$L^2$-index formula for proper cocompact group actions

L 2 -INDEX FORMULA FOR PROPER COCOMP A CT GROUP A CTIONS HANG W ANG A B S T R AC T . W e study inde x theory of G -inv ariant elliptic pseudo-dif ferential o perators actin g on a complete Riemanni an manifold, where a unimodular , loc ally compact group G acts properly , co compactl y and isometric ally . An L 2 -inde x formula is obtained using the heat kernel method. Mathematics Subject Classifacation (201 0): 19K56, 58J35, 58J40 K eywor d s :L 2 -index, K -theoretic index, G -trace, he at kernel. C O N T E N T S 1. Introd uction. 1 1.1. Main result. 1 1.2. Remarks on the result. 2 1.3. Idea of the proof . 3 1.4. Acknowledgement. 3 2. Preliminaries. 3 3. The G -trace and the L 2 -index. 7 3.1. The G -trace of operators on X . 8 3.2. The L 2 -index of elliptic operators on X . 12 4. The connection of the L 2 -index to the K -theoretic index. 14 5. Reduction to the L 2 -index of a Dirac type operator . 17 6. Local index formula. 22 6.1. L 2 -index of Dirac type operators. 22 6.2. Conclusion. 29 6.3. L 2 -index theorem for homogeneo us spaces of Lie groups. 31 References 33 1. I N T RO D U C T I O N . 1.1. Main result. Let X be a com plete R iemannian ma nifold acted on pr operly , coco m- pactly and isomertrically by a locally compact unimodular group G an d let E b e a Z / 2 Z - graded G -vector b u ndle over X . Let P =  0 P ∗ 0 P 0 0  : L 2 ( X , E ) → L 2 ( X , E ) be a 0 -order prop erly sup ported elliptic pseudo -dif ferential op erator inv arian t u nder the group action. Such an operator has a real-valued L 2 -index deﬁned as the difference 1 2 HANG W ANG of the von Neum ann traces of the projections onto the closed G -in variant subspaces Ker P 0 , Ker P ∗ 0 of L 2 ( X , E ) : ind P = tr G P K er P 0 − tr G P K er P ∗ 0 . The pap er i s to p rov e that the L 2 -index of P is calcu lated by the following topologica l formu la: (1.1) ind P = Z T X ( c ◦ π ) · ( ˆ A ( X )) 2 ch ( σ P ) . Here c ∈ C ∞ c ( X ) is a non-negative function s atisfying Z G c ( g − 1 x ) d g = 1 for all x ∈ X , and π : T X → X is the pr ojection. 1.2. Remarks on the result. Th e formu la (1.1) generalizes the L 2 -index f ormula f or free cocomp act grou p a ctions d ue to Atiyah [2] an d the L 2 -index f ormula f or hom oge- neous spaces of un imodular Lie groups due to Con nes and Moscovici [10]. T he study of L 2 -indices in gene ral h as implicatio ns in other are as of mathem atics. For example, the non-vanishing of th e L 2 -index for the sign ature op erator on X in dicates th e existence of L 2 -harmo nic forms on X . T he L 2 -index is of intere s t in the stu dy of discrete series r ep- resentations [1 0] and h as been m odiﬁed for use in a p roof of the N o vikov conjectur e for hyperb olic groups [11]. Our index fo rmula (1.1) is analog ous to the type II theory in von Neum ann alg ebra. The key featu re o f a type II in dex theor y is that the elliptic o perators bein g inv estigated are n o longer F redho lm, b ut using some techn iques analogous to those used in type II von Neumann theory , say , by form ulating some trace, on e may obtain gen eralized Fredholm indices associated to the elliptic ope rators. Refer to [28, 29] for anoth er example of this type. When the orbit space X / G is an orbifold, the L 2 -index discussed in this paper is no t the same as th e index for X / G as a compact orb ifold [23, 24]. For examp le, Dirac operator s on a go od or bifold are Fredh olm and hav e integer indices, reﬂecting the info rmation of the or bit space, wh ile the L 2 -indices of the Dirac opera tors lifted to the universal cover of the o rbifold are rational nu mbers by d eﬁnition. T he integer ind ices and the r ational indices are d if fer ent in genera l [12]. They co incide on spacial cases, for exam ple, when the orbit space is a smooth man ifold [2]. Another example is that when b oth X and G are co mpact, (1.1) i s the same as the Atiyah-Singer index form ula for compact manifolds, regardless of the group G [4], while the ind e x fo rmula correspon ding to the orbifold X / G in volves gr oup action [24]. Our for mula is expected to have interesting applica tions when the group is not compact. W e also notice the existence of L 2 -index formula (in some special cases) when X / G is a nonco mpact orbif old but has ﬁnite volume [ 34], where the an alysis on the strata of o f X / G is heavily used . It is inter esting to stud y the L 2 -index (if exists) where th e quotient is nonc ompact. Howe ver , o ur o perator algeb raic app roach in ﬁnding the f ormula of L 2 - index does not work fo r th e case o f n oncompact quo tient. Th e reaso n is tha t when G acts proper ly , cocomp actly and isometrically on X , the gr oup G and the manifold X are coarse equiv alence, then we may use G , mo re precisely , C ∗ ( G ) to study the ellip tic operators on X in variant under the action of G [25, 22]. Howe ver, when X / G is not com pact, the gro up G has nothing to do with the L 2 -index for G -in variant elli ptic ope rators on X . L 2 -INDEX FORM ULA FOR PR OPER COC OMP ACT GROUP ACTIONS 3 Finally , (1. 1 ) ﬁt s into the fr ame work of the high er ind e x formula taking v alues in cyclic theory . In [ 27], a g eneral form ula was p rov ed and the indices of Dir ac opera tors take values in the en tire cyclic homology of some subalgebr a o f the grou p C ∗ -algebra C ∗ ( G ) . Formally , for Dirac operators, (1.1) is obtained from [27] corollary 1.2 by taking g ∈ G to be th e g roup iden tity and by taking n = 0 . W e would like to have a deep er in vestigation on the connectio n o f the two results in futur e. 1.3. Idea of the proof. T o prove (1.1), regard P as an element in the K -homolog y gr oup K 0 G ( C 0 ( X )) , from which P has a higher index in K 0 ( C ∗ ( G )) , wher e C ∗ ( G ) is the maximal group C ∗ -algebra. The L 2 -index of P d epends only on th e equiv alence class of its h igher index in K 0 ( C ∗ ( G )) . This is proved in section 4 b y deﬁning a trace on a den s e h olomor- phic closed ideal S ( E ) in K ( E ) , where E is a Hilbert C ∗ ( G ) -module having the same K -theor y as C ∗ ( G ) . The trace is the von Neumann trace of a type II von Neumann alg ebra in the sense o f Breu er [8]. A co mprehensive discussion o n t he link betwee n the L 2 -index and the higher index may be found in [31]. Secondly , in section 5, we reduce the proble m o f ﬁn ding in d P into ﬁnd ing ind D for some Dirac type o perator D , which has the same higher index as P . Kasparov’ s K- theoretic in de x f ormula [17] is essential in the argum ent. T he for mulation of Dirac type operator s ou t of elliptic opera tors is also related to the vector bundle modiﬁcation con- struction in the deﬁnition of geometric K -homo logy [ 5 ]. The ﬁn al step is to calculate in d D using the h eat kernel method . When D is a ﬁrst order G -inv a riant operato r of Dirac type on X , we have the McKean-Singer for mula fo r the L 2 -index: (1.2) ind D = tr G e − t D ∗ D − tr G e − t D D ∗ , t > 0 . In the case o f a c ompact ma nifold withou t group action, a c ohomological fo rmula was obtained by studying the lo cal inv ariants of metrics and conne ctions [ 3 , 14]. The pro of of the lo cal index fo rmula was simpliﬁed by a rescalin g argument of Getzler [13] on the asymptotic expan sion of the heat kernel e − t D 2 around t = 0. Since the ind e x ind D in (1.2) is local wh en t → 0 + , the group action does not af fe ct the calculation. The proof is based on a modiﬁcation of the proof s in [30, 6] and is complete in section 6. 1.4. Acknowledgement. Th e work is modiﬁed after my PHD thesis and is fund ed by NSF , V anderbilt Un i versity and I HES. Special tha nks go to Pr ofessor Gennadi Kasparov for pro posing this topic and fo r h is adv ice. After writing up m y p aper , I received many comments as well as warm he lps. I wish to express my sincere gr atitude to all th e p ro- fessors wh o hav e he lped m e. Fin ally , I would like to thank the r eferee for the helpfu l remarks. 2. P R E L I M I N A R I E S . Let G be a lo cally compact and unimod ular gro up, that is, ther e is a b i-in variant Haar measure µ on G . For example, co mpact group s and discre te groups ar e unimo dular . Set d g . = d µ ( g ) and we have d ( t g ) = d g , d ( gt ) = d g and d ( g − 1 ) = d g fo r any g , t ∈ G . 4 HANG W ANG Let X be a com plete Riemann ian manif old, on which G acts proper ly, coco mpactly and isometrically, that is, the pre-image of any compact set under th e continuous map G × X → X × X : ( g , x ) 7→ ( g · x , x ) is compact, the quotient space X / G is compact, and G respects the metric < · , · > : < x , y > = < gx , gy > for all x , y ∈ X , g ∈ G . The reason to con sider prope r co compact actions is the existence o f a cu tof f function o n X . Deﬁnition 2.1. A nonnegative functio n c ∈ C ∞ c ( X ) is a cutoff function if for all x ∈ X , Z G c ( g − 1 x ) d g = 1 . Remark 2.2 . A proper cocompact G -space has a cutoff fun ction c ∈ C ∞ c ( X ) given by c ( x ) = h ( x ) R G h ( g − 1 x ) d g , where h ( x ) ∈ C ∞ c ( X ) is no nnegati ve an d has no n-empty intersection with each or bit. Example 2.3 . Let G be a Lie group with a compact subgr oup H , and let X = G / H be the homoge neous spac e con sis ting of all the left cosets of H in G . The action of G on X is prop er . Fu rther , let E be a represen tation space of H . Th e induced represen tation Y = G × H E , wh ich forms a G -vector bundle over X , is a prope r G -space. Acc ording to the slice theorem, e very proper space has such a local structure. Theorem 2. 4 ( Sli ce theorem ) . Let G be a loca ll y compac t gr oup a nd X b e a p r op er G- space. Then for a ny x ∈ X a nd for any neig hborhood O of x in X , ther e exis ts a compa ct subgr o up K of G with G x . = { g ∈ G | gx = x } ⊂ K and a K -slice S su c h that x ∈ S ⊂ O . Recall that A K -invariant subset S ⊂ X is a K -slice in X if (1) The union G ( S ) (tubular set) of all orbits inters ecting S is ope n; (2) Ther e is a G-equ ivariant map f : G ( S ) → G / K (the slicing map) , such th at S = f − 1 ( eK ) . An introduction to the slice theorem may be foun d in [1] section 2. According to [7] Ch.II Theor em 4.2, the tubular set G ( S ) ⊂ X with a co mpact slicing su bgroup K is G -homeo morphic to G × K S . Remark 2.5 . Since X is cov ered by G -in variant neigh borhoods a nd since X / G is comp act, then X ad mits a ﬁnite sub- co ver , that is, (2.1) X = ∪ N i = 1 G × K i S i = ∪ N i = 1 G ( S i ) . The local structure (2 .1) o f X deﬁnes a G -in variant measure d x on X . In f act, The measure of a set in G ( S i ) is calculated from the measure on G and on S i divided by the measure of K i . Then the measure of a set T ⊂ X is deﬁn ed using a partition of un ity argument. The 1-density on the Riemannian manifold X also d eﬁnes the same measur e. In order to introduce ellipticity , we recall the following d eﬁnitions con cerning pseudo - differential op erators. Let ( E , p ) be a ﬁnite dimension al c omplex G -vector bundle over X , that is, there is a smooth G action on E such that p ( gv ) = g p ( v ) for v ∈ E and th e maps of the ﬁb ers g : E x → E gx are lin ear . L et π : T ∗ X → X be the projection m ap and π ∗ E over L 2 -INDEX FORM ULA FOR PR OPER COC OMP ACT GROUP ACTIONS 5 T ∗ X be th e p ull-back bundle of E . Her e, E = E 0 ⊕ E 1 is Z / 2 Z -graded and the G -action is