊕ F ) . Proof.
The first point follows from the explicit constructions in Section 3.1 of [32]. Hereone uses the fact that the spinor bundle associated to G × H P E is G × H S E , where S E → N is the spinor bundle associated to P E . This is compatible with the grading operators.The second point can be proved in a similar way, or deduced from the first point, byusing the fact that destabilisation is the inverse of stabilisation, as in (2.5) and (2.6). Remark 3.16.
We have only considered the principle Spin c ( r ) -bundle part P E → X of aSpin c -structure on a vector bundle E → X of rank r over a manifold X , not the isomor-phism P E × Spin c ( r ) R r ∼ = E. If E is the tangent bundle to X , then this isomorphism determines the Riemannian metricon X induced by the Spin c -structure. For cocompact actions, where we will apply thematerial in this subsection, the index of the Spin c -Dirac operator is independent of thismetric, however. 16 art II Cocompact actions
The main result on cocompact actions is Theorem 4.7, which states that that Spin c -quantisationcommutes with reduction at K -theory generators. In this section, we state Paradan andVergne’s result for compact groups and manifolds in [28], and Theorem 4.7 for cocompactactions. We will deduce Theorem 4.7 from Paradan and Vergne’s result in Section 5.We keep using the notation of Section 2. First of all, we define Spin c -quantisation of sufficiently regular reduced spaces, whichwill always be compact in the settings we consider. Let ξ be a Spin c -regular value of µ ∇ . Then by Lemma 3.3, the reduced space M ξ is a Spin c -orbifold. Suppose that M ξ iscompact and even-dimensional. Let D M ξ be the Spin c -Dirac operator on M ξ , defined withthe connection on the determinant line bundle L ξ → M ξ induced by a given connectionon the determinant line bundle L → M . Definition 4.1.
The Spin c -quantisation of M ξ is the index of D M ξ : Q Spin c ( M ξ ) := index ( D M ξ ) ∈ Z . Now suppose that G = K is compact and connected. Suppose that M is even-dimensional,and also compact and connected. Since M is even-dimensional, the spinor bundle S splitsinto even and odd parts, sections of which are interchanged by the Spin c -Dirac operator D . Because M is compact, this Dirac operator has finite-dimensional kernel, and one candefine(4.1) Q Spin c K ( M ) := K -index ( D ) = [ ker D + ] − [ ker D − ] ∈ R ( K ) , where D ± are the restrictions of D to the even and odd parts of S , repectively, and R ( K ) isthe representation ring of K .Let T < K be a maximal torus, with Lie algebra t ⊂ k . Let t ∗ + ⊂ t ∗ be a choice of (closed)positive Weyl chamber. Let R be the set of roots of (k C , t C ) , and let R + be the set of positiveroots with respect to t ∗ + . Set ρ K := X α ∈ R + α. Let F be the set of relative interiors of faces of t ∗ + . Then t ∗ + = [ σ ∈ F σ,
17 disjoint union. For σ ∈ F , let k σ be the infinitesimal stabiliser of a point in σ . Let R σ bethe set of roots of (cid:0) (k σ ) C , t C (cid:1) , and let R + σ := R σ ∩ R + . Set ρ σ := X α ∈ R + σ α. Note that, if σ is the interior of t ∗ + , then ρ σ = .For any subalgebra h ⊂ k , let (h) be its conjugacy class. Set H k := { (k ξ ); ξ ∈ k } . For (h) ∈ H k , write F (h) := { σ ∈ F ; (k σ ) = (h) } . Let (k M ) be the conjugacy class of the generic (i.e. minimal) infinitesimal stabiliser k M ofthe action by K on M . Note that by Lemma 2.4, one has (k M ) = if µ ∇ has regular values.Let Λ + ⊂ i t ∗ be the set of dominant integral weights. In the Spin c -setting, it is natu-ral to parametrise the irreducible representations by their infinitesimal characters, ratherthan by their highest weights. For λ ∈ Λ + + ρ K , let π Kλ be the irreducible representation of K with infinitesimal character λ , i.e. with highest weight λ − ρ K . Then one has, for such λ , Q Spin c ( K · λ ) = π Kλ , see Lemma 2.1 in [28].Write Q Spin c K ( M ) = M λ ∈ Λ + + ρ K m λ [ π Kλ ] , with m λ ∈ Z . Then Paradan and Vergne proved the following expression for m λ in termsof reduced spaces. Theorem 4.2 ([28], Theorem 3.4) . Suppose ([k M , k M ]) = ([h , h]) , for (h) ∈ H k . Then (4.2) m λ = X σ ∈ F (h) s.t. λ − ρ σ ∈ σ Q Spin c ( M λ − ρ σ ) . Here the quantisation Q Spin c ( M λ − ρ σ ) of the reduced space M λ − ρ σ is defined in Section4 of [28], which includes cases where Lemma 3.3 does not apply, and reduced spaces aresingular.If the generic stabiliser k M is Abelian , Theorem 4.2 simplifies considerably. As notedabove, this occurs in particular if µ ∇ has a regular value. Corollary 4.3. If k M is Abelian, then m λ = Q Spin c ( M λ ) . for ξ ∈ i k ∗ , we write M ξ := M ξ/i . roof. If one takes h = t in Theorem 4.2, then F (h) only contains the interior of t ∗ + . Hence ρ σ = , for the single element σ ∈ F (h) .In particular, if is a regular value of µ ∇ , then the invariant part of the Spin c -quantisationof M is(4.3) Q Spin c K ( M ) K = Q Spin c ( M ρ K ) , since π Kρ K is the trivial representation. Now suppose M and G may be noncompact, but M/G is compact. Then Landsman [14,20] defined geometric quantisation via the analytic assembly map from the Baum–Connesconjecture [2]. This takes values in the K -theory of the maximal or reduced group C ∗ -algebra C ∗ G or C ∗ r G of G . Landsman’s definition extends directly to the Spin c case. Definition 4.4. If M/G is compact, the Spin c -quantisation of the action by G on M is(4.4) Q Spin c G ( M ) := G -index ( D ) ∈ K ∗ ( C ∗ G ) , where G -index denotes the analytic assembly map.In this definition, the maximal C ∗ -algebra C ∗ G of G was used. By applying the map r ∗ : K ∗ ( C ∗ G ) → K ∗ ( C ∗ r G ) induced by the natural map r : C ∗ G → C ∗ r G , one obtains the reduced Spin c -quantisation Q Spin c G ( M ) r := r ∗ (cid:0) Q Spin c G ( M ) (cid:1) ∈ K ∗ ( C ∗ r G ) . (This is equal to (4.4), if G -index denotes the assembly map for C ∗ r G , but we include themap r ∗ to make the distinction clear.) If G is compact, then K ∗ ( C ∗ G ) and K ∗ ( C ∗ r G ) equalthe representation ring R ( G ) of G . Then the above definitions of Spin c -quantisation andreduced Spin c -quantisation both reduce to (4.1).Landsman used the reduction map R : K ∗ ( C ∗ G ) → Z induced on K -theory by the continuous map C ∗ G → C , Note that the word ‘reduced’ and the map r ∗ used here have nothing to do with reduction; this is justan unfortunate clash of terminology. C c ( G ) ⊂ C ∗ G is given by integration over G . If G is compact, then R : R ( G ) → Z is taking the multiplicity of the trivial representation. Landsman conjectured that(4.5) R (cid:0) Q G ( M ) (cid:1) = Q ( M ) , in the symplectic case (if M is smooth). Here quantisation is defined as in Definition 4.4,where D is a Dirac operator coupled to a prequantum line bundle.This conjecture was proved by Hochs and Landsman [14] for a specific class of groups G , and by Mathai and Zhang [23] for general G , where one may need to replace the pre-quantum line bundle by a tensor power. As a special case of Theorem 6.8, we will obtaina generalisation to the Spin c -setting of Mathai and Zhang’s result on the Landsman con-jecture (see Corollary 10.1). This asserts that (4.5) still holds for Spin c -quantisation, for awell-chosen Spin c -structure on M and a connection on its determinant line bundle. (SeeSubsection 6.3 for questions about ρ -shifts in this context.) µ ∇ Landsman’s conjecture was extended to reduction at K -theory classes corresponding tonontrivial representations in [17, 18]. Here one works with reduced quantisation, withvalues in K ∗ ( C ∗ r G ) .Because we will deduce the result in this subsection from Paradan and Vergne’s resultin [28], we now adopt their convention concerning the definition of the momentum map: − i2 µ ∇ X = ∇ X M − L LX . I.e., the factor in (2.2), which was chosen for consistency with [15, 33], is replaced by − i/2 . We use this convention in the present subsection, and in Section 5.Suppose G is almost connected, and let K < G be a maximal compact subgroup. Withnotation as in Subsection 4.1, one has R ( K ) = M λ ∈ Λ + + ρ K Z [ π Kλ ] . Set d := dim ( G/K ) . By the Connes–Kasparov conjecture, proved in [9] for almost con-nected groups, the Dirac induction mapD-Ind GK : R ( K ) → K d ( C ∗ r G ) is an isomorphism of Abelian groups, while K d + ( C ∗ r G ) = . In other words, the K -theorygroup K ∗ ( C ∗ r G ) is the free Abelian group generated by(4.6) [ λ ] := D-Ind GK [ π Kλ ] , for λ ∈ Λ + + ρ K , and these generators have degree d . For G semisimple with discreteseries, ‘most’ of the generators [ λ ] are associated to discrete series representations [19]. If G
20s complex-semisimple, they are associated to families of principal series representations[31]. See also [18].Since K d + ( C ∗ r G ) = , it follows that Q Spin c G ( M ) r = if d M and d have different parities.(Recall that we set d M := dim ( M ) .) So assume d M − d is even. In [18], the case where M carries a (pre)symplectic form was considered. It was conjectured that quantisationcommutes with reduction at any λ ∈ Λ + + ρ K , in the sense that(4.7) Q Spin c G ( M ) r = X λ ∈ Λ + + ρ K Q ( M λ )[ λ ] ∈ K d ( C ∗ r G ) . It was assumed that the momentum map image has nonzero intersection with the interiorof a positive Weyl chamber, to simplify the ρ -shifts that occur (analogously to the wayTheorem 4.2 simplifies to Corollary 4.3). We will not make this assumption in Theorem4.7.In the symplectic setting, a formal version of quantisation, defined as the right handside of (4.7), was extended to non-cocompact actions and studied in [16].Replacing G by a double cover if necessary, we may assume the lift (3.3) of the ad-joint action by K on p exists. Let the slice N ⊂ M and the Spin c -structure P N → N beas in Proposition 3.10. Since M/G is compact, N is compact in this case. We choose aconnection ∇ M on L M such that (3.8) holds.To quantise singular reduced spaces, we extend Definition 4.1 by using the homeo-morphism of Lemma 3.13 and Paradan and Vergne’s definition in the singular case. Recallthat µ ∇ N is the Spin c -momentum map for the action by K on N . Definition 4.5. If ξ ∈ k ∗ is a singular value of µ ∇ N , then Q Spin c ( M ξ ) := Q Spin c ( N ξ ) , where Q Spin c ( N ξ ) is defined as in Section 4 of [28].Note that different choices of N lead to homeomorphic reduced spaces by Lemma 3.13.If ξ is a regular value of µ ∇ N , then Definition 4.1 applies by Proposition 3.12. Because ofProposition 3.14, one has Q Spin c ( M ξ ) = Q Spin c ( N ξ ) in that case, so Definitions 4.1 and 4.5are consistent. Remark 4.6.
