Higher localized analytic indices and strict deformation quantization

Higher lo calized analytic indices and strict deformation quan tization P aulo Carrillo Ro u se Abstract. This paper is conce r ned with the lo cali zation of hi gher analytic indices for Lie groupoids. Let G b e a Lie group oid with Lie algebroid A G . Let τ be a (p erio dic) cyclic cocycle ov er the conv olution algebra C ∞ c ( G ). W e sa y that τ can be localized if there is a corresp ondence K 0 ( A ∗ G ) I nd τ − → C satisfying I nd τ ( a ) = h ind D a , τ i (Connes pair ing). In this case, w e call I nd τ the higher lo calized index asso ciated to τ . In [ CR08a ] we use the algebra of functions ov er the tangent group oid introduced in [ CR08 b ], which is in fact a strict deformation quan tization of the Sc hw artz algebra S ( A G ), to prov e the following r esults: • Ev ery b ounded contin uous cyclic co cycle can b e localized. • If G i s ´ etale, every cyclic cocycle can be lo calized. W e will recall this results with the difference that in this paper , a formula for higher lo calized indices will b e given i n terms of an asymptotic li mit of a pair ing at the level of the deformation algebra mentioned ab o ve. W e wi ll discuss ho w the higher index formulas of Connes-Moscovici, Gorokho vsky-Lott fit in this unifying setting. Contents 1. Int r oductio n 1 2. Index theory for Lie group oids 4 3. Index theory and strict deformation quantization 11 4. Higher lo calized indices 15 References 20 1. In tro duction This paper is conc e r ned with the lo ca liz ation of higher analytic indices for Lie group oids. In [ CM90 ], Connes and Mo s covici defined, for any smoo th manifold M and every Alexander-Spanier class [ ¯ ϕ ] ∈ ¯ H ev c ( M ), a lo calized index mo rphism (1) I nd ϕ : K 0 c ( T ∗ M ) − → C . which has as a particular case the a nalytic index mor phism of A tiyah-Singer fo r [1] ∈ ¯ H 0 ( M ). 1 2 P AULO CARRILLO ROUSE Indeed, given an Alexander-Spa nier co cycle ϕ on M , Connes-Moscovici con- struct a cyclic co cycle τ ( ϕ ) over the alg e br a of smo othing op era tors, Ψ −∞ ( M ) (lemma 2.1, ref.cit.). Now, if D is a n elliptic pseudo differential op erator over M , it de fines an index class ind D ∈ K 0 (Ψ −∞ ( M )) ≈ Z . Then they show ed (theor em 2.4, r ef.cit.) that the pairing (2) h ind D , τ ( ϕ ) i only dep ends on the principal symbol clas s [ σ D ] ∈ K 0 ( T ∗ M ) a nd on the clas s o f ϕ , a nd this defines the loca lized index mor phism (1). Connes-Mosc o vic i go further to pr o ve a lo calized index formula genera lizing the Atiy ah-Singer theorem. They used this for m ula to prove the so called Hig he r index theor em for cov er ings whic h served for proving the Novik ov co njecture for Hyp erb olic groups. W e discuss now the Lie group oid case. This concept is central in non commu- tative geo metr y . Group oids ge ne r alize the concepts of spac e s, gro ups and equiv- alence relatio ns. In the la te 7 0’s, ma inly with the work of Ala in Connes, it be- came c lear tha t group oids app eared natur ally as substitutes of singula r spaces [ Con79, Mac87, Ren 8 0, Pa t99 ]. Many p eople hav e contributed to realizing this idea. W e can find for instance a gr o upoid- lik e tr e a tmen t in Dixmier’s works on transformatio n groups, [ Dix ], or in Brown-Green-Rieffel’s work on orbit classifica- tion of r e la tions, [ BGR77 ]. In foliation theory , several mo dels for the lea f space of a folia tion were r ealized using g roup oids, mainly by pe o ple like Haefliger ([ Hae84 ]) and Wilkelnk emp er ([ Win83 ]), to ment io n some of them. There is also the ca se of Orbifolds, these ca n be seen indeed as ´ etale group oids, (see for example Mo erdijk’s pap er [ Mo e 0 2 ]). There are a ls o some particular g roup oid mo dels for manifolds with corners and conic manifolds w o rked by peo ple like Mon thubert [ Mon03 ], Deb ord- Lescure-Nistor ([ DLN06 ]) and Aastrup-Melo- Mo n thubert-Schrohe ([ AM M S ]) for example. F urthermore, Co nnes show ed that many gro upoids and algebras as s o- ciated to them a ppear ed as ‘non commut a tiv e analogues ‘ o f smo oth manifolds to which many to ols of geometry such as K-theor y and Charac teristic cla sses could be applied [ Con79, Con94 ]. Lie group oids b ecame a very na tural place where to per form pseudo differential calculus and index theor y , [ Con79, MP97, N WX99 ]. The study of the indices in the gr oupo id case is, as we will see, more delicate than the clas sical case. There a re ne w phenomena app earing. If G is a Lie group oid, a G - pseudodiffer en tial op erato r is a differentiable fa mily (see [ MP97 , NWX99 ]) of op erator s. Let P b e such a n op erato r, the index of P , ind P , is an ele men t of K 0 ( C ∞ c ( G )). W e hav e a lso a Co nnes-Chern pa iring K 0 ( C ∞ c ( G )) × H C even ( C ∞ c ( G )) h , i − → C . W e w o uld like to compute the pa irings o f the form (3) h ind D , τ i for D a G -pseudo differen tial elliptic oper ator. F or instance, the Connes-Moso covici Higher index theorem gives a formula for the ab ov e pairing when the group oid G is the group oid asso ciated to a Γ-covering and for cyclic group co cycles. Now, the fir st s tep in o rder to give a for m ula fo r the pair ing (3) a b ov e is to lo calize the pairing, that is, to show that it only depends on the principal s ym b ol class in K 0 ( A ∗ G ) (this would b e the analog of theorem 2.4, [ CM90 ]). HIGHER LOCALIZED ANAL YTIC INDICES AND STRICT DEFORMA TION QUANTIZA TION 3 Let τ b e a (p erio dic) cyclic co cycle ov er C ∞ c ( G ). W e say that τ can b e lo ca liz e d if the corres pondence (4) E l l ( G ) ind K 0 ( C ∞ c ( G )) h ,τ i C factors through the principal symbol cla s s mo r phism, where E l l ( G ) is the set of G - pseudo different ia l elliptic op erator s. In other words, if there is a unique morphism K 0 ( A ∗ G ) I nd τ − → C which fits in the following commutativ e diagram (5) E l l ( G ) ind [ psy m b ] K 0 ( C ∞ c ( G )) h ,τ i C K 0 ( A ∗ G ) I nd τ i.e. , satisfying I nd τ ( a ) = h ind D a , τ i , and hence completely characterized by this prop erty . In this case, we ca ll I nd τ the higher loca lized index asso ciated to τ . In