grading preserving. The G -ac ti ons on E 0 , E 1 giv e rise to a G -bundle Hom ( π ∗ E 0 , π ∗ E 1 ) over T ∗ X . A symbol functio n σ of order m is a co ntinuous section of this G -bundle satisfying (2.2) | ∂ a ∂ x | a | ∂ b ∂ ξ | b | σ ( x , ξ ) | ≤ C a , b , K ( 1 + k ξ k ) m −| b | for x in any co mpact set K ⊂ X an d ξ in th e ﬁbe r T x X , where C a , b , K is a con st ant dep ending on a , b , K . Here a = ( a 1 , . . . , a n ) , b = ( b 1 , . . . , b n ) and | a | = n ∑ i = 1 a i , | b | = n ∑ i = 1 b i ( dim X = n ) . The set o f all or der m symbols is den oted by S m ( X ; E 0 , E 1 ) and a principal symbo l of order m is an element in the quotient S m ( X ; E 0 , E 1 ) / S m − 1 ( X ; E 0 , E 1 ) . W e shall omit the word “principal” from no w on. Each symbol σ has an amplitud e p deﬁn ed by p ( x , y , ξ ) = α ( x , y ) σ ( q ( y , ( x , ξ x ))) , where α ∈ C ∞ ( X × X ) h as support co ntained in a small neig hborhood of the diag onal so that α ( x , x ) = 1 and α ( x , y ) ≥ 0 fo r all x , y ∈ X and q : X × T ∗ X → T ∗ X : ( y , ( x , ξ x )) 7→ ( y , ξ y ) ( ξ y is the parallel transport of ξ x from x to y ). Conversely , σ ( x , ξ ) = p ( x , x , ξ ) . Denote by C ∞ c ( X , E ) the set of smooth section s of E with compact sup port in X and G ac ts on C ∞ c ( X , E ) by ( g · f )( x ) = g ( f ( g − 1 x )) , fo r all g ∈ G , f ∈ C ∞ c ( X , E ) . T o each amplitude p ( x , y , ξ ) , we may construct a pseu do-differ ential operator P 0 : C ∞ c ( X , E 0 ) → C ∞ ( X , E 1 ) by (2.3) P 0 u ( x ) = Z X × T ∗ x X e i Φ ( x , y , ξ ) p ( x , y , ξ ) u ( y ) d y d ξ x , where Φ ( x , y , ξ ) = < exp − 1 x ( y ) , ξ x > is the phase fu nction. The Schwartz kernel K P 0 ( x , y ) ∈ Hom ( E 0 y , E 1 x ) of P 0 , that is, P 0 u ( x ) = Z X K P 0 ( x , y ) u ( y ) d y f or all u ( x ) ∈ C ∞ c ( X , E 0 ) , is expressed in the follo wing distrib utional sense, (2.4) K P ( x , y )( w ) = Z X × T ∗ X e i Φ ( x , y , ξ ) p ( x , y , ξ ) w ( x , y ) d x d y d ξ , w ∈ C ∞ c ( X × X ) . W e assume P 0 to be G-in variant , that is, P 0 ( g f ) = gP 0 ( f ) , f ∈ C ∞ c ( X , E 0 ) , for all g ∈ G . Clearly , the Schwartz k ernel of a G -in variant opera tor P 0 satisﬁes that (2.5) K P 0 ( x , y ) = K P 0 ( gx , gy ) for all x , y ∈ X , g ∈ G . In ad dition, assum e P 0 to be pr o perly su pported , th at is, for any compact sub set K ⊂ X , the su bsets supp K P ∩ ( K × X ) and sup p K P ∩ ( X × K ) in X × X are co mpact. Proper supportn ess of P 0 in particu lar implies that P 0 maps C ∞ c ( X , E 1 ) to itself. 6 HANG W ANG Choose a G -inv ariant Herm itian structure o n E and let L 2 ( X , E ) be th e comp letion of C c ( X , E ) und er inner pro duct, < f , g > L 2 = Z X < f ( x ) , g ( x ) > E x d x . Let P b e an essentially self-adjoint operato r o n L 2 ( X , E ) with odd grading , in the for m of P =  0 P ∗ 0 P 0 0  . W itho ut lo ss of generality , P is assumed to be o f ord er 0 and then P exten ds to b e a bound ed self-ad joint op erator o n L 2 ( X , E ) . W e shall use the f ollo win g n otations and we omit E , F o r X when it is clear in th e context. • Ψ n ( X ; E , F ) : the set of order n pseudo -dif ferential operator s from C ∞ c ( X , E ) to C ∞ ( X , F ) ; • Ψ n G ( X ; E , F ) : the sub set of G -inv ariant elements in Ψ n ( X ; E , F ) ; • Ψ n G , p ( X ; E , F ) : the subset of proper ly supported elements in Ψ n G ( X ; E , F ) ; • Ψ n c ( X ; E , F ) : the subset of Ψ n ( X ; E , F ) h a vin g co mpactly supp orted Schwartz kernels. The symbo l of an operator P ∈ Ψ ∗ G , p is G -invariant. C onv ersely , if σ ( x , ξ ) is a G - in variant symb ol, then there is an operator in Ψ ∗ G , p with symbol σ ( x , ξ ) . T o do th is we construct P using (2.3) and use the a veraging operation from [10]: (2.6) A v G : Ψ ∗ c → Ψ ∗ G , p : P 7→ Z G gPg − 1 d g . Then A v G ( cP ) ∈ Ψ ∗ G , p , where c is a cutoff fu nction for X , has the symb ol σ ( x , ξ ) . Deﬁnition 2.6. [19] A pseudo- dif f erntial operato r P ∈ Ψ m ( X ; E , F ) is elliptic if there exists Q ∈ Ψ − m ( X ; F , E ) so tha t (2.7) k σ P ( x , ξ ) σ Q ( x , ξ ) − I k → 0 and k σ Q ( x , ξ ) σ P ( x , ξ ) − I k → 0 unifor mly in x ∈ K as ξ → ∞ in T ∗ x X for any com pact subset K in X . Without loss of gen er - ality , we will co nsider order-0 elliptic pseudo-differential operators P 0 ∈ Ψ 0 G , p ( X ; E 0 , E 1 ) with the condition (2.7) replaced by (2.8) k σ P 0 ( x , ξ ) σ P ∗ 0 ( x , ξ ) − I k → 0 a nd k σ P ∗ 0 ( x , ξ ) σ P 0 ( x , ξ ) − I k → 0 . Proposition 2. 7. (1) If P ∈ Ψ n c , then A v G ( P ) ∈ Ψ n G , p . (2) If P ∈ Ψ n G , p ( X ) is elliptic, then there e xists a pa r ametrix Q ∈ Ψ − n G , p ( X ) such that (2.9) 1 − PQ = S 1 ∈ Ψ − ∞ G , p ( X ) , 1 − QP = S 2 ∈ Ψ − ∞ G , p ( X ) , wher e Ψ − ∞ G , p ( X ) = ∩ n ∈ R Ψ n G , p ( X ) is the set of smoo thing operators. (3) If S ∈ Ψ − ∞ G , p ( X ) , then K S ( · , · ) is smooth and pr operly supp orted. Pr oof. ( 1) Clearly , A v G ( P ) ∈ Ψ ∗ G , p ( X ) . If p ( x , y , ξ ) ∈ S m ( X × T ∗ X ) is the amplitude then P ∈ Ψ n c implies that K = { ( x , y ) ∈ X × X | p ( x , y , ξ ) 6 = 0 } is co mpact. Usin g th e fact th at the Riemannian metric on T ∗ X is G -inv arian t an d the measure on X is G -in variant, we calculate the amplitude for A v G ( P ) as Z G p ( g − 1 x , g − 1 y , ξ g − 1 x ) d g L 2 -INDEX FORM ULA FOR PR OPER COC OMP ACT GROUP ACTIONS 7 which is o f o rder n becau se the in te gral is taken over a set { g ∈ G | ( g − 1 x , g − 1 y ) ∈ K } which is compact. (2) Let P ∈ Ψ n G , p ( X ) be elliptic and c ∈ C ∞ c ( X ) be a cu tof f fu nction for X . Cover X by ﬁnitely many bo unded op en balls { U i } N i = 1 such th at sup p ( c ) ⊂ ∪ N i = 1 U i . Let { a i } N i = 1 be a partition o f unity subordinate to the ﬁnite co ver . Since P is elliptic, which implies that for any com pact K ⊂ X , there e xists a constant C K such that | σ P | ≥ C ( 1 + | ξ | ) n unifor mly for all | ξ | ≥ C K , then there exist Q i ∈ Ψ − n c ( U i ) , 1 ≤ i ≤ N so that PQ i − a i = R 1 , i , Q i P − a i = R 2 , i are elements in Ψ − ∞ c ( U i ) . E xtend the elem ents in Ψ ∗ c ( U i ) to Ψ ∗ c ( X ) an d then c N ∑ i = 1 Q i P − c = c N ∑ i = 1 R 2 , i . Since N ∑ i = 1 Q i ∈ Ψ − n c ( X ) , N ∑ i = 1 R 2 , i ∈ Ψ − ∞ c ( X ) , we set Q = Z G g ( c N ∑ i = 1 Q i ) d g ∈ Ψ − n G , p ( X ) an d S = Z G g ( c N ∑ i = 1 R 2 , i ) d g ∈ Ψ − ∞ G , p ( X ) . Th en QP = Z G g ( c N ∑ i = 1 Q i ) P d g = Z G g ( c ) g ( N ∑ i = 1 Q i P ) d g = Z G g ( c ) d g + Z G g ( c ) g ( n ∑ i = 1 R 2 , i ) d g = I + S . Similarly , there is a Q ′ = Z G g ( N ∑ i = 1 Q i c ) d g ∈ Ψ − n G , p ( X ) and S ′ ∈ Ψ − ∞ G , p ( X ) so that PQ ′ − I = S ′ . Since Q ′ + S Q ′ − Q = ( 1 + S ) Q ′ − Q = Q ( PQ ′ − 1 ) = QS ′ , then Q ′ − Q ∈ Ψ − ∞ G , p ( X ) . Hence there are S 1 , S 2 = S ∈ Ψ − ∞ G , p ( X ) such that PQ = 1 + S 1 , QP = 1 + S 2 . (3) If S ∈ Ψ − ∞ G , p ( X ) , then cS ∈ Ψ − ∞ c ( X ) . W e kn o w that cS ∈ Ψ − ∞ c ( X ) is equ i valent to the fact th at K cS ( x , y ) is smooth and com- pactly supported in X × X . T herefore th e statemen t fo ll ows fr om the fact that K S ( x , y ) = K A v G ( cS ) ( x , y ) = Z G K cS ( g − 1 x , g − 1 y ) d g and the fact that the inte gral vanishes o utside a com pact set in G .  3. T H E G - T R AC E A N D T H E L 2 - I N D E X . When X is co mpact and when G is trivial, the dimension s of Ker P 0 and Ker P ∗ 0 are ﬁnite and their d if fe rence deﬁnes the ind e x of P . In o ur case we measure the size of Ker P 0 or Ker P ∗ 0 by a real n umber in terms of von Neum ann dimension. An L 2 -index of P , analogou s to the Fr edholm ind e x is deﬁned, m oti vated by the L 2 -index deﬁned by Atiyah [2] and modiﬁed upon [10]. 8 HANG W ANG 3.1. The G -tra ce of opera tors on X . Reca ll that a bo unded oper ator T o n a Hilbert space H is of t race class if ∞ ∑ i = 1 | < | T | e i , e i > | < ∞ , where { e i } ∞ i = 1 is an o rthonormal basis of the Hilbert space and its trace calculated by tr ( T ) = ∞ ∑ i = 1 < T e i , e i > is independ ent of the orth onormal basis . Deﬁnition 3 .1. A bou nded operator S : L 2 ( X , E ) → L 2 ( X , E ) , which commu tes with the action of G , is of G-trace class if φ | S | ψ is of trace class for all φ , ψ ∈ C ∞ c ( X ) . If S is a G -trace class op erator , we calculate the G -trace by the formula (3.1) tr G ( S ) = tr ( c 1 Sc 2 ) , where c 1 , c 2 ∈ C ∞ c ( X ) are non ne gative, satisfyin g c 1 c 2 = c for so me cutoff f unction c o n X . Remark 3.2 . When G is discrete, Deﬁn ition 3.1 is essentially the deﬁnition of the G -tra ce class operator appearing in [2]. Similarly to Lem ma 4.9 of [2], we pr o ve in the fo llo wing propo si tion that tr G is well deﬁned , that is, tr G is in dependent of the cho ice of c 1 , c 2 and c . Proposition 3.3. Let S (bou nded, G-invariant and positive) b e a G-trace c lass op er a tor and c 1 , c 2 , d 1 , d 2 ∈ C ∞ c ( X ) b e non ne gative functions satisfying Z G c 1 ( g − 1 x ) c 2 ( g − 1 x ) d g = 1 and Z G d 1 ( g − 1 x ) d 2 ( g − 1 x ) d g = 1 , which means that c = c 1 c 2 and d = d 1 d 2 ar e cutoff functions on X . Then tr ( c 1 Sc 2 ) = tr ( d 1 Sd 2 ) . Pr oof. L et K = { g ∈ G | supp ( g · ( d 1 d 2 )) ∩ supp c 6 = / 0 } an d th en K is compact by the proper ness o f the gr oup actio n. Henc e, tr ( c 1 Sc 2 ) = tr ( Z G [ g · ( d 1 d 2 )] c 1 Sc 2 d g ) = tr ( Z K [ g · ( d 1 d 2 )] c 1 Sc 2 d g ) = Z K tr ([ g · d 1 ][ g · d 2 ] c 1 Sc 2 ) d g = Z K tr ( c 1 [ g · d 1 ] D [ g · d 2 ] c 2 ) d g = Z K tr ([ g − 1 · c 1 ] d 1 Sd 2 [ g − 1 · c 2 ]) d g = tr ([ Z G g ( c 1 c 2 ) d g ] d 1 Sd 2 ) = tr ( d 1 Dd 2 ) .  Using the fact that tr is a well-deﬁned trace on compac tly sup ported operators o n X , it is easy to s ee that tr G is linear , faithful, norma l and semi-ﬁnite. The tracial pr operty of tr G is proved in the following proposition together with some other properties of tr G . Proposition 3. 