A local equivariant index theorem for sub-signature operators
aa r X i v : . [ m a t h . DG ] D ec A local equivariant index theorem for sub-signature operators
Kaihua Bao, Jian Wang, Yong Wang ∗ School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, P.R.China
Abstract
In this paper, we prove a local equivariant index theorem for sub-signature operators which generalizes theZhang’s index theorem for sub-signature operators.
Keywords:
Sub-signature operator; equivariant index.
1. Introduction
The Atiyah-Bott-Segal-Singer index formula is a generalization with group action of the Atiyah-Singerindex theorem. In [1], Berline and Vergne gave a heat kernel proof of the Atiyah-Bott-Segal-Singer indexformula. In [2], Lafferty, Yu and Zhang presented a simple and direct geometric proof of the Lefschetzfixed point formula for an orientation preserving isometry on an even dimensional spin manifold by Cliffordasymptotics of heat kernel. In [3], Ponge and Wang gave a different proof of the equivariant index formula bythe Greiner’s approach of the heat kernel asymptotics. In [4], in order to prove family rigidity theorems, Liuand Ma proved the equivariant family index formula. In [5], Wang gave another proof of the local equivarintindex theorem for a family of Dirac operators by the Greiner’s approach of the heat kernel asymptotics. In[6], using the Greiner’s approach of heat kernel asymptotics, Wang gave proofs of the equivariant Gauss-Bonnet-Chern formula and the variation formulas for the equivariant Ray-Singer metric, which are originallydue to J. M. Bismut and W. Zhang.In parallel, Freed[7] considered the case of an orientation reversing involution acting on an odd dimen-sional spin manifold and gave the associated Lefschetz formulas by the K-theretical way. In [8], Wangconstructed an even spectral triple by the Dirac operator and the orientation-reversing involution and com-puted the Chern-Connes character for this spectral triple. In [9], Liu and Wang proved an equivariant oddindex theorem for Dirac operators with involution parity and the Atiyah-Hirzebruch vanishing theorems forodd dimensional spin manifolds.In [10] and [11], Zhang introduced the sub-signature operators and proved a local index formula for theseoperators. In [12] and [13], by computing the adiabatic limit of eta-invariants associated to the so-calledsub-signature operators, a new proof of the Riemann-Roch-Grothendieck type formula of Bismut-Lott wasgiven. The motivation of the present article is to prove a local equivariant index formula for sub-signatureoperators.This paper is organized as follows: In Section 2, we present some notations of sub-signature operators.In Section 3.1, we prove a local equivariant even dimensional index formula for sub-signature operators. InSection 3.2, we prove a local equivariant odd dimensional index formula for sub-signature operators withthe orientation-reversing involution. ∗ Corresponding author (Yong Wang);
Email address: [email protected] (Yong Wang)
Preprint submitted to Elsevier April 16, 2018 . The sub-signature operators
Firstly we give the standard setup (also see Section 1 in [10]). Let M be an oriented closed manifold ofdimension n . Let E be an oriented sub-bundle of the tangent vector bundle T M . Let g T M be a metric on
T M . Let g E be the induced metric on E . Let E ⊥ be the sub-bundle of T M orthogonal to E with respectto g T M . Let g E ⊥ be the metric on E ⊥ induced from g T M . Then (