this pap er, we prov e a lo calization r esult using an a ppropriate str ict defor- mation quantization algebra. F or stating the main theor em we need to introduce some terms. Let G b e a Lie group oid. It is known that the top olog ical K -theory group K 0 ( A ∗ G ), enco des the clas ses of principal symbols of all G -pseudo differential el- liptic o p era tors, [ AS68 ]. On other ha nd the K - theory of the Sch wartz algebra of the Lie alg ebroid satisfies K 0 ( S ( A G )) ≈ K 0 ( A ∗ G ) (see fo r ins tance [ CR08a ] Prop osition 4.5). In [ CR08b ], we constructed a strict deformatio n quantization of the algebr a S ( A G ). This algebr a is based on the notion of the tangent group oid which is a deformation group oid a sso ciated to any Lie group oid: Indeed, asso ciated to a Lie group oid G ⇒ G (0) , there is a Lie gro upoid G T := A G × { 0 } G G × (0 , 1 ] ⇒ G (0) × [0 , 1 ] , compatible with A G a nd G , called the tangent group oid o f G . W e can now recall the main theorem in ref.cit. Theorem. Ther e exists an interme diate algebr a S c ( G T ) c onsisting of smo oth func- tions over the tangent gr oup oid C ∞ c ( G T ) ⊂ S c ( G T ) ⊂ C ∗ r ( G T ) , such that it is a field of algebr as over [0 , 1] , whose fib ers ar e S ( A G ) at t = 0 , and C ∞ c ( G ) for t 6 = 0 . Let τ b e a ( q + 1 ) − m ultilinear functional over C ∞ c ( G ). F or each t 6 = 0, we let τ t be the ( q + 1)-multilinear functional over S c ( G T ) defined by (6) τ t ( f 0 , ..., f q ) := τ ( f 0 t , ..., f q t ) . It is immediate that if τ is a (pe rio dic) cyclic co cyc le ov er C ∞ c ( G ), then τ t is a (per iodic) c yclic co cyc le over S c ( G T ) for each t 6 = 0. The main result of this pap er is the following: 4 P AULO CARRILLO ROUSE Theorem 1. Every b ounde d cyclic c o cycle c an b e lo c alize d. Mor e over, in this c ase, the fol lowing formula for the higher lo c alize d index holds: (7) I nd τ ( a ) = li m t → 0 h e a, τ t i , wher e e a ∈ K 0 ( S c ( G T )) is such that e 0 ( e a ) = a ∈ K 0 ( A ∗ G ) . In fact the p airing ab ove is c onstant for t 6 = 0 . Where, a multilinear map τ : C ∞ c ( G ) × · · · × C ∞ c ( G ) | {z } q +1 − times → C is bo unded if it extends to a contin uous multilinear map C k c ( G ) × · · · × C k c ( G ) | {z } q +1 − times τ k − → C , fo r some k ∈ N . The r estriction o f taking b ounded contin uo us cyclic co cycles in the last theorem is not at all restrictive. In fact, all the geometr ical co cycles are of this kind (Group co cycle s, The transverse fundamental cla ss, Go dbillon-V ey and all the Gelfand-F uc hs co cycles for instance). Moreov er , for the ca se of ´ etale gro upoids, the explicit calculations of the Peri- o dic co homologies spaces developed in [ BN94, Cra99 ] allow us to conclude that the formula (7) ab ov e ho lds for every cyclic co cycle (Corollar y 4.8). A t the end of this work w e will discuss how the hig her index formulas o f Co nnes- Moscovici, Goro khovsky-Lott ([ CM9 0, GL03 ]) fit in this unifying setting. Ac kno wle dgmen ts I would like to thank Georg es Sk andalis for reading an earlier version of this pap er and for the very useful comments a nd remarks he did to it. I would also like to thank the referee for his remarks to improv e this w or k. 2. Index theory for Lie group oids 2.1. Lie group oids. Le t us recall what a group oid is: Definition 2. 1. A gr oup oid consis ts o f the fo llowing data : tw o sets G and G (0) , and maps · s, r : G → G (0) called the source and target map resp ectively , · m : G (2) → G called the pro duct map (where G (2) = { ( γ , η ) ∈ G × G : s ( γ ) = r ( η ) } ), such that there e x ist tw o maps , u : G (0) → G (the unit map) and i : G → G (the inv erse map), such tha t, if w e denote m ( γ , η ) = γ · η , u ( x ) = x and i ( γ ) = γ − 1 , we hav e 1. r ( γ · η ) = r ( γ ) and s ( γ · η ) = s ( η ). 2. γ · ( η · δ ) = ( γ · η ) · δ , ∀ γ , η , δ ∈ G when this is p ossible. 3. γ · x = γ and x · η = η , ∀ γ , η ∈ G with s ( γ ) = x and r ( η ) = x . 4. γ · γ − 1 = u ( r ( γ )) and γ − 1 · γ = u ( s ( γ )), ∀ γ ∈ G . Generally , we denote a group oid by G ⇒ G (0) . Along this pap er we will only deal with Lie g roup oids, that is, a gro upoid in which G and G (0) are smo oth manifolds (p ossibly with b oundary), and s, r, m, u a re smo oth maps (with s and r submersions, see [ Mac87, P at99 ]). F or A, B subsets of G (0) we use the notation G B A for the subset { γ ∈ G : s ( γ ) ∈ A, r ( γ ) ∈ B } . Our first example of Lie gr oupo ids will b e the Lie g roups, we will give o ther examples b elow. HIGHER LOCALIZED ANAL YTIC INDICES AND STRICT DEFORMA TION QUANTIZA TION 5 Example 2. 2 (Lie Groups) . Let G b e a Lie g roup. Then G ⇒ { e } is a Lie gr oupo id with pro duct given by the gro up pro duct, the unit is the unit element of the gro up and the inv ers e is the group inv erse Lie group oids gener alize Lie groups. Now, for Lie gro upoids there is also a notion playing the role of the Lie algebr a: Definition 2.3 (The Lie a lgebroid of a Lie group oid) . Let G → G (0) be a Lie group oid. The Lie a lgebroid o f G is the vector bundle A G → G (0) given by definition as the normal vector bundle ass o cia ted to the inclusion G (0) ⊂ G (w e identify G (0) with its image by u ). F or the cas e when a Lie gr oupoid is given by a Lie gro up a s a bove G ⇒ { e } , we recover AG = T e G . Now, in the Lie theory is very imp ortant that this vector space, T e G , ha s a Lie algebr a structure. In the s e tting o f Lie gro upoids the Lie algebroid A G has a structure of Lie alg e broid. W e will not need this in this pap er. Let us put some classic a l examples of Lie gro upoids . Example 2.4 (Manifolds) . Let M b e a C ∞ -manifold. W e can consider the group oid M ⇒ M where every morphism is the identit y ov er M . Example 2 .5 (Group oid asso ciated to a manifold) . Le t M b e a C ∞ -manifold. W e can consider the group oid M × M ⇒ M with s ( x, y ) = y , r ( x, y ) = x a nd the pro duct given b y ( x, y ) ◦ ( y , z ) = ( x, z ). W e denote this group oid by G M . Example 2.6. [Fib er pro duct g roup oid asso ciated to a submersion] This is a gen- eralization of the example ab ov e. Let N p → M b e a submersio