4. (1) A p r op erly supp orted smooth ing op er ator A ∈ Ψ − ∞ G , p is o f G- trace class. If K A : X × X → Hom E is the kernel o f A, th en its G-trace is ca lcu- lated by (3.2) tr G ( A ) = Z X c ( x ) Tr K A ( x , x ) d x , wher e c is a cuto f f functio n a nd T r is the matrix trace o f Hom E . In fact, this formula holds for all G-in variant operators having smooth integr al k ernel. L 2 -INDEX FORM ULA FOR PR OPER COC OMP ACT GROUP ACTIONS 9 (2) If A ∈ Ψ ∗ G is of G-trace class, s o is A ∗ . (3) If A ∈ Ψ ∗ G is o f G-trace class and B ∈ Ψ ∗ G is bou nded, then AB a nd B A ar e of G-trace class. (4) If AB and BA are of G-tr ace class, then tr G ( AB ) = tr G ( BA ) Pr oof. L et φ , ψ ∈ C ∞ c ( X ) and let { α 2 i } N i = 1 be th e G -inv a riant partition of unity in Pro po- sition 2.7. (1) Prop osition 2.7 (3) shows that A ∈ Ψ − ∞ G , p has smooth kernel. T hen K φ A ψ ( x , y ) = φ ( x ) K A ( x , y ) ψ ( y ) , is smooth and compactly suppo rted, which means that φ A ψ is of t race class. The integral formula for smoothing operators is class ical. A proof may be found at [32] Section 2.21. (2) Because ¯ ψ A ¯ φ has ﬁnite trace by deﬁnition, then φ A ∗ ψ = ( ¯ ψ A ¯ φ ) ∗ is of trace class. (3) Assume we have a G -trace class operator A ∈ Ψ ∗ G , p . Sin ce supp ψ is compac t then as A is proper ly supp orted, th ere is a compact set K so that supp A ψ ⊂ K . Choose η , ζ ∈ C ∞ c ( X ) with K ⊂ supp η and η ζ = η . Then η A ψ = A ψ and fo r a bounded B ∈ Ψ ∗ G , we have th at φ BA ψ = φ B ζ η A ψ = ( φ B ζ )( η A ψ ) . Since φ B ζ is bo unded o perator with compact suppo rt and η A ψ is trace class operator then their prod uct is also a trace-c las s operator . So BA is of G -tr ace class. AB is of G -tr ace class bec ause B ∗ A ∗ is of G -trace class. If A ∈ Ψ ∗ G , then we have A = A 1 + A 2 so that A 1 ∈ Ψ ∗ G , p and A 2 has smooth kern el (which follows fr om a classical statement saying that the Schwartz kern el is smo oth o f f the diago nal). Th en the s tatement follo ws f rom th e fact that φ A 1 ψ has smoo th, co mpactly supported Schwartz kernel. (4) W e ﬁrst pr o ve a special case when AB and BA have smoo th integral kernels. Use the slice theorem (2.1) to g et { G × K i S i = G ( S i ) } N i = 1 , G -inv ariant tubular open sets covering X . Then there exist G - in variant map s α i : X → [ 0 , 1 ] with supp α i ⊂ G ( S i ) such that N ∑ i = 1 α 2 i = 1. In fact, let ˜ α i 2 be a p artition of unity of X / G su bordinate to the o pen sets G ( S i ) / G . Lift ˜ α i to α i on X , then { α 2 i } is a G -inv ar iant p artition o f u nity of X . Th en: tr G ( AB ) = Z X Z X c ( x ) Tr ( K A ( x , y ) K B ( y , x )) d y d x = ∑ i , j Z G × K i S i Z G × K j S j α 2 i ( x ) α 2 j ( y ) c ( x ) T r ( K A ( x , y ) K B ( y , x )) d y d x = ∑ i , j 1 µ ( K i ) µ ( K j ) Z S i Z S j α 2 i ( ¯ s ) α 2 j ( ¯ t ) Z G Z G c ( ht ) T r ( K A ( gs , ht ) K B ( ht , gs )) d g d s d h d t = ∑ i , j 1 µ ( K i ) µ ( K j ) Z S i Z S j α 2 i ( ¯ s ) α 2 j ( ¯ t ) Z G T r ( K B ( ht , ¯ s ) K A ( ¯ s , ht )) d g d s d h d t = tr G ( BA ) . Note that in the third equality , ¯ gs . = ( g , s ) K i = x ∈ G × K i S i and ¯ ht . = ( h , t ) K j = y ∈ G × K j S j and by deﬁn ition α i ( ¯ s ) = α i ( ¯ gs ) , α j ( ¯ t ) = α j ( ¯ ht ) . Also, we have used (2.5), d h − 1 = d h , d ( h − 1 g ) = d g , and chang e of variable in the fourth equality . 10 HANG W ANG If either A o r B are prop erly supported, (say A ), then tr G ( AB ) = tr ( c 1 ABc 2 ) = tr ( Z G c 1 Ag · ( c 1 c 2 ) Bc 2 ) . So the set { g ∈ G | c 1 Ag · c 1 6 = 0 } is compact in K , which allows us to inter- change tr and Z K , and to use tracial prop erty of tr an d G -inv ariance of A and B to prove tr G ( AB ) = tr G ( BA ) In gener al let A = A 1 + A 2 and B = B 1 + B 2 where A 1 , B 1 are p roperly sup ported and A 2 , B 2 are b ounded an d have smooth kern el. Then tr G ( AB ) = tr G ( BA ) using the special cases discussed above.  Remark 3.5 . Let S be a bound ed G -inv ariant operator with smooth integral kernel and deﬁne S i . = α i S α i ∈ Ψ − ∞ c ( X ; E , E ) . Then α 2 i S is o f G -trac e class by Pr oposition 3.4 (3). W e may calculate tr G ( S ) as f ollo ws, tr G ( S ) = tr G ( N ∑ i = 1 α 2 i S ) = N ∑ i = 1 Z G × K i S i α i ( x ) c ( x ) T r K S ( x , x ) α i ( x ) d x = N ∑ i = 1 Z G × K i S i c ( x ) Tr K S i ( x , x ) d x = N ∑ i = 1 µ ( K i ) − 1 Z G × S i c (( g , s )) T r K S i (( g , s ) , ( g , s )) d g d s = N ∑ i = 1 µ ( K i ) − 1 Z G × S i c (( g , s )) T r K S i (( e , s ) , ( e , s )) d g d s = N ∑ i = 1 µ ( K i ) − 1 Z S i T r K S i ( s , s ) d s . The above trace formula coin cides wit h the trace formulas in the special cases. (1) If th e action is fr ee and cocomp act, then X = G × U , and f or a bou nded positi ve self-adjoint operato r S with smoo th kernel, we obtain tr G ( S ) = Z U T r K S ( x , x ) d x . (2) For a h omogeneous space of a Lie grou p X = G / H , and fo r S ∈ Ψ − ∞ G , p ( X ) , we have tr G ( S ) = K S ( e , e ) , whe re e is the gro up iden tity . Proposition 3.6. If P 0 ∈ Ψ m G , p is an elliptic operator , th en P K er P 0 ∈ Ψ − ∞ G is of G-trace class. Pr oof. By Proposition 2 .7 , ther e is a Q ∈ Ψ − m G , p so that 1 − QP 0 = S ∈ Ψ − ∞ G , p . Then apply it to P K er P 0 and get P K er P 0 = S P K er P 0 ∈ Ψ − ∞ G . The statemen t is pr o ved using (1 ) and (3) of Proposition 3.4.  Remark 3.7 . Let { α 2 i } N i = 1 be the G -in variant partition of unity in the proof of Proposition 3.4 (4). Then by th e same pro perty and for any bound ed op erator T ∈ Ψ − ∞ G , we have tr G T = N ∑ i = 1 tr G ( α i T α i ) where e very summand α i T α i is G -in variant an d restricts to a slice G × K i S i in X . L 2 -INDEX FORM ULA FOR PR OPER COC OMP ACT GROUP ACTIONS 11 The action o f G on the vecto r b undle E is induce d b y the action o f its subgr oup K i on V . = E | S i , the restriction of the bundle E over a subset { ( e , s ) K i | s ∈ S i } of X . V = E | S i k ∈ K i − − − − → V = E | S i   y   y { e } × S i k ∈ K i − − − − → { e } × S i Then we have th e iden ti ﬁcation of the Hilber t spa ces L 2 ( G × K i S i , E ) = ( L 2 ( G ) ⊗ L 2 ( S i , V )) K i , which con sis ts of th e elem ents of L 2 ( G ) ⊗ L 2 ( S , V ) inv a riant under the action of K i , where k ∈ K i acts by k ( f ( g ) , h ( s )) = ( f ( gk − 1 ) , k · h ( s )) , g ∈ G , s ∈ S i , f ∈ L 2 ( G ) , h ∈ L 2 ( S i , V ) . The G -in variance of ker P 0 implies that α i P ker P 0 α i is an element of (3.3) R ( L 2 ( G )) ⊗ B ( L 2 ( S i , V )) , and this elem ent comm utes with the action of the group K i on R ( L 2 ( G )) ⊗ B ( L 2 ( S i , V )) . Here R ( L 2 ( G )) is the weak closur e of the rig ht regular representation of G ( L 1 ( G ) mor e precisely) r epresented on L 2 ( G ) . On this set there is a natu ral von Neuman n trace d eter - mined by τ ( R ( f ) ∗ R ( f )) = Z G | f ( g ) | 2 d g , where f ∈ L 2 ( G ) ∩ L 1 ( G ) and R ( f ) = Z G f ( g ) R ( g ) d g . Here R ( g ) is th e rig ht regular representatio n of g ∈ G on L 2 ( G ) . Also B ( L 2 ( S i , V )) also has a subset where an operato r trace tr can be deﬁned. Ther e is a natural normal, semi- ﬁnite and faithful tra ce deﬁned on R ( L 2 ( G )) ⊗ B ( L 2 ( S i , V )) given by τ ⊗ tr on algebraic tensors. Refer to [26] Section 2 for a detailed description. This trace coincides with the G -tra ce in Deﬁnition 3.1 on the set o f bounde d G - in variant op erators with smooth kernel. I n fact, by a partition of unity argum ent, su ch an oper ator is ﬁnite su m of operators of for m S = A ⊗ B ∈ R ( L 2 ( G )) ⊗ B ( L 2 ( S i , V )) , which commu tes with the a ction of K i , wher e A an d B have smoo th kernel. In [1 0 ] , it has be en sho wn th at τ ( A ) = K A ( e , e ) . L et d ∈ C ∞ c ( G ) be any cutoff function fo r G . Then τ ( A ) = Z G d ( g ) K A ( g , g ) d g . Hence , τ ( A ) tr ( B ) = Z G d ( g ) K A ( g , g ) d g Z S i T r K B ( s , s ) d s = Z G × S i 1 µ ( K i ) c (( g , s )) T r K S (( g , s ) , ( g , s )) d g d s = Z G × K i S i c ( x ) T r K S ( x , x ) d x . Therefo re we have proved the following pr oposition. Proposition 3 .8. On Ψ − ∞ G , p ( X ; E , E ) , the G-trace equals the natural von Neuman n trace on the von Neuma nn algebra R ( L 2 ( X , E )) , the weak clo s ur e of all the n atural bou nded operators on L 2 ( X , E ) wh ic h commu te with the action of G. The L 2 -index is the differ ence of the von Neumann trace of P K er P 0 and P K er P ∗ 0 . 12 HANG W ANG Example 3.9 . When G is a discrete group acting on itself by left translations, deﬁne c ( g ) = ( 1 g = e 0 g 6 = e , then tr cT = ∑ g ∈ G < cT δ g , δ g > = < ∑ g ∈ G g − 1 ( cT ) g δ e , δ e > = < A v ( cT ) δ e , δ e > = tr G A v ( cT ) . In general, A v ( c · ) : B ( L 2 ( G )) → R ( L 2 ( G )) extends the map Ψ ∗ c → Ψ ∗ G , p : cT → A v ( cT ) which preser v es the correspon ding trace. When T is G -inv ar iant, T = A v ( cT ) an d then tr G T = tr G A v ( cT ) = tr cT motiv ates the tr G formu la. 3.2. The L 2 -index of elliptic operato rs on X . Accor ding to Proposition 3.6, we deﬁne a real v alued G-dimension of K , a closed G -inv ar iant su bspace of L 2 ( X , E ) , by dim G K = tr G P K where P K is the projection from L 2 ( X , E ) onto K , and is G -inv ar iant. Deﬁnition 3.10. The L 2 -index o f the elliptic ope rator P ∈ Ψ ∗ G , p is (3.4) ind P = d im G Ker P 0 − dim G Ker P ∗ 0 . An immediate compu tation of the L 2 -index is gi ven by the f ollo wing p roposition, Proposition 3. 11. Let P ∈ Ψ m G , p be elliptic and Q b e an ope r a tor so that 1 − QP 0 = S 1 , 1 − P 0 Q = S 2 ar e of G-trace class, th en ind P = tr G S 1 − tr G S 2 . Pr oof. T he proof is similar to the on e in [2]. W e have S 1 P K er P 0 = P K er P 0 and P K er P ∗ 0 S 2 = P K er P ∗ 0 by c omposing QP 0 = 1 − S 1 with P K er P 0 and by comp osing P K er P ∗ 0 with 1 − S 2 = P 0 Q respectively . Also, P 0 ( QP 0 ) = ( P 0 Q ) P 0 implies that P 0 S 1 = S 2 P 0 . Set R = δ 0 ( P ∗ 0 P 0 ) P ∗ 0 where δ 0 ( 0 ) = 1 , δ 0 ( x ) = 0 for x 6 = 0, so RP 0 = 1 − P K er P 0 , P 0 R = 1 − P K er P ∗ 0 . On on e hand tr G S 1 − tr G P K er P 0 = tr G S 1 ( 1 − P K er P 0 ) = tr G ( S 1 RP 0 ) . On the other han d tr G S 2 − tr G P K er P ∗ 0 = tr G S 2 ( 1 − P K er P ∗ 0 ) = tr G ( S 2 P 0 R ) = tr G ( P 0 S 1 R ) . The refore tr G S 1 − tr G S 2 = tr G P K er P 0 − tr G P K er P ∗ 0 by Proposition 3.4.  From the last propo sit ion we deri ve t he following McKean-Singer f ormula . Corollary 3.1 2. If D =  0 D ∗ 0 D 0 0  ∈ Ψ 1 G ( X ; E , E ) is a ﬁrst or d er essentially self-adjoint elliptic differ ential operator , then (3.5) ind D = tr G ( e − t D ∗ 0 D 0 ) − tr G ( e − t D 0 D ∗ 0 ) for all t > 0 , which in pa rticular means that ind D is inde pendent of t > 0 . T o prove (3.5) we need the follo wing lemma. Lemma 3.13. Let D 0 be as above, then e − t D 0 D ∗ 0 and e − t D ∗ 0 D 0 ar e of G-trace class. L 2 -INDEX FORM ULA FOR PR OPER COC OMP ACT GROUP ACTIONS 13 Pr oof of lemma 3.13. It is sufﬁcient to prove the case when t = 1 . Th e proof is based on the ideas in [ 15, 10]. Also refer to the heat kernel estimate fo r a Riemannian manifold in [21] Appendix B. If λ ∈ C − [ 0 , ∞ ) , then λ I − D ∗ 0 D 0 is invertible. Let L = { λ ∈ C | d ( λ , R + ) = 1 } b e clock-wise oriented. Th en e − D ∗ 0 D 0 = 1 2 π i Z L e − λ λ I − D ∗ 0 D 0 d λ . Let φ , ψ ∈ C ∞ c ( X ) be su pported in a compa ct set K ⊂ X an d let { α i } N i = 1 be a partitio n of unity sub ordinated to an open cover of K o f local coo rdinate charts. W e approx imate φ e − D ∗ 0 D 0 ψ by an o perator in Ψ − ∞ c (with sm ooth and comp actly supported Sch w artz ker- nel) by in verting λ I − D ∗ 0 D 0 “locally”. Let p i be the full symbo l of α i φ ( λ I − D ∗ 0 D 0 ) − 1 ψ , having the asym ptotic sum (3.6) p i ∼ ∞ ∑ j = 2 a − j on a local coord inate, that is, Op ( p i − m ∑ j = 2 a − j ) ∈ Ψ − m − 1 c , ∀ m > 1 , where Op means the operator correspon ding to th e local symb ol. For any l > 0 an d n > 0 , choose a large enough M and set the op erator ap proximating α i φ ( λ I − D ∗ 0 D 0 ) − 1 ψ to be (3.7) P i ( λ ) = Op ( M ∑ j = 2 a − j ) , in the sense that P i ( λ ) is ana lytic in λ and for any ﬁxed u ∈ L 2 ( X , E ) , (3.8) k ( P k ( λ ) − α i φ ( λ I − D ∗ 0 D 0 ) − 1 ψ ) u k l ≤ C ( 1 + | λ | ) − n , where the norm is the Soble v l -norm k · k l . Th e estimate (3 .8 ) is mad e possible by the as- ymptotic s um (3.6). I n fact, let r ( x , ξ ) be the symb ol of R . = P i ( λ ) − α i φ ( λ I − D ∗ 0 D 0 ) − 1 ψ which is in S − M − 1 and then the left han d s ide of ( 3.8 ) is k Ru k l = R ( 1 + | ξ | 2 ) l | c Ru ( ξ ) | d ξ , where Ru ( x ) = R e < x − y , ξ > r ( x , ξ ) u ( y ) d y d ξ can b e controlled by th e rig ht hand sid e o f ( 3.8) when M >> 2 l + 2 n . T his is because b y the d eﬁnition of r ( x , ξ ) there is a con st ant C so that | r ( x , ξ ) | < C ( 1 + | ξ | ) − M − 1 . Set (3.9) E ( λ ) = N ∑ i = 1 E i ( λ ) = N ∑ i = 1 1 2 π i Z L e − λ P i ( λ ) d λ . Then the following two observations prove that φ e − D ∗ 0 D 0 ψ is of trace class. (1) The operator E ( λ ) is a compactly supp orted o perator with smooth Schwartz ker- nel. Pr oof of cla im. W e ne ed to show th at the Schwartz kerne l of E k ( λ ) is smooth. In view o f ( 3.7) and ( 3.9 ), it is sufﬁcient to show that Op ( a j ) , j ≤ − 2 has smooth kernel and Z L e − λ ∂ β ( Op ( a j ) u ) d λ is integrable for all β . This claim can be proved by the symb olic calculu s ([1 5 ]) . The crucial p art in the argu ment is that by the sy mbolic calculus, all a j , j ≤ − 2 contain the f actor e − σ 2 ( D ∗ 0 D 0 ) and the fact that e − t σ 2 ( D ∗ 0 D 0 ) is rapidly decreasing in ξ .  