Another way of phrasing Definition 4.5 is that for any (singular) value ξ ∈ k ∗ of µ ∇ N , the shifted element ξ + ε ∈ k ∗ is a regular value for generic ε in a linear sub-space of k ∗ , and hence a Spin c -regular value of µ ∇ M by Proposition 3.12. Furthermore,the quantisation Q Spin c ( M ξ + ε ) is independent of such ε close enough to . Hence one candefine Q Spin c ( M ξ ) := Q Spin c ( M ξ + ε ) for such ε .To see that this is true, note that, as mentioned above, Proposition 3.14 implies that Q Spin c ( M ξ + ε ) = Q Spin c ( N ξ + ε ) ξ + ε is a regular value of µ ∇ N . The claim therefore follows from Theorem 5.4 in [28] if K is a torus, and from the arguments in Section 5.3 in [28] for general K .Paradan and Vergne’s result generalises to the cocompact setting in the following way. Theorem 4.7 (Spin c quantisation commutes with reduction; cocompact case) . If M and G are connected, and d M − d is even, then (4.8) Q Spin c G ( M ) r = X λ ∈ Λ + + ρ K m λ [ λ ] , with m λ given by (4.2) . This result will be proved in Section 5. We will use the constructions in Subsections3.2 and 3.3 and a quantisation commutes with induction result to deduce it from Paradanand Vergne’s result. In the symplectic setting, an additional assumption was needed in[17] to apply a similar kind of reasoning. The authors view this as a sign that it is verynatural to study the quantisation commutes with reduction problem in the Spin c -setting. Spin c -structures on reduced spaces and fibred products In this section, we prove the statements in Section 3. Together with a generalisation of the quantisation commutes with induction results in [17, 18], this allows us to deduce Theorem4.7 from Paradan and Vergne’s result, Theorem 4.2. Note that in Subsections 3.1 and 3.2,group actions were not asumed to be cocompact. So the statements made there applymore generally (and many will also be used in Part III). The cocompactness assumptionwill only be made in Subsection 5.5.Proposition 3.5, Lemma 3.13 and Propositions 3.12 and 3.14 are proved in Subsections5.1–5.4. In Subsection 5.5, we show that quantisation commutes with induction in theSpin c -setting, and use this to prove Theorem 4.7. Spin c -reduction at regular values We start by proving Proposition 3.5. Suppose ξ ∈ g ∗ is a regular value of µ ∇ . Then byLemma 2.4, G ξ acts locally freely on ( µ ∇ ) − ( ξ ) . Let q : ( µ ∇ ) − ( ξ ) → M ξ be the quotientmap. The restriction of TM to ( µ ∇ ) − ( ξ ) decomposes as follows. Lemma 5.1.
There is a G ξ -equivariant isomorphism of vector bundles (5.1) TM | ( µ ∇ ) − ( ξ ) = q ∗ TM ξ ⊕ g ∗ ⊕ g ξ , where G ξ acts on the right hand side by g (cid:0) ( m, v ) , η, X (cid:1) = (cid:0) ( gm, v ) , Ad ∗ ( g ) η, Ad ( g ) X (cid:1) , for g ∈ G ξ , m ∈ ( µ ∇ ) − ( ξ ) , v ∈ T G ξ · m M ξ , η ∈ g ∗ and X ∈ g ξ . roof. See (5.6) in [12] for the case where G is a torus. In general, since ξ is a regular valueof µ ∇ , we have the short exact sequence(5.2) → ker ( Tµ ∇ ) → TM | ( µ ∇ ) − ( ξ ) Tµ ∇ −− → ( µ ∇ ) − ( ξ ) × g ∗ → Now ker ( Tµ ∇ ) = T (cid:0) ( µ ∇ ) − ( ξ ) (cid:1) fits into the short exact sequence(5.3) → ker ( Tq ) → T (cid:0) ( µ ∇ ) − ( ξ ) (cid:1) Tq − → TM ξ → Since ker ( Tq ) is the bundle of tangent spaces to G ξ -orbits, and g ξ acts locally freely on ( µ ∇ ) − ( ξ ) by Lemma 2.4, we have(5.4) ker ( Tq ) ∼ = ( µ ∇ ) − ( ξ ) × g ξ , via the map ( m, X ) → X Mm , for ( m, X ) ∈ ( µ ∇ ) − ( ξ ) × g ξ .Combining (5.2), (5.3) and (5.4), we obtain the desired vector bundle isomorphism.Because of Lemma 5.1, Proposition 3.5 follows from the following fact. Lemma 5.2.
If the conditions in Proposition 3.5 hold, then the sub-bundle (5.5) ( µ ∇ ) − ( ξ ) × (g ∗ ⊕ g ξ ) → ( µ ∇ ) − ( ξ ) of (5.1) has a G ξ -equivariant Spin -structure.Proof.
Using the given Ad ( G ξ ) -invariant, nondegenerate bilinear form on g , and the sub-space V ⊂ g , we obtain an Ad ( G ξ ) -equivariant isomorphism(5.6) g ∗ ⊕ g ξ ∼ = (g ξ ⊕ g ξ ) ⊕ V. Identifying g ξ ⊕ g ξ ∼ = g ξ + i g ξ = (g ξ ) C , and using the given complex structure on V ,one gets an Ad ( G ξ ) -invariant complex structure on (5.6). This induces a G ξ -equivariantSpin c -structure on the vector bundle (5.5), with determinant line bundle(5.7) ( µ ∇ ) − ( ξ ) × ^ top C (cid:0) (g ξ ) C ⊕ V (cid:1) → ( µ ∇ ) − ( ξ ) . Since G and G ξ are unimodular, the adjoint action by G ξ on g , g ξ and hence V , has deter-minant one. Therefore, G ξ acts trivially on ^ top C (g ξ ) C ⊗ ^ top C V = ^ top C (cid:0) (g ξ ) C ⊕ V (cid:1) , so that the determinant line bundle (5.7) is equivariantly trivial. Hence the Spin c -structureon (5.5) is induced by a G -equivariant Spin-structure. (Compare this with the fact that thenatural embedding of U ( n ) into Spin c ( ) maps SU ( n ) into Spin ( ) .)23 .2 Induced connections and momentum maps In the rest of this section, we fix a slice N ⊂ M and a K -equivariant Spin c -structure P N → N as in Proposition 3.10.To prove Lemma 3.13, we will choose the connection ∇ M in such a way that the Spin c -momentum maps are related as in (3.8). Let ∇ N be a K -equivariant Hermitian connectionon the determinant line bundle L N → N . We will use the connection ∇ M on L M = G × K L N induced by ∇ N , as discussed in Section 3.1 in [17]. We briefly review the construction ofthis connection.Let p N : G × N → N be projection onto the second factor. For a K -invariant section s ∈ Γ ∞ ( G × N, p ∗ N L N ) K , one has the section σ ∈ Γ ∞ ( L M ) given by(5.8) σ [ g, n ] = [ g, s ( g, n )] . (Here s is viewed as a map G × N → L N .) For such an s , and for g ∈ G and n ∈ N , write s g ( n ) := s ( g, n ) =: s n ( g ) ∈ L Nn . This defines s g ∈ Γ ∞ ( L N ) and s n ∈ C ∞ ( G, L Nn ) ∼ = C ∞ ( G ) .Let q : G × N → M be the quotient map. Note that q ∗ L M ∼ = p ∗ N L N ∼ = G × L N → G × N, and that under this isomorphism, q ∗ σ corresponds to s . For X ∈ g , n ∈ N and v ∈ T n N ,one has Tq ( X, v ) ∈ T [ g,n ] M. Write X = X k + X p according to the decomposition g = k ⊕ p . Then the connection ∇ M isdefined by the properties that it is G -invariant, and satisfies(5.9) (cid:0) ∇ MTq ( X,v ) σ (cid:1) [ e, n ] = (cid:2) e, ( ∇ Nv s e )( n ) + X ( s n )( e ) + ∇ N X k ( n ) s ( e, n ) (cid:3) , for X ∈ g , n ∈ N , v ∈ T n N , and σ and s as above.Let µ ∇ N : N → k ∗ be the Spin c -momentum map associated to ∇ N , and let µ ∇ M : M → g ∗ be the Spin c -momentum map for the induced connection ∇ M . Lemma 3.13 followsdirectly from the relation (3.8) between µ ∇ N and µ ∇ M , which holds because of the fourthpoint in Lemma 3.11 and the following fact. Lemma 5.3.
For all n ∈ N , one has µ ∇ M ( n ) ∈ k ∗ . Recall that we consider k ∗ as a subspace of g ∗ by identifying it with the annihilator of p . Proof.
As in (5.8), let s ∈ Γ ∞ ( G × N, p ∗ N L N ) K , and let σ ∈ Γ ∞ ( L M ) be the associated sectionof L M . Let X ∈ g , and n ∈ N . Then one has ( L L M X σ )[ e, n ] = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = exp ( tX )[ exp (− tX ) , s ( exp (− tX ) , n )]= ddt (cid:12)(cid:12)(cid:12)(cid:12) t = [ e, s ( exp (− tX ) , n )]= [ e, X ( s n )( e )] ∈ L M [ e,n ] . Tq ( X, 0 ) = X M in (5.9), one therefore has ( ∇ MX M σ )[ e, n ] = ( L L M X σ )[ e, n ] + ∇ N X k ( n ) σ [ e, n ] . Here X = X k + X p according to the decomposition g = k ⊕ p . The claim follows. Spin c -reduction for fibred products We now turn to a proof of Proposition 3.12. For any group H acting on a manifold Y , weuse the notation q H for the quotient map Y → Y/H . If
H < K , we will write p Y for thetrivial bundle Y × p → Y , on which H acts via the adjoint representation on p . Proof of Proposition 3.12.
Let ξ ∈ k ∗ be a regular value of µ ∇ N . Since (3.8) holds, we have(5.10) ( µ ∇ M ) − ( Gξ ) = G × K ( µ ∇ N ) − ( Kξ ) . Because of this relation, it will be convenient to initially consider the restriction of TM to ( µ ∇ M ) − ( Gξ ) , rather than to ( µ ∇ M ) − ( ξ ) . Let(5.11) TN | ( µ ∇ N ) − ( Kξ ) = q ∗ K TN ξ ⊕ N KξN be a K -invariant splitting. By Lemma 5.4 below, we have a G -invariant splitting TM | ( µ ∇ ) − ( Gξ ) = q ∗ G TM ξ ⊕ N GξM , with N GξM = (cid:0) G × K N KξN (cid:1) ⊕ (cid:0) G × K p ( µ ∇ N ) − ( Kξ ) (cid:1) . By Lemma 5.5, the vector bundles G × K N KξN and G × K p ( µ ∇ N ) − ( Kξ ) over ( µ ∇ M ) − ( Gξ ) = G × K ( µ ∇ N ) − ( Kξ ) have G -equivariant Spin-structures. By Lemma2.5 and Remark 2.6, these induce a G -equivariant Spin c -structure on N GξM with equivari-antly trivial determinant line bundle, i.e. a G -equivariant Spin-structure. Restricting allbundles from ( µ ∇ M ) − ( Gξ ) to ( µ ∇ M ) − ( ξ ) , and group actions from G to G ξ , we obtain a G ξ -equivariant splitting TM | ( µ ∇ ) − ( ξ ) = q ∗ G ξ TM ξ ⊕ N ξM , where N ξM has a G ξ -equivariant Spin-structure. (cid:3) It remains to prove Lemmas 5.4 and 5.5, used in the proof of Proposition 3.12.25 emma 5.4.
One has (5.12) TM | ( µ ∇ ) − ( Gξ ) = q ∗ G TM ξ ⊕ N GξM , with N GξM = (cid:0) G × K N KξN (cid:1) ⊕ (cid:0) G × K p ( µ ∇ N ) − ( Kξ ) (cid:1) , and N KξN as in (5.11) .Proof.
Because of (3.6) and (5.10), we see that TM | ( µ ∇ M ) − ( Gξ ) = G × K (cid:0) TN | ( µ ∇ N ) − ( Kξ ) ⊕ p ( µ ∇ N ) − ( Kξ ) (cid:1) = G × K (cid:0) q ∗ K TN ξ ⊕ N KξN ⊕ p ( µ ∇ N ) − ( Kξ ) (cid:1) = q ∗ G TM ξ ⊕ (cid:0) G × K N KξN (cid:1) ⊕ (cid:0) G × K p ( µ ∇ N ) − ( Kξ ) (cid:1) . Lemma 5.5.
For a choice of the bundle N KξN as in (5.11) , and hence for any such bundle, the vectorbundles G × K N KξN and G × K p ( µ ∇ N ) − ( Kξ ) over ( µ ∇ M ) − ( Gξ ) = G × K ( µ ∇ N ) − ( Kξ ) have G -equivariant Spin -structures.Proof.