n. W e co ns ider the fiber pro duct N × M N := { ( n, n ′ ) ∈ N × N : p ( n ) = p ( n ′ ) } ,which is a manifold bec ause p is a submersion. W e can then take the group oid N × M N ⇒ N which is only a subgroup oid of N × N . Example 2.7 (G-spa c es) . Let G b e a Lie g r oup ac ting by diffeomo rphisms in a manifold M . The tra ns formation g r oupo id as so ciated to this action is M ⋊ G ⇒ M . As a set M ⋊ G = M × G , and the ma ps are given b y s ( x, g ) = x · g , r ( x, g ) = x , the pro duct given by ( x, g ) ◦ ( x · g , h ) = ( x, g h ), the unit is u ( x ) = ( x, e ) and with inv erse ( x, g ) − 1 = ( x · g , g − 1 ). Example 2.8 (V ector bundles) . Let E p → X b e a s mo oth vector bundle ov er a manifold X . W e consider the gr oupo id E ⇒ X 6 P AULO CARRILLO ROUSE with s ( ξ ) = p ( ξ ), r ( ξ ) = p ( ξ ), the pro duct uses the vector space structure and it is given by ξ ◦ η = ξ + η , the unit is z e r o sectio n and the in verse is the additiv e inv erse at each fib er. Example 2.9 (Haefliger’s group oid) . L e t q b e a p ositive integer. The Haefliger’s group oid Γ q has as space of ob jects R q . A morphism (or arr ow) x 7→ y in Γ q is the ger m of a (lo cal) diffeo morphism ( R q , x ) → ( R q , y ). This Lie gr o upoid and its classifying space play a vey imp ortant ro le in the theo ry of folia tions, [ H ae84 ]. Example 2.10 (Orbifolds) . An O r bifold is an ´ etale group oid for which ( s, r ) : G → G (0) × G (0) is a prop er ma p. See [ M o e02 ] for further details. Example 2.11 (Groupo id ass ocia ted to a c o vering) . Let Γ b e a discr et gro up acting freely and prop erly in f M with compact quotient f M / Γ := M . W e denote by G the quotient f M × f M by the diagona l action of Γ. W e have a Lie group oid G ⇒ G (0) = M with s ( e x, e y ) = y , r ( e x, e y ) = x and pro duct ( e x, e y ) ◦ ( e y , e z ) = ( e x , e z ). A par ticula r of this situatio n is when Γ = π 1 ( M ) and f M is the universal cov ering. This group oid played a main role in the Novik ov’s conjecture pro of for hyperb olic groups given by Connes and Moscovici, [ CM90 ]. Example 2.12 (Holonomy group oid of a F oliation) . Let M b e a compact ma nifold of dimension n . Le t F b e a subv ector bundle of the ta ng en t bundle T M . W e say that F is integrable if C ∞ ( F ) := { X ∈ C ∞ ( M , T M ) : ∀ x ∈ M , X x ∈ F x } is a Lie subalgebra of C ∞ ( M , T M ). This induces a partition in embedded subma nifolds (the leav es of the folia tion), given by the solutio n of integrating F . The holonomy gro upoid of ( M , F ) is a Lie gr o upoid G ( M , F ) ⇒ M with L ie algebr oid A G = F and minimal in the following sense: any Lie gro up oid int eg rating the foliation 1 contains an o pen subgr oupo id whic h maps onto the ho- lonomy group oid by a smo oth morphism of Lie group oids. The holono m y g roup o id was constructed by Ehres mann [ Ehr65 ] and Winkelnk em- per [ W in83 ] (see also [ CC00 ], [ Go d91 ], [ P at99 ]). 2.1.1. The c onvolution algebr a of a Lie gr oup oid. W e reca ll how to define a n algebra structure in C ∞ c ( G ) using smo oth Haar systems. Definition 2.13. A smo oth Haar syst em ov er a Lie gro upoid is a family of measure s µ x in G x for each x ∈ G (0) such that, • fo r η ∈ G y x we have the following compatibility condition: Z G x f ( γ ) d µ x ( γ ) = Z G y f ( γ ◦ η ) dµ y ( γ ) • fo r ea c h f ∈ C ∞ c ( G ) the map x 7→ Z G x f ( γ ) d µ x ( γ ) belo ngs to C ∞ c ( G (0) ) 1 ha ving F as Lie algebroid HIGHER LOCALIZED ANAL YTIC INDICES AND STRICT DEFORMA TION QUANTIZA TION 7 A Lie gr oupoid always po sses a smoo th Haar sys tem. In fact, if we fix a smo oth (po sitiv e) section o f the 1-dens ity bundle a sso ciated to the Lie alg ebroid we obtain a smo oth Ha ar system in a canonica l w ay . W e suppo s e for the rest of the paper a given smo oth Haar sy s tem given by 1-densities (for complete details see [ Pat99 ]). W e can no w define a convolution pro duct o n C ∞ c ( G ): Let f , g ∈ C ∞ c ( G ), we set ( f ∗ g )( γ ) = Z G s ( γ ) f ( γ · η − 1 ) g ( η ) dµ s ( γ ) ( η ) This gives a well defined asso ciative pr oduct. Remark 2.14. There is a wa y to define the co nvolution a lgebra using ha lf densities (see Connes bo ok [ Con94 ]). 2.2. Analytic indices for Lie group oids. As we men tioned in the intro duc- tion, we are going to conside r some elements in the K -theory group K 0 ( C ∞ c ( G )). W e recall how these elemen ts are usually defined (See [ NWX99 ] for complete de- tails): First we recall a few facts ab out G -Pseudo differential calculus: A G - Pseudo differ ential op er ator is a family of pseudo differential op erators { P x } x ∈ G (0) acting in C ∞ c ( G x ) such that if γ ∈ G a nd U γ : C ∞ c ( G s ( γ ) ) → C ∞ c ( G r ( γ ) ) the induced op erator, then we hav e the following compatibility co ndition P r ( γ ) ◦ U γ = U γ ◦ P s ( γ ) . W e also admit, as usual, op erato rs acting in sections of vector bundles E → G (0) . There is also a differentiabilit y conditio n with r espect to x that can b e found in [ NWX99 ]. In this work we are going to work exclusively with uniformly supp o rted op er- ators, let us r e call this notion. Le t P = ( P x , x ∈ G (0) ) be a G -o per a tor, we denote by k x the Sch wartz kernel pf P x . L e t supp P := ∪ x supp k x , and supp µ P := µ 1 ( supp P ) , where µ 1 ( g ′ , g ) = g ′ g − 1 . W e s a y that P is uniformly supp orted if supp µ P is com- pact. W e denote by Ψ m ( G , E ) the space o f uniformly supp orted G -op erators, a cting on sections of a vector bundle E . W e denote also Ψ ∞ ( G , E ) = S m Ψ m ( G , E ) et Ψ −∞ ( G , E ) = T m Ψ m ( G , E ) . The comp osition of tw o suc h op erators is again o f this k ind (lemma 3, [ NWX99 ]). In fa ct, Ψ ∞ ( G , E ) is a filtered algebr a (theorem 1, rf.cit.), i . e . , Ψ m ( G , E )Ψ m ′ ( G , E ) ⊂ Ψ m + m ′ ( G , E ) . In pa rticular, Ψ −∞ ( G , E ) is a bila teral ideal. Remark 2.15. The choice on the supp ort justifies on the fact that Ψ −∞ ( G , E ) is ident ified with C ∞ c ( G , E nd ( E )), thank s the Sch wartz kernel theorem (theorem 6 [ NWX99 ]). The notion of principal symbol extends a lso to this setting. Let us deno te by π : A ∗ G → G (0) the pro jection. F or P = ( P x , x ∈ G (0) ) ∈ Ψ m ( G , E , F ), the princ ipa l symbol of P x , σ m ( P x ), is a C ∞ section of the vector bundle E nd ( π ∗ x r ∗ E , π ∗ x r ∗ F ) 8 P AULO CARRILLO ROUSE ov er T ∗ G x (where π x : T ∗ G x → G x ), s uc h that at each fiber the mo rphism is homogeneous of deg ree m (see [ AS68 ] for more deta ils). Ther e is a section σ m ( P ) of E nd ( π ∗ E , π ∗ F ) ov er A ∗ G s uch that (8) σ m ( P )( ξ ) = σ m ( P x )( ξ ) ∈ E nd ( E x , F x ) si ξ ∈ A ∗ x G Hence (8) ab ove, induces a unique surjective linea r ma p (9) σ m : Ψ m ( G , E ) → S m ( A ∗ G , E nd ( E , F )) , with kernel Ψ m − 1 ( G , E ) (s e e for instance pro pos itio n 2 [ NWX99 ]) and where S m ( A ∗ G , E nd ( E , F )) denotes the sec tio ns o f the fibe r E nd ( π ∗ E , π ∗ F ) ov er A ∗ G homogeneous of degree m at each fib e r . Definition 2. 