14 HANG W ANG (2) The function ( E ( λ ) − φ e − D ∗ 0 D 0 ψ ) u is in H l for any ﬁxed u ∈ L 2 . Pr oof of cla im. Using (3.8), and ﬁxing a u ∈ L 2 ( X , E ) k ( E ( λ ) − φ e − D ∗ 0 D 0 ψ ) u k l ≤ 1 2 π N ∑ i = 1 Z L e − λ k ( P i ( λ ) − α i φ ( D ∗ 0 D 0 − λ I ) − 1 ψ ) u k l d λ ≤ C Z L e − λ ( 1 + | λ | ) − n d λ → 0 as n → ∞ .  Note that E ( λ ) depend s on the num ber M , wh ich is chosen based o n l , n , and it ha s a com pactly suppor ted smo oth kernel b y the ﬁrst claim an d henc e E ( λ ) u ∈ C ∞ c ⊂ H l . The second claim shows th at φ e − D ∗ 0 D 0 ψ is in H l . (When n → ∞ , there is a sequence of E ( λ ) ∈ H l approa ching φ e − D ∗ 0 D 0 ψ in k · k l norm.) Let l → ∞ , then by the Sob ole v Emb edding Theor em ( φ e − D ∗ 0 D 0 ψ ) u is smooth for all u ∈ L 2 . The refore φ e − D ∗ 0 D 0 ψ has a compactly suppo rted smooth kernel an d is a trace-class operator .  Pr oof of Cor olla ry 3.5 . Let Q = Z t 0 e − sD ∗ 0 D 0 D ∗ 0 d s , which is the param etrix of D 0 because 1 − QD 0 = e − t D ∗ 0 D 0 , I − D 0 Q = e − t D 0 D ∗ 0 which is of G - trace c las s by the lemma. The statement follows f rom Prop osition 3.11. The indep endence of t can be carried out by a modiﬁca ti on of the seco nd proof of [6] Theorem 3.50.  4. T H E C O N N E C T I O N O F T H E L 2 - I N D E X T O T H E K - T H E O R E T I C I N D E X . Let f ∈ C 0 ( X ) be identiﬁed as an operato r on L 2 ( X , E ) by p oint-wise mu ltiplication. Let A ∈ Ψ 0 p ( X ; E , E ) be elliptic in the sense o f D eﬁnition 2. 6 . Using the Rellich lemm a one may check that A 0 : L 2 ( X , E 0 ) → L 2 ( X , E 1 ) satisﬁes th e fo llo wing condition s : • ( A 0 A ∗ 0 − I ) f ∈ K ( L 2 ( X , E 1 )) , ( A ∗ 0 A 0 − I ) f ∈ K ( L 2 ( X , E 0 )) , ; • A f − f A ∈ K ( L 2 ( X , E )) ; • A 0 − g · A 0 ∈ K ( L 2 ( X , E 1 ) , L 2 ( X , E 2 )) fo r all g ∈ G . Hence A represents an element in the K -homolo gy grou p K 0 G ( C 0 ( X )) . T opolo gically , the K -theoretic inde x of [ A ] ∈ K 0 G ( C 0 ( X )) , accordin g to [1 7 ], is d eﬁned by Ind t A . = [ p ] ⊗ C ∗ ( G , C 0 ( X )) j G ([ A ]) ∈ K 0 ( C ∗ ( G )) , which is the image of [ A ] un der the descen t map j G : K K G ( C , C 0 ( X )) → K K ( C ∗ ( G ) , C ∗ ( G , C 0 ( X ))) composed with the intersection produ ct with [ p ] ∈ K K ( C , C ∗ ( G , C 0 ( X ))) , [ p ] ⊗ C ∗ ( G , C 0 ( X )) : K K ( C ∗ ( G , C 0 ( X )) , C ∗ ( G )) → K K ( C , C ∗ ( G )) . Here p . = ( c · g ( c )) 1 2 an idempo tent in C c ( G , C 0 ( X )) ,being the image of 1 un der the ∗ - homom orphism C → C ∗ ( G , C 0 ( X )) and d eﬁning an eleme nt in K 0 ( C ∗ ( G , C 0 ( X ))) . L 2 -INDEX FORM ULA FOR PR OPER COC OMP ACT GROUP ACTIONS 15 Analytically , the K -theoretic index of A is co nstructed e xplicitly as follows [18]. First of all, embed C c ( X , E ) in a larger Hilbert C ∗ ( G ) -module C ∗ ( G , L 2 ( X , E )) and af ter com- pletion un der the norm of th e Hilb ert m odule, we obtain a C ∗ ( G ) -module E co ntaining C c ( X , E ) as a d ense subalgebra. Note that E is a direct summand of C ∗ ( G , L 2 ( X , E )) and is obtained by compressing the C ∗ ( G ) -module C ∗ ( G , L 2 ( X , E )) with the idemp otent p . Then the operato r ¯ A . = A v ( cA ) : C c ( X , E ) → C c ( X , E ) in Ψ 0 G , p ( X ; E , E ) extend s to two boun ded maps ¯ A : L 2 ( X , E ) → L 2 ( X , E ) and ¯ A : E → E with k ¯ A k E ≤ k ¯ A k L 2 ( X , E ) . Denote b y B ( E ) the C ∗ -algebra of all b ounded operators on E having an adjoint and being C ∗ ( G ) -module maps. The n ¯ A : E → E deﬁnes an element in B ( E ) acco rding to [19]. On the Hilber t C ∗ ( G ) mo dule E , fo r e , e 1 , e 2 ∈ C c ( X , E ) , a rank one operator is deﬁned by θ e 1 , e 2 ( e )( x ) = e 1 ( e 2 , e )( x ) = Z X ( Z G θ g ( e 1 )( x ) , g ( e 2 )( y ) d g ) e ( y ) d y , ∀ x ∈ X . The closure o f the th e linear combinatio ns of the rank one operato rs under the no rm of B ( E ) is the set of compact operators , den oted by K ( E ) . The elemen ts of K ( E ) can be id entiﬁed with the in te gral ope rators with G -inv ariant continuo us kerne l and with proper support. The following propo s ition indicates some features of elements from B ( E ) , K ( E ) . Proposition 4 .1. [19] If the symbol of the G- in variant pr o perly su pported o per ator P of or der 0 is boun ded in th e co tangent d ir ection by a constant, then th e norm of P in B ( E ) / K ( E ) does not exceed that con st ant. The op er a tor P is comp act i.e . P ∈ K ( E ) , if the symbol of P is 0 at inﬁnity (in the cotangent dir ection). Since ¯ A is elliptic, which means that k σ ¯ A ( x , ξ ) 2 − 1 k → 0 as ξ → 0 , x ∈ K uniform ly for any com pact set K ⊂ X , then according to Propo sition 4.1 we ha ve that ¯ A 2 − I d ∈ K ( E ) . Let us set ¯ A =  0 ¯ A 0 ∗ ¯ A 0 0  , then [ ¯ A 0 ] ∈ K 1 ( B ( E ) / K ( E )) . The analytical K -theo r etic index , Ind a A , is image of th is class in the K -theory of the quotien t algeb ra u nder the bound ary map ∂ : K ∗ ( B ( E ) / K ( E )) → K ∗ + 1 ( K ( E )) of the six term exact sequ ence associated to the short exact sequence 0 → K ( E ) → B ( E ) → B ( E ) / K ( E ) → 0 . Remark 4.2 . The set of ﬁnite ran k K ( E ) - v alu ed p rojections f orms a ﬁnite gen erated projective C ∗ ( G ) -module. Then Theroem 3 of section 6 in [16] implies that K ∗ ( K ( E )) ≃ K ∗ ( C ∗ ( G )) . Hen ce, Ind a P ∈ K 0 ( C ∗ ( G )) . As a generalization of the Atiyah-Sing er index theor em, Kaspa rov proved that Ind a and Ind t coincide [17, 19]. W e will simp ly use Ind to deno te the K -theor etic index. In sum - mary , the K -theo retic in de x under the h omomorphism In d : K 0 G ( C 0 ( X )) → K K ( C , C ∗ ( G )) ≃ K 0 ( K ( E )) is calculated by (4.1) [( L 2 ( X , E ) , A )] 7→ [( E , ¯ A )] 7→ [   ¯ A 0 ¯ A ∗ 0 ¯ A 0 q 1 − ¯ A ∗ 0 ¯ A 0 q 1 − ¯ A ∗ 0 ¯ A 0 ¯ A ∗ 0 1 − ¯ A ∗ 0 ¯ A 0   ] − [  1 0 0 0  ] . Note that the second arrow is the Fredh olm pictu re of K K ( C , C ∗ ( G )) via bounda ry map . Giv en th e K -theo retic index I nd A ∈ K 0 ( K ( E )) , we will d eﬁne the a homo morphism K 0 ( K ( E )) → R . T o do this we ﬁnd a de nse subalgebr a S ( E ) of K ( E ) o n which 16 HANG W ANG a “tr ace” can be deﬁned and which is closed u nder holomor phic fu nctional calculus. Since K ( E ) is generated b y G -inv ariant o perators with continuo us and p roperly sup- ported kernel, we de ﬁne S ( E ) to be the subset of the bo unded G -inv ar iant operato rs with smooth kern els. Let S : C ∞ c ( X , E ) → C ∞ c ( X , E ) be a G -inv ariant smooth ing operator . Ex- tend S to an oper ator ¯ S ∈ B ( E ) and the n ¯ S ∈ S ( E ) . Deﬁne the trace o n ¯ S ∈ S ( E ) by tr G ( S ) and still denote by tr G . The tr ace is well deﬁned f or all the elements of S ( E ) . An element of S ( E ) is viewed as matrice s with C ∗ ( G ) -entries. The tr ace on such a matrix is the Bre uer v on Neumann trace [8] o n th e image of the f ollo wing map S ( E ) 7→ S ( E ⊗ C ∗ ( G ) R ( L 2 ( G ))) ⊂ R ( L 2 ( X , E )) . Here S ( E ⊗ C ∗ ( G ) R ( L 2 ( G ))) is a subset of all G -trace class op erators and its elemen ts are represented as matrices with R ( L 2 ( G )) -entries. Recall (Remark 3. 7 ) that tr G is deﬁned on a dense sub set of the G - in variant operato rs on L 2 ( X , E ) ,which can be represented as elements of R ( L 2 ( G )) ⊗ ( ⊕ i , j B ( L 2 ( U i , E ) , L 2 ( U j , E )) , and an element of this set can be expressed in terms of a R ( L 2 ( G )) -valued ma trix. Proposition 4. 3. • W e have a canonica l isomo r phism K 0 ( K ( E )) ≃ K 0 ( S ( E )) . • The G-trace tr G on S deﬁ nes a gr oup homomorph ism tr G ∗ : K 0 ( K ( E )) → R . Pr oof. Pr oposition 3. 4 (4) shows tha t S ( E ) is an idea l of B ( E ) . Since S ( E ) co ntains the ra nk one operato rs, then K ( E ) is the C ∗ -closure of S ( E ) . L et J = K ( E ) , J = S ( E ) and let ˜ J , ˜ J 0 be o btained by adjoining a u nit. Note that ˜ J = B ( E ) . W e claim that J 0 is stable u nder h olomorphic fu nctional c alculus. T o show the claim we essentially need to prove that if a ∈ ˜ J 0 is inv ertible in ˜ J , the n a − 1 ∈ ˜ J 0 [9]. Let a − 1 = λ I + r , wh ere λ ∈ C , I is th e u nit a nd r ∈ J . Choose an s ∈ J 0 so that k a − 1 − λ I − s k < m in { 1 k a k , 1 } . T hen k 1 − λ a − as k < 1 implies that a ( λ I + s ) is invertible. So λ I + s is also inv ertible and s − 1 ∈ ˜ J , the n a − 1 = ( λ I + s )[ a ( λ I + s )] − 1 . Since J 0 is an idea l of ˜ J we o nly need to show that a ( λ I + s ) − 1 ∈ ˜ J 0 . Let x = a ( λ I + s ) ∈ J 0 , then k 1 − x k < 1, then x − 1 = [ 1 − ( 1 − x )] − 1 = ∞ ∑ i = 0 ( 1 − x ) i ∈ ˜ J 0 . The claim i s proved. H ence S ( E ) is a den s e subalgeb ra of K ( E ) c losed under holomorp hic fun ctional calcu lus, which implies that K ∗ ( K ( E )) = K ∗ ( S ( E )) . An element of K 0 ( S ( E )) is represen ted by projectio n m atrix with entries in S ( E ) , on which the re is a natural trac e consisting of the compo sition of the m atrix trace with τ on S ( E ) . No te that if the e lement was r epresented by the d if fer ence o f two classes of matrices with entries in S ( E ) + , the algeb ra deﬁned by adding a unit, then we d eﬁne the trace of this extra unit to be 0 . Hence we o btain a homom orphism tr G ∗ : K ∗ ( S ( E )) → R by the proper ties of the trace τ .  Composing with the K -theor etic index, P h as a nu merical ind e x given by the image of the map K 0 G ( C 0 ( X )) K-theoretic index − − − − − − − − − → K 0 ( S ) tr G ∗ − − → R and this numb er d epends o nly on the symbo l class and the man ifold acco rding to Kas- parov’ s K -th eoretic index formula (T heorem 5.1). W e show that this number is in fact the L 2 -index. L 2 -INDEX FORM ULA FOR PR OPER COC OMP ACT GROUP ACTIONS 17 Proposition 4 .4. Let P ∈ Ψ 0 G , p ( X ; E , E ) be elliptic, then its L 2 -index co incides with the trace of its K -theor etic index, i.e. ind P = tr G ∗ ( Ind [ P ]) . Pr oof. L et P = A and then P = ¯ A = A v ( cA ) in 4.1. Th en Ind P = [  P 0 P ∗ 0 P 0 p 1 − P ∗ 0 P 0 p 1 − P ∗ 0 P 0 P ∗ 0 1 − P ∗ 0 P 0  ] − [  1 0 0 0  ] . W e shall alter the m atrix rep resentati ves witho ut chan ging the eq ui valence class, so that we may apply tr G to the 2 × 2-matr ices. Giv en P 0 ∈ Ψ 0 G , p ( X ; E 0 , E 1 ) and using Pro position 2.7, there is a Q ∈ Ψ 0 G , p so that 1 − QP = S 0 , 1 − PQ = S 1 . Accor ding to the bound ary map con struction in [11] section 2, we lift  0 − Q P 0  which is invertible in M 2 ( B ( E ) / S ( E )) to an invertible elem ent u =  S 0 − ( 1 + S 0 ) Q P S 1  in M 2 ( B ( E )) and then Ind P . = [ u  1 0 0 0  u − 1 ] − [  0 0 0 1  ] = [  S 2 0 S 0 ( 1 + S 0 ) Q P 0 S 1 1 − S 2 1  ] − [  0 0 0 1  ] . Therefo re, tr G ∗ ( Ind P ) = tr G ( S 2 0 ) + tr G ( 1 − S 2 1 ) − τ ( 1 ) = tr G ( S 2 0 ) − tr G ( S 2 1 ) . Choose ano ther Q ′ . = 2 Q − QPQ , then 1 − Q ′ P 0 = S 2 0 , 1 − PQ ′ = S 2 1 with S 2 0 , S 2 1 being smo othing ope r - ators. Then using Pro position 3 .11 , we conclud e that tr G ( S 2 0 ) − tr G ( S 2 1 ) = in d P . Hence tr G ∗ ( Ind P ) = ind P .  