T M, g
T M ) has the following orthogonalsplittings
T M = E ⊕ E ⊥ , (2.1) g T M = g E ⊕ g E ⊥ . (2.2)Clearly, E ⊥ carries a canonically induced orientation. We identify the quotient bundle T M/E with E ⊥ .Let Ω( M ) = L n Ω i ( M ) = L n Γ( ∧ i ( T ∗ M )) be the set of smooth sections of ∧ ( T ∗ M ). Let ∗ be theHodge star operator of g T M . Then Ω( M ) inherits the following standardly induced inner product h α, β i = Z M α ∧ ∗ β, α, β ∈ Ω( M ) . (2.3)We use g T M to identify
T M and T ∗ M . For any e ∈ Γ( T M ), let e ∧ and i e be the standard notation forexterior and interior multiplications on Ω( M ). Let c ( e ) = e ∧ − i e , ˆ c ( e ) = e ∧ + i e be the Clifford actions onΩ( M ) verifying that c ( e ) c ( e ′ ) + c ( e ′ ) c ( e ) = − h e, e ′ i g T M , (2.4)ˆ c ( e )ˆ c ( e ′ ) + ˆ c ( e ′ )ˆ c ( e ) = 2 h e, e ′ i g TM , (2.5) c ( e )ˆ c ( e ′ ) + ˆ c ( e ′ ) c ( e ) = 0 . (2.6)Denote k = dim E . Let { f , · · · , f k } be an oriented (local) orthonormal basis of E . Setˆ c ( E, g E ) = ˆ c ( f ) · · · ˆ c ( f k ) , (2.7)where ˆ c ( E, g E ) does not depend on the choice of the orthonormal basis.Let ǫ = Id ∧ even ( T ∗ M ) − Id ∧ odd ( T ∗ M ) be the Z -grading operator of ∧ ( T ∗ M ) = ∧ even ( T ∗ M ) ⊕ ∧ odd ( T ∗ M ) . Set τ ( M, g E ) = ǫ ˆ c ( E, g E ) . (2.8)One verifies easily that τ ( M, g E ) = ( − k ( k +1)2 . (2.9)Let ∧ ± ( T ∗ M, g E ) = n ω ∈ ∧ ∗ ( T ∗ M ) , τ ( M, g E ) ω = ± ω o the (even/odd) eigen-bundles of τ ( M, g E ) and by Ω ± ( M, g E ) the corresponding set of smooth sections.Let δ = d ∗ be the formal adjoint operator of the exterior differential operator d on Ω( M ) with respectto the inner product (2.3).Set D E = 12 (cid:0) ˆ c ( E, g E )(d + δ ) + ( − k (d + δ )ˆ c ( E, g E ) (cid:1) . (2.10)2hen one verifies easily that D E τ ( M, g E ) = − τ ( M, g E ) D E , (2.11) D ∗ E = ( − k ( k +1)2 D E , (2.12)where D ∗ E is the formal adjoint operator of D E with respect to the inner product (2.3).Set ˜ D E = ( √− k ( k +1)2 D E . From (2.11), ˜ D E is a formal self-adjoint first order elliptic differential operator on Ω( M ) interchangingΩ ± ( M, g E ). Definition 2.1.
The sub-signature operator ˜ D E, + with respect to ( E, g
T M ) is the restriction of ˜ D E on Ω + ( M, g E ) . If we denote the restriction of ˜ D E on Ω − ( M, g E ) by ˜ D E, − , then one has clearly˜ D ∗ E, ± = ˜ D E, ∓ . Recall that E is the subbundle of T M and that we have the orthogonal decomposition (2.1) of
T M andthe metric g T M . Let P E (resp. P E ⊥ ) be the orthogonal projection from T M to E (resp. E ⊥ ).Let ∇ T M be the Levi-Civita connection of g T M . We will use the same notation for its lifting on Ω( M ).Set ∇ E = P E ∇ T M P E , (2.13) ∇ E ⊥ = P E ⊥ ∇ T M P E ⊥ . (2.14)Then ∇ E (resp. ∇ E ⊥ ) is a Euclidean connection on E (resp. E ⊥ ), and we will use the same notation for itslifting on Ω( E ∗ )(resp. Ω( E ⊥ , ∗ )).Let S be the tensor defined by ∇ T M = ∇ E + ∇ E ⊥ + S. Then S takes values in skew-adjoint endomorphisms of T M , and interchanges E and E ⊥ .Let { e , · · · , e n } be an oriented(local) orthonormal base of T M . To specify the role of E , set { f , · · · , f k } be an oriented (local) orthonormal basis of E . We will use the greek subscripts for the basis of E . Then byProposition 1.4 in [10], we have Proposition 2.2.