Since K is compact, and ξ is a regular value of µ ∇ N , Proposition 3.5 and Example3.7 imply that TN | ( µ ∇ N ) − ( ξ ) = q ∗ K ξ TN ξ ⊕ N ξN , where N ξN has a K ξ -equivariant Spin-structure P ξN . Set N KξN := K · N ξ . Then we have a K -equivariant vector bundle isomorphism K × K ξ N ξN ∼ = N KξN , given by [ k, v ] → T n k ( v ) , for n ∈ ( µ ∇ N ) − ( ξ ) , v ∈ ( N ξN ) n and k ∈ K . This extends to a G -equivariant isomorphism(5.13) G × K ξ N ξN ∼ = G × K N KξN
Now P GξN := G × K ξ P ξN → G × K ξ ( µ ∇ N ) − ( ξ ) ∼ = ( µ ∇ M ) − ( Gξ ) defines a Spin-structure on (5.13). 26urthermore, since the adjoint action by K on p lifts to Spin (p) , the vector bundle p ( µ ∇ N ) − ( Kξ ) has a K -equivariant Spin-structure ( µ ∇ N ) − ( Kξ ) × Spin (p) . As above, this induces a G -equivariant Spin-structure on G × K p ( µ ∇ N ) − ( Kξ ) → ( µ ∇ M ) − ( Gξ ) . Spin c -structures on N ξ and M ξ The last statement from Section 3 we prove is Proposition 3.14. As before, let ξ ∈ k ∗ be aregular value of µ ∇ N , and let let the Spin c -structure P N → N be as in Proposition 3.10. Toprove Proposition 3.14, we must show that the Spin c -structures induced on N ξ and M ξ ,induced by P N and P M respectively, via Propositions 3.5 and 3.12, coincide.We first give a slightly different description of Spin c -structures induced on reducedspaces from the expression (3.2). Lemma 5.6.
In the setting of Lemma 3.3, the
Spin c -structure P M ξ induced on M ξ equals P M ξ = Destab N Gξ (cid:0) P M | ( µ ∇ ) − ( Gξ ) (cid:1) /G, where N Gξ → ( µ ∇ ) − ( Gξ ) is a vector bundle with the property of N GξM in (5.12) , and with a G -equivariant Spin -structure.Proof.
By (3.2) and Lemma 3.15, we have P M ξ = Destab N ξ (cid:0) P M | ( µ ∇ ) − ( ξ ) (cid:1) /G ξ = (cid:0) G × G ξ Destab N ξ (cid:0) P M | ( µ ∇ ) − ( ξ ) (cid:1)(cid:1) /G = Destab G × Gξ N ξ (cid:0) G × G ξ (cid:0) P M | ( µ ∇ ) − ( ξ ) (cid:1)(cid:1) /G. Here N ξ → ( µ ∇ ) − ( ξ ) has a G ξ -equivariant Spin-structure P N ξ .Similarly to the proof of Lemma 5.5, set N Gξ := G · N ξ . Then G × G ξ N ξ ∼ = N Gξ . The left hand side has the G -equivariant Spin-structure G × G ξ P N ξ . Since also G × G ξ (cid:0) P M | ( µ ∇ ) − ( ξ ) (cid:1) ∼ = P M | ( µ ∇ ) − ( Gξ ) . the claim follows. 27 roof of Proposition 3.14. Let P N ξ → N ξ be the Spin c -structure on N ξ induced by P N becauseof Proposition 3.5, and let P M ξ → M ξ be the Spin c -structure on M ξ induced by P M becauseof Proposition 3.12. We saw in Proposition 3.10 that P M = G × K Stab p N ( P N ) . Let N GξM and N KξN be as in Lemma 5.4. Then, by Lemma 5.6, P M ξ = Destab N GξM (cid:0) P M | ( µ ∇ M ) − ( Gξ ) (cid:1) /G = Destab N GξM (cid:0)(cid:0) G × K Stab p N ( P N ) (cid:1) | ( µ ∇ M ) − ( Gξ ) (cid:1) /G = Destab N GξM (cid:0)
Stab G × K p N (cid:0) G × K ( P N | ( µ ∇ N ) − ( Kξ ) ) (cid:1)(cid:1) /G = Destab G × K N KξN (cid:0) G × K ( P N | ( µ ∇ N ) − ( Kξ ) ) (cid:1) /G. In the third equality, we have used the first point of Lemma 3.15 and (5.10). In the lastequality, we applied Lemmas 2.8 and 5.4. By the second point of Lemma 3.15, we con-clude that P M ξ = G × K (cid:0) Destab N KξN ( P N | ( µ ∇ N ) − ( Kξ ) ) (cid:1) /G = (cid:0) Destab N KξN ( P N | ( µ ∇ N ) − ( Kξ ) ) (cid:1) /K = P N ξ , by Lemma 5.6 (applied to the action by K on N ). (cid:3) Together with the constructions of Spin c -structures proved so far in this section, the quan-tisation commutes with induction techniques of [17, 18] allow us to deduce Theorem 4.7from Paradan and Vergne’s result, Theorem 4.2.We now suppose that M/G , and hence N is compact . The connections ∇ N and ∇ M induce Dirac operators on N and M , which can be used to define the quantisations ofthese manifolds. After the quantisation commutes with induction results of [17] (in thesymplectic setting) and [18] (in the presymplectic setting), the following Spin c -version ofthis principle is perhaps the most natural and general. Theorem 5.7 (Spin c -quantisation commutes with induction) . In the setting of Proposition3.10, the Dirac induction map
D-Ind GK maps the Spin c -quantisation of N to the Spin c -quantisationof M : D-Ind GK (cid:0) Q Spin c K ( N ) (cid:1) = Q Spin c G ( M ) r ∈ K ∗ ( C ∗ r G ) . Proof.
Let K K ∗ ( N ) and K G ∗ ( M ) be the equivariant K -homology groups [2] of N and M , re-spectively. In Theorem 4.6 in [17] and Theorem 4.5 in [18], a mapK-Ind GK : K K ∗ ( N ) → K G ∗ ( M )
28s constructed, such that the following diagram commutes: K G ∗ ( M ) r ∗ ◦ G -index / / K ∗ ( C ∗ r G ) K K ∗ ( N ) K-Ind GK O O K -index / / R ( K ) . D-Ind GK O O Here, as before, G -index is the analytic assembly map. The map K -index is the analyticassembly map for the action by K on N , which coincides with the usual equivariant index.In Section 6 of [17], it is shown that the map K-Ind GK maps the class [ D N ] ∈ K K0 ( N ) of to the Spin c -Dirac operator D N on N , to the class [ D M ] ∈ K Gd ( M ) of the Spin c -Dirac operator D M on M . Although in [17] the symplectic setting is con-sidered, the arguments in Section 6 of that paper are stated purely in terms of Spin c -structures. Hence they apply in this more general setting, and we conclude thatD-Ind GK (cid:0) Q Spin c K ( N ) (cid:1) = D-Ind GK (cid:0) K -index [ D N ] (cid:1) = r ∗ ◦ G -index (cid:0) K-Ind GK [ D N ] (cid:1) = r ∗ ◦ G -index [ D M ]= Q Spin c G ( M ) r . Theorem 4.7 follows by combining Theorem 5.7, Proposition 3.10, Proposition 3.14,and Paradan and Vergne’s Theorem 4.2.
Proof of Theorem 4.7.
By Proposition 3.10, Theorem 5.7 and Theorem 4.2, we have Q Spin c G ( M ) r = D-Ind GK (cid:0) Q Spin c K ( N ) (cid:1) = X λ ∈ Λ + + ρ K m λ [ λ ] , with m λ as in (4.2), where Q Spin c ( M ξ ) is replaced by Q Spin c ( N ξ ) for all ξ that occur. ByDefinition 4.5, these two quantisations are equal if ξ is a singular value of µ ∇ N . If ξ is aregular value of this map, they are equal by Proposition 3.14, and the claim follows. (cid:3) art III Non-cocompact actions
The main result in this paper for untwisted Spin c -Dirac operators, for possibly non-cocompactactions and reduction at zero, is Theorem 6.8. We state it in Subsection 6.2, and prove itin Sections 7 and 8. The generalisation of this result to Spin c -Dirac operators twisted byvector bundles, Theorem 6.12, is stated in Subsection 6.4. It is proved in Section 9.While the proof of Theorem 4.7 in Section 5 was based on Paradan and Vergne’s resultin [28], our proofs of Theorems 6.8 and 6.12 are independent of their result.To state a Spin c -quantisation commutes with reduction result without assuming that M/G is compact, we recall some facts about the G -invariant, transversally L -index in-troduced in Section 4 of [15]. We now suppose that G is unimodular , and fix a left- andright-invariant Haar measure dg on G . L -index The definition of the invariant, transversally L -index involves cutoff functions . Definition 6.1.
Let G be a unimodular locally compact group acting properly on a locallycompact Hausdorff space X . A cutoff function is a continuous function f on X such that thesupport of f intersects every G -orbit in a compact set, and for all x ∈ X , one has Z G f ( gx ) dg = with respect to a Haar measure dg on G .It is shown in Proposition 8 in Section 2.4 of Chapter 7 in [5] that cutoff functions exist.Let E → M be a G -equivariant vector bundle, equipped with a G -invariant metric. Let L ( E ) be the L -space of sections of E , with respect to this metric, and the density on M associated to the Riemannian metric induced by the Spin c -structure. Definition 6.2.
The space L ( E ) of transversally L -sections of E is the space of measurablesections s of E such that fs ∈ L ( E ) for all cutoff functions f on M , up to equality almosteverywhere.One can show that for a G -invariant transversally L -section s ∈ L ( E ) G , the L -normof fs does not depend on the cutoff function f (see Lemma 4.4 in [15]). This turns the G -invariant part L ( E ) G of L ( E ) into a Hilbert space.Let D be a G -equivariant (differential) operator on Γ ∞ ( E ) . Suppose E is Z -graded,and that D is odd with respect to this grading.30 efinition 6.3. The transversally L -kernel of D isker L ( D ) := ker ( D ) ∩ L ( E ) . If the G -invariant part ker L ( D ) G of ker L ( D ) is finite-dimensional, then the G -invariant,transversally L -index of D is the integerindex GL ( D ) := dim (cid:0) ker L ( D + ) G (cid:1) − dim (cid:0) ker L ( D − ) G (cid:1) , where D ± is the restriction of D to the even or odd part of Γ ∞ ( E ) . Remark 6.4. If G is compact, then the transversally L -index of D is the G -invariant partof its L -index. If M/G is compact, then the transversally L -index of D is the index of D restricted to G -invariant smooth sections. As shown in [15], the transversally L -index of Definition 6.3 allows one to make sense ofquantisation and reduction without assuming M , G or M/G to be compact. There willonly be a cocompactness assumption on the set of zeros of a vector field on M . This vectorfield is defined in terms of the momentum map and a family of inner products on g ∗ , bywhich we mean a metric on the vector bundle g ∗ M := M × g ∗ → M, with a certain G -invariance property. Using such a family of inner products, rather thana single one, allows us to define a suitable G -invariant vector field, despite the fact that g does not admit an Ad ( G ) -invariant inner product in general.Let { (− , −) m } m ∈ M be a G -invariant metric on the vector bundle g ∗ M , with respect to the G -action given by g · ( m, ξ ) = ( g · m, Ad ∗ ( g ) ξ ) , for g ∈ G , m ∈ M and ξ ∈ g ∗ . Such a metric exists by Lemma 2.1 in [15]. Consider themap ( µ ∇ ) ∗ : M → g defined by(6.1) h ξ, ( µ ∇ ) ∗ ( m ) i = (cid:0) ξ, µ ∇ ( m ) (cid:1) m , for all ξ ∈ g ∗ and m ∈ M . This induces a G -invariant vector field v ∇ on M , given by(6.2) v ∇ m := (cid:0) ( µ ∇ ) ∗ ( m ) (cid:1) Mm = (cid:12)(cid:12)(cid:12)(cid:12) t = exp (cid:0) t ( µ ∇ ) ∗ ( m ) (cid:1) m, for m ∈ M . (The factor was included for consistency with [15, 33].) A central assump-tion we make is that the critical set Crit ( v ∇ ) of zeros of v ∇ is cocompact . This implies that M is compact.Recall the definition of the Dirac operator D p in Subsection 2.1, for a p ∈ N . We willapply the invariant, transversally L -index to a Witten-type deformation of D p .31 efinition 6.5. For p ∈ N and t ∈ R , the deformed Dirac operator D p,t is the operator D p,t := D p + it2 c ( v ∇ ) on Γ ∞ ( S p ) .Note that D = D + i2 c ( v ∇ ) . In general, D p,t is G -equivariant, by G -invariance of v ∇ . Suppose that M is even-dimensional.Then S p is Z -graded, and D p,t is odd with respect to this grading.Suppose M is complete in the Riemannian metric induced by the Spin c -structure. Itturns out that in this non-cocompact setting, the invariant, transversally L -index of D p,t is well-defined for large enough t . Theorem 6.6.