Remark 4.5 . Let X = G / H be a ho mogeneous space of a unimodu lar Lie group G (where H is a co mpact sub group). I n [ 10 ] sectio n 3 , it was sh o wn directly that the L 2 -index depend s only on th e symbol class [ σ P ] of P in K G 0 ( C 0 ( T ∗ X )) . Plus, there exists a homo- morph is m i : K G 0 ( C 0 ( T ∗ X )) → R so th at i [ σ P ] = ind P . Note that th e Poincar ´ e duality b e- tween K-ho mology and K-the ory gi ves rise to K G 0 ( C 0 ( T ∗ X )) ≃ K 0 G ( C 0 ( X )) . So L 2 -index essentially giv es a homomorph ism: (4.2) ind : K 0 G ( C 0 ( X )) → R . Remark 4.6 . In th is section we work on the cycles in K 0 G ( C 0 ( X )) determin ed b y odd self-adjoint elliptic p seudo-differential op erators on X . I f Y is another pr oper cocompact G -manifo ld an d if E is a G -bundle where L 2 ( Y , E ) admits a C 0 ( X ) -representatio n, so th at [( L 2 ( Y , E ) , Q )] ∈ K 0 G ( C 0 ( X )) with Q ∈ Ψ 0 G , p ( Y ; E , E ) , we may carry out similar construc- tions to those in the sectio n e asil y and th ere is no p roblem to deﬁn e the L 2 -index of Q . Howe ver, it is no t clear how to d eﬁne L 2 -index for an arbitrary re presentati ve ( A , F ) in a general r epresenting cycle [( A , F )] ∈ K 0 G ( C 0 ( X )) , where A is a C 0 ( X ) -algebra a nd F is a general e lli ptic oper ator . Because we d o no t know th e way to d eﬁne pseud o-differential calculus fo r the C ∗ -algebra A and we do not have Propo sition 2.7 fo r F , u s ing which we calculate the L 2 -index. But it shou ld be p ossi ble to ﬁnd a p roper cocomp act G -manifold Y a nd pseu do-differential ope rator Q on Y so tha t [( A , F )] = [( L 2 ( Y , E ) , Q )] ∈ K 0 G ( C 0 ( X )) . 5. R E D U C T I O N T O T H E L 2 - I N D E X O F A D I R AC T Y P E O P E R ATO R . W e shall show in this section that f or any elliptic op erator P ∈ Ψ 0 G , p ( X ; E , E ) , there is a Dirac type operator ˜ D satisfying ind P = ind ˜ D . T o do this, we sh o w th at P and ˜ D have the same K -theoretic index and then apply Proposition 4.4. 18 HANG W ANG Theorem 5.1. [17, 19] Let X be a complete Rieman nian ma nifold and let G b e a locally compact gr oup a cting on X pr op erly an d iso metr ically . Let P be a G-invariant elliptic operator on X o f order 0 . Then (5.1) [ P ] = [ σ P ] ⊗ C 0 ( T ∗ X ) [ D ] ∈ K ∗ G ( C 0 ( X )) , wher e [ D ] is the equivalen ce c lass deﬁned by the Dolbeault operator on T ∗ X . Remark 5 .2 . In (5.1), th e ellipticity of P =  0 P ∗ 0 P 0 0  ∈ Ψ 0 G , p ( X ; E , E ) (Deﬁnition 2. 6 ) implies that the symbol σ P =  0 σ P 0 σ P 0 0  deﬁnes an elem ent of K K G ( C 0 ( X ) , C 0 ( T ∗ X )) . In fact, using the Hermitian structure on E = E 0 ⊕ E 1 , we obtain C 0 ( T ∗ X , π ∗ E ) , a Hilbert module over C 0 ( T ∗ X ) , and the set of “comp act operators” is C 0 ( T ∗ X , Hom ( π ∗ E , π ∗ E )) . Also C 0 ( X ) acts on C 0 ( T ∗ X , π ∗ E 0 ⊕ π ∗ E 1 ) by p ointwise multip li cation. Hence fo r all f ∈ C 0 ( X ) , ( σ 2 P − I ) f is comp act by (2 .8 ) and [ σ P , f ] = 0. Ther efore, the symbo l σ P : π ∗ E → π ∗ E deﬁnes the following elem ent in K K -theory: [( C 0 ( T ∗ X , π ∗ E 0 ⊕ π ∗ E 1 ) ,  0 σ ∗ P 0 σ P 0 0  )] ∈ K K G ( C 0 ( X ) , C 0 ( T ∗ X )) . In (5.1), the Dolbea ult operator D is a ﬁrst ord er differential operator D = √ 2 ( ¯ ∂ + ¯ ∂ ∗ ) acting on smooth sections of Λ 0 , ∗ ( T ∗ ( T ∗ X )) , wher e ¯ ∂ = ∂ ∂ ¯ z = 1 2 ( ∂ ∂ ξ + i ∂ ∂ x ) . De- note by H the Hilbert space of L 2 -forms of bi-degre e ( 0 , ∗ ) on T ∗ X graded by th e od d and even forms. Then D is an order 1 essentially self-adjoint ope rator on H . The C ∗ - algebra C 0 ( T ∗ X ) ac ts on H by poin t-wise mu ltiplication. The Dolbeault element is th e K -hom ological cycle given b y [( H , D √ 1 + D 2 )] ∈ K 0 G ( C 0 ( T ∗ X )) = K K G ( C 0 ( T ∗ X ) , C ) . Remark 5.3 . Theor em 5.1 says that [ P ] is giv en b y the index pa iring of the symbol with some f undamental (Dolbeu lt ) oper ator on T ∗ X . This is the essence o f the Atiya h-Singer index theorem. When X is com pact with trivial group action, apply the map C ∗ : K 0 ( C ( X )) → K 0 ( C ) induced by the con stant map C : C ( X ) → C : f 7→ f ( pt ) to both sides of (5.1). The left han d side of (5. 1 ) is then the Fredholm index of P and the r ight hand side is the intersection produ ct of [ σ P ] ∈ K 0 ( C 0 ( T ∗ X )) with [ D ] ∈ K 0 ( C 0 ( T ∗ X )) . I t is classical fact that [ σ P ] is viewed as some eq ui valence class o f vector b u ndle V . The n th e in tersection pr oduct is th e well-known Fred holm in de x o f the Dirac op erator D with coefﬁcients in V . The following K -theoretic ind e x formula serves as an imp ortant corollary to Theo rem 5.1. Theorem 5. 4. [ 17 ] Let X be a comp lete Riemannian ma nifold, o n which a locally co m- pact gr ou p G acts pr operly an d isometrica lly wi th comp act q uotient. Let P be a pr operly supported G-in va riant elliptic operator on X of o r d er 0 . Then Ind P = [ p ] ⊗ C ∗ ( G , C 0 ( X )) j G ([ P ]) = [ p ] ⊗ C ∗ ( G , C 0 ( X )) j G ([ σ P ]) ⊗ C ∗ ( G , C 0 ( T ∗ X )) j G ([ D ]) ∈ K ∗ ( C ∗ ( G )) . L 2 -INDEX FORM ULA FOR PR OPER COC OMP ACT GROUP ACTIONS 19 Wher e p is th e idemp otent in C ∗ ( G , L 2 ( X , E )) de ﬁned b y p = ( c · g ( c )) 1 2 and [ D ] is the Dolbeault element. Analogou s to the vector b undle constru ction me ntioned in Remar k 5.3 (See also [ 3 ] section 7), we deﬁne a G -bundle V ( σ P ) using the sym bol σ P as follows. Let B ( X ) ⊂ T ∗ X be th e unit b all bundle with its b oundary , that is, the sph ere bundle S ( X ) ⊂ T ∗ X . A new manifold Σ X is ob tained by gluin g two copies of B ( X ) alo ng their bo undaries: (5.2) Σ X = B ( X ) ∪ S ( X ) B ( X ) . The action of G o n T ∗ X extends n aturally to Σ X because G acts on X isometr ically . The ellipticity of P imp lies the in vertibility of σ P | S ( X ) , the symbol restricted to S ( X ) . Deﬁne a G -vector b u ndle over Σ X by the gluing map σ P on the bound ary , that is, (5.3) V ( σ P ) = π ∗ E | B ( X ) ∪ σ P | S ( X ) π ∗ E | B ( X ) . Here V ( σ P ) deﬁne s an elemen t in the repr esentable K K - theory RK K 0 G ( X ; C 0 ( X ) , C 0 ( Σ X )) . V ( σ P ) is Z / 2 Z -grade d and is the direct sum of two b u ndles: V ( σ P 0 ) = π ∗ E 0 | B ( X ) ∪ σ P 0 | S ( X ) π ∗ E 1 | B ( X ) and V ( σ P ∗ 0 ) = π ∗ E 1 | B ( X ) ∪ σ P ∗ 0 | S ( X ) π ∗ E 0 | B ( X ) . The re is a natural homomorp hism RK K 0 G ( X ; C 0 ( X ) , C 0 ( Σ X )) → K K G ( C 0 ( X ) , C 0 ( Σ X )) . Denote by [ V ( σ P )] as the equiv a- lence class of V ( σ P ) eithe r in RK K 0 G ( X ; C 0 ( X ) , C 0 ( Σ X )) and or in K K G ( C 0 ( X ) , C 0 ( Σ X )) . W e sh all not d isti nguish the no tations wh en it is clear from the co nte xt. In the proo f Proposition 5.6 we shall see that as a K K -cycle, [ V ( σ P )] = [( C 0 ( Σ X , V ( σ P )) , 0 )] in K K G ( C 0 ( X ) , C 0 ( Σ X )) . Remark 5.5 . When X is com pact an d when G = { e } , the inclusion C → C ( X ) fur ther reduces σ P to an elemen t o f K K ( C , C 0 ( T ∗ X )) by “forgetting ” the actio n of C ( X ) on the Hilbert- C ( X ) mo dule C 0 ( T ∗ X ) . Ther efore, [ σ P ] ∈ K K ( C , C 0 ( T ∗ X ))) ≃ K 0 ( C 0 ( T ∗ X )) maps to a vector bundle, tri vial at inﬁnity in T ∗ X . The bundle is constru cted b y gluing π ∗ E | B ( X ) and π ∗ E | T ∗ X − B ( X ) ◦ along the boundaries u s ing the in vertible map σ P | S ( X ) and is the restriction of V ( σ P ) to T ∗ X . Proposition 5. 6. The homo morphism K K G ( C 0 ( X ) , C 0 ( T ∗ X )) → K K G ( C 0 ( X ) , C 0 ( Σ X )) [( C 0 ( T ∗ X , π ∗ E ) , σ P )] 7→ [( C 0 ( Σ X , V ( σ P )) , 0 )] is induced by the inclusion map i : C 0 ( T ∗ X ) → C 0 ( Σ X ) . Pr oof. Fir s t of all, the cycle ( C 0 ( Σ X , V ( σ P )) , 0 ) d eﬁnes an element of K K G ( C 0 ( X ) , C 0 ( Σ X )) , because f · ( 0 2 − Id C 0 ( Σ X , V ( σ P )) ) is co mpact in the Hilber t- C 0 ( Σ X ) -module C 0 ( Σ X , V ( σ P )) . Here, the com pactness of the ﬁbe r o f Σ X over X is important. The argumen t fails wh en replacing Σ X by T ∗ X . F or example, ( C 0 ( Σ X , V ( σ P ) | T ∗ X ) , 0 ) d oes not d eﬁne a n ele ment in K K G ( C 0 ( X ) , C 0 ( T ∗ X )) . W itho ut l oss of gener ality , we may assum e σ P satisﬁes that σ 2 P = 1 on S ( X ) an d k σ P k ≤ 1 . Using the standar d b oundary map construction in the exact sequen ce o f K -theory , we obtain the fo llo win g pr ojection Q using the unitary u =   σ P − q 1 − σ 2 P q 1 − σ 2 P σ P   ∈ 20 HANG W ANG M 2 ( C 0 ( T ∗ X , π ∗ E )) : Q . = u  1 0 0 0  u − 1 =   σ 2 P σ P q 1 − σ 2 P q 1 − σ 2 P σ P 1 − σ 2 P   . Recall that the bundle ( V ( σ P ) | T ∗ X glued by σ P is giv en by the image of the projectio n Q on π ∗ E ⊕ π ∗ E , that is, C 0 ( T ∗ X , V ( σ P ) | T ∗ X ) = Q C 0 ( T ∗ X , π ∗ E ⊕ π ∗ E ) . Let w = u  σ P 0 0 1  u ∗ . Then QwQ = w , ( 1 − Q ) w ( 1 − Q ) = 1 − Q , and Qw ( 1 − Q ) = ( 1 − Q ) wQ = 0 . The n [( C 0 ( T ∗ X , π ∗ E ) , σ P )] = [( C 0 ( T ∗ X , π ∗ E ) , σ P )] + [( C 0 ( T ∗ X , π ∗ E ) , 1 )] =[( C 0 ( T ∗ X , π ∗ E ⊕ π ∗ E ) ,  σ P 0 0 1  )] = [( C 0 ( T ∗ X , π ∗ E ⊕ π ∗ E ) , w )] =[( Q C 0 ( T ∗ X , π ∗ E ⊕ π ∗ E ) , x )] + [(( 1 − Q ) C 0 ( T ∗ X , π ∗ E ⊕ π ∗ E ) , 1 − Q )] =[( Q C 0 ( T ∗ X , π ∗ E ⊕ π ∗ E ) , x )] → [( C 0 ( Σ X , V ( σ P )) , ˜ x )] = [( C 0 ( Σ X , V ( σ P )) , 0 )] . Here, the arrow in the last line comes fr om the following fact. The Hilbe rt C 0 ( T ∗ X ) - module C 0 ( T ∗ X , V ( σ P ) | T ∗ X ) maps to the H il bert C 0 ( Σ X ) -module C 0 ( Σ X , V ( σ P )) under the m ap i ∗ : K K G ( C 0 ( X ) , C 0 ( T ∗ X )) → K K G ( C 0 ( X ) , C 0 ( Σ X )) induc ed from the in clusion i . The last equ ality follows fro m the op erator ho motopy t → t ˜ x and th e ob serv a tion that ( C 0 ( Σ X , V ( σ P )) , t ˜ x ) is a Kasparov ( C 0 ( X ) , C 0 ( Σ X )) mod ule for all t ∈ [ 0 , 1 ] . The proof is complete.  The Dolbeault op erator on T ∗ X extends to the proper coco mpact G -manifold Σ X , which also ha s an almo st complex structure. W e ju s t glue two Dolbeau lt operators on B ( X ) ⊂ T ∗ X along the bou ndary (th e nor mal directio ns of S ( X ) in B ( X ) need to switch signs on different p ieces). T he new Do lbeault o perator ¯ D is clearly G -inv ariant an d de- ﬁnes an element (5.4) [ ¯ D ] = [( L 2 ( Σ X , Λ 0 , ∗ ( T ∗ ( Σ X ))) , ¯ D √ 1 + ¯ D 2 )] in K K G ( C 0 ( Σ X ) , C ) . (In section 6 we shall not distinguish [ ¯ D ] a nd [ D ] . ) The following propo si tion is obvious. Proposition 5. 