The following identity holds, ˜ D E = ( √− k ( k +1)2 (cid:0) ˆ c ( E, g E )(d + δ ) + 12 X i c ( e i )( ∇ T Me i ˆ c ( E, g E )) (cid:1) . (2.15)Similar to Lemma 1.1 in [10], we have Lemma 2.3.
For any X ∈ Γ( T M ) , the following identity holds, ∇ T MX ˆ c ( E, g E ) = − ˆ c ( E, g E ) X α ˆ c ( S ( X ) f α )ˆ c ( f α ) . (2.16)Let ∆ T M , ∆ E be the Bochner Laplacians∆ T M = n X i ( ∇ T M, e i − ∇ T M ∇ T Mei e i ) , (2.17)∆ E = k X i ( ∇ E, e i − ∇ E ∇ Eei e i ) . (2.18)3et K be the scalar curvature of ( M, g
T M ). Let R T M , R E , R E ⊥ be the curvatures of ∇ T M , ∇ E , ∇ E ⊥ respectively. Let { h , · · · , h n − k } be an oriented(local) orthonormal base of E ⊥ . Now we can state thefollowing Lichnerowicz type formula for ˜ D E . From Theorem 1.1 in [10], we have Theorem 2.4. [10] The following identity holds, ˜ D E = − ∆ T M + K X ≤ i,j ≤ n X ≤ α,β ≤ k h R E ( e i , e j ) f β , f α i c ( e i ) c ( e j )ˆ c ( f α )ˆ c ( f β )+ 18 X ≤ i,j ≤ n X ≤ s,t ≤ n − k h R E ⊥ ( e i , e j ) h t , h s i c ( e i ) c ( e j )ˆ c ( h s )ˆ c ( h t ) + 12 X α ˆ c (cid:0) (∆ T M − ∆ E ) f α (cid:1) ˆ c ( f α )+ X i,α (cid:16) ˆ c ( S ( e i ) f α )ˆ c ( f α ) ∇ T Me i − ˆ c ( S ( e i ) ∇ Ee i f α )ˆ c ( f α ) + 12 ˆ c (cid:0) ∇ E ( ∇ TMei −∇ Eei ) e i f α (cid:1) ˆ c ( f α ) + 34 k S ( e i ) f α ) k (cid:17) + 14 X i,α = β ˆ c ( S ( e i ) f α )ˆ c ( S ( e i ) f β )ˆ c ( f α )ˆ c ( f β ) . (2.19)
3. A local equivariant index Theorem for sub-signature operators
Let M be a closed even dimensional n oriented Riemannian manifold and φ be an isometry on M preserving the orientation. Then φ induces a map ˜ φ = φ − , ∗ : ∧ T ∗ x M → ∧ T ∗ φ ( x ) M on the exterior algebrabundle ∧ T ∗ x M . Let ˜ D E be the sub-signature operator. We assume that d φ preserves E and E ⊥ and theirorientations, then ˜ φ ˆ c ( E, g E ) = ˆ c ( E, g E ) ˜ φ . Then ˜ φ ˜ D E = ˜ D E ˜ φ . We will compute the equivariant indexInd φ ( ˜ D + E ) = Tr( ˜ φ | ker ˜ D + E ) − Tr( ˜ φ | ker ˜ D − E ) . (3.1)We recall the Greiner’s approach of heat kernel asymptotics as in [3] and [14], [16]. Define the operatorgiven by ( Q u )( x, s ) = Z ∞ e − s ˜ D E [ u ( x, t − s )] dt, u ∈ Γ c ( M × R , ∧ T ∗ M ) , (3.2)maps continuously u to D ′ ( M × R , ∧ T ∗ M )) which is the dual space of Γ c ( M × R , ∧ T ∗ M )) . We have( ˜ D E + ∂∂t ) Q u = Q ( ˜ D E + ∂∂t ) u = u, u ∈ Γ c ( M × R , ∧ T ∗ M )) . (3.3)Let ( ˜ D E + ∂∂t ) − is the Volterra inverse of ˜ D E + ∂∂t as in [14]. Then( ˜ D E + ∂∂t ) Q = I − R ; Q ( ˜ D E + ∂∂t ) = 1 − R , (3.4)where R , R are smooth operators. Let( Q u )( x, t ) = Z M × R K Q ( x, y, t − s ) u ( y, s ) dyds, (3.5)and k t ( x, y ) is the heat kernel of e − t ˜ D E . We get K Q ( x, y, t ) = k t ( x, y ) when t > , when t < , K Q ( x, y, t ) = 0 . (3.6) Definition 3.1.