One can choose the metric on g ∗ M in such a way that for all t ≥ , the G -invariantpart of ker L ( D p,t ) is finite-dimensional, for all p ∈ N . This allows us to define the G -invariant part of Spin c -quantisation. Definition 6.7.
The G -invariant Spin c -quantisation of M with respect to the given Spin c -structure, and the connection ∇ on L , is Q Spin c ( M ) G := index GL ( D ) . Suppose is a Spin c -regular value of µ ∇ . By Proposition 3.5 and Example 3.6, this istrue for example if is a regular value of µ ∇ and G is semisimple or Abelian. Alternatively,by Proposition 3.12, it is enough that is a regular value of a Spin c -moment map µ ∇ N : N → k ∗ on a Spin c -slice N . Since M is compact by cocompactness of Crit ( v ∇ ) , Definition4.1 applies, and one has Q Spin c ( M ) = index ( D M ) . Analogously to the symplectic case [15] and the compact case (4.3), one expects Spin c -quantisation to commute with reduction in this non-cocompact setting. We will prove thefollowing version of this statement. Theorem 6.8 (Spin c -quantisation commutes with reduction; non-cocompact case) . Sup-pose G acts freely on ( µ ∇ ) − ( ) (rather than just locally freely). Then there exists a G -equivariant Spin c -structure on M and a connection on the corresponding determinant line bundle, such that,for these choices, (6.3) Q Spin c ( M ) G = Q Spin c ( M ) ∈ Z . It will turn out that, for a natural choice of ∇ ′ on the determinant line bundle of the Spin c -structureused, the Spin c -momentum maps for ∇ and ∇ ′ differ by a nonzero factor, so that the condition that G actsfreely on ( µ ∇ ) − ( ) is the same for the two connections. emark 6.9. The choice of Spin c -structure in Theorem 6.8 amounts to taking large enoughtensor powers of the determinant line bundle of a given Spin c -structure. I.e. one startswith an initial Spin c -structure P → M with determinant line bundle L → M , and the resultholds for Spin c -structures with determinant line bundle L p → M , for p large enough. Soif L is not a torsion class in H ( M ; Z ) , then the result holds for infinitely many Spin c -structures.The connection on the determinant line bundle L p used can be any connection inducedby a connection on L (and the minimal value of p depends on this inital connection on L ). Remark 6.10.
We could prove Theorem 6.6 by referring to [7] and using the elliptic reg-ularity arguments in [15]. We will give an independent proof of finite-dimentionalityof ker L ( D p,t ) G , however, as a by-product of the localisation arguments needed to proveTheorem 6.8. ρ -shifts and asymptotic results If M and G are compact, one may take t = in Definition 6.7. Then Q Spin c ( M ) G isthe invariant part of (4.1), which by (4.3) equals Q ( M ρ K ) . On the other hand, Theorem6.8 states that, for a certain G -equivariant Spin c -structure on M and a connection on itsdeterminant line bundle, Q Spin c ( M ) G = Q Spin c ( M ) . Hence, apparently, one has(6.4) Q ( M ) = Q ( M ρ K ) for this choice of Spin c -structure and connection.This potential contradiction can be resolved, by noting that, for the Spin c -structureand the connection ∇ ′ used, one has µ ∇ ′ = pµ ∇ , for a connection ∇ on the determinant line bundle of a Spin c -structure initially given, anda large enough integer p . (See (8.9) in the proof of Proposition 8.6.) For any ξ ∈ g ∗ , let M ξ and M ′ ξ be the reduced spaces at ξ for the momentum maps µ ∇ and µ ∇ ′ , respectively.Then M ′ ξ = M ξ/p . In particular, M ′ = M , and M ′ ρ K = M ρ K /p .The statement (6.4) is therefore that Q ( M ρ K /p ) = Q ( M ) , for p large enough. In the symplectic setting, this follows from the fact that Q ( M ξ ) is inde-pendent of small variations of ξ (see Theorem 2.5 in [25] if the action is free on ( µ ∇ ) − ( ξ ) ,33r [38] for a holomorphic version). More generally, if M is of the form M = G × K N as inSubsection 4.3, then by Proposition 3.14, one has Q ( M ξ ) = Q ( N ξ ) , which is independent of small variations of ξ if N is a compact Hamiltonian K -manifold(but M is not necessarily symplectic).In the general non-cocompact setting of Subsection 6.2, this leads one to expect that, if µ ∇ is G -proper (in the sense that the preimage of any cocompact set is cocompact), thereis an open neighbourhood U of in g ∗ , such that for all Spin c -regular values ξ ∈ U of µ ∇ , Q ( M ξ ) = Q ( M ) . The above arguments show that, for ‘asymptotic’ quantisation commutes with reduc-tion results, reduction at zero (or possibly a nearby regular value of the momentum map)is really the only natural case to consider.
Spin c -Dirac operators The main results on Spin c -Dirac operators in the non-cocompact case, Theorems 6.6 and6.8, generalise to Spin c -Dirac operators twisted by arbitrary vector bundles. We use this toobtain an index formula for Braverman’s analytic index of such operators, Theorem 6.12,expressing it in terms of characteristic classes on M . A potentially interesting feature ofthis formula is that it involves localisation to ( µ ∇ ) − ( ) . In the setting we consider, wherethe manifold M , the group G acting on it, and the quotient M/G may all be noncompact,it is unlikely that there is a topological expression for the index of (twisted) Spin c -Diracoperators in terms of characteric classes on M . However, localisation to ( µ ∇ ) − ( ) allowsus to still define a meaningful topological index, as an integral over the compact space M .In the compact setting, the index of any elliptic operator on a Spin c -manifold equalsthe index of a twisted Spin c -Dirac operator. Hence index formulas for the latter kind ofoperators immediately generalise to the former. In the noncompact setting we considerhere, such a principle is not (yet) available. Still, the index formula we obtain for twistedSpin c -Dirac operators strongly suggests a more general underlying equality of topologicaland analytic indices.Fix p ∈ N . We retain all other notation used previously. In particular, we have the con-nection ∇ S p on S p , and the Spin c -moment map µ ∇ : M → g ∗ induced by a connection ∇ on the determinant line bundle L → M . In addition, consider a Hermitian, G -equivariantvector bundle E → M . Let ∇ E be a Hermitian, G -invariant connection on E . Consider theconnection ∇ S p ⊗ E := ∇ S p ⊗ E + S p ⊗ ∇ E on S p ⊗ E . 34 efinition 6.11. The twisted
Spin c -Dirac operator associated to ∇ and ∇ E is the composition D Ep : Γ ∞ ( S p ⊗ E ) ∇ S p ⊗ E −−−− → Ω ( M ; S p ⊗ E ) c ⊗ E −−− → Γ ∞ ( S p ⊗ E ) . For t ∈ R , the deformed Spin c -Dirac operator twisted by E via ∇ E is the operator D Ep,t := D Ep + it2 c ( v ∇ ) ⊗ E , Theorems 6.6 and 6.8 generalise to the operator D Ep as follows. Theorem 6.12.
Suppose that is a Spin c -regular value of µ ∇ , and that G acts freely on ( µ ∇ ) − ( ) .Then there are a G -invariant metric on g ∗ M and a p E ∈ N such that if p ≥ p E , then (cid:0) ker L D Ep,1 (cid:1) G is finite-dimensional, and one has index GL D Ep,1 = index D E M = Z M ch ( E ) e p2 c ( L ) ^ A ( M ) . Here E := ( E | ( µ ∇ ) − ( ) ) /G and L := ( L | ( µ ∇ ) − ( ) ) /G . In the compact case, results analogous to Theorem 6.12 were obtained in [30, 34]. The-orem 6.12 will be proved in Section 9. Some applications are given in Subsection 10.4.
We now turn to proving Theorems 6.6 and 6.8. As in [15, 33], the starting point is anexplicit formula, given in Theorem 7.1, for the square of the deformed Dirac operator D p,t of Definition 6.5. This is the basis of the localisation estimates, Propositions 8.1 and 8.2,that will be used to prove Theorems 6.6 and 6.8.We continue using the notation of Section 2 and Subsection 6.2. We will also write d M and d G for the dimensions of M and G , respectively. We denote the Riemannian metric on M induced by the given Spin c -structure by (− , −) . The associated Levi–Civita connectionon TM will be denoted by ∇ TM . Let us fix some notation that will be used in the expression for D . Let { h , . . . , h d G } be anorthonormal frame for g ∗ M with respect to a given G -invariant metric. (Such a frame canbe obtained for example by applying the Gram-Schmidt procedure to a constant frame.)35et { h ∗ , . . . , h ∗ d G } be the dual frame of M × g → M . Let µ ∇ , . . . , µ ∇ d G ∈ C ∞ ( M ) be thefunctions such that(7.1) µ ∇ = d G X j = µ ∇ j h j , so that ( µ ∇ ) ∗ = d G X j = µ ∇ j h ∗ j , and(7.2) v ∇ = d G X j = µ ∇ j V j , where V j is the vector field given by(7.3) V j ( m ) = (cid:0) h ∗ j ( m ) (cid:1) Mm , at a point m ∈ M . Consider the norm-squared function H ∇ of µ ∇ , given by(7.4) H ∇ ( m ) = k µ ∇ ( m ) k = d G X j = µ ∇ j ( m ) . Here k · k m is the norm on g ∗ induced by (− , −) m .We will use the operators L S p h ∗ j on Γ ∞ ( S p ) given by (cid:0) L S p h ∗ j s (cid:1) ( m ) = (cid:0) L S p h ∗ j ( m ) s (cid:1) ( m ) . Finally, for any vector field u on M , consider the commutator vector field [ u, ( h ∗ j ) M ] , givenby [ u, ( h ∗ j ) M ]( m ) = (cid:2) u, h ∗ j ( m ) M (cid:3) ( m ) . Here h ∗ j ( m ) M is the vector field induced by h ∗ j ( m ) ∈ g , and [− , −] is the Lie bracket ofvector fields. Importantly, for fixed m , the vector fields V j and h ∗ j ( m ) M are equal at thepoint m , but not necessarily at other points.The square of D p,t has the following form. Theorem 7.1.
One has D = D + tA + ( + ) H ∇ + t k v ∇ k − d G X j = µ ∇ j L S p h ∗ j , here A is a vector bundle endomorphism of S p , given in terms of a local orthonormal frame { e , . . . , e d M } of TM by (7.5) A := i4 d M X k = c ( e k ) c (cid:0) ∇ TMe k v ∇ (cid:1) + i2 d G X j = c ( grad µ ∇ j ) c ( V j )− i2 d G X j = M X k = µ ∇ j c ( e k ) c (cid:0) [ e k , ( h ∗ j ) M − V j ] (cid:1) . An important ingredient of the proof of Theorem 7.1 is an expression for the Lie derivativeof sections of S p . Lemma 7.2.
Let X ∈ g . Then, as operators on Γ ∞ ( S p ) , one has L S p X = ∇ S p X M − B X − ( + ) πiµ ∇ X , where, in terms of a local orthonormal frame { e , . . . , e d M } of TM , B X := d M X k,l = (cid:0) ∇ e k X M , e l (cid:1) c ( e k ) c ( e l ) . Proof.