7. The inclusion i : C 0 ( T ∗ X ) → C 0 ( Σ X ) ind uces th e na tur al map (5.5) i ∗ : K K G ( C 0 ( Σ X ) , C ) → K K G ( C 0 ( T ∗ X ) , C ) : [ ¯ D ] 7→ [ D ] . Corollary 5.8 . A ss uming the same nota tions and con ditions in the K -ho mological for- mula in Theor em 5.1, we have (1) The elliptic p seudo-differ ential operator P is in the same K-homology class a s the intersection pr oduct [ V ( σ P )] ⊗ [ ¯ D ] in the image of the map K K G ( C 0 ( X ) , C 0 ( Σ X )) × K K G ( C 0 ( Σ X ) , C ) → K K G ( C 0 ( X ) , C ) . L 2 -INDEX FORM ULA FOR PR OPER COC OMP ACT GROUP ACTIONS 21 (2) The op er a tor P r elates to a Dirac typ e op er a tor ¯ D V ( σ P ) , tha t is, the Dolb eault operator ¯ D on Σ X twisted b y the bundle V ( σ P ) over Σ X , in the following sense: (5.6) [ P ] = j ∗ [ ¯ D V ( σ P ) ] wher e j ∗ : K K G ( C 0 ( Σ X ) , C ) → K K G ( C 0 ( X ) , C ) is indu ced by th e inclusion j : C 0 ( X ) → C 0 ( Σ X ) . Pr oof. T he ﬁrst statem ent is a re s ult o f T heorem 5.1 as well as the f unctorality o f inter- section produ cts [ P ] = [ σ P ] ⊗ C 0 ( T X ) [ D ] = [ σ P ] ⊗ C 0 ( T X ) i ∗ [ ¯ D ] = i ∗ [ σ P ] ⊗ C 0 ( Σ X ) [ ¯ D ] = [ V ( σ P )] ⊗ C 0 ( Σ X ) [ ¯ D ] . T o prove the second stateme nt, we calcu late (5.7) [ V ( σ P )] ⊗ C 0 ( Σ X ) [ ¯ D ] = [( C 0 ( Σ X , V ( σ P )) , φ 1 , 0 )] ⊗ C 0 ( Σ X ) [( L 2 ( Σ X , Λ 0 , ∗ ( T ∗ ( Σ X ))) , φ 2 , F )] , where F . = ¯ D √ 1 + ¯ D 2 . W e d enote b y [( H , η , I )] the K K -product appeared in (5 .7 ). According to the deﬁnition of K K -p roduct, H = C 0 ( Σ X , V ( σ P )) ⊗ C 0 ( Σ X ) L 2 ( Σ X , Λ 0 , ∗ ( T ∗ ( Σ X ))) and the operator I needs to satisfy the following tw o conditions [33]: (1) I is an F -connec tion; (2) I has the prop erty η ( a )[ 0 ⊗ 1 , I ] η ( a ) ≥ 0 mo dulo K ( H ) . By Kasparov’ s stabilization theorem, there is a C 0 ( Σ X ) -valued p rojection Q such that C 0 ( Σ X , V ( σ P )) = Q ( ⊕ ∞ 1 C 0 ( Σ X )) . Th erefore, H = Q ( ⊕ ∞ 1 C 0 ( Σ X )) ⊗ C 0 ( Σ X ) L 2 ( Σ X , Λ 0 , ∗ ( T ∗ ( Σ X ))) = φ 2 ( Q )( ⊕ ∞ 1 L 2 ( Σ X , Λ 0 , ∗ ( T ∗ ( Σ X ))) , where, φ 2 ( Q ) , by deﬁnition, acts by matrix multiplication and point-wise multiplication. W e claim that (5.8) I = φ 2 ( Q )( ⊕ ∞ 1 F ) φ 2 ( Q ) The statement is proved if (5. 8 ) is true. In fact, one needs only to observe that H = φ 2 ( Q )( ⊕ ∞ 1 L 2 ( Σ X , Λ 0 , ∗ ( T ∗ ( Σ X ))) = L 2 ( Σ X , Λ 0 , ∗ ( T ∗ ( Σ X )) ⊗ V ( σ P )) and φ 2 ( Q )( ⊕ ∞ 1 ¯ D ) φ 2 ( Q ) = ¯ D V ( σ P ) on H . T o prove the claim ( 5.8 ), it is sufﬁcient to show the following ob serv atio ns. • ( I 2 − 1 ) η ( f ) ∈ K ( H ) , for all f ∈ C 0 ( X ) ; • [ I , η ( f )] ∈ K ( H ) , f or all f ∈ C 0 ( X ) ; • [ ˜ T ξ , F ⊕ I ] ∈ K ( L 2 ( Σ X , Λ ∗ ( Σ X )) ⊕ H ) , ∀ ξ ∈ C 0 ( Σ X , V ( σ P )) , wher e ˜ T ξ =  0 T ∗ ξ T ξ 0  ∈ B ( L 2 ( Σ X , Λ 0 , ∗ ( T ∗ ( Σ X )) ⊕ H ) , T ξ ∈ B ( L 2 ( Σ X , Λ 0 , ∗ ( T ∗ ( Σ X )) , H ) is deﬁned by T ξ ( η ) = ξ ˆ ⊗ η ∈ H .  22 HANG W ANG Proposition 5.9. Let P be a pr operly supported G- in variant ellip tic pseudo- dif fer en tial operator of or d er 0 , ¯ D b e the Dolbeau lt op er ator on Σ X deﬁned in (5 .4 ) an d V ( σ P ) be the G-vector bundle o ver Σ X deﬁn ed in (5.3) Then P and D V ( σ P ) have the same L 2 -index, that is, (5.9) ind P = in d D V ( σ P ) . Pr oof. I n vie w of Corollar y 5.8, the cycle [( L 2 ( Σ X , Λ 0 , ∗ ( T ∗ ( Σ X )) ⊗ V ( σ P )) , ¯ D V ( σ P ) )] represents as th e intersection prod uct [ V ( σ P )] ⊗ [ ¯ D ] , wh ich is the same as [( L 2 ( X , E ) , P )] in K 0 G ( C 0 ( X )) . This implies that I nd P = Ind D V ( σ P ) and the statemen t is proved b y taking the trace of the K -theoretic indices.  6. L O C A L I N D E X F O R M U L A . 6.1. L 2 -index o f Dirac t ype operators. Using Pro position 5.9, to ﬁnd a coh omological formu la fo r th e L 2 -index o f P , it is sufﬁcient to ﬁgu re out a f ormula for Dirac ty pe op- erators. Let M b e an e ven-dimension al (d im M = n ) prope r cocompac t G -manifold with a G -Clifford bundle V , which is a C l ( T ∗ M ) -module via Clifford multiplica tion. Here C l ( T ∗ M ) = Cl ( T ∗ M ) ⊗ C is the co mplex Clifford algebra genera ted b y T ∗ M . W e con - struct D , a Dirac type operator acting on sections in V . Let ∇ be the G -in variant Levi- Ci vita connection on T M , which can be extend ed to Cl ( T ∗ M ) . Let ∇ V be th e G -inv ar iant Cliffor d con nection on V , i.e. [ ∇ V , c ( a )] = c ( ∇ a ) , a ∈ C ∞ c ( M , Cl ( T ∗ M )) . A Dirac op er a tor D : C ∞ c ( M , V ) → C ∞ c ( M , V ) is deﬁned as th e compo si tion of the conn ection ∇ V and th e Clif ford multiplication c : C ∞ c ( M , T ∗ M × V ) → C ∞ c ( M , V ) by D = ∑ i c ( e i ) ∇ V e i , where { e i } form s an ortho normal basis of the bundle T M and { e i } is the dual basis of T ∗ M . Here, V = V 0 ⊕ V 1 is Z / 2 Z graded and D is essentially self-ad joint with an odd grading , in particu lar , D =  0 D ∗ 0 D 0 0  : L 2 ( M , V ) → L 2 ( M , V ) . T he L 2 -index o f D is expressed b y t he McKean-Singer formula (3.5) which is independen t of t : (6.1) ind D = str G ( e − t D 2 ) , where str G (  a b c d  ) = tr G ( a ) − tr G ( d ) an d D 2 =  D ∗ 0 D 0 0 0 D 0 D ∗ 0  . Let R V = ( ∇ V ) 2 ∈ Λ 2 ( M , Hom V ) be the curvature tensor of the Clif ford conn ection ∇ V , then D 2 = − ∑ i ( ∇ V e i ) 2 + ∑ i ∇ V ∇ e i e i + ∑ i < j c ( e i ) c ( e j ) R V ( e i , e j ) . = ∆ V + ∑ i < j c ( e i ) c ( e j ) R V ( e i , e j ) is a gener alized Laplacian. Let S be the spinor ( irreducible) rep resentation of Cl ( T ∗ x M ) . It is a standard fact that Hom S = S ⊗ S ∗ = C l ( T ∗ x M ) . The ﬁber of the Clifford mo dule V at x has the deco mposition V x = S ⊗ W . Her e W is the set of vectors in V x that co mmute with the action of C l ( T ∗ x M ) . Theref ore on the endom orphism lev el we ha ve (6.2) Hom V x = C l ( T ∗ x M ) ⊗ Hom W . L 2 -INDEX FORM ULA FOR PR OPER COC OMP ACT GROUP ACTIONS 23 Here Hom C l ( T ∗ x M ) ( V x ) . = Hom W is m ade of the tr ansformations of V x that co mmute with C l ( T ∗ x M ) . Accord ing to [ 6] Prop osition 3.43 , the c urv ature R V decomp oses under the isomorph ism (6 .2 ) into (6.3) R V = R S + F V / S where R S ( e i , e j ) = 1 4 ∑ kl ( R ( e i , e j ) e k , e l ) c k c l is the action of t he R iemannian curvature R . = ∇ 2 of M o n the bundle V and F V / S ∈ Λ 2 ( M , Hom C l V ) is the twisting cu rv a ture of the Clif ford con nection ∇ V . Accor ding to the L ichnerowicz Formu la, [6] Proposition 3.52 , the generalized Laplacian is calculated by: (6.4) D 2 = − n ∑ i = 1 ( ∇ V e i ) 2 + ∑ i ∇ V ∇ e i e i + 1 4 r M + ∑ i < j F V / S ( e i , e j ) c ( e i ) c ( e j ) , where F V / S ( e i , e j ) ∈ Hom C l V are the coef ﬁcients of the twisting curvature F V / S . Let the heat kernel k t be the Schwartz kernel of the solutio n o perator e − t D 2 of the heat equation ∂ ∂ t u ( t , x ) + D 2 u ( t , x ) = 0. It is a smooth map M × M → Ho m ( V , V ) satisfying e − t D 2 f ( x ) = Z M k t ( x , y ) f ( y ) d y . Hence ind D = Z M c ( x ) str k t ( x , x ) d x . W e have the fo llo wing properties of the heat kernel. Lemma 6.1. (1 ) F or f ( x ) ∈ L 2 ( M ) , e − t D 2 f is a smooth section ; (2) The kernel k t ( x , y ) of e − t D 2 tends to the δ function weakly , i.e. e − t D 2 s ( x ) = Z M k t ( x , x 0 ) s ( x 0 ) d x 0 → s ( x ) un if ormly on a comp act set in M as t → 0 . Pr oof. W e have proved that the Schwartz kernel of ce − t D 2 is smooth in Lemma 3.13. So ( e − t D 2 f )( x ) = Z G × M c ( g − 1 x ) k t ( x , y ) f ( y ) d y d g . = Z G h g ( x ) d g , where h g ( x ) = R M c ( g − 1 x ) k t ( x , y ) f ( y ) d y is smo oth in x ∈ M for ﬁxed g ∈ G . Using th e fact that e − t D 2 is a bound ed op erator an d that c ( x ) is smoo th a nd compactly supp orted, we con clude th at h g ( x ) depen ds smo othly o n g ∈ G . Let K b e any compact n eighborhoo d of x , then by the proper ness o f the gr oup actio n, the set Z . = { g ∈ G | c ( g − 1 x ) 6 = 0 , x ∈ K , g ∈ G } is compact and then ( e − t D 2 f )( x ) = Z Z h g ( x ) d g is smooth for x ∈ K . Therefo re th e ﬁrst statement is proved. T o prove the second on e, let u be a s mooth function with nor m 1 . T hen < e − t D 2 u , u > = Z λ ∈ sp ( D ) e − t λ 2 d P u , u , where sp ( D ) mean s the spectrum of D . Since the set of integrals for 0 < t ≤ 1 is bound ed by 1, then by the dominated con vergence theorem , < e − t D 2 u , u > → Z λ ∈ sp ( D ) 1d P u , u = < u , u > as t → 0 .  24 HANG W ANG The heat kernel on R n of u t − n ∑ i = 1 ∂ 2 ∂ 2 x i = 0 , which is (6.5) p t ( x , y ) = 1 ( 4 π t ) n / 2 e − d ( x , y ) 2 / 4 t , suggests a ﬁrst ap proximation fo r th e h eat kern el on M . Th e small time behavior of the heat kern el k t ( x , y ) fo r x near y depen ds on th e local geo metry of x n ear y . Th is is made precise by the asymptotic e xpansion for k t ( x , y ) . Deﬁnition 6.2 ([3 0 ]) . Let B be a Banach space with nor m k · k and f : R + → B : t 7→ f ( t ) be a fu nction. A fo rmal s eries ∞ ∑ k = 0 a k ( t ) with a k ( t ) ∈ E is ca lled an asympto tic expansion for f , d enoted by f ( t ) ∼ ∞ ∑ i = 0 a k ( t ) , if fo r an y m > 0, the re are M m and ε m > 0 . So that f or all l ≥ M m , t ∈ ( 0 , ε m ] , we have k f ( t ) − l ∑ k = 0 a k ( t ) k ≤ Ct m . When M is com pact and when B = C 0 ( M , Hom ( V , V )) has C 0 -norm k f k = sup x ∈ M | f ( x ) | , it is the standard fact that the heat kernel k t ( x , x ) of e − t D 2 has an asympto ti c e xpansion k t ( x , x ) ∼ 1 ( 4 π t ) n / 2 ∞ ∑ j = 0 t j a j ( x ) where a j ( x ) ∈ Hom ( V x , X x ) , x ∈ M are smooth section s ([3 0 ] Theorem 7.15) . In o ur case, this theorem is formulated as follows. Theorem 6.3. Let M be a pr oper cocompa ct Riemann ian G-manifo ld and D be an eq ui- variant Dirac type ope r a tor ac ting on the sections of a Cliffor d b undle V , and k t be the heat kernel o f D . There is an asymptotic e xpansion for c ( x ) k t ( x , x ) u nder the C 0 -norm k f k = sup x ∈ M | f ( x ) | : (6.6) c ( x ) k t ( x , x ) ∼ c ( x ) 1 ( 4 π t ) n / 2 ∞ ∑ j = 0 t j a j ( x ) wher e a j ∈ C ∞ ( M , Hom V ) and a j ( x ) depe nds only o n the the geometry at x (involving metrics, connectio n coefﬁcients an d th eir de r ivatives). In pa rticular a 0 ( x ) = 1 . The as- ymptotic expansion works for any C l -norm for l ≥ 0 . (W e only need and pr ove th e ca s e when l = 0 .) T o prove Theo rem 6.3 we c onstructe an “appro ximating heat kern el”. T he proof is a mod iﬁcation o f th e case of op erators o n co mpact m anifold ([30] Theo rem 7. 15 or [6] Chapter 2). Now k t ( x , y ) satisﬁes the heat equation : (6.7) ∂ ∂ t k t ( x , y ) + D 2 k t ( x , y ) = 0 , k 0 ( x , y ) = δ y ( x ) where D operates on the x -coo rdinate on ly . W e ﬁx y and d enote it by x 0 and so lv e this equation locally on a coordin ate neighbo rhood O x 0 of x 0 with x ∈ O x 0 . W e app roximate L 2 -INDEX FORM ULA FOR PR OPER COC OMP ACT GROUP ACTIONS 25 the heat kernel k t ( x , x 0 ) , x ∈ O x 0 locally by lookin g f or a fo rmal solutio n (6.8) p t ( x , x 0 ) ∞ ∑ i = 0 t i b i ( x ) to the equation (6 .7 ), where p t ( x , x 0 ) = 1 ( 4 π t ) n 2 e − r 2 4 t with r = | x | = d ( x , x 0 ) is the heat kernel on E uclidean space (6. 