The operator P is called the Volterra Ψ DO if(i) P has the Volterra property,i.e. it has a distribution kernel of the form K P ( x, y, t − s ) where K P ( x, y, t ) vanishes on the region t < . (ii) The parabolic homogeneity of the heat operator P + ∂∂t , i.e. the homogeneity with respect to thedilations of R n × R given by λ · ( ξ, τ ) = ( λξ, λ τ ) , ( ξ, τ ) ∈ R n × R , λ = 0 . (3.7)4n the sequel for g ∈ S ( R n +1 ) and λ = 0, we let g λ be the tempered distribution defined by h g λ ( ξ, τ ) , u ( ξ, τ ) i = | λ | − ( n +2) (cid:10) g λ ( ξ, τ ) , u ( λ − ξ, λ − τ ) (cid:11) , u ∈ S ( R n +1 ) . (3.8) Definition 3.2.
A distribution g ∈ S ( R n +1 ) is parabolic homogeneous of degree m, m ∈ Z, if for any λ = 0 ,we have g λ = λ m g. Let C − denote the complex halfplane { Im τ < } with closure C − . Then: Lemma 3.3. [14] Let q ( ξ, τ ) ∈ C ∞ (( R n × R ) / be a parabolic homogeneous symbol of degree m such that:(i) q extends to a continuous function on ( R n × C − ) / in such way to be holomorphic in the last variablewhen the latter is restricted to C − .Then there is a unique g ∈ S ( R n +1 ) agreeing with q on R n +1 / so that:(ii) g is homogeneous of degree m ;(iii) The inverse Fourier transform ˘ g ( x, t ) vanishes for t < . Let U be an open subset of R n . We define Volterra symbols and Volterra Ψ DO s on U × R n +1 / Definition 3.4. S mV ( U × R n +1 ) , m ∈ Z , consists in smooth functions q ( x, ξ, τ ) on U × R n × R with anasymptotic expansion q ∼ P j ≥ q m − j , where:(i) q l ∈ C ∞ ( U × [( R n × R ) / is a homogeneous Volterra symbol of degree l , i.e. q l is parabolic homogeneousof degree l and satisfies the property (i) in Lemma 2.3 with respect to the last n + 1 variables;(ii) The sign ∼ means that, for any integer N and any compact K, U, there is a constant C NKαβk > such that for x ∈ K and for | ξ | + | τ | > we have | ∂ αx ∂ βξ ∂ kτ ( q − X j Let q m ( x, ξ, τ ) ∈ C ∞ ( U × ( R n +1 / be a homogeneous Volterra symbol of order m and let g m ∈ C ∞ ( U ) ⊗ S ′ ( R n +1 ) denote its unique homogeneous extension given by Lemma 2.3. Then:(i) ˘ q m ( x, y, t ) is the inverse Fourier transform of g m ( x, ξ, τ ) in the last n + 1 variables;(ii) q m ( x, D x , D t ) is the operator with kernel ˘ q m ( x, y − x, t ) . Definition 3.7. The following properties hold.1) Composition. Let Q j ∈ Ψ m j V ( U × R ) , j = 1 , have symbol q j and suppose that Q or Q is properlysupported. Then Q Q is a Volterra Ψ DO of order m + m with symbol q ◦ q ∼ P α ! ∂ αξ q D αx q . Q is the order m Volterra Ψ DO with the paramatrix P then QP = 1 − R , P Q = 1 − R (2 . where R , R are smooth operators. Definition 3.8. The differential operator ˜ D E + ∂ t is invertible and its inverse ( ˜ D E + ∂ t ) − is a Volterra Ψ DO of order − M φ the fixed-point set of φ , and for a = 0 , · · · , n, we let M φ = S ≤ a ≤ n M φa , where M φa isan a -dimensional submanifold. Given a fixed-point x in a component M φa , consider some local coordinates x = ( x , · · · , x a ) around x . Setting b = n − a, we may further assume that over the range of the domain ofthe local coordinates there is an orthonormal frame e ( x ) , · · · , e b ( x ) of N φz . This defines fiber coordinates v = ( v , · · · , v b ) . Composing with the map ( x, v ) ∈ N φ ( ε ) → exp x ( v ) we then get local coordinates x , · · · , x a , v , · · · , v b for M near the fixed point x . We shall refer to this type of coordinates as tubularcoordinates. Then N φ ( ε ) is homeomorphic with a tubular neighborhood of M φ . Set i M φ : M φ ֒ → M be an inclusion map. Since d φ preserves E and E ⊥ , considering the oriented (local) orthonormal basis { f , · · · , f k , h , · · · , h n − k } , set d φ x = (cid:18) exp( L ) 00 exp( L ) (cid:19) , (3.10)where L ∈ s o ( k ) and L ∈ s o ( n − k )Let b A ( R M φ ) = det R M φ / π sinh( R M φ / π ) ! ; ν φ ( R N φ ) := det − (1 − φ N e − RNφ π ) . (3.11)By (3.38) and Lemma 2.15 (2), and Lemma 9.13 in [3], we get the main Theorem in this section. Theorem 3.9. ( Local even dimensional equivariant index Theorem for sub-signature operators )Let x ∈ M φ , then lim t → Str h ˜ φ ( x ) K t ( x , φ ( x )) i = ( 1 √− k n n b A ( R M φ ) ν φ ( R N φ ) i ∗ M φ h det (cid:16) cosh (cid:0) R E π − L (cid:1)(cid:17) × det (cid:16) sinh( R E ⊥ π − L ) R E ⊥ π − L (cid:17) Pf (cid:16) R E ⊥ π − L (cid:17)io ( a, ( x ) . (3.12)Next we give a detailed proof of Theorem 3.9.Let Q = ( ˜ D E + ∂ t ) − . For x ∈ M φ and t > I Q ( x, t ) := e φ ( x ) − Z N φx ( ε ) φ (exp x v ) K Q (exp x v, exp x ( φ ′ ( x ) v ) , t ) dv. (3.13)Here we use the trivialization of ∧ ( T ∗ M ) about the tubular coordinates. Using the tubular coordinates,then I Q ( x, t ) = Z | v | <ε e φ ( x, − e φ ( x, v ) K Q ( x, v ; x, φ ′ ( x ) v ; t ) dv. (3.14)Let q ∧ ( T ∗ M ) m − j ( x, v ; ξ, ν ; τ ) := e φ ( x, − e φ ( x, v ) q m − j ( x, v ; ξ, ν ; τ ) . (3.15)Recall Proposition 3.10. [3] Let Q ∈ Ψ mV ( M × R , ∧ ( T ∗ M )) , m ∈ Z . Uniformly on each component M φa I Q ( x, t ) ∼ X j ≥ t − ( a +[ m ]+1) I jQ ( x ) as t → + , (3.16) where I jQ ( x ) is defined by I ( j ) Q ( x ) := X | α |≤ m − [ m ]+2 j Z v α α ! (cid:16) ∂ αv q ∧ ( T ∗ M )2[ m ] − j + | α | (cid:17) ∨ ( x, 0; 