16 ( G -Elliptic o pera tors) . Let P = ( P x , x ∈ G (0) ) b e a G -pseudo differential op erator. W e will say that P is elliptic if ea c h P x is elliptic. W e denote by E ll ( G ) the set o f G -pseudo differential e lliptic oper ators. The linear map (9) defines a principal symbol clas s [ σ ( P )] ∈ K 0 ( A ∗ G ): (10) E ll ( G ) σ − → K 0 ( A ∗ G ) . Connes, [ Con79 ], prov ed tha t if P = ( P x , x ∈ G (0) ) ∈ E ll ( G ), then it exists Q ∈ Ψ − m ( G , E ) such that I E − P Q ∈ Ψ −∞ ( G , E ) et I E − QP ∈ Ψ −∞ ( G , E ), where I E denotes the identit y oper ator over E . In other words, P defines a quasi- isomorphism in (Ψ + ∞ , C ∞ c ( G )) and thus an elemen t in K 0 ( C ∞ c ( G )) explicitly (when E is tr ivial) g iv en by (11)  T  1 0 0 0  T − 1  −  1 0 0 0  ∈ K 0 ( ^ C ∞ c ( G )) , where 1 is the unit in ^ C ∞ c ( G ) (unitarisation of C ∞ c ( G )), and where T is given by T =  (1 − P Q ) P + P P Q − 1 1 − QP Q  with inv erse T − 1 =  Q 1 − QP P Q − 1 (1 − P Q ) P + P  . If E is no t trivial w e obtain in the sa me wa y an element of K 0 ( C ∞ c ( G , E nd ( E , F ))) ≈ K 0 ( C ∞ c ( G )) since C ∞ c ( G , E nd ( E , F ))) is Morita equiv alent to C ∞ c ( G ). Definition 2.17 ( G -Index ) . Let P b e a G -pseudo differential elliptic op erator. W e denote b y ind P ∈ K 0 ( C ∞ c ( G )) the element defined b y P as ab ov e. It is called the index of P . It defines a cor resp ondence (12) E l l ( G ) ind − → K 0 ( C ∞ c ( G )) . Example 2.1 8 (The principa l symbo l clas s a s a Group oid index) . Let G b e a Lie gro up oid. W e can co nsider the Lie alg e broid a s Lie group oid with its vector bundle structure A G ⇒ G (0) . Let P b e a G -pseudo differential elliptic op erato r, then the principa l symbol σ ( P ) is a A G -pseudo differential elliptic op era to r. Its index, i nd ( σ ( P )) ∈ K 0 ( C ∞ c ( A G )) ca n be pushfor w ar d to K 0 ( C 0 ( A ∗ G )) using the inclusion of algebra s C ∞ c ( A G ) j ֒ → C 0 ( A ∗ G ) (mo dulo F ourier), the re s ulting ima g e gives precisely the map (10) ab ov e, i.e. , j ∗ ( ind σ ( P )) = [ σ ( P )] ∈ K 0 ( C 0 ( A ∗ G )) ≈ K 0 ( A ∗ G ). HIGHER LOCALIZED ANAL YTIC INDICES AND STRICT DEFORMA TION QUANTIZA TION 9 W e hav e a diagram E l l ( G ) ind σ K 0 ( C ∞ c ( G )) K 0 ( A ∗ G ) , where the p ointed ar row do es not alwa ys exist. It do es in the clas sical ca s es, but not for general Lie group oids as shown by the next example ([ Con94 ] pp. 1 4 2). Example 2.19. Let R → { 0 } be the g roup o id g iven by the group structure in R . In [ C o n94 ] (pr opo sition 1 2, I I.10 . γ ), Connes shows that the map D 7→ in d D ∈ K 0 ( C ∞ c ( R )) defines an injection of the pro jective spa c e of no zero p o lynomials D = P ( ∂ ∂ x ) int o K 0 ( C ∞ c ( R )). W e could consider the morphism (13) K 0 ( C ∞ c ( G )) j − → K 0 ( C ∗ r ( G )) induced by the inclusion C ∞ c ( G ) ⊂ C ∗ r ( G ), then the comp osition E l l ( G ) ind − → K 0 ( C ∞ c ( G )) j − → K 0 ( C ∗ r ( G )) do es factors thro ugh the pr inc ipa l symbol cla s s. In other words, we hav e the fol- lowing commutativ e diagram E l l ( G ) ind σ K 0 ( C ∞ c ( G )) j K 0 ( A ∗ G ) ind a K 0 ( C ∗ r ( G )) . Indeed,, ind a is the index mo r phism asso ciated to the exa ct sequence o f C ∗ -algebra s ([ Con79 ], [ CS84 ], [ MP97 ], [ NWX99 ]) (14) 0 → C ∗ r ( G ) − → Ψ 0 ( G ) σ − → C 0 ( S ∗ G ) → 0 where Ψ 0 ( G ) is a certain C ∗ − completion of Ψ 0 ( G ), S ∗ G is the sphere vector bundle of A ∗ G a nd σ is the extension of the principal symbol. Definition 2. 20. [ G -Analytic index] Let G ⇒ G (0) be a Lie gr oupo id. The mor- phism (15) K 0 ( A ∗ G ) ind a − → K 0 ( C ∗ r ( G )) is called the analytic index of G . The K -theor y of C ∗ -algebra s has very go o d cohomolo gical pr o per ties, how ever as we ar e going to discuss in the next subsection, it is sometimes preferable to work with the indices at the level of C ∞ c -algebra s. 10 P AULO CARRILLO ROUSE 2.2.1. Pairing with Cyclic c ohomolo gy: Index formulas. The interest to keep track on the C ∞ c -indices is b ecause at this level we c an make explicit calcula tions via the Cher n-W eil-Connes theory . In fact there is a pairing [ Con8 5, Con94, Kar87 ] (16) h , i : K 0 ( C ∞ c ( G )) × H P ∗ ( C ∞ c ( G )) → C There are several known co cycle s ov er C ∞ c ( G ). An impor tan t pr o blem in Noncom- m utative Geometry is to c o mpute the ab ov e pairing in order to obtain numerical inv ariants from the indices in K 0 ( C ∞ c ( G )), [ Co n9 4, CM90, GL06 ]. Let us illus- trate this affirmation with the following exa mple. Example 2.21 . [ CM90, Co n94 ] Let Γ b e a discr ete g roup acting prop erly and freely on a smo oth manifold ˜ M with compact quotient ˜ M / Γ := M . Let G ⇒ G (0) = M b e the Lie group oid quotient of ˜ M × ˜ M by the dia gonal a ction o f Γ. Let c ∈ H ∗ (Γ) := H ∗ ( B Γ). Connes- Mo scovici showed in [ C M 90 ] that the higher No vikov signature, S ig n c ( M ), can b e o btained with the pairing of the sig- nature op erator D sign and a cyclic co cycle τ c asso ciated to c : (17) h τ c , i nd D sgn i = S ig n c ( M , ψ ) . The Novik ov conjecture states that these higher signature s a r e orie nted homotopy inv ariants of M . Hence, if ind D sign ∈ K 0 ( C ∞ c ( G )) is a homotopy inv ariant of ( M , ψ ) then the Novik ov conjecture would follow. W e only know