Let X ∈ g be given. We give a local argument on an open subset U ⊂ M , using thedecomposition (2.1) of S P | U . Let ∇ L | be the connection on L | → U induced by ∇ . Wefirst note that(7.6) L L | X = ∇ L | X M − iπµ ∇ X | U . Indeed, if t , t ∈ Γ ∞ (cid:0) L | (cid:1) , then by definition of µ ∇ , (cid:0) L L | t (cid:1) ⊗ t + t ⊗ (cid:0) L L | t (cid:1) = L L | U X ( t ⊗ t )= (cid:0) ∇ X M − ∇ X (cid:1) ( t ⊗ t )= (cid:18)(cid:0) ∇ L | X M − iπµ ∇ X (cid:1) t (cid:19) ⊗ t + t ⊗ (cid:18)(cid:0) ∇ L | X M − iπµ ∇ X (cid:1) t (cid:19) . Let s ∈ Γ ∞ ( S Spin U ) . Then(7.7) L S Spin U X s = ∇ S Spin U X M s − B X s. Let t , . . . , t + ∈ Γ ∞ (cid:0) L | (cid:1) . Then s ⊗ t ⊗ · · · ⊗ t + ∈ Γ ∞ (cid:0) S Spin U ⊗ L | p + (cid:1) = Γ ∞ ( S p | U ) . L S p X ( s ⊗ t ⊗ · · · ⊗ t + ) = (cid:0) L S Spin U X s (cid:1) ⊗ t ⊗ · · · ⊗ t + + s ⊗ + X j = t ⊗ · · · ⊗ (cid:0) L L | X t j (cid:1) ⊗ · · · ⊗ t + ! = (cid:0) ∇ S Spin U X M s (cid:1) ⊗ t ⊗ · · · ⊗ t + + s ⊗ + X j = t ⊗ · · · ⊗ (cid:0) ∇ L | X M t j (cid:1) ⊗ · · · ⊗ t + ! − (cid:0) B X + ( + ) πiµ X (cid:1) s ⊗ t ⊗ · · · ⊗ t + = (cid:16) ∇ S p X M − B X − ( + ) πiµ ∇ X ) (cid:17) s ⊗ t ⊗ · · · ⊗ t + . Using Lemma 7.2, we can prove Theorem 7.1.As in the equality (1.26) in [33], the fact that ∇ S p satisfies a Leibniz rule with respect tothe Clifford action (see e.g. Proposition 4.11 in [21]) implies that(7.8) D = D + it2 d M X k = c ( e k ) c ( ∇ TMe k v ∇ ) − it ∇ S p v ∇ + t k v ∇ k . The main part of the proof of Theorem 7.1 is a computation of an expression for the first-order term ∇ S p v ∇ .By (7.2), we have ∇ S p v ∇ = d G X j = µ ∇ j ∇ S p V j . By Lemma 7.2, one has for all s ∈ Γ ∞ ( S p ) , all m ∈ M and all j , (cid:0) ∇ S p V j s (cid:1) ( m ) = (cid:0) ∇ S p h ∗ j ( m ) M s (cid:1) ( m ) = (cid:16)(cid:16) L S p h ∗ j ( m ) + B h ∗ j ( m ) + ( + ) πiµ ∇ j (cid:17) s (cid:17) ( m ) . Multiplying this identity by ∇ j ( m ) and summing over j , we obtain(7.9) (cid:0) ∇ S p v ∇ s (cid:1) ( m ) = (cid:18) d G X j = µ ∇ j L S p h ∗ j (cid:19) s ! ( m )+ (cid:18) d G X j = µ ∇ j B h ∗ j ( m ) (cid:19) s ! ( m ) + (cid:0) ( + ) H ∇ s (cid:1) ( m ) . (cid:18) d G X j = µ ∇ j B h ∗ j ( m ) (cid:19) s ! ( m ) = d G X j = µ ∇ j d M X k,l = (cid:0) ∇ e k h ∗ j ( m ) M , e l (cid:1) c ( e k ) c ( e l )= (cid:18) d M X k = c ( e k ) c (cid:0) ∇ TMe k v ∇ (cid:1) − d G X j = c ( grad µ ∇ j ) c ( V j )+ d G X j = M X k = µ ∇ j c ( e k ) c (cid:0) [ e k , ( h ∗ j ) M − V j ] (cid:1)(cid:19) s ! ( m )= i (cid:18) A − it2 d M X k = c ( e k ) c (cid:0) ∇ TMe k v ∇ (cid:1)(cid:19) s ! ( m ) . Theorem 7.1 follows from this equality and (7.8) and (7.9).
Remark 7.3.
Lemma B.3 in [15] does not apply in the general Spin c -case, so that grad µ ∇ j ,which appears in the expression for the operator A , cannot be worked out further in thepresent setting. A To prepare for the localisation estimates in Section 8, we show that the operator A inTheorem 7.1 satisfies a certain estimate with respect to a rescaling of the metric on g ∗ M bya function.For any positive, G -invariant smooth function ψ ∈ C ∞ ( M ) G , consider the metric(7.10) { ψ ( m )(− , −) m } m ∈ M on g ∗ M . Let A ψ be the operator in Theorem 7.1, defined with respect to this metric. In thechoice of the metric on g ∗ M in Proposition 8.3, we will use the following property of thedependence of the operator A ψ on ψ . Lemma 7.4.
There are G -invariant, positive, continuous functions F , F ∈ C ( M ) G such that forall G -invariant, positive smooth functions ψ ∈ C ∞ ( M ) , one has the pointwise estimate (7.11) k A ψ k ≤ F ψ + F k dψ k . Proof.
Let ψ ∈ C ∞ ( M ) G be a G -invariant, positive smooth function. With respect to themetric (7.10) rescaled by ψ , we use the orthonormal frame of g ∗ M made up of the functions h ψj := h j . The dual frame of M × g → M consists of the functions ( h ψj ) ∗ = ψ h ∗ j . ( µ ∇ j ) ψ be defined like the functions µ ∇ j in (7.1), with h j replaced by h ψj . Analogously,let V ψj be the vector field defined like V j in (7.3), with the same replacement. Then ( µ ∇ j ) ψ = ψ µ ∇ j ; V ψj = ψ V j . (7.12)It follows for example from the latter two equalities and (7.2) that the vector field ( v ∇ ) ψ ,defined like v ∇ with the metric on g ∗ M rescaled by ψ , equals(7.13) ( v ∇ ) ψ = ψv ∇ . We start with some local computations for each term in the definition (7.5) of the op-erator A ψ . Let { e , . . . , e d M } be a local orthonormal frame for TM . By (7.13), we have forall k , ∇ TMe k ( v ∇ ) ψ = ψ ∇ TMe k v ∇ + e k ( ψ ) v ∇ . Hence (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) i4 d M X k = c ( e k ) c (cid:0) ∇ TMe k ( v ∇ ) ψ (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ d M X k = (cid:0) ψ k∇ TMe k v ∇ k + k e k ( ψ ) kk v ∇ k (cid:1) ≤ a ψ + a k dψ k , with a := d M X k = k∇ TMe k v ∇ k ; a :=
14 d M k v ∇ k . Note that the function a is not defined globally, and is not G -invariant on its domain ingeneral. We will come back to this later.Secondly, because of (7.12), we have(7.14) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) i2 d G X j = c (cid:0) grad ( µ ∇ j ) ψ (cid:1) c ( V ψj ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ d G X j = (cid:0) ψ k grad µ ∇ j k k V j k + | µ ∇ j | ψ k grad ψ k k V j k (cid:1) . Since ψ k grad ψ k = k dψ k , (7.14) is at most equal to b ψ + b k dψ k , b := d G X j = k grad µ ∇ j k k V j k ; b := d G X j = | µ ∇ j | k V j k . Finally, Lemma C.8 in [15] implies that (cid:2) e k , (cid:0) ( h ∗ j ) ψ (cid:1) M − V ψj (cid:3) = ψ [ e k , ( h ∗ j ) M − V j ] − e k ( ψ ) V j . Therefore,(7.15) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − i2 d G X j = M X k = ( µ ∇ j ) ψ c ( e k ) c (cid:0)(cid:2) e k , (cid:0) ( h ∗ j ) ψ (cid:1) M − V ψj (cid:3)(cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ d G X j = M X k = (cid:0) ψ | µ ∇ j | (cid:13)(cid:13) [ e k , ( h ∗ j ) M − V j ] (cid:13)(cid:13) + ψ k e k ( ψ ) k | µ ∇ j | k V j k . (cid:1) Since ψ k e k ( ψ ) k = k e k ( ψ ) k ≤ k dψ k , we find that (7.15) is at most equal to c ψ + c k dψ k , with c := d G X j = M X k = | µ ∇ j | (cid:13)(cid:13) [ e k , ( h ∗ j ) M − V j ] (cid:13)(cid:13) ; c := d M d G X j = | µ ∇ j | k V j k . The functions a j , b j and c j are not all defined globally and/or G -invariant. To geta global estimate for A , let W ⊂ M be an open subset that intersects all G -orbits innonempty, relatively compact sets. By Lemmas C.1 and C.2 in [15], there are G -invariant,positive, continuous functions F and F on M , and local orthonormal frames of TM around each point in W , such that on W , with respect to these frames, one has a + b + c ≤ F ; a + b + c ≤ F . Then the estimate (7.11) holds on W . Since both sides of (7.11) are G -invariant, and thedefinition of A is independent of the local orthonormal frame chosen, we get the desiredestimate on all of M . 41 Localisation estimates
Two localisation estimates are at the cores of the proofs of Theorems 6.6 and 6.8. Theseare Propositions 8.1 and 8.2 below. In the proofs of these estimates, we will not use theassumption that is a Spin c -regular value of µ ∇ . They therefore also hold in the singularcase. The regularity assumption is only needed to apply the arguments near ( µ ∇ ) − ( ) toobtain Theorem 6.8.The localisation estimates are stated in terms of certain Sobolev norms. D p,t Theorem 6.6 follows from the fact that for large t , the operator D p,t induces a Fredholmoperator between certain Sobolev spaces. By an elliptic regularity argument, the indexof this operator is precisely the G -invariant transversally L -index index GL of D p,t . TheseSobolev spaces and the index theory on them that we will use, were introduced in Section4 of [15]. We will not need to go into the details of these spaces, but will refer to therelevant results in [15]. We do need certain ingredients of the definition of these spaces.One of these is a smooth cutoff function f on M (see Definition 6.1). We will alsoconsider transversally compactly supported sections of vector bundles, by which we meansections whose support is mapped to a compact set by the quotient map M → M/G . Let Γ ∞ tc ( S p ) G be the space of G -invariant, smooth, transversally compactly supported sectionsof S p . For k ∈ N , and s, s ′ ∈ Γ ∞ tc ( S p ) G , we set(8.1) ( fs, fs ′ ) k := k X j = ( fD jp s, fD jp s ′ ) L ( S p ) . (Note that fD jp s and fD jp s ′ are compactly supported for all j .) By Lemma 4.4 in [15], thisinner product is independent of f , since s and s ′ are G -invariant. We will write k · k k forthe induced norm on fΓ ∞ tc ( S p ) G .These Sobolev norms allow us to state the localisation estimates we will use. Fix a G -invariant open neighbourhood V of the set Crit ( v ∇ ) of zeros of v ∇ . We assumed thatCrit ( v ∇ ) is cocompact, so we may assume that V is relatively cocompact, in the sense that V/G is a relatively compact subset of
M/G . Proposition 8.1.
There is a G -invariant metric on g ∗ M , and there are t , C, b > 0 , such that forall t ≥ t , all p ∈ N , and all G -invariant s ∈ Γ ∞ tc ( S p ) G with support disjoint from V , one has (8.2) k fD p,t s k ≥ C (cid:0) k fs k + ( t − b ) k fs k (cid:1) . Proposition 8.2.
The metric on g ∗ M used in Proposition 8.1 can be chosen such that, in additionto the conclusions of that proposition, for every G -invariant open neighbourhood U of ( µ ∇ ) − ( ) ,there are p ∈ N and t , C, b > 0 , such that for all t ≥ t and p ≥ p , and all G -invariant s ∈ Γ ∞ tc ( S p ) G with support disjoint from U , the estimate (8.2) holds.