0 , (1 − φ ′ ( x )) v ; 1) dv. (3.17)6imilar to Theorem 1.2 in [4] and Section 2 (d) in [19], we haveStr τ [ ˜ φ exp( − t ˜ D E )] = ( √− k Z M Str ǫ h ˆ c ( E, g E ) k t ( x, φ ( x )) i d x = ( √− k Z M Str ǫ [ˆ c ( E, g E ) K ( ˜ D E + ∂ t ) − ( x, φ ( x ) , t )]d x. (3.18)We will compute the local index in this trivialization.Let ( V, q ) be a finite dimensional real vector space equipped with a quadratic form. Let C ( V, q ) be theassociated Clifford algebra, i.e. the associative algebra generated by V with the relations v · w + w · v = − q ( v, w ) for v, w ∈ V . Let { e , · · · , e n } be the orthomormal basis of ( V, q ), Let C ( V, q ) ˆ ⊗ C ( V, − q ) be thegrading tensor product of C ( V, q ) and C ( V, .q ) and ∧ ∗ V ˆ ⊗ ∧ ∗ V be the grading tensor product of ∧ ∗ V and ∧ ∗ V . Define the symbol map: σ : C ( V, q ) ˆ ⊗ C ( V, − q ) → ∧ ∗ V ˆ ⊗ ∧ ∗ V ; (3.19)where σ ( c ( e j ) · · · c ( e j l ) ⊗ 1) = e j ∧ · · · ∧ e j ⊗ σ (1 ⊗ ˆ c ( e j ) · · · ˆ c ( e j l )) = 1 ⊗ ˆ e j ∧ · · · ∧ ˆ e j . Using theinterior multiplication ι ( e j ) : ∧ ∗ V → ∧ ∗− V and the exterior multiplication ε ( e j ) : ∧ ∗ V → ∧ ∗ +1 V , wedefine representations of C ( V, q ) and C ( V, − q ) on the exterior algebra: c : C ( V, q ) → End ∧ V, e j c ( e j ) : ε ( e j ) − ι ( e j ); (3.20) c : C ( V, − q ) → End ∧ V, e j ˆ c ( e j ) : ε ( e j ) + ι ( e j ); (3.21)The tensor product of these representations yields an isomorphism of superalgebras c ⊗ ˆ c : C ( V, q ) ˆ ⊗ C ( V, − q ) → End ∧ V (3.22)which we will also denote by c . We obtain a supertrace (i.e. a linear functional vanishing on supercommu-tators) on C ( V, q ) ˆ ⊗ C ( V, − q ) by setting Str( a ) = Str End ∧ V [ c ( a )] for a ∈ C ( V, q ) ˆ ⊗ ( V, − q ), where Str End ∧ V isthe canonical supertrace on End V . Lemma 3.11. For ≤ i < · · · < i p ≤ n , ≤ j < · · · < j q ≤ n , when p = q = n , Str[ c ( e i ) · · · c ( e i n )ˆ c ( e i ) · · · ˆ c ( e i n )] = ( − n ( n +1)2 n (3.23) and otherwise equals zero. We will also denote the volume element in ∧ V ˆ ⊗ ∧ V by ω = e ∧ · · · ∧ e n ∧ ˆ e ∧ · · · ∧ ˆ e n . For a ∈ ∧ V ˆ ⊗ ∧ V ,let T a be the coefficient of ω . The linear functional T : ∧ V ˆ ⊗ ∧ V → R is called the Berezin trace. Then fora a ∈ C ( V, q ) ˆ ⊗ ( V, .q ) , one has Str s ( a ) = ( − n ( n +1)2 n (T σ )( a ). We define the Getzler order as follows:deg ∂ j = 12 deg ∂ t = − deg x j = 1 , deg c ( e j ) = 1 , degˆ c ( e j ) = 0 . (3.24)Let Q ∈ Ψ ∗ V ( R n × R , ∧ ∗ T ∗ M ) have symbol q ( x, ξ, τ ) ∼ X k ≤ m ′ q k ( x, ξ, τ ) , (3.25)where q k ( x, ξ, τ ) is an order k symbol. Then taking components in each