that j ( ind D sign ) ∈ K 0 ( C ∗ r ( G )) is a homo top y inv ar iant . But then we hav e to extend the action of τ c to K 0 ( C ∗ r ( G )). Co nnes-Moscovici show that this action extends for Hyp erbo lic groups. The pa ir ing (16) ab ov e is not interesting for C ∗ -algebra s. Indeed, the Cyclic cohomolog y for C ∗ -algebra s is trivia l (see [ C ST04 ] 5.2 for an explanatio n). In fact, as shown by the example ab ov e, a very interesting pro blem is to compute the pairing at the C ∞ c -level and then extend the action of the cyclic co cycle s to the K -theor y of the C ∗ -algebra . This problem is known as the extensio n problem and it was solved by Connes for some c y clic co cycle s as soc iated to foliations, [ Con86 ], and by Connes-Moscovici, [ CM90 ], for gr oup co cycles when the gr oup is hyper b olic. The most general formula for the pairing (16), known until these days (as far the author is aw are), is the one of Goro khovsky-Lott for F oliatio n group oids ([ GL06 ], theorem 5.) which gener alized a previous Co nnes formula for ´ etale gr oupo ids ([ Con94 ], theor em 12, I I I.7. γ , see also [ GL03 ] for a sup erco nnec tion pro of ). It basically says the following: Let G ⇒ M be a foliation g r oupo id (Morita equiv a len t to an ´ etale gr oupo id). It carrie s a foliation F . Let ρ b e a closed ho lonomy-in v ariant transverse curr e n t on M . Supp ose G acts freely , prop erly and co compa ctly on a manifold P . Let D b e a G - e lliptic different ia l o p era tor on P . Then the following formula holds: (18) h I nd D , ρ i = Z P / G ˆ A ( T F ) ch ([ σ D ]) ν ∗ ( ω ρ ) , where ω ρ ∈ H ∗ ( B G , o ) is a universal class asso ciated to ρ and ν : P / G → B G is a classifying map. Now, we can exp e ct an easy (top ological) calcula tion only if the map D 7→ h D , τ i ( τ ∈ H P ∗ ( C ∞ c ( G )) fix) factors thro ugh the sy m b ol cla ss of D , [ σ ( D )] ∈ HIGHER LOCALIZED ANAL YTIC INDICES AND STRICT DEFORMA TION QUANTIZA TION 11 K 0 ( A ∗ G ): we w ant to hav e a diag ram of the following kind: E l l ( G ) ind σ K 0 ( C ∞ c ( G )) h ,τ i C K 0 ( A ∗ G ) τ . This pap er is conc e rned w ith the so lution of the factor ization pr o blem justed describ ed. Our approach will use a geo metrical deformation asso ciated to a ny Lie group oid, known as the tang en t group oid. W e will discuss this in the next section. 3. Index theory and strict deformation quan tization 3.1. Deformation to the normal cone. The tangent gr oupo id is a partic- ular c ase of a geometric construction that we describ e here. Let M b e a C ∞ manifold and X ⊂ M b e a C ∞ submanifold. W e denote by N M X the normal bundle to X in M , i . e . , N M X := T X M /T X . W e define the following set D M X := N M X × 0 G M × R ∗ (19) The purpo se o f this sectio n is to re c all how to define a C ∞ -structure in D M X . T his is more or less classical, for example it was extensively used in [ HS87 ]. Let us firs t cons ider the case where M = R p × R q and X = R p × { 0 } (wher e we iden tify canonica lly X = R p ). W e denote by q = n − p and by D n p for D R n R p as ab ov e. In this case we clearly hav e that D n p = R p × R q × R (as a s e t). Consider the bijection ψ : R p × R q × R → D n p given by (20) ψ ( x, ξ , t ) =  ( x, ξ , 0) if t = 0 ( x, tξ , t ) if t 6 = 0 which inv erse is given explicitly by ψ − 1 ( x, ξ , t ) =  ( x, ξ , 0) if t = 0 ( x, 1 t ξ , t ) if t 6 = 0 W e can consider the C ∞ -structure on D n p induced by this bijection. W e pass now to the genera l cas e. A lo cal chart ( U , φ ) in M is said to b e a X -slice if 1) φ : U ∼ = → U ⊂ R p × R q 2) If U ∩ X = V , V = φ − 1 ( U ∩ R p × { 0 } ) (we de no te V = U ∩ R p × { 0 } ) With this notation, D U V ⊂ D n p as an op en subset. W e may define a function (21) ˜ φ : D U V → D U V in the following way: F or x ∈ V w e hav e φ ( x ) ∈ R p × { 0 } . If we write φ ( x ) = ( φ 1 ( x ) , 0), then φ 1 : V → V ⊂ R p is a diffeomorphism. W e set ˜ φ ( v , ξ , 0 ) = ( φ 1 ( v ) , d N φ v ( ξ ) , 0) and ˜ φ ( u, t ) = ( φ ( u ) , t ) for t 6 = 0 . Her e d N φ v : N v → R q is the norma l comp onent of the deriv ative dφ v for v ∈ V . It is clear that ˜ φ is a ls o a bijection (in particular it induces a C ∞ structure on D U V ). Now, let us co ns ider a n atlas { ( U α , φ α ) } α ∈ ∆ of M consisting of 12 P AULO CARRILLO ROUSE X − slices. Then the co llection { ( D U α V α , ˜ φ α ) } α ∈ ∆ is a C ∞ -atlas o f D M X (prop osition 3.1 in [ CR08b ]). Definition 3 .1 (Deforma tion to the normal cone) . Le t X ⊂ M b e as above. The set D M X equipp e d with the C ∞ structure induced by the atlas des cribe d in the last prop osition is called ” The deformatio n to n ormal c one asso ciate d t o X ⊂ M ”. Remark 3.2 . F ollowing the s a me steps , we can define a defo rmation to the no r mal cone asso ciated to a n injectiv e immer sion X ֒ → M . One imp ortant featur e ab out this construction is that it is in so me sense func- torial. More explicitly , let ( M , X ) and ( M ′ , X ′ ) b e C ∞ -pairs as ab ov e a nd let F : ( M , X ) → ( M ′ , X ′ ) b e a pa ir mor phism, i.e., a C ∞ map F : M → M ′ , with F ( X ) ⊂ X ′ . W e define D ( F ) : D M X → D M ′ X ′ by the following for m ulas: D ( F )( x, ξ , 0) = ( F ( x ) , d N F x ( ξ ) , 0) and D ( F )( m, t ) = ( F ( m ) , t ) for t 6 = 0, where d N F x is by definition the map ( N M X ) x d N F x − → ( N M ′ X ′ ) F ( x ) induced by T x M dF x − → T F ( x ) M ′ . Then D ( F ) : D M X → D M ′ X ′ is a C ∞ -map (prop osition 3.4 in [ CR08b ]). 3.2. The tangent group o id. Definition 3. 