42o the estimate holds for all s supported outside V for all p , and for all s supportedoutside the smaller set U for large p .It is important that the metric on g ∗ M used in Propositions 8.1 and 8.2 is the same. Theytherefore actually form one result, with two conclusions.In addition, note that the condition that t ≥ t can be absorbed into the choice ofthe metric on g ∗ M , since multiplying this metric by a constant results on multiplying thevector field v ∇ by the same constant. The parameter t was just introduced to make thearguments that follow clearer. M × g ∗ One advantage of using a family of inner products on g ∗ , i.e. a metric on g ∗ M , is that thisallows us to define the G -invariant vector field v ∇ and the G -invariant function H ∇ . An-other advantage that is very important for our arguments is that choosing this metricin a suitable way allows us to control the terms that appear in the Bochner formula inTheorem 7.1.To make this precise, consider the G -invariant, positive, continuous function η on M defined by (8.3) η ( m ) = Z G f ( gm ) k df k ( gm ) dg, for m ∈ M . Proposition 8.3.
The G -invariant metric on the bundle g ∗ M can be chosen in such a way that forall m ∈ M \ V , H ∇ ( m ) ≥ ; (8.4) k v ∇ m k ≥ + η ( m ) , (8.5) and there is a positive constant C , such that for all m ∈ M , the operator A m on ( S p ) m is boundedbelow by (8.6) A m ≥ − k v ∇ m k − C. Proof.
Fix any G -invariant metric { (− , −) m } m ∈ M on g ∗ M . Let the G -invariant, positive, con-tinuous functions F and F be as in Lemma 7.4. Set ϕ := min (cid:18) H ∇ , k v ∇ k + η , k v ∇ k (cid:19) ϕ := k v ∇ k . What follows holds for any G -invariant, positive, continuous function η . G -invariant, continuous functions ϕ and ϕ on M , which are positive out-side Crit ( v ∇ ) . Since Crit ( v ∇ ) /G is compact, the functions ϕ j have uniform lower boundsoutside the neighbourhood V of Crit ( v ∇ ) . Hence there are positive, G -invariant, continu-ous functions ˜ ϕ j on M , such that ˜ ϕ j | M \ V = ϕ j | M \ V , for j =
1, 2 . By Lemma C.3 in [15], there is a G -invariant, positive, smooth function ψ on M , such that ψ − ≤ ˜ ϕ ; k d ( ψ − ) k ≤ ˜ ϕ . Consider the metric { ψ ( m )(− , −) m } m ∈ M on g ∗ M , obtained by rescaling the given metric by ψ . We claim that this metric has the desired properties.First of all, the function H ∇ ψ and the vector field ( v ∇ ) ψ associated to this metric satisfy,outside V , H ∇ ψ = ψ H ∇ ≥ ϕ − H ∇ ≥ ; k ( v ∇ ) ψ k = ψ k v ∇ k ≥ ϕ − k v ∇ k ≥ + η. Furthermore, by Lemma 7.4, the operator A ψ in Theorem 7.1, associated to the metricon g ∗ M rescaled by ψ , satisfies, outside V , k A ψ kk ( v ∇ ) ψ k ≤ F ψ + F k dψ k ψ k v ∇ k = F k v ∇ k ψ − + F k v ∇ k k d ( ψ − ) k≤ Hence k A ψ k ≤ k ( v ∇ ) ψ k , on M \ V . Since V is relatively cocompact and A ψ is G -equivarant,it is bounded on V . So A ψ ≥ − C on V , for a certain C > 0 . We conclude that A ψ ≥ − k ( v ∇ ) ψ k − C on all of M . Remark 8.4.
A priori, the choice of metric on g ∗ M could influence index GL ( D p,t ) , if Crit ( v ∇ ) changes (while staying cocompact). Multiplying a metric by a function ψ as in Proposi-tion 8.3 does not change Crit ( v ∇ ) , however, and the second point in Theorem 2.15 in [7]implies that index GL ( D p,t ) is independent of ψ . It follows from Theorem 6.8 that this index44s independent of the metric in general, as long as Crit ( v ∇ ) is cocompact, for large enough p . Also note that one may take t = in Theorem 6.6, since, in the notation of the proofof Proposition 8.3, it2 c (cid:0) ( v ∇ ) ψ (cid:1) = i2 c (cid:0) ( v ∇ ) tψ (cid:1) . Proposition 8.3 allows us to prove Propositions 8.1 and 8.2. Fix a G -invariant metric on g ∗ M as in Proposition 8.3, and a smooth cutoff function f . It will be useful to consider theoperator e D p,t : fΓ ∞ tc ( S p ) G → fΓ ∞ tc ( S p ) G , defined by(8.7) e D p,t fs = fD p,t s, for s ∈ Γ ∞ tc ( S p ) G . We will write e D p := e D p,0 .We need some arguments to account for the fact that, unlike D p,t , the operator e D p,t isnot symmetric with respect to the L -inner product. Let e D ∗ p,t be its formal adjoint. Com-bining Theorem 7.1 and Proposition 8.3, one obtains the following key estimate for theoperator e D ∗ p,t e D p,t . Corollary 8.5.
One has e D ∗ p,t e D p,t = f D p ∗ f D p + tB + ( + ) H ∇ + t k v ∇ k , where B is a vector bundle endomorphism of S p for which there is a constant C > 0 such that onehas the pointwise estimate B ≥ − C (cid:0) k v ∇ k + (cid:1) . Proof.
This was proved in the symplectic setting in Proposition 6.7 in [15]. The argumentsremain the same, however. References to Theorem 5.1 and to Proposition 6.6 in the proofof Proposition 6.7 in [15] should be replaced by references to Theorem 7.1 and Proposition8.3 in the present paper, respectively. Note that the last term in the Bochner formula ofTheorem 7.1 vanishes on G -invariant sections.The proofs of Propositions 8.1 and 8.2 are now the same as the proofs of Propositions6.1 and 6.3 in [15], with Corollary 8.5 playing the role of Proposition 6.7 in [15].45 .4 Proofs of Theorems 6.6 and 6.8 Theorem 6.6 follows from Proposition 8.1, in the way that Theorem 3.4 in [15] followsfrom Proposition 6.1 in [15]. Indeed, for t ≥ b + and any p in Proposition 8.1, one has k fD p,t k ≥ C k fs k , for G -invariant sections s ∈ Γ ∞ tc ( S p ) G with support disjoint from the set V . By Proposition4.7 in [15], the operator e D p,t therefore extends to a Fredholm operator between Sobolevspaces. By Proposition 4.8 in [15], ker L ( D p,t ) G is finite-dimensional, and the index of theFredholm operator induced by e D p,t equals index GL ( D p,t ) . It is noted in part 2 of Theorem2.15 in [7] that this index is independent of t , so that Theorem 6.6 follows.To prove Theorem 6.8, we apply Proposition 8.2. This proposition shows that thearguments in Sections 6.5 and 7 of [15] apply to the operator D p,t , for large enough p and t . Therefore, the techniques from Sections 8 and 9 in [4] can be used as in [15, 23, 33]. Itfollows that, for large enough p and t ,(8.8) index GL ( D p,t ) = index ( D ∇ M ) , where D ∇ M is the Spin c -Dirac operator on the reduced space M associated to the Spin c -structure of Lemma 3.3, and the connection ∇ on the line bundle L + → M induced bythe connection ∇ on L . Theorem 6.8 therefore follows from the proposition below. Proposition 8.6.
For all p ∈ N , there exists a G -equivariant Spin c -structure on M , and a con-nection on the associated determinant line bundle, such that the corresponding invariant Spin c -quantisation is Q Spin c ( M ) G = index GL ( D p,t ) , for t large enough, and Q Spin c ( M ) = index ( D ∇ M ) . Proof.
Let P → M be the given G -equivariant principal Spin c -structure on M . Let P ′ → M be the G -equivariant Spin c -structure with determinant line bundle L ′ = L + . Explicitly, P ′ = P × U ( ) UF ( L p ) , where UF denotes the unitary frame bundle. (See e.g. part (2) of Proposition D.43 in [12].)Let ∇ ′ be the connection on L ′ induced by ∇ .Let S ′ → M be the spinor bundle associated to P ′ . Then S ′ = S p (see e.g. (D.15) in [21]).Hence the connection ∇ S ′ on S ′ induced by ∇ ′ and the Levi–Civita connection on TM equals the connection on S p used to define the Dirac operator D p . Therefore, the Spin c -Dirac operator D ′ on S ′ equals the operator D p . Furthermore, the Spin c -momentum map µ ∇ ′ : M → g ∗ associated to ∇ ′ is given by(8.9) ∇ ′ X = ∇ ′ X M − L L + X = ( + ) µ ∇ X , X ∈ g . It follows that the induced vector field v ∇ ′ equals v ∇ ′ = ( + ) v ∇ . We conclude that the deformed Dirac operator on S ′ associated to ∇ ′ is D ′ = D ′ + it2 c ( v ∇ ′ ) = D p + ( + ) it2 c ( v ∇ ) = D p, ( + ) t . Let t , t ′ ∈ R be as in Theorem 6.6, for he operators D p,t and D ′ p,t , respectively. Thistheorem states that index GL ( D p,t ) does not depend on t ≥ t . Hence, if t ≥ t ; t ′ ≥ t ′ ; and ( + ) t ′ ≥ t , then, with respect to the Spin c -structure P ′ and the connection ∇ ′ , Q Spin c ( M ) G = index GL ( D ′ ′ ) = index GL ( D p, ( + ) t ′ ) = index GL ( D p,t ) . Finally, by (8.9), one has M = ( µ ∇ ′ ) − ( ) /G = ( µ ∇ ) − ( ) /G. And the connection ( ∇ ′ ) on L ′ = L + is the one induced by the connection ∇ on L , sothe second claim follows as well. Spin c -Dirac operators Theorem 6.12 can be proved by generalising the steps in the proofs of Theorems 6.6 and6.8 to twisted Spin c -Dirac operators. As in the case for untwisted Dirac operators, the proof of Theorem 6.12 starts with anexpression for the square of the deformed Dirac operator D Ep,t . This expression will bededuced from Theorem 7.1 by comparing the square of D Ep,t to the square of D p,t . Themain difference between these two involves the generalised moment map µ E ∈ g ∗ ⊗ End ( E ) , defined by EX = L EX − ∇ EX M ∈ End ( E ) , for all X ∈ g , where L EX is the Lie derivative of sections of E with respect to X . Using themetric on g ∗ M , we obtain ( µ ∇ , µ E ) ∈ End ( E ) . roposition 9.1. On G -invariant sections of S p ⊗ E , one has ( D Ep,t ) = ( D Ep ) + tA ⊗ E + ( + ) H ∇ + t k v ∇ k + S ⊗ ( µ ∇ , µ E ) , with A ∈ End ( S p ) as in Theorem 7.1. The first step in the proof of Proposition 9.1 is a simple relation between the operators D Ep and D p . Fix a local orthonormal frame { e j } d M j = of TM . The operator P E := d M X j = c ( e j ) ⊗ ∇ Ee j on Γ ∞ ( S p ⊗ E ) is independent of this frame, and hence globally defined. Lemma 9.2.
One has D Ep = D p + P E . Proof.
In terms of the frame { e j } d M j = , we have D Ep = d M X j = ( c ( e j ) ⊗ E ) (cid:0) ∇ S p e j ⊗ E + S p ⊗ ∇ Ee j (cid:1) = d M X j = c ( e j ) ∇ S p e j ⊗ E + d M X j = c ( e j ) ⊗ ∇ Ee j = D p + P E . Lemma 9.3.
For all vector fields v on M , ( c ( v ) ⊗ E ) ◦ P E + P E ⊗ ( c ( v ) ⊗ E ) = − ( S p ⊗ ∇ Ev ) . Proof.
Since c ( v ) c ( e j ) + c ( v ) c ( e j ) = − ( v, e j ) for all j , we see that ( c ( v ) ⊗ E ) ◦ P E + P E ⊗ ( c ( v ) ⊗ E ) = d M X j = (cid:0) c ( v ) c ( e j ) + c ( e j ) c ( v ) (cid:1) ⊗ ∇ Ee j = − d M X j = ( v, e j ) S p ⊗ ∇ Ee j = − ( S p ⊗ ∇ Ev ) . ( µ ∇ ) ∗ : M → g be dual to µ ∇ with respect to a given metric on g ∗ M . For any G -equivariant vector bundle F → M , consider the Lie derivative operator L F ( µ ∇ ) ∗ on Γ ∞ ( F ) ,defined by ( L F ( µ ∇ ) ∗ s )( m ) := ( L F ( µ ∇ ) ∗ ( m ) s )( m ) for s ∈ Γ ∞ ( F ) and m ∈ M . Proposition 9.4.