subspace ∧ j T ∗ M ⊗ ∧ l T ∗ M of ∧ T ∗ M ⊗ ∧ T ∗ M and using Taylor expansions at x = 0 give formal expansions σ [ q ( x, ξ, τ )] ∼ X j,k σ [ q k ( x, ξ, τ )] ( j,l ) ∼ X j,k,α x α α ! σ [ ∂ αx q k (0 , ξ, τ )] ( j,l ) . (3.26)The symbol x α α ! σ [ ∂ αx q k (0 , ξ, τ )] ( j,l ) is the Getzler homogeneous of k + j − | α | . So we can expand σ [ q ( x, ξ, τ )]as σ [ q ( x, ξ, τ )] ∼ X j ≥ q ( m − j ) ( x, ξ, τ ) , q ( m ) = 0 , (3.27)where q ( m − j ) is a Getzler homogeneous symbol of degree m − j .7 efinition 3.12. The integer m is called as the Getzler order of Q . The symbol q ( m ) is the principle Getzlerhomogeneous symbol of Q . The operator Q ( m ) = q ( m ) ( x, D x , D t ) is called as the model operator of Q . Let e , . . . , e n be an oriented orthonormal basis of T x M such that e , · · · , e a span T x M φ and e a +1 , · · · , e n span N φx . This provides us with normal coordinates ( x , · · · , x n ) → exp x ( x e + · · · + x n e n ) . Moreoverusing parallel translation enables us to construct a synchronous local oriented tangent frame e ( x ) , ..., e n ( x )such that e ( x ) , · · · , e a ( x ) form an oriented frame of T M φa and e a +1 ( x ) , · · · , e n ( x ) form an (oriented) frame N τ (when both frames are restricted to M φ ) . This gives rise to trivializations of the tangent and exterioralgebra bundles. Write φ ′ (0) = (cid:18) φ N (cid:19) = exp( A ij ) , (3.28)where A ij ∈ s o ( n ).Let ∧ ( n ) = ∧ ∗ R n be the exterior algebra of R n . We shall use the following gradings on ∧ ( n ) ˆ ⊗ ∧ ( n ), ∧ ( n ) ˆ ⊗ ∧ ( n ) = M ≤ k , k ≤ a ≤ l , l ≤ b ∧ k ,l ( n ) ˆ ⊗ ∧ k ,l ( n ) , (3.29)where ∧ k,l ( n ) is the space of forms dx i ∧ · · · ∧ dx i k + l with 1 ≤ i < · · · < i k ≤ a and a + 1 ≤ i k +1 < · · ·
Proposition 3.14. The model operator of F is F (2) = − n X r =1 (cid:0) ∂ r + 18 X ≤ i,j,l ≤ n h R T M ( e i , e j ) e l , e r i y l e i ∧ e j (cid:1) + 18 X ≤ i,j ≤ n X ≤ α,β ≤ k h R E ( e i , e j ) f β , f α i e i ∧ e j ˆ c ( f α )ˆ c ( f β )+ 18 X ≤ i,j ≤ n X ≤ s,t ≤ n − k h R E ⊥ ( e i , e j ) h t , h s i e i ∧ e j ˆ c ( h s )ˆ c ( h t ) . (3.34)From the representation of F (2) , we get the model operator of ∂∂t + ˜ D E is ∂∂t + F (2) . And we have( ∂∂t + F (2) ) K Q ( − ( x, y, t ) = 0 . (3.35)Similar to Lemma 2.9 in [3], we get 9 emma 3.15. Let Q ∈ Ψ ( − ( R n × R , , ∧ ( T ∗ M )) be a parametrix for ( F (2) + ∂ t ) − . Then(1)Q has Getzler order -2 and its model operator is ( F (2) + ∂ t ) − ;(2) For all t > , ( √− k ˆ c ( E, g E ) I ( F (2) + ∂ t ) − (0 , t )= ( √− k ˆ c ( E, g E ) (4 πt ) − a det (1 − φ N ) det (cid:16) tR ′ sinh( tR ′ ) (cid:17) det − (1 − φ N e − tR ′′ )exp (cid:0) t ( ˜˙ R + ˜¨ R ) (cid:1) . (3.36)Similar to Lemma 3.6 in [Wa]. we have Lemma 3.16. Q ∈ Ψ ∗ V ( R n × R , ∧ ( T ∗ M )) has the Getzler order m and model operator Q ( m ) . Then as t → + (1) σ [ I Q (0 , t )] ( j,l ) = O ( t j − m − a − ) , if m − j is odd.