3 (T angent g roup o id) . Let G ⇒ G (0) be a Lie gr oupo id. The tangent gr oup oid asso ciated to G is the group oid that has D G G (0) as the s e t o f arrows and G (0) × R as the units, with: · s T ( x, η , 0) = ( x, 0) a nd r T ( x, η , 0) = ( x, 0) a t t = 0. · s T ( γ , t ) = ( s ( γ ) , t ) and r T ( γ , t ) = ( r ( γ ) , t ) at t 6 = 0 . · The pro duct is given by m T (( x, η , 0) , ( x, ξ , 0)) = ( x, η + ξ , 0 ) et m T (( γ , t ) , ( β , t )) = ( m ( γ , β ) , t ) if t 6 = 0 and if r ( β ) = s ( γ ). · The unit map u T : G (0) → G T is given by u T ( x, 0) = ( x, 0) and u T ( x, t ) = ( u ( x ) , t ) for t 6 = 0. W e denote G T := D G G (0) and A G := N G G (0) . As w e ha ve seen a bove G T can b e considered as a C ∞ manifold with b oundary . As a consequence of the functoriality of the Deforma tion to the normal cone, one can show that the tangent gro upoid is in fa c t a Lie gro upoid. Indeed, it is easy to chec k that if we identif y in a canonical w ay D G (2) G (0) with ( G T ) (2) , then m T = D ( m ) , s T = D ( s ) , r T = D ( r ) , u T = D ( u ) where we are consider ing the following pair mor phisms: m : (( G ) (2) , G (0) ) → ( G , G (0) ) , s, r : ( G , G (0) ) → ( G (0) , G (0) ) , u : ( G (0) , G (0) ) → ( G , G (0) ) . Remark 3.4. Finally , le t { µ x } be a smo oth Haar system on G , i.e. , a choice of G -inv ariant Leb esgue measures. In par ticular we have a n asso ciated s moo th Haar system on A G (gro upoid given b y the vector bundle structure), which we denote HIGHER LOCALIZED ANAL YTIC INDICES AND STRICT DEFORMA TION QUANTIZA TION 13 again by { µ x } . Then the following family { µ ( x,t ) } is a smo oth Haar system for the tangent gro up oid of G (deta ils may b e found in [ P at99 ]): • µ ( x, 0) := µ x at ( G T ) ( x, 0) = A x G a nd • µ ( x,t ) := t − q · µ x at ( G T ) ( x,t ) = G x for t 6 = 0, where q = dim G x . In this a r ticle, w e are only going to consider thes e Ha ar systems for the ta ng en t group oids. 3.2.1. Analytic indic es for Lie gr oup oids as deformations. Let G ⇒ G (0) be a Lie group oid and K 0 ( A ∗ G ) ind a − → K 0 ( C ∗ r ( G )) , its ana lytic index. This mor phism ca n also be constructed using the tangent group oid a nd its C ∗ -algebra . It is easy to chec k that the ev a luation mo rphisms extend to the C ∗ -algebra s: C ∗ r ( G T ) ev 0 − → C ∗ r ( A G ) a nd C ∗ r ( G T ) ev t − → C ∗ r ( G ) for t 6 = 0 . Moreov er , since G × (0 , 1 ] is an op en sa tur ated subset of G T and A G a n op en saturated closed subset, we hav e the following exa c t sequence ([ HS87 ]) (22) 0 → C ∗ r ( G × (0 , 1]) − → C ∗ r ( G T ) ev 0 − → C ∗ r ( A G ) → 0 . Now, the C ∗ -algebra C ∗ r ( G × (0 , 1 ]) ∼ = C 0 ((0 , 1 ] , C ∗ r ( G )) is co ntractible. This im- plies tha t the groups K i ( C ∗ r ( G × (0 , 1])) v anish, for i = 0 , 1. Then, a pply ing the K − theory functor to the e x act s e quence ab ov e, we o btain that K i ( C ∗ r ( G T )) ( ev 0 ) ∗ − → K i ( C ∗ r ( A G )) is an is omorphism, for i = 0 , 1 . In [ MP97 ], Month ub ert-Pierr ot show that (23) ind a = ( ev 1 ) ∗ ◦ ( ev 0 ) − 1 ∗ , mo dulo the F our ier isomorphism identifying C ∗ r ( A G ) ∼ = C 0 ( A ∗ G ) (see also [ HS87 ] and [ NWX99 ]). Putting this in a commut a tiv e diagr am, we hav e (24) K 0 ( C ∗ r ( G T )) e 0 ≈ e 1 K 0 ( A ∗ G ) ind a K 0 ( C ∗ r ( G )) . Compare the last diagram with (31) ab ov e. The algebra C ∗ r ( G T ) is a strict deforma tion qua n tization of C 0 ( A ∗ G ), and the analytic index morphism o f G ca n b e cons tructed by means of this deformatio n. In the next section we are going to discuss the existence o f a strict defor mation quantization alg ebra a sso ciated the tang en t g roup oid but in more primitive level, that is, not a C ∗ -algebra but a Sch wartz type a lgebra. W e will use afterwards to define other index morphisms as deformations. 14 P AULO CARRILLO ROUSE 3.3. A Sc h wart z alge bra for the tangent group oid. In this section we will reca ll how to co ns truct the deformation algebr a ment io ned at the intro duction. F or complete details, we refer the rea der to [ CR08b ]. The Sch wartz a lgebra for the T angent g roup oid will b e a particular c a se of a construction asso ciated to any deformation to the normal cone. Definition 3. 5. Let p, q ∈ N and U ⊂ R p × R q an op en subset, and let V = U ∩ ( R p × { 0 } ). (1) Let K ⊂ U × R b e a co mpact subset. W e say that K is a conic compac t subset of U × R r e la tiv e to V if K 0 = K ∩ ( U × { 0 } ) ⊂ V (2) Let Ω U V = { ( x, ξ , t ) ∈ R p × R q × R : ( x, t · ξ ) ∈ U } , which is an o pen subset of R p × R q × R a nd th us a C ∞ manifold. Let g ∈ C ∞ (Ω U V ). W e say that g has compact conic supp ort, if there exists a conic compact K of U × R relative to V such that if ( x, tξ , t ) / ∈ K then g ( x , ξ , t ) = 0. (3) W e de no te b y S c (Ω U V ) the s et of functions g ∈ C ∞ (Ω U V ) that hav e compac t conic suppo rt and that satisfy the following co nditio n: ( s 1 ) ∀ k , m ∈ N , l ∈ N p and α ∈ N q there exist C ( k,m,l,α ) > 0 such that (1 + k ξ k 2 ) k k ∂ l x ∂ α ξ ∂ m t g ( x, ξ , t ) k ≤ C ( k,m,l,α ) Now, the spaces S c (Ω U V ) a re inv aria n t under diffeomorphis ms . More pre c is ely: Let F : U → U ′ be a C ∞ -diffeomorphism such tha t F ( V ) = V ′ ; let ˜ F : Ω U V → Ω U ′ V ′ be the induced map. Then, for e very g ∈ S c (Ω U ′ V ′ ), we hav e that ˜ g := g ◦ ˜ F ∈ S c (Ω U V ) (prop osition 4.2 in [ CR08b ]). This compatibility res ult allows to give the following definition. Definition 3.6. Let g ∈ C ∞ ( D M X ). (a) W e say that g has conic c o mpact supp ort K , if there exists a compact subset K ⊂ M × R with K 0 := K ∩ ( M × { 0 } ) ⊂ X (conic compac t relative to X ) such that if t 6 = 0 and ( m, t ) / ∈ K then g ( m, t ) = 0. (b) W e say that g is rapidly decaying at zero if for ev er y ( U , φ ) X -slice c ha rt and for every χ ∈ C ∞ c ( U × R ), the map g χ ∈ C ∞ (Ω U V ) (Ω U V as in definition 3.5.) g iv en by g χ ( x, ξ , t ) = ( g ◦ ϕ − 1 )( x, ξ , t ) · ( χ ◦ p ◦ ϕ − 1 )( x, ξ , t ) is in S c (Ω U V ), where · p is the deformation of the pair ma p ( M , X ) I d − → ( M , M ), i.e. , p : D M X → M × R is g iv en by ( x, ξ , 0) 7→ ( x, 0), a nd ( m, t ) 7→ ( m, t ) for t 6 = 0, and · ϕ := ˜ φ − 1 ◦ ψ : Ω U V → D U V , where ψ and ˜ φ are defined at (20) and (21) ab ov e. Finally , we denote by S c ( D M X ) the set o f functions g ∈ C ∞ ( D M X ) that are rapidly decaying at zero with conic compact supp ort. Remark 3.7. (a) Obviously C ∞ c ( D M X ) is a subspace of S c ( D M X ). HIGHER LOCALIZED ANAL YTIC INDICES AND STRICT DEFORMA TION QUANTIZA TION 15 (b) Let { ( U α , φ α ) } α ∈ ∆ be a family of X − slices cov ering X . W e have a decom- po sition of S c ( D M X ) as follows (se e remark 4.5 in [ CR08b ] and discussion below it): S c ( D M X ) = X α ∈ Λ S c ( D U α V α ) + C ∞ c ( M × R ∗ ) . (25) The main theorem in [ CR08b ] (Theorem 4.10 ) is the following Theorem 3.8. The sp ac e S c ( G T ) is stable under c onvolution, and we have the fol lowing inclusions of algebr as C ∞ c ( G T ) ⊂ S c ( G T ) ⊂ C ∗ r ( G T ) Mor e over, S c ( G T ) is a field of algebr as over R , whose fib ers ar e S ( A G ) at t = 0 , and C ∞ c ( G ) for t 6 = 0 . In the statement of this theorem, S ( A G ) deno tes the Sch wartz algebra ov er the Lie algebro id. Let us briefly re c all the notio n of Sch wartz spa ce a sso ciated to a vector bundle: F or a trivial bundle X × R q → X , S ( X × R q ) := C ∞ c ( X, S ( R q )) (see [ T r ` e 06 ]). In genera l, S ( E ) is defined using lo cal charts. More precisely , a partition o f the unity argument, allows to see tha t if we take a covering of X , { ( V α , τ α ) } α ∈ ∆ , cons isting on trivia lizing charts, then we hav e a decomp osition o f the following kind: (26) S ( E ) = X α S ( V α × R q ) . The ”Sc hw a rtz a lg ebras” hav e in ge ner al the go od K − theory gr oups. As we said in the in tr oduction, we are interested in the group K 0 ( A ∗ G ) = K 0 ( C 0 ( A ∗ G )). It is not enough to take the K − theor y of C ∞ c ( A G ) (see example 2.1 9). As we s ho wed in [ CR08a ] (prop osition 4.5), S ( A ∗ G ) has the wanted K -theory , i.e. , K 0 ( A ∗ G ) ∼ = K 0 ( S ( A G )). In particular, our deformation algebra restr icts at zero to the r ight algebra. F rom now on it w ill b e imp ortant to res trict our functions on the tangent group oid to the clos ed interv al [0 , 1]. W e keep the notation S c ( D M X ) for the r e- stricted s pace. All the re sults ab ov e remain true. So for instance S c ( G T ) is an algebra which is a field of a lgebras over the clo sed interv al [0 , 1 ] with 0-fib er S ( A G ) and C ∞ c ( G ) otherwise. W e have the following short exact sequence of a lgebras ([ CR08 a ], prop o sition 4.6): 0 − → J − → S c ( G T ) e 0 − → S ( A G ) − → 0 , (27) where J = K er ( e 0 ) by definition. 4. Highe r lo calized i n di ces Definition 4. 1. Let τ b e a (p erio dic) cy c lic co cycle ov er C ∞ c ( G ). W e s a y that τ can be loc a lized if the c o rresp ondence (28) E l l ( G ) ind K 0 ( C ∞ c ( G )) h ,τ i C 16 P AULO CARRILLO ROUSE factors through the principa l symbol cla ss morphism. In other w o rds, if there is a unique morphism K 0 ( A ∗ G ) I nd τ − → C which fits in the following commutative diagram (29) E l l ( G ) ind [ psy m b ] K 0 ( C ∞ c ( G )) h ,τ i C K 0 ( A ∗ G ) I nd τ i.e. , satisfying I nd τ ( a ) = h ind D a , τ i , and hence completely characterized by this prop erty . In this case, we ca ll I nd τ the higher loca lized index asso ciated to τ . Remark 4.2. If a cyclic co cycle can b e lo calized then the higher lo calize d index I nd τ is completely characterized by the prop erty: I nd τ ([ σ D ]) = h ind D , τ i , ∀ D ∈ E l l ( G ). W e a r e going to prov e first a lo caliza tion result for Bounded cyclic co cycles, w e recall its definition. Definition 4.3. A multilinear map τ : C ∞ c ( G ) × · · · × C ∞ c ( G ) | {z } q +1 − times → C is b ounded if it extends to a contin uous multilinear map C k c ( G ) × · · · × C k c ( G ) | {z } q +1 − times τ k − → C , for some k ∈ N . W e ca n r e - state theo rem 6.9 in [ CR 0 8a ] in the following wa y: Theorem 4.4 . L et G ⇒ G (0) b e a Lie gr oup oid, then (i) Every b ounde d cyclic c o cycle over C ∞ c ( G ) c an b e lo c alize d. (ii) Mor e over, if the gr oup oid is ´ etale, then every cyclic c o cycle c an b e lo c alize d. W e will reca ll the main steps for proving this result. F o r this purp ose we need to define the int er mediate g roup K 0 ( C ∞ c ( G )) → K B 0 ( G ) → K 0 ( C ∗ r ( G )) . Let us denote, for each k ∈ N , K h,k 0 ( G ) the quotient group of K 0 ( C k c ( G )) b y the equiv a lence re lation induced by K 0 ( C k c ( G × [0 , 1])) e 0 ⇒ e 1 K 0 ( C k c ( G )). Let K F 0 ( G ) = lim ← − k K h,k 0 ( G ) b e the pro jective limit relative to the inclusions C k c ( G ) ⊂ C k − 1 c ( G ). W e ca n take the inductiv e limit lim − → m K F 0 ( G × R 2 m ) induced by K F 0 ( G × R 2 m ) B ott − → K F 0 ( G × R 2( m +1) ) (the Bott morphis m). W e denote this group b y (30) K B 0 ( G ) := lim − → m K F 0 ( G × R 2 m ) , Now, theorem 5.4 in [ CR08a ] establish the following t wo a ssertions: HIGHER LOCALIZED ANAL YTIC INDICES AND STRICT DEFORMA TION QUANTIZA TION 17 (1) Ther e is a unique group mo rphism ind B a : K 0 ( A ∗ G ) → K B 0 ( G ) that fits in the following commutativ e diagr am (31) K 0 ( S c ( G T )) e 0 e B 1 K 0 ( A ∗ G ) ind B a K B 0 ( G ) , where e B 1 is the ev a luation at one K 0 ( S c ( G T )) e 1 − → K 0 ( C ∞ c ( G )) followed by the canonical map K 0 ( C ∞ c ( G )) → K B 0 ( G ). (2) This morphism also fits in the following commutativ e diagram (32) E l l ( G ) σ ind K 0 ( C ∞ c ( G )) K 0 ( A ∗ G ) id ind B a K B 0 ( G ) K 0 ( A ∗ G ) ind a K 0 ( C ∗ r ( G )) . Next, it is very ea sy to check (see [ CR08a ] Prop osition 6.7) tha t if τ is a bo unded cyclic co cycle, then the pair ing morphism K 0 ( C ∞ c ( G )) h , τ i − → extends to K B 0 ( G ), i.e. , we hav e a commutativ e diagram of the following type: (33) K 0 ( C ∞ c ( G )) <,τ > ι C K B 0 ( G ) τ B Now, theorem 4.4 follows immediately b ecause we can put tog e ther dia grams (32) and (33) to get the following commutativ e diagr am (34) E l l ( G ) ind σ K 0 ( C ∞ c ( G )) h ,τ i C K 0 ( A ∗ G ) ind B a K B 0 ( G ) τ B . 4.1. Higher lo calized index formula. In this section we will give a formula for the Higher lo calized indices in terms of a pairing in the strict defor mation quantization algebra S c ( G T ). W e have first to in tro duce some notation : Let τ b e a ( q + 1 ) − m ultilinear functional over C ∞ c ( G ). F or each t 6 = 0, we let τ t be the ( q + 1)-multilinear functional over S c ( G T ) defined by (35) τ t ( f 0 , ..., f q ) := τ ( f 0 t , ..., f q t ) . In fact, if we consider the ev aluation morphisms e t : S c ( G T ) → C ∞ c ( G ) , 18 P AULO CARRILLO ROUSE for t 6 = 0, then it is o bvious that τ t is a ( b, B )-co cycle (per iodic cyclic c ocy cle) ov er S c ( G T ) if τ is a ( b, B )-cocyc le over C ∞ c ( G ). Indeed, τ t = e ∗ t ( τ ) by definition. W e ca n now state the main theorem of this article. Theorem 4.5. L et τ b e a b ounde d cyclic c o cycle t hen the higher lo c alize d index of τ , K 0 ( A ∗ G ) I nd τ − → C , is given by (36) I nd τ ( a ) = li m t → 0 h e a, τ t i , wher e e a ∈ K 0 ( S c ( G T )) is such that e 0 ( e a ) = a ∈ K 0 ( A ∗ G ) . In fact the p airing ab ove is c onstant for t 6 = 0 . Remark 4. 