5 ). Deno te by s t ( x , x 0 ) = ∞ ∑ i = 0 t i b i ( x ) in (6.8) an d so the heat kernel is written as (6.9) k t ( x , x 0 ) = p t ( x , x 0 ) s t ( x , x 0 ) . According to [30] equation 7.16 , D 2 in (6.4) on O x 0 is calculated by (6.10) [ ∂ ∂ t + D 2 ]( p t s t ) = p t [ ∂ ∂ t + D 2 + r 4 gt ∂ g ∂ r + 1 t ∇ r ∂ ∂ r ] s t . when operating on (6 .9 ) , wh ere r = | x | , g = det ( g i j ) and ( g i j ) is the Riemannia n metric on M . T o ﬁnd th e formal solution (6.8), set the right hand side o f (6 .10 ) to be 0. Then the compariso n of th e coefﬁcients of terms c ontaining t i for each i ≥ 0 en ables us to ﬁnd b i inductively via [30] equation (7.1 7): (6.11) ∇ ∂ ∂ r ( r i g 1 4 b i ( x )) = ( 0 i = 0 − r i − 1 g 1 4 D 2 b i − 1 ( x ) i > 0 (1) (So lv e α 0 ( x ) ) It is trivial to see that p t ( x , x 0 ) = 1 ( 4 π t ) n 2 e − r 2 4 t → δ x 0 ( x ) u niformly as t → 0 + . From L emma 6 .1 , k t ( x , x 0 ) → δ x 0 ( x ) unifo rmly as t → 0 + for all x ∈ K , where K ⊂ X is any com pact subset. Therefor e b 0 ( x 0 ) = 1 necessarily . The ﬁrst lin e in (6.11) indicates th at g 1 4 b 0 ( x ) = g ( x 0 ) 1 4 b 0 ( x 0 ) = 1, and then b 0 ( x ) = g − 1 4 ( x ) is determin ed b y b 0 ( x 0 ) . (2) (So lv e b i ( x ) , i > 0) Ind ucti vely the smo othness of b i implies the u niqueness of the smooth solu tion b i + 1 . In fact, when solving the eq uation in (6 .11 ), th e constan t term has to be 0 other wise b i + 1 is n ot smooth at r = 0. T hen b i + 1 is smo oth except that it may blow up at 0 . Bu t by setting r = 0 in the second line in (6 .11 ) we have b i + 1 ( x 0 ) = − 1 j ( D 2 b i )( x 0 ) which makes sense if b i is smooth . T herefore, there e xists a sequence of smooth sections { b i ( x ) } in Ho m ( V x 0 , V x ) un iquely determined by b 0 ( x 0 ) = 1 . Note that b i s are deﬁn ed on a coord inate neighbo rhood O x 0 and dep end smoothly on the local geometry around x 0 . For example, b 1 ( x ) = 1 6 k ( x ) − K ( x ) , where k is scalar curvature an d K satisﬁes D 2 = ∆ + K . Denote b i ( x ) by b i ( x , x 0 ) , x ∈ O x 0 . Now for any x 0 . = y ∈ M , we o btain a form al solu tion b i ( x , y ) which smooth ly d epends on both x and y for x ∈ O y . Choose O ′ ⊂ M × M such that { ( x , x ) | x ∈ M } ⊂ O ′ ⊂ ∪ y ∈ M O y and choose φ ( x , y ) ∈ C ∞ ( M × M ) so th at φ ( x , y ) = ( 1 ( x , y ) ∈ O ′ 0 ( x , y ) / ∈ ∪ y ∈ M O y . This deﬁnition is based on a cutoff function used to deﬁne the approxim ate heat kernel in [6] Deﬁnition 2.28. 26 HANG W ANG Deﬁnition 6.4. Let (6.9) be the true heat kernel. The approx imating heat kernel is (6.12) h n t ( x , y ) = p t ( x , y ) n ∑ i = 0 t i a i ( x , y ) , where a i ( x , y ) = φ ( x , y ) b i ( x , y ) ∈ C ∞ ( M × M ) and suppor ted in a neigh borhood o f the diagona l. W ith the previous set up we may state the following lem ma, which implies Th eorem 6.3 when setting x = y . Lemma 6 .5. Let k t ( x , y ) b e the heat kernel and h n t ( x , y ) be the o ne in (6.12). Let c ∈ C ∞ c ( M ) be a cutoff functio n of the pr oper co compact G-manifo ld M . Choose ¯ c ∈ C ∞ c ( M ) satisfying c ( x ) ¯ c ( x ) = c ( x ) , x ∈ M . F or all m > 0 , ther e is a N m , so that fo r all l > N m and t ∈ ( 0 , 1 ] , (6.13) k c ( x ) h l t ( x , y ) ¯ c ( y ) − c ( x ) k t ( x , y ) ¯ c ( y ) k < Ct m wher e k f k = sup x , y ∈ M | f ( x , y ) | . Pr oof. For a ll m , let N m > ma x { n + 1 , m + n 2 } , where n = d im M . By d eﬁnition h N m t ( x , y ) approx imately satisﬁes th e he at equ ation in the sense that (6.14) ( ∂ ∂ t + D 2 ) h N m = t N m p t ( x , y ) D 2 a N m ( x , y ) + O ( t ∞ ) . = r t ( x , y ) , where the ﬁr s t term in (6 .14 ) co mes from the calculation of the formal solution. In fact, using (6.10), (6.1 1 ), we have ( ∂ ∂ t + D 2 )[ p t ( x , y ) N m ∑ j = 0 t j b j ( x , y )] = t N m p t ( x , y ) D 2 b N m ( x , y ) . What remains O ( t ∞ ) is of order t ∞ , because t his term contains the deri vativ es of φ , which are o f 0-value for x n ear y , and p t ( x , y ) , ( x 6 = y ) , which de creases faster than any positiv e power t k as t → 0 + . r t ( x , y ) has the following p roperties: (1) Th e rema inder r t ( x , y ) is smoo th f or a n y ﬁxed t > 0. This is b ecause p t ( x , y ) in (6.5) and a i ( x , y ) s in Deﬁnitio n 6.4 are smoo th functio ns, for all t > 0. (2) Denote the k th Sobolev norm on C m ( M × M ) by k · k k . Then fo r all ﬁxed t > 0 and fo r all k : k c ( x ) r t ( x , y ) ¯ c ( y ) k k exists. This is bec ause c ( x ) r t ( x , y ) ¯ c ( y ) is smooth an d compactly supported on M × M . (3) W e have the es timate k c ( x ) r t ( x , y ) ¯ c ( y ) k n 2 + 1 < Ct m unifor mly for all t ∈ ( 0 , 1 ] . In fact, in the ﬁrst term c ( x ) t N m p t ( x , y )( D 2 a N m ( x , y )) ¯ c ( y ) of c ( x ) r t ( x , y ) ¯ c ( y ) , on ly t N m p t ( x , y ) depend s on t , it is sufﬁcient to kn o w the order of t in the k th derivati ve (in x or y ) of t N m p t ( x , y ) , where k ≤ n 2 + 1 and the order is: t N m t − n 2 t − k = t N m − n 2 − k . So k c ( x ) t N m p t ( x , y )( D 2 a N m ( x , y )) ¯ c ( y ) k n 2 + 1 ≤ n 2 + 1 ∑ k = 0 c k t N m − n 2 − k . Since N m > n + 1, ther e are no term s of non- positi ve order in t on the rig ht hand side. In addition, since N m > n 2 + m , then for all t ∈ ( 0 , 1 ] , there is a constan t C 1 so that k c ( x ) t N m p t ( x , y )( D 2 a N m ( x , y )) ¯ c ( y ) k n 2 + 1 ≤ C 1 t N m − n 2 ≤ C 1 t m . L 2 -INDEX FORM ULA FOR PR OPER COC OMP ACT GROUP ACTIONS 27 The deri vatives of c ( x ) O ( t ∞ ) ¯ c ( y ) do not h a ve an y terms co ntaining negativ e power of t so k c ( x ) O ( t ∞ ) ¯ c ( y ) k n 2 + 1 < C 2 t m for all t ∈ ( 0 , 1 ] . So pr operty (3) is proved. Next, we use r t ( x , y ) to relate k t ( x , y ) and h N m t ( x , y ) in the following claim : Claim: Th ere is a un ique smoo th solution fo r the fo llo win g e quation: (6.15) ( ( ∂ ∂ t + D 2 ) u t ( x , y ) = r t ( x , y ) u 0 ( x , y ) = 0 Here, u t ( x , y ) is regard as a function of t and x . In fact, It is trivial to check that u 1 = R t 0 e − ( t − τ ) D 2 r τ ( x , x 0 ) d τ is smooth and satisﬁes the eq uation. I f u 2 is another smo oth solution, then u = u 1 − u 2 satisﬁes ( ∂ ∂ t + D 2 ) u = 0 , u 0 = u ( t = 0 ) = 0 . He nce d d t k u k 2 L 2 = d d t < u , u > = − < u , D 2 u > − < D 2 u , u > = − 2 k D u k 2 L 2 implies tha t k u k 2 is non-decrea s ing in t , and so k u ( t = 0 ) k = 0 forces u = u 1 − u 2 = 0 . So the claim is proved. Since h N m t ( x , y ) − k t ( x , y ) is also a solution to the equatio n (6.1 5 ) , by the uniqu eness o f solution we have that h N M t ( x , y ) − k t ( x , y ) = Z t 0 e − ( t − τ ) D 2 r τ ( x , y ) d τ . Then for all t ∈ ( 0 , 1 ] , k c ( x ) k t ( x , y ) ¯ c ( y ) − c ( x ) h N m t ( x , y ) ¯ c ( y ) k n 2 + 1 ≤ t sup {k c ( x ) r τ ( x , y ) ¯ c ( y ) k n 2 + 1 | 0 ≤ τ ≤ t } ≤ Ct m , where the second inequality is because of property (3). By the Sob ole v emb edding the orem, f or all p > n 2 , k u k ≤ C 0 k u k p for u ∈ H p , where k · k is th e C 0 sup nor m and k · k p is the Sobolev p - norm. Th erefore, k c ( x ) k t ( x , y ) ¯ c ( y ) − c ( x ) h N m t ( x , y ) ¯ c ( y ) k ≤ C ′ k c ( x ) k t ( x , y ) ¯ c ( y ) − c ( x ) h N m t ( x , y ) ¯ c ( y ) k n 2 + 1 ≤ C ′ Ct m . In fact, since c ( x ) an d ¯ c ( x 0 ) are c ompactly sup ported, th e fun ction in the nor m is supp orted in a compact set in M × M , where the theorem can be applied .  Remark 6 .6 . From (6.6) lim t → 0 + c ( x ) str k t ( x , x ) = lim t → 0 + c ( x ) 1 ( 4 π t ) n / 2 l ∑ j = 0 t j str a j ( x ) for sufﬁ- ciently large l . T o calculate the left hand si de it is sufﬁcient to inv estigate a j s on the right hand side. If a ∈ H om V x then a has a decomposition a = b ⊗ c , b ∈ C l ( T ∗ x M ) , c ∈ Hom W as in ( 6 . 2 ) . T he super-trace str a is th en calculated b y str ( b ⊗ c ) = τ ( b ) · str V / S ( c ) wh ere str V / S is the super-trace on C -lin ear endom orphisms o f W under th e identiﬁcation Ho m C l ( T ∗ x M ) ( V x ) = Hom C ( W ) and τ s is th e the super-trace on Ho m S = S ⊗ S ∗ = C l ( T ∗ x M ) . The super-trace τ s on C l ( T ∗ x M ) is explicitly calculated by [6] Proposition 3.21 . Let c = ∑ c i 1 i 2 ··· i k e i 1 e i 2 · · · e i k be an element in C l ( T ∗ x M ) = Ho m ( S ) , where c i 1 i 2 ··· i k , 1 ≤ i 1 ≤ i 2 ≤ · · · ≤ i k ≤ n is th e coefﬁcient o f the eleme nt e i 1 e i 2 · · · e i k in C l ( T ∗ x M ) . Then (6.16) τ s ( c ) = ( − 2 i ) n 2 c 12 ··· n . 28 HANG W ANG The Clifford algebra C l ( T ∗ x M ) is a ﬁltered algeb ra, mo re speciﬁcally , C l ( T ∗ x M ) = C l ( R n ) = ∪ n i = 0 C l i . Here C l i is th e linear comb ination of e j 1 · · · e j k , k ≤ i . In proving The orem 6.3, the following lemma is ob tained as a coro llary . Lemma 6.7. Let a i ( x ) be the ith term in the asymp totic expansion. Then (6.17) a i ( x ) ∈ C l 2 i ⊗ Hom C l ( T ∗ x M ) ( S x ) . Pr oof. W e d eﬁne a i ( x ) = a i ( x , x ) to be α ( x , x ) . W e need to show that α i ( y , y ) ∈ C l 2 i ⊗ Hom C l ( V y ) . Set x = y in (6.11), then α 0 ( y , y ) = 1 and α j ( y , y ) = − 1 j ( D 2 α j − 1 )( y , y ) . with α 0 ( y , y ) = 1 ∈ C l 0 ⊗ Ho m C l ( V y ) . Inducti vely , the fact that D 2 contains the factor c ( e i ) c ( e j ) , makes sure that the d e gree o f α i ( x ) d oes n ot incre as e by more than 2 compared to that of α i − 1 ( x ) .  Remark 6.8 . As a co nsequence of (6.16) an d (6. 17 ) we h a ve str a i ( x ) = 0 for i ≤ n 2 . Ther e- fore ind D = 1 ( 4 π t ) n 2 ∑ i ≥ n 2 t i Z M c ( x ) str ( a i ( x )) d x . Furthermor e, since th e in de x is indepen- dent of t and n is even, we have the follo wing theorem. Theorem 6.9. The index of the gr aded Dirac operator D is equa l to (6.18) ind D = 1 ( 4 π ) n 2 Z M c ( x ) str ( a n / 2 ( x )) d x . The element str ( a n 2 ( x )) in (6.18) can be c alculated an alytically in terms of differential forms on M . T o calcu late str ( a n / 2 ( x )) ∈ Hom ( V x ) , we localize the o perator D and the heat kern el k t ( x , y ) at a p oint x . Because the local calculation is irrelev an t to M bein g compact or not, we u s e the classical calcula ti on of str a n 2 on a compac t man ifold without modiﬁcation .Therefore, str ( a n 2 ( x )) is the n form par t of det 1 2 ( R / 2 sinh R / 2 ) tr V / S ( e − F ) . For details, p lease refer to [6] Chapter 4. Finally we obtain the following main theo rem of this subsection. Theorem 6.10. Let R be the curva tur e 2-form with res pect to th e Le vi-Civita conn ection on the manifold (on T M ). Then, (6.19) ind D = Z M c ( x ) ˆ A ( M ) · ch ( V / S ) . wher e ˆ A ( M ) = d et 1 2 ( R / 4 π i sinh R / 4 π i ) is the ˆ A-class of T M a nd ch ( V / S ) = tr V / S ( e − F V / S ) is the r ela tive Chern character , i.e. Chern character of the twisted curvature F V / S of the bundle S. L 2 -INDEX FORM ULA FOR PR OPER COC OMP ACT GROUP ACTIONS 29 6.2. Conclusion. In th is subsection we will ﬁgure out ind D V ( σ P ) where D is the Dol- beault op erator on Σ X , and wher e V ( σ P ) is a bundle over Σ X . D V ( σ P ) is a g eneralized Dirac op erator and we c alculate the case when D = D V ( σ P ) , M = Σ X in the previous subsections. Firstly we have the follo wing proposition, as a corollary to Theorem 6.10. Proposition 6.11. Let G b e a locally compact u nimodular gr oup and let M be p r op er cocompa ct G -manifold of dimen s ion n havin g an almo st complex structure , curvatur e R, a cutoff function c ∈ C ∞ c ( M ) a nd a G-bundle E with cu r vature F . Let D : L 2 ( M , Λ 0 , ∗ T ∗ M ) → L 2 ( M , Λ 0 , ∗ T ∗ M ) be the Dolbeau lt operator on M . Then the L 2 -index of the tw isted Dirac operator D E is, ind D E = Z M c Td ( M ) ch ( E ) , wher e Td ( M ) = d et ( R 1 − e R ) an d ch ( E ) = tr s ( e − F ) . Both Td ( M ) and ch ( E ) are G - in variant forms. So the in te gral does n ot depend on the choice o f the cu tof f fu nction. I f M = Σ X , then the cutoff function on M can be ob tained from the cutoff fu nction on X b y setting the v alues of t he elements in the same ﬁber to be the same. Th e fo llo wing index formula is immediate assuming the proposition. Theorem 6.12. Let X be a c omplete Riemannian manifold wher e a lo cally compact uni- modular gr oup G acts pr operly , coco mpactly and isometrically . I f P is a zer o order pr o p- erly supported elliptic pseud o-differ ential operator , then the L 2 index of P is given by th e formula (6.20) ind P = Z T X c ( x )( ˆ A ( X )) 2 ch ( σ P ) . Pr oof. Set M = Σ X , V = V ( σ P ) . Clearly , M has an almo st complex structu re. By Propo- sition 5.9 and Proposition 6.11, ind P = Z Σ X c ( x ) Td ( Σ M ) ch ( V σ P ) = Z T X c ( x ) Td ( T X ⊗ C ) ch ( σ P ) . Observe that Td ( T X ⊗ C ) = ( ˆ A ( X )) 2 , th en the statemen t follows.  