One has ( D Ep,t ) = ( D Ep ) + tA ⊗ E + ( + ) H ∇ + t k v ∇ k + S ⊗ ( µ ∇ , µ E )− (cid:0) L S p ( µ ∇ ) ∗ ⊗ E + S p ⊗ L E ( µ ∇ ) ∗ (cid:1) , with A ∈ End ( S p ) as in Theorem 7.1. Since for all X ∈ g , L S p X ⊗ E + S p ⊗ L EX is the Lie derivative on S p ⊗ E with respect to X , the operator L S p ( µ ∇ ) ∗ ⊗ E + S p ⊗ L E ( µ ∇ ) ∗ equals zero on G -invariant sections. Hence Proposition 9.4 implies Proposition 9.1. Proof of Proposition 9.4.
First note that ( D Ep,t ) = ( D Ep ) + (cid:0) it2 c ( v ∇ ) ⊗ E (cid:1) + it2 (cid:0) D Ep ◦ ( c ( v ∇ ) ⊗ E ) + ( c ( v ∇ ) ⊗ E ) ◦ D Ep (cid:1) . Because of Lemma 9.2 and 9.3, we have D Ep ◦ ( c ( v ∇ ) ⊗ E ) + ( c ( v ∇ ) ⊗ E ) ◦ D Ep = (cid:0) D p ◦ c ( v ∇ ) + c ( v ∇ ) ◦ D p (cid:1) ⊗ E + ( c ( v ∇ ) ⊗ E ) ◦ P E + P E ⊗ ( c ( v ∇ ) ⊗ E ) = (cid:0) D p ◦ c ( v ∇ ) + c ( v ∇ ) ◦ D p (cid:1) ⊗ E − ( S p ⊗ ∇ Ev ∇ ) . Furthermore, (cid:0) it2 c ( v ∇ ) (cid:1) + it2 (cid:0) D p ◦ c ( v ∇ ) + c ( v ∇ ) ◦ D p (cid:1) = D − D . The right hand side of this equality was computed in Theorem 7.1. Using the expressionobtained there and the above computations, we find that ( D Ep,t ) = ( D Ep ) + tA ⊗ E + ( + ) H ∇ + t k v ∇ k − L S p ( µ ∇ ) ∗ ⊗ E − it1 S p ⊗ ∇ Ev ∇ . Now for all m ∈ M , ∇ Ev ∇ m = ∇ E ( µ ∇ ) ∗ ( m ) Mm = (cid:0) L E ( µ ∇ ) ∗ ( m ) − E ( µ ∇ ) ∗ ( m ) ( m ) (cid:1) . Since µ E ( µ ∇ ) ∗ ( m ) ( m ) = ( µ ∇ , µ E )( m ) , the claim follows. (cid:3) .2 Localisation Proposition 8.2, which is the key step in the proof of Theorem 6.8, generalises to twistedDirac operators in the following way.
Proposition 9.5.
There is a metric on g ∗ M such that for every G -invariant open neighbourhood U of ( µ ∇ ) − ( ) , there are p E ∈ N and t , C, b > 0 , such that for all t ≥ t and p ≥ p E , and all G -invariant s ∈ Γ ∞ tc ( S p ⊗ E ) G with support disjoint from U , k fD Ep,t s k ≥ C (cid:0) k fs k + ( t − b ) k fs k (cid:1) . Here k · k k denotes the Sobolev norm defined by the operator D Ep , as in (8.1) . Theorem 6.12 follows from Proposition 9.5 in the same way that Theorems 6.6 and6.8 follows from Propositions 8.1 and 8.2, as described in Subsection 8.4. The topologicalexpression for the index of D M then follows from the Atiyah–Singer index theorem. Wenow do not use an analogue of Proposition 8.1 (localisation to neighbourhoods of Crit ( v ∇ ) for p = ), because for twisted Dirac operators we always use large enough powers of L .It therefore remains to prove Proposition 9.5. This proof is based on a generalisationof Proposition 8.3. Lemma 9.6.
There is a G -invariant metric on g ∗ M such that, in addition to the properties inProposition 8.3, there is a C ′ > 0 such that the operator ( µ ∇ , µ E ) satisfies the pointwise estimate (9.1) S p ⊗ ( µ ∇ , µ E ) ≥ − k v ∇ k − C ′ (for any p ∈ N ).Proof. As in Section 8, choose a relatively cocompact, G -invariant neighbourhood V ofCrit ( v ∇ ) . Choose a G -invariant, positive function ψ E ∈ C ∞ ( M ) G such that, outside V , k S p ⊗ ( µ ∇ , µ E ) k ≤ ψ E k v ∇ k . Fix any G -invariant metric { (− , −) m } m ∈ M on g ∗ M . Consider the metric { ψ E ( m )(− , −) m } m ∈ M rescaled by ψ E , and let ( v ∇ ) ψ E = ψ E v ∇ be the vector field associated to this metric. Then,outside V , k S p ⊗ ψ E ( µ ∇ , µ E ) k ≤ k ( v ∇ ) ψ E k Furthermore, the function k S p ⊗ ψ E ( µ ∇ , µ E ) k is G -invariant, and hence bounded on V . Sothere is a C ′ > 0 such that, on all of M , k S p ⊗ ψ E ( µ ∇ , µ E ) k ≤ k ( v ∇ ) ψ E k + C ′ Let ψ ∈ C ∞ ( M ) G be as in the proof of Proposition 8.3. Choose a positive function˜ ψ ∈ C ∞ ( M ) G such that ˜ ψ − ≤ min ( ψ − , ψ − ); k d ˜ ψ − k ≤ k dψ − k . (This is possible by Lemma C.3 in [15].) Then the metric { ˜ ψ ( m )(− , −) m } m ∈ M has the prop-erties in Proposition 8.3, and also satisfies (9.1).50et f ∈ C ∞ ( M ) be a cutoff function. Analogously to (8.7), we define the operator e D Ep,t on fΓ ∞ tc ( S p ⊗ E ) G by e D Ep,t fs = fD Ep,t s for all s ∈ Γ ∞ tc ( S p ⊗ E ) G . Corollary 8.5 now generalises as follows. Corollary 9.7.
One has ( e D Ep,t ) ∗ e D Ep,t = ( e D Ep ) ∗ e D Ep + tB + ( + ) H ∇ + t k v ∇ k , where B is a vector bundle endomorphism of S p ⊗ E for which there is a constant C > 0 such thatone has the pointwise estimate B ≥ − C (cid:0) k v ∇ k + (cid:1) . Proof.
As in Lemma 6.8 of [15], one has for s ∈ Γ ∞ tc ( S p ⊗ E ) G , ( e D Ep,t ) ∗ fs = e D Ep,t fs + ( c ( df ) ⊗ E ) s. Hence, as in Lemma 6.9 of [15], one deduces from Proposition 9.1 that for such s , ( e D Ep,t ) ∗ e D Ep,t fs =( e D Ep ) ∗ e D Ep fs + t (cid:0) A ⊗ E + ( + ) H ∇ + S ⊗ ( µ ∇ , µ E ) (cid:1) fs + t k v ∇ k fs + it ( c ( df ) c ( v ∇ ) ⊗ E ) s with A ∈ End ( S p ) as in Theorem 7.1.Write Bfs := (cid:0) A ⊗ E + S ⊗ ( µ ∇ , µ E ) (cid:1) fs + i ( c ( df ) c ( v ∇ ) ⊗ E ) s, for s as above. By Lemma 9.6, the metric on g ∗ M can be chosen such that there is a C ′ > 0 for which A ⊗ E + S ⊗ ( µ ∇ , µ E ) ≥ −( + ) k v ∇ k − C ′ . By Lemma 6.10 in [15], there is a C ′′ > 0 such that for all s ∈ Γ ∞ tc ( S p ⊗ E ) G Re (cid:0) i ( c ( df ) c ( v ∇ ) ⊗ E ) s, fs (cid:1) ≥ − C ′′ (cid:0) ( k v ∇ k + ) fs, fs (cid:1) . This implies that B ≥ −( C ′ + C ′′ + + )( k v ∇ k + ) . The proof of Proposition 9.5 (and hence of Theorem 6.12) can now be finished as inthe proof of Proposition 6.3 in Section 6.4 of [15], with Corollary 9.7 playing the role ofProposition 6.7 in [15]. 51
Let us mention some applications and examples of Theorems 4.7, 6.8 and 6.12. We will seethat Theorem 6.8 reduces to a Spin c -version of the result in [23] in the cocompact case, anddiscuss how to generate examples of Theorems 4.7 and 6.12. We show how formal degreesof classes in K ∗ ( C ∗ r G ) , generalising formal degrees of discrete series representations, andrelated to certain charcteristic classes on M . Finally, we use the index formula for twistedSpin c -Dirac operators in Theorem 6.12 to draw conclusions about Braverman’s analyticindex of such operators.As before, we assume G is unimodular. Spin c -manifolds As noted in Subsection 4.2, Theorem 6.8 implies that the main result in [23] generalises tothe Spin c -setting. Corollary 10.1.
In the situation of Theorem 6.8, suppose that
M/G is compact . Then, in thenotation of Subsection 4.2, R (cid:0) Q Spin c G ( M ) (cid:1) = Q ( M ) , for the Spin c -structures on M and a connections on their determinant line bundles for whichTheorem 6.8 holds.Proof. If M/G is compact, one may take t = in Theorem 6.6. (By using V = M inProposition 8.1.) As noted in Remark 6.4, the fact that all smooth sections are transversally L in this case implies that Q Spin c ( M ) G = dim ( ker D + ) G − dim ( ker D − ) G . Bunke shows in the appendix to [23] that this equals R (cid:0) Q Spin c G ( M ) (cid:1) .In other words, an extension of Landsman’s conjecture (4.5) to the Spin c -case holds forsuitable choices of Spin c -structures and connections. In fact, Theorem 6.12 can be used togeneralise this result to twisted Spin c -Dirac operators. Using the constructions in Subsection 3.2, one can generate a large class of examples ofTheorems 4.7 and 6.12 from cases where the group acting is compact. Indeed, let K be acompact, connected Lie group, and let N be a manifold equipped with an action by K anda K -equivariant Spin c -structure. Let µ ∇ N : N → k ∗ be the Spin c -momentum map associ-ated to a K -invariant Hermitian connection ∇ N on the determinant line bundle L N → N of the Spin c -structure on N . Let v ∇ N be the vector field on N associated to µ ∇ N as in (6.2),with respect to a single Ad ∗ ( K ) -invariant inner product on k ∗ . Suppose it has a compact52et Crit ( v ∇ N ) of zeros. As noted in Lemma 3.24 in [27], and on page 4 of [35], this is true if N is real-algebraic and µ ∇ N is algebraic and proper. (And also, of course, if N is compact.)Let G be a connected, unimodular Lie group containing K as a maximal compact sub-group. Suppose the lift f Ad in (3.3) exists, which is true if one replaces G by a doublecover if necessary. We saw in Subsections 3.2 and 3.3 that the manifold M := G × K N has a G -equivariant Spin c -structure with determinant line bundle L M = G × K L N . Furthermore,by Proposition 3.10, all G -equivariant Spin c -manifolds arise in this way (though possiblynot all Riemannian metrics on such manifolds). In Subsection 5.2, a connection ∇ M on L M was constructed, such that the associated Spin c -momentum map µ ∇ M is given by (3.8).If N is compact and even-dimensional, then Theorem 4.7 applies, and yields a decom-position of Q Spin c G ( M ) r ∈ K ∗ ( C ∗ r G ) . If N is possibly noncompact, then Theorem 6.12 appliesfor a suitable metric on g ∗ M . Corollary 10.2.
Suppose the dimension of M is even. Let E → M be a G -equivariant, Hermitianvector bundle, equipped with a G -invariant, Hermitian connection. If ∈ k ∗ is a regular value of µ ∇ N , and K acts freely on ( µ ∇ N ) − ( ) , then there are a metric on g ∗ M and a p E ∈ N such that forall p ≥ p E , index GL D Ep,1 = Z M ch ( E ) e p2 c ( L ) ^ A ( M ) . Proof.