(2) σ [ I Q (0 , t )] ( j,l ) = O ( t j − m − a − ) I Q ( m ) (0 , ( j,l ) + O ( t j − m − a ) , if m − j is even.In particular, for m = − and j = a and a is even we get σ [ I Q (0 , t )] (( a, , ( a,b − l )) = I Q ( − (0 , (( a, , ( a,b − l )) + O ( t ) . (3.37)With all these preparations, we are going to prove the local even dimensional equivariant index theoremfor sub-signature operators. Substituting (3.33), (3.36) into (3.30), we obtainlim t → Str ε h ˜ φ ( x )( √− k ˆ c ( E, g E ) I ( F + ∂ t ) − ( x , t ) i = ( − n n ( 12 ) n − a (4 π ) − a ( √− k (cid:12)(cid:12) b A ( R M φ ) ν φ ( R N φ ) σ (cid:2) ˆ c ( f ) · · · ˆ c ( f k )exp( ˜˙ R + ˜¨ R ) (cid:3)(cid:12)(cid:12) (( a, ,n ) = ( 1 √− k n n b A ( R M φ ) ν φ ( R N φ ) i ∗ M φ h det (cid:16) cosh (cid:0) R E π − L (cid:1)(cid:17) × det (cid:16) sinh( R E ⊥ π − L ) R E ⊥ π − L (cid:17) Pf (cid:16) R E ⊥ π − L (cid:17)io ( a, ( x ) . (3.38)Where we have used the algebraic result of Proposition 3.13 in [17] , and the Berezin integral in the righthand side of (3.38) is the application of the following lemma. Lemma 3.17. | σ (cid:2) ˆ c ( f ) · · · ˆ c ( f k )exp( ˜˙ R + ˜¨ R ) (cid:3) | ( n ) = ( − n − k det (cid:16) cosh (cid:0) R E − L (cid:1)(cid:17) det (cid:16) sinh( R E ⊥ − L )( R E ⊥ − L ) / (cid:17) Pf (cid:16) R E ⊥ − L (cid:17) . (3.39) Proof. In order to compute this differential form, we make use of the Chern root algorithm (see [9]). Assumethat n = dim M and k = dim E are both even integers. As in [5], we write R E − L = (cid:18) − θ θ (cid:19) . . . − θ − k θ − k ! , R E ⊥ − L = (cid:18) − ˆ θ ˆ θ (cid:19) . . . − ˆ θ n − k ˆ θ n − k ! (3.40)10hen we obtain14 X ≤ α,β ≤ k h ( R E − L ) f α , f β i ˆ c ( f α )ˆ c ( f β ) = 12 X ≤ α<β ≤ k h ( R E − L ) f α , f β i ˆ c ( f α )ˆ c ( f β )= 12 X ≤ j ≤ k θ j ˆ c ( f j − )ˆ c ( f j ); (3.41)14 X ≤ s,t ≤ n − k h ( R E ⊥ − L ) h s , h t i ˆ c ( h s )ˆ c ( h t ) = 12 X ≤ s ReferencesReferences [1] N. Berline and M. Vergne, A computation of the equivariant index of the Dirac operators. Bull. Soc. Math. Prance113(1985) 305-345.[2] J. D. Lafferty, Y. L. Yu and W. P. Zhang, A direct geometric proof of Lefschetz fixed point formulas, Trans. AMS. 329(1992), 571-583.[3] R. Ponge and H. 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