Pr oof of Pr o position 6 .11. Let J be an almost complex s tructure on M . Say x i , y i , 1 ≤ i ≤ m are a loca l fram e o f T M and J ( x i ) = y i , J ( y i ) = − x i . J extends C -linearly to T M ⊗ C = T M 1 , 0 ⊕ T M 0 , 1 where T M 1 , 0 = { v − iJ v | v ∈ T M } is the set of ho lomorphic tangen t vectors of the form z i . = x j − iy j and T M 0 , 1 = { v + iJ v , v ∈ T M } is the set of anti-ho lomorphic tangent vector s of form ¯ z j . = x j + iy j . W e have real isomorp hisms π 1 , 0 : T M → T M 1 , 0 , v 7→ v 1 , 0 = 1 2 ( v − iJ v ) and π 0 , 1 : T M → T M 0 , 1 , v 7→ v 0 , 1 = 1 2 ( v + iJ v ) . Th erefore ( T M , J ) ≃ T M 1 , 0 ≃ T M 0 , 1 as an almost comp le x bundle. Similarly , the com plexiﬁed cotangen t b undle d ecomposes a s T ∗ M ⊗ C = T ∗ M 1 , 0 ⊕ T ∗ M 1 , 0 where T ∗ M 1 , 0 = { η ∈ T ∗ M ⊗ C | η ( J v ) = i η ( v ) } , con s isting of covectors o f form z j . = x j + iy j , is the C -dual of T M 1 , 0 (notation : x j ( x i ) = δ i j , y j ( y i ) = δ i j ) and T ∗ M 0 , 1 = { η ∈ T ∗ M ⊗ C | η ( J v ) = − i η ( v ) } , con sis ting o f covectors o f form ¯ z j . = x j − iy j , is the C -dual of T M 0 , 1 . Let Ω ∗ M be the set of smooth section s of Λ ∗ M , w hich splits into typ es ( p , q ) with Λ p , q T ∗ M = ( Λ p T ∗ M 1 , 0 ) ⊗ ( Λ q T ∗ M 0 , 1 ) . If α ∈ Ω p , q ( M ) , then the differential decomp oses 30 HANG W ANG into d α = p + q + 1 ∑ i = 0 ( d α ) i , p + q + 1 − i and set ∂ α = ( d α ) p + 1 , q , ¯ ∂ α = ( d α ) p , q + 1 . The Dolbeau lt operator ¯ ∂ : Ω 0 , q → Ω 0 , q + 1 is the o rder 1 differential operator given by ¯ ∂ = ∂ ∂ y + i ∂ ∂ x in the local co ordinate ( x , y ) ∈ M . If we use the grading, the Dolb eault o perator is ¯ ∂ + ¯ ∂ ∗ on Ω 0 , ∗ M . The Dolbea ult oper ator “is” the canon ical Dirac op erator on M in th e sen s e that th e y have the sam e symbol. The can onical Dirac o perator on M is deﬁned as fo llo ws. The bundle S = Λ 0 , ∗ T ∗ M has an action of the cotang ent vecto rs via Clifford multiplication: c ( η ) s = √ 2 ( ε ( η 0 , 1 )( s ) − ι ( η 1 , 0 ) s ) , η ∈ T ∗ M , s ∈ Λ 0 , ∗ T ∗ M . Here, c ( x i ) = 1 √ 2 ( ε ( ¯ z ) − ι ( z )) , c ( x i ) c ( x j ) + c ( x j ) c ( x i ) = − 2 δ i j and ε is the exterior multi- plication and ι is the C -linear compression by a vector . The canonica l Dirac oper ator is deﬁn ed to be D = ∑ c ( e i ) ∇ L e i where { e i } fo rms a lo cal orthon ormal basis o f T M and ∇ L is the Levi-Ci vita c onnection o n S . Now if there is an auxiliary complex G -vector bundle E → M , with a G -inv ariant Her mitian metric and G - in variant conn ection ∇ E , the Dolbeault operator D E acting on V = S ⊗ E with coefﬁcients in E ca n b e represented by (up t o a lower order term): D E = ∑ c ( e i ) ∇ V e i , where ∇ V = ∇ L ⊗ 1 + 1 ⊗ ∇ E . Let ∇ be the Levi-Civita connec ti on on M (on ( T M ) 0 , 1 , be ing more p recise) and R = ∇ 2 ∈ Λ 2 ( M , so ( T M )) be Riemannian curvature, the matrix with coefﬁcients of two fo rms representin g th e cu rv atu re of M , R ( X , Y ) = ∇ X ∇ Y − ∇ Y ∇ X − ∇ [ X , Y ] , X , Y ∈ C ∞ ( M , T M ) . In the orth onormal fr ame e i of T M , R ( e i , e j ) = − ∑ k < l ( R ( e i , e j ) e l , e k ) e k ∧ e l , where we identify so ( T M ) with the bundle of two f orms on M . Now we have a Clifford modu le S , where C l ( T ∗ M ) = Hom ( S ) , on wh ich T ∗ M acts by Clifford multiplication. On S there is a Clifford con nection ∇ S so that the Clifford mu ltiplication by u nit vectors preserves the metric and ∇ S is co mpatible with th e conn ection on M . L et R S = ( ∇ S ) 2 be th e curvature associated to ∇ S . It is well known that the Lie algebra isomorphism spin n ≃ so n giv en by 1 4 [ v , w ] 7→ v ∧ w imp lies that R S ( e i , e j ) = 1 2 ∑ k < l ( R ( e i , e j ) e k , e l ) c ( e k ) c ( e l ) = 1 4 ∑ k , l ( R ( e i , e j ) e k , e l ) c ( e k ) c ( e l ) . On S , there is also a Levi-Civita connection, denoted by ∇ L . Th e associated cu rv atu re R L = ( ∇ L ) 2 ∈ Hom S is written as R L = R S + F where R S ( · , · ) = 1 4 ∑ k , l ( R ( · , · ) ¯ z k , z l ) c ( ¯ z k ) c ( z l ) + 1 4 ∑ k , l ( R ( · , · ) z k , ¯ z l ) c ( z k ) c ( ¯ z l ) ∈ Cl ( T M ) and F ∈ H om Cl V is the twisting curvature. L 2 -INDEX FORM ULA FOR PR OPER COC OMP ACT GROUP ACTIONS 31 Recall that the cu rv a ture of the L e vi-Civita connection on Λ V ∗ is the d eri vation of the algebra Λ V ∗ which coincides with R ( e i , e j ) on V and is giv en by the formula ∑ k , l < e k , R ( e i , e j ) e l > ε ( e k ) ι ( e l ) = ∑ k , l ( R ( e i , e j ) e l , e k ) ε ( e k ) ι ( e l ) . Let R − be the cu rv atu re of the Levi-Ci v ita co nnection on T 0 , 1 M . Note that R = R − . The n the curvature of ∇ L on S is given by R L ( · , · ) = 1 4 ∑ i , j ( R − ( · , · ) z i , ¯ z j ) ε ( ¯ z j ) ι ( z i ) = − 1 8 ∑ i , j ( R − ( · , · ) z i , ¯ z j ) c ( ¯ z j ) c ( z i ) . Using th e fact that c ( z i ) 2 = 0 , c ( ¯ z i ) 2 = 0 , c ( z i ) c ( ¯ z j ) + c ( ¯ z j ) c ( z i ) = − 4 δ i j , whe re c ( z ) = c ( x ) + i c ( y ) , c ( ¯ z ) = c ( x ) − i c ( y ) . we have F V / S = R V − R S = 1 2 ∑ k ( Rz k , ¯ z k ) = 1 2 T r R + F E and a direct calculation shows that ˆ A ( M ) e F V / S = d et R / 2 sinh R / 2 e 1 2 Tr R ( e F E ) = det R e R − 1 ( e F E ) = Td ( M ) T r ( e − F E ) .  The following theorem is an imm ediate coro llary to Theo rem 6.12. Theorem 6.13 (Atiy ah’ s L 2 -index theorem) . Let D be a n elliptic operator on a compact manifold X a nd ˜ D be th e π 1 ( X ) -in variant op er a tor deﬁned on the universal cover spa ce ˜ X as the lift of D. Then ind ˜ D = ind D . 6.3. L 2 -index theorem for homogeneous spaces o f Lie groups. Let G be a unimodu lar Lie g roup and H b e a co mpact subgrou p. Consider the h omogenous space M = G / H of left cosets of H in G , a G -bundle ¯ E over M and a G -inv ariant elliptic o perator D on ¯ E . The ﬁber of ¯ E at eH , den oted by E = ¯ E | eH , is an H -space, so that ¯ E = G × H E . Similarly , set V = T eH M , then T M = G × H V . Let Ω ∈ Λ 2 ( T M ) ∗ ⊗ gl ( T M ) be th e curvature of M , associated to th e G -inv a riant Levi-Ci vita co nnection on T M . Then we have the G -in variant ˆ A -class ˆ A ( M ) = det 1 2 Ω / 4 π i sinh Ω / 4 π i . Let Ω E ∈ Λ 2 ( Σ M ) ∗ ⊗ gl ( V ( σ A )) be a curvature form ass ociated to some G -in variant con- nection on V ( σ D ) over Σ M . Th en ch ( σ D ) = T r e Ω E | T M is the Chern cha racter of V ( σ A ) r es tricted to T M . Let Ω V be the curvature tensor Ω re- stricted to V = T eH M and Ω E V be th e curvature tensor Ω E restricted to V . Then we d eﬁne the correspon ding ˆ A -class and Chern character as ˆ A ( M ) V . = d et 1 2 Ω V / 2 sinh Ω V / 2 and ch ( σ D ) V . = Tr e Ω E V . W e have as a coro llary the L 2 -index theorem for homogeneo us spaces. 32 HANG W ANG Corollary 6.14 . The L 2 -index of a G-in varian t elliptic operator D : L 2 ( M , ¯ E ) → L 2 ( M , ¯ E ) is (6.21) ind D = Z V ˆ A 2 ( M ) V ch ( σ D ) V . Pr oof. T he L 2 -index theorem of D says that ind D = Z T M c ˆ A 2 ( M ) ch ( σ D ) . Since T M = G × H V , the integration o f the form c ˆ A 2 ( M ) ch ( σ A ) on T M can be computed by lifting to an H -in variant form on G × V and th en integratin g over the group part and then the tang ent space at eH . Since ˆ A 2 ( M ) ch ( σ D ) is G -inv ariant, then at any g ∈ G , the form will be the same as its value at th e un it e of G : ˆ A 2 ( M ) V ch ( σ A ) V . Hen ce, Z T M c ˆ A 2 ( M ) ch ( σ D ) = Z V ˆ A 2 ( M ) V ch ( σ D ) V Z G c ( g − 1 v ) vol = Z V ˆ A 2 ( M ) V ch ( σ D ) V , where v ol is the volume form on G .  Remark 6 .15 . The formu la (6.21) is essentially the L 2 -index for mula in [ 10 ]. The com- ponen ts o f the form ula in 6 .21 are sketched as follows. On the Lie algeb ra g of G there is an H -in variant splitting g = h ⊕ m where h is the L ie algebra of H and m is an H -in variant comp lement. V = T eH ( G / H ) is a cand idate for m . Th ere is a curvature form o n m deﬁn ed by Θ ( X , Y ) = − 1 2 θ ([ X , Y ]) , X , Y ∈ m wh ere θ is th e conn ection form giv en by the pr ojection θ : g → h . Θ co mposed with r : h → gl ( E ) , the d if fer ential o f a unitar y rep resentation o f H on so me vecto r spac e E , is an H -inv ariant curvature f orm Θ r ( X , Y ) = r ( Θ ( X , Y )) , X , Y ∈ m . Then ch : R ( H ) → H ∗ ( g , H ) : r 7→ Tr e Θ r is a well-deﬁned Cher n ch aracter ([10] page 3 09). Also, co mpose th e cur v atu re form (6.15), with h → gl ( V ) , the differential of th e H -m odule structure of V . An d a cur v atu re form Θ V ∈ Λ 2 m ∗ ⊗ gl ( V ) on V is construc ted a nd the ˆ A -class is deﬁned as ˆ A ( g , H ) = d et 1 2 Θ V / 2 sinh Θ V / 2 . The L 2 -index formula of D in [10] is (6.22) ind D = Z V ch ( a ) ˆ A ( g , H ) , where a is an element o f the rep resentation ring R ( H ) , speciﬁcally a is the pre-image of V ( σ D ) | V + under the Thom isomor phism R ( H ) → K H ( V ) . Her e, V + is the space built from V by ad ding one point at inﬁnity . It is the ball ﬁber in Σ M at eH . No te that the Thom isom orphism exists o nly for th e case wh en the action of H on V , lifts to Spin ( V ) . The g eneral case was done by introducin g a do uble covering o f H and by redu cing th e problem to this situation [10] page 307. T o see that 6.2 1 and 6.22 are the same form ula, we prove the fo llo wing assertions. (1) ˆ A ( M ) V = ˆ A ( g , H ) . In fact, since T M = G × H V is a pr incipal G -bundle over V / H an d V is a principal H - b u ndle over V / H , then by [20] II Prop. 6.4, the con nection fo rm on T M r estricted to V is also a connection form. Also, on G / H , the restriction of any G -in variant tensor on T M L 2 -INDEX FORM ULA FOR PR OPER COC OMP ACT GROUP ACTIONS 33 to V is an H -in variant tenso r on V . Therefo re Ω V is an H -in variant cu rv atu re form on V and the restriction ˆ A ( M ) V is th e ˆ A -class deﬁned by curv ature Ω V . By deﬁnition ˆ A ( g , H ) is the ˆ A -class of the cur v atu re Θ V on V , ˆ A -class of another co nnection on th e same V . The statement is p rov ed because ˆ A is a top ological in variant and is indepen dent of the c hoice of connectio n o n V . (2) ch ( σ D ) V = ch ( a ) . Similarly to th e last proo f, Ω E V is an H -in variant curvature form of V ( σ D ) | V + restricted to V . Recall that V ( σ D ) is glued by th e G - in variant symb ol σ D and there fore it is deter- mined by its restriction at the ball ﬁbe r , V + . By d eﬁnition V ( σ D ) | V + is glu ed t wo cop ies of B V × E on the bound ary by σ D | S V . 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(Current Address) R OO M 1 4 3 , M AT HE M AT I CA L S C I E N C ES C E N T E R , J I N C H U N Y UA N W E S T B U I L D I N G , T S I N G H UA U N I V E R S I T Y , H AI D I A N D I S T R I C T , B E I J I N G , C H I NA 1 0 0 0 8 4 . E-mail addre ss : hwang@math.tsin ghua.edu .cn, hang.wang@v anderbilt .edu.

$L^2$-index formula for proper cocompact group actions

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