By Proposition 3.12, zero is a Spin c -regular value of µ ∇ M . By (5.10), G acts freely on ( µ ∇ M ) − ( ) . To apply Theorem 6.12, it therefore only remains to show that the vector field v ∇ M on M , induced by the momentum map µ ∇ M as in (6.2), has a cocompact set Crit ( v ∇ M ) of zeros. This follows from the fact thatCrit ( v ∇ M ) = G × K Crit ( v ∇ N ) , for a suitable metric on g ∗ M . This is proved in Lemma 10.4 below. Therefore, Theorem 6.12implies the claim. Remark 10.3.
In the setting of Corollary 10.2, Proposition 3.14 implies that Q Spin c ( M ) G = Q Spin c ( M ) = Q Spin c ( N ) = Q Spin c ( N ) K . Lemma 10.4.
There is a G -invariant metric on g ∗ M such that the set of zeros of the vector field v ∇ M on M , used in the proof of Corollary 10.2, equals (10.1) Crit ( v ∇ M ) = G × K Crit ( v ∇ N ) . Proof.
Let (− , −) K be an Ad ∗ ( K ) -invariant inner product on g ∗ that extends the inner prod-uct on k ∗ used to define v ∇ N . Consider the G -invariant metric on g ∗ M defined by ( ξ, ξ ′ ) [ g,n ] := (cid:0) Ad ∗ ( g ) − ξ, Ad ∗ ( g ) − ξ ′ (cid:1) K , ξ, ξ ′ ∈ g ∗ , g ∈ G and n ∈ N . Let v ∇ M be defined via this metric. We will show that v ∇ M | N = v ∇ N , where we embed N into M via the map n → [ e, n ] . Then (10.1) follows by G -invariance of both sides.The dual map ( µ ∇ M ) ∗ : M → g , defined with respect to the above metric on g ∗ M ,satisfies ( µ ∇ M ) ∗ [ e, n ] = ( µ ∇ N ) ∗ ( n ) , for all n ∈ N , where ( µ ∇ N ) ∗ is the map dual to µ ∇ N with respect to the restriction of (− , −) K to k ∗ . Here k ∗ is embedded into g ∗ via the inner product (− , −) K (i.e. p ⊂ g is defined asthe orthogonal complement to k with respect to the induced inner product on g ). Hence v ∇ M [ e,n ] = (cid:0) ( µ ∇ N ) ∗ ( n ) (cid:1) M [ e,n ] = (cid:12)(cid:12)(cid:12)(cid:12) t = h exp (cid:0) t ( µ ∇ N ) ∗ ( n ) (cid:1) , n i = (cid:12)(cid:12)(cid:12)(cid:12) t = h e, exp (cid:0) t ( µ ∇ N ) ∗ ( n ) (cid:1) n i = v ∇ N n , so the claim follows. Theorem 4.7 is stated in terms of K -theory of C ∗ -algebras, but it has purely geometricconsequences. In particular, it yields an expression for the formal degrees of discreteseries representations of semisimple groups in trems of characteristic classes on coadjointorbits.Let τ : C ∗ r G → C be the von Neumann trace, determined by τ (cid:0) R ( ϕ ) ∗ R ( ϕ ) (cid:1) = Z G | ϕ ( g ) | dg, for ϕ ∈ L ( G ) ∩ L ( G ) , where R denotes the right regular representation. This induces amorphism τ ∗ : K ∗ ( C ∗ r G ) → R . Wang showed in Proposition 4.4 and Theorem 6.12 in [37]that τ ∗ (cid:0) Q Spin c G ( M ) r (cid:1) = Z M fe c ( L ) ^ A ( M ) . Here f is a cutoff function as in Definition 6.1. For λ ∈ Λ + + ρ K , let [ λ ] ∈ K d ( C ∗ r G ) be as in(4.6). (As before, d is the dimension of G/K .) We define the formal degree of [ λ ] as d λ := τ ∗ [ λ ] ∈ R . Theorem 4.7 has the following consequence.54 orollary 10.5.
In the setting of Theorem 4.7, we have Z M fe c ( L M ) ^ A ( M ) = X λ ∈ Λ + + ρ K m λ d λ , where m λ ∈ Z is given by the quantisation commutes with reduction relation (4.2) . This corollary is a noncompact generalisation of the equality Z N e c ( L N ) ^ A ( N ) = X λ ∈ Λ + + ρ K m λ dim ( V λ ) , in the compact case. Here V λ is the representation space of π Kλ .Now suppose G is semisimple with discrete series, and let λ ∈ Λ + + ρ K . Let d π be theformal degree of the discrete series representation π with Harish–Chandra parameter λ .Then by (5.3) in [18] and the remarks in Section 2.3 in [19], we have d π = (− ) d/2 d λ . (This motivates the term ‘formal degree’ for the number d λ in general.) In part (iii) ofProposition 7.3.A in [10], Connes and Moscovici gave a decomposition of the L -index ofthe Spin c -Dirac operator on a homogeneous space of G into the formal degrees d π . Theleft-hand side of the equality in Corollary 10.5 is the L -index of the Spin c -Dirac operatoron M by Theorem 6.12 in [37]. Therefore, Corollary 10.5 is a version of quantisationcommutes with reduction for an index as in Connes and Moscovici’s result, if M is ahomogeneous space.For specific choices of such homogeneous spaces, one actually only picks up a singleformal degree. Using Proposition 4.4 in [37] along with Theorem 6.12 in that paper, orConnes and Moscovici’s index theorem, Theorem 5.3 in [10], one obtains d π = (− ) d/2 Z G/K f ch ( G × K V λ ) ^ A ( G/K ) . (Also compare this with Corollary 7.3.B in [10].) In a similar way, Corollary 2.8 in [18]implies that d π = (− ) d/2 Z G · λ fe c ( L ) ^ A (cid:0) G · λ ) . Corollary 10.5 is a generalisation of the latter equality from strongly elliptic coadjointorbits to arbitrary manifolds (satisfying the hypotheses of Theorem 4.7).
The index formula for twisted Spin c -Dirac operators in Theorem 6.12 implies some prop-erties of the index of such operators, which are not a priori clear from Braverman’s ana-lytic definition of this index. 55raverman’s cobordism invariance result, Theorem 3.6 in [7], implies the excisionproperty that the index only depends on data near Crit ( v ∇ ) , as in Lemma 3.12 in [7].Because of Theorem 6.12, the index has a more refined excision property for twistedSpin c -Dirac operators, namely that it only depends on data near the subset ( µ ∇ ) − ( ) of Crit ( v ∇ ) . Corollary 10.6 (Excision) . For j =
1, 2 , let M j be a G -equivariant Spin c -manifold, with spinorbundle S M j → M j . Let ∇ L j be a G -invariant Hermitian connection on the determinant line bundle L j → M j . Let E j → M j be a G -equivariant Hermitian vector bundle, equipped with a G -invariantHermitian connection. Suppose these data satisfy the conditions of Theorem 6.12 for j =
1, 2 .In addition, suppose there are G -invariant neighbourhoods U j of ( µ ∇ Lj ) − ( ) , and a G -equivariant,isometric diffeomorphism ϕ : U → U , such that ϕ (cid:0) ( µ ∇ L1 ) − ( ) (cid:1) = ( µ ∇ L2 ) − ( ); ϕ ∗ (cid:0) S M | U (cid:1) = S M | U ; ϕ ∗ ( ∇ L | U ) = ∇ L | U ; ϕ ∗ ( E | U ) = E | U . Then there are G -invariant metrics on g ∗ M and g ∗ M , and there is a p ∈ N such that for all p ≥ p , (10.2) index GL D E p,1 = index GL D E p,1 . Proof.
Under the conditions stated, one has ( µ ∇ L1 ) − ( ) /G ∼ = ( µ ∇ L2 ) − ( ) /G =: M ; (cid:0) L | ( µ ∇ L1 ) − ( ) (cid:1) /G ∼ = (cid:0) L | ( µ ∇ L2 ) − ( ) (cid:1) /G =: L ; (cid:0) E | ( µ ∇ L1 ) − ( ) (cid:1) /G ∼ = (cid:0) E | ( µ ∇ L2 ) − ( ) (cid:1) /G =: E . Furthermore, because S M and S M coincide on a neighbourhood of ( µ ∇ L1 ) − ( ) = ( µ ∇ L2 ) − ( ) ,the Spin c -structures on M defined by these spinor bundles are equal. Hence by Theorem6.12, if p ≥ max ( p E , p E ) , both sides of (10.2) equal Z M ch ( E ) e p2 c ( L ) ^ A ( M ) . A direct consequence of Theorem 6.12 is that index GL D Ep,1 , when defined, dependspolynomially on p . 56 orollary 10.7. In the setting of Theorem 6.12, there is a p E ∈ N such that for all p ≥ p E , index GL D Ep,1 = ( dim M ) /2 X k = a k p k , with rational coefficients a k := k k ! Z M ch ( E ) c ( L ) k ^ A ( M ) . In particular, index GL D Ep,1 − ( dim M ) /2 X k = a k p k = Z M ch ( E ) ^ A ( M ) is independent of p . Finally, certain topological invariants of M can be recovered as indices on M . Weillustrate this for a twisted version of the signature.Let γ be the involution of V T ∗ M ⊗ C equal to γ := i ( dim M + j ( j − )) /2 ∗ on V j T ∗ M ⊗ C , where ∗ is the Hodge operator. Consider the de Rham operator B := d + d ∗ on Γ ∞ ( V T ∗ M ⊗ C ) . It satisfies Bγ = − γB , and hence defines the signature operator B : Γ ∞ (cid:0)^ + T ∗ M ⊗ C (cid:1) → Γ ∞ (cid:0)^ − T ∗ M ⊗ C (cid:1) where the + and − signs denote the + and − eigenspaces of γ . (See e.g. Example 6.2 in[21].)For any integer p , let B L p be the signature operator B , twisted by L p via the givenconnection on L . Write B L p v ∇ := B L p + i2 c ( v ∇ ) . Let N → ( µ ∇ ) − ( ) be the normal bundle to q ∗ TM in TM | ( µ ∇ ) − ( ) . If is a Spin c -regularvalue of µ ∇ , then N has a G -equivariant Spin-structure, with spinor bundle S N → ( µ ∇ ) − ( ) .Let S N = ( S N | ( µ ∇ ) − ( ) ) /G → M be the induced vector bundle over M . Then Theorem6.12 implies a version of Hirzebruch’s signature theorem in this setting. Corollary 10.8 (Twisted signature theorem) . Suppose that is a Spin c -regular value of µ ∇ ,and that G acts freely on ( µ ∇ ) − ( ) . Then there is a G -invariant metric on g ∗ M such that for largeenough integers p , index GL B L p v ∇ = Z M ch ( S N ) e ( p − ) c ( L ) L ( M ) . Here L ( M ) is the L -class of M . roof. For all Spin c -manifolds U , with spinor bundles S U → U , one has ^ T ∗ U ∼ = Cl ( TU ) ∼ = End ( S U ) ∼ = S U ⊗ S ∗ U ∼ = S U ⊗ S U . If U is Spin, then under this identification, the signature operator B U equals the Spin-Diracoperator twisted by S U : B U = D S U U . (See e.g. below Proposition 3.62 in [3].) In our setting, M is only Spin c . But as in (2.1), wehave on small enough open sets U ⊂ M , S p | U = S Spin U ⊗ L | p/2U , where S Spin U is the spinor bundle of a local Spin-structure. Hence, locally, we have for all p ∈ N , D p | U = ( D Spin U ) L | p/2U , the local Spin-Dirac operator D Spin U coupled to L | p/2U via the given connection. Twisting D p by S , we therefore obtain D S p | U = ( D Spin U ) S| U ⊗ L | p/2U = ( D Spin U ) S Spin U ⊗ L | ( p + ) /2U = ( B | U ) L | ( p + ) /2U . If p + is even, then B L ( p + ) /2 is defined globally, and by the above local argument, itequals D S p . This means that for all k ∈ N , in the notation of Definition 6.11, B L k v ∇ = D S − . Under the conditions stated, Theorem 6.12 therefore yields the equalityindex GL B L k v ∇ = Z M ch ( S ) e ( k − ) c ( L ) ^ A ( M ) , for k large enough. Since S = S M ⊗ S N and ch ( S M ) ^ A ( M ) = L ( M ) , the claim follows. References [1] H. Abels,
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