6 . Hence, if τ is a b ounded c y clic co cycle a nd D is a G -pseudo differen tia l elliptic op erator, then we hav e the following formula for the pa iring: (37) h ind D , τ i = h f σ D , τ t i , for ea c h t 6 = 0, and wher e f σ D ∈ K 0 ( S c ( G T )) is suc h that e 0 ( e σ ) = σ D . In particular, (38) h i nd D , τ i = lim t → 0 h f σ D , τ t i . F or the pr oo f of the theo rem a bove w e will need the following le mma . Lemma 4.7. F or s, t ∈ (0 , 1] , τ s and τ t define the same p airing map K 0 ( S c ( G T )) − → C . Proof. Let p b e an idemp otent in ^ S c ( G T ) = S c ( G T ) ⊕ C . It defines a smo oth family of idemp oten ts p t in ^ S c ( G T ). W e set a t := dp t dt (2 p t − 1). Hence, a s imple calculation shows d dt h τ , p t i = 2 n X i =0 τ ( p t , ..., [ a t , p t ] , ..., p t ) =: L a t τ ( p t , ..., p t ) . Now, the Lie derivatives L x t act trivially o n H P 0 ( S c ( G T )) (see [ Con85, Go o 85 ]), then h τ , p t i is cons tan t in t . Finally , by definition, h τ t , p i = h τ , p t i . Hence t 7→ h τ t , p i is a constant function for t ∈ (0 , 1].  Proof o f theorem 4.5. P utting together diagr ams (31) and (34), we get the following commutativ e diagra m (39) E l l ( G ) ind σ K 0 ( C ∞ c ( G )) h ,τ i K 0 ( A ∗ G ) ind B a K B 0 ( G ) τ B C K 0 ( S c ( G T )) e B 1 τ 1 e 0 . In other words, for a ∈ K 0 ( A ∗ G ), I nd τ ( a ) = h e a, τ 1 i . Now, by lemma 4.7 we can conclude that I nd τ ( a ) = h e a, τ t i , for each t 6 = 0. In par ticular the limit when t tends to zero is given by I nd τ ( a ) = li m t → 0 h e a, τ t i .  HIGHER LOCALIZED ANAL YTIC INDICES AND STRICT DEFORMA TION QUANTIZA TION 19 F or ´ e ta le gro up oids, we can state the following corollary . Corollary 4. 8. If G ⇒ G (0) is an ´ etale gr oup oid, then formula (36) holds for every cyclic c o cycle. Proof. Thanks to the works of B ur ghelea, Br ylinski-Nistor and Crainic ([ Bur85, BN94, Cra99 ]), we k no w a very explicit description of the Perio dic cyclic coho- mology for ´ etale g r oupo ids. F or instance , we have a decomp osition o f the following kind (see for example [ Cra99 ] theorems 4.1.2 . and 4.2.5) (40) H P ∗ ( C ∞ c ( G )) = Π O H ∗ + r τ ( B N O ) , where N O is an ´ etale gr oupo id as so c ia ted to O (the norma lizer of O , see 3.4.8 in ref.cit.). F or insta nc e , when O = G (0) , N O = G . Now, all the cyclic co cycles coming fro m the cohomolog y o f the class ifying space are b ounded. Indeed, we know that ea c h factor o f H P ∗ ( C ∞ c ( G )) in the decomp osition (40) consists of b ounded cyclic co cycles (see last section of [ CR08 a ]). Now, the pairing H P ∗ ( C ∞ c ( G )) × K 0 ( C ∞ c ( G )) − → C is well defined. In particular, the r e s triction to H P ∗ ( C ∞ c ( G )) | O v a nishes for almost every O . The conclusion is now immediate fro m the theor em ab ov e.  Once we hav e the formula (36) ab ov e, it is well worth it to re c all why the ev a luation morphism (41) K 0 ( S c ( G T )) e 0 − → K 0 ( A ∗ G ) is sur jective. Let [ σ ] ∈ K 0 ( S ( A G )) = K 0 ( A ∗ G ). W e know from the G - pseudo differ en tial calculus that [ σ ] c an be repr esen ted by a smo oth homoge ne o us elliptic symbol (see [ AS68, CH90, MP97, NW X99 ]). W e can consider the symbol over A ∗ G × [0 , 1] that coincides with σ for all t , we denote it by ˜ σ . Now, s ince A G T = A G × [0 , 1], we can take ˜ P = ( P t ) t ∈ [0 , 1] a G T -elliptic pseudo different ia l op era tor asso ciated to σ , that is, σ ˜ P = ˜ σ . Let i : C ∞ c ( G T ) → S c ( G T ) b e the inclusion (which is an al- gebra morphism), then i ∗ ( ind ˜ P ) ∈ K 0 ( S c ( G T )) is such that e 0 , ∗ ( i ∗ ( ind ˜ P )) = [ σ ]. Hence, the lifting of a pr incipal s ym b ol c lass is given by the index of ˜ P = ( P t ) t ∈ [0 , 1] and theor em 4 .5 says that the pairing with a b o unded cy c lic co cycle do es not de- pend on the choice of the ope rator P . Now, for compute this index, as in formula (11), one should find first a parametrix for the family ˜ P = ( P t ) t ∈ [0 , 1] . F or insta nce, in [ CM90 ] (section 2), Connes-Mo scovici consider elliptic differ- ent ia l op erators over co mpa ct manifolds, let us say an o per ator D ∈ D O r ( M ; E , F ) − 1 . Then they cons ider the family of op era to rs tD (multiplication by t in the normal direction) fo r t > 0 a nd they construct a fa mily of par ametrix e Q ( t ). The cor re- sp onding idempotent is then homotopic to W ( tD ), wher e (42) W ( D ) = e − D ∗ D e − 1 2 D ∗ D ( I − e − D ∗ D D ∗ D ) 1 2 D ∗ e − 1 2 DD ∗ ( I − e − DD ∗ DD ∗ ) 1 2 D I − e − D ∗ D ! is the W asser man idempotent. In the langua g e o f the tangent group oid, the family ˜ D = { D t } t ∈ [0 , 1] where D 0 = σ D and D t = tD for t > 0, defines a G T -differential elliptic op erator. What Connes and Mos c o vic i compute is precisely the limit o n right hand side of formula (36). Also, in [ MW94 ] (section 2 ), Mos covici-W u pro ceed in a similar wa y b y using the finite propa gation spee d pro perty to cons truct a parametrix for op erators ˜ D = 20 P AULO CARRILLO ROUSE { D t } t ∈ [0 , 1] ov er the tangent group oid. Then they obtain as asso ciated idempo ten t the so called graph pro jector. What they compute after is again a particular case of the right hand side of (36). Finally , in [ GL03 ] (section 5.1), Gor okhovsky-Lott use the same tec hnics as the tw o pre vious examples in order to obtain their index formula. Remark 4.9. As a final remark is interesting to mention that in the formula (36) bo th sides make alwa ys s ense. In fact the pair ing h e a, τ t i is constant for t 6 = 0. W e co uld then consider the differences η τ ( D ) := h ind D , τ i − l im t → 0 h f σ D , τ t i for any D pseudo differen tial elliptic G -op erator. 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