An Index Formula for Groups of Isometric Linear Canonical Transformations
aa r X i v : . [ m a t h . OA ] A ug An Index Formula for Groups of Isometric LinearCanonical Transformations
Anton Savin, Elmar Schrohe
Abstract
We define a representation of the unitary group U ( n ) by metaplectic operators actingon L ( R n ) and consider the operator algebra generated by the operators of the repre-sentation and pseudodifferential operators of Shubin class. Under suitable conditions, weprove the Fredholm property for elements in this algebra and obtain an index formula. Contents
Given a representation of a group G on a space of functions on a manifold M , we consider theclass of operators equal to linear combinations of the form D = X g ∈ G D g Φ g , (1)where the Φ g are the operators of the representation, the D g are pseudodifferential operatorson M , and we assume that the sum is finite, i.e., only a finite number of D g is nonzero.Operators with shifts (or functional differential operators) are the most widely knownexamples of operators of the form (1). Indeed, suppose that G acts on M by diffeomor-phisms x g ( x ) , x ∈ M, g ∈ G . Then we define a representation of G by shift operators1 g u ( x ) = u ( g − ( x )). The set of all operators of the form (1) is closed under taking sumsand compositions. The theory of C ∗ -algebras was applied to define the notion of ellipticityand to prove the Fredholm property for such operators, see e.g. [1]; also index formulas wereobtained [2–6]. Let us mention that operators with shifts arise in noncommutative geome-try [7–11], mechanics [12–14], etc.Recently, operators of type (1) associated with representations by quantized canonical trans-formations on closed manifolds were considered [15,16]. A Fredholm criterion was obtained andan approach to the computation of the index based on algebraic index theory was proposed. Ina similar vein, an algebraic index theorem was established [17]. Note that operators associatedwith quantized canonical transformations arise for example when reducing hyperbolic problemsto the boundary [18, 19].So far, the efforts were limited to the case of compact manifolds. In this article, we studyoperators of type (1) on R n for a particularly interesting class of quantized canonical transfor-mations, namely metaplectic operators. More precisely, we define a unitary representation ofthe unitary group U ( n ) on L ( R n ) by metaplectic operators. For a subgroup G of U ( n ), weconsider operators of the form (1), where the Φ g are the metaplectic operators in the represen-tation and the D g are pseudodifferential operators on R n of Shubin type, see [24] or Section 3,below, for details.There are many equivalent definitions of the metaplectic group, see e.g. [20–22]. For in-stance, it is the group generated by the following three types of operators on L ( R n ):( i ) f ( x ) f ( Ax ) √ det A , where A is a real nonsingular n × n matrix;( ii ) f ( x ) f ( x ) e i ( Bx,x ) , where B is a real symmetric n × n matrix;( iii ) f ( x ) ( f )( x ), where F is the Fourier transform.Elements of the metaplectic group arise in quantum mechanics as solution operators of nonsta-tionary Schr¨odinger equations with quadratic Hamiltonians [22], also fractional Fourier trans-forms [23] are elements of the metaplectic group.Somewhat surprisingly, the theory becomes rather transparent for this situation. Thereis a natural notion of ellipticity that implies the Fredholm property. Moreover – and this isthe main result in this article – we obtain an index formula valid for all groups G ⊂ U ( n ) ofpolynomial growth in the sense of Gromov [25].This index formula represents the Fredholm index as a sum of contributions over conjugacyclasses in G , cf. [3]. Each contribution is defined in the framework of noncommutative geometryusing a certain closed twisted trace (cf. [9,26]). The proof of the index formula itself is based ontwo facts. First, the standard index one Euler operator on R n , defined in terms of the creationand annihilation operators, see [27], is actually equivariant with respect to the action of U ( n )by metaplectic transformations. Second, this operator can be used to derive an equivariantBott periodicity in the following form K ∗ ( C ( C n ) ⋊ G ) ≃ K ∗ ( C ∗ G ) , (2)where now G ⊂ U ( n ) is an arbitrary subgroup, C ∗ G is the maximal group C ∗ -algebra of G , C ( C n ) ⋊ G is the maximal C ∗ -crossed product associated with the natural action of G ⊂ U ( n )2n C n and K ∗ stands for the K -theory of C ∗ -algebras. Note that the isomorphism (2) firstappeared in [27] in terms of Z / C ∗ -algebras. Here we define this isomorphism in termsof symbols of elliptic operators and give an independent proof of the periodicity isomorphism.The isomorphism (2) enables us to reduce the proof of the index formula to the special case ofthe Euler operator twisted by projections over C ∗ G , where a direct computation of both sidesof the index formula is possible. Acknowledgment . We thank Gennadi Kasparov for pointing out the Bott periodicitytheorem in [27] to us. The work of the first author was partly supported by RUDN Universityprogram 5-100; that of the second by DFG through project SCHR 319/8-1.
Let us recall the necessary facts about the symplectic and metaplectic groups from [20], seealso [28, 29].
The symplectic and the metaplectic groups and their Lie algebras.
The metaplecticgroup Mp( n ) ⊂ B L ( R n ) is the group generated by unitary operators of the formexp( − i b H ) ∈ Mp( n ) , where b H is the Weyl quantization of a homogeneous real quadratic Hamiltonian H ( x, p ), ( x, p ) ∈ T ∗ R n . In its turn, the complex metaplectic group Mp c ( n ) is similarly generated by unitariesassociated with Hamiltonians H ( x, p ) + c , where H ( x, p ) is as above, while c is a real constant.The symplectic group Sp( n ) ⊂ GL (2 n, R ) is the group of linear canonical transformations of T ∗ R n ≃ R n . We consider the faithful representation of this group on L ( R n ) by shiftoperators u ( x, p ) u ( A − ( x, p )) , where u ∈ L ( R n ) and A ∈ Sp( n ) and identify Sp( n ) withits image in B L ( R n ) under this representation. One can show that this group is generated bythe unitary shift operators exp (cid:18) − (cid:18) H p ∂∂x − H x ∂∂p (cid:19)(cid:19) ∈ Sp( n )associated with the canonical transformation equal to the evolution operator for time t = 1 ofthe Hamiltonian system ˙ x = H p , ˙ p = − H x , where H ( x, p ) is a homogeneous real quadratic Hamiltonian as above.It is well known that Mp( n ) is a nontrivial double covering of Sp( n ). The projection takes ametaplectic operator to the corresponding canonical transformation. Hence, their Lie algebras, i.e., linear transformations that preserve the symplectic form dx ∧ dp . n ) and sp( n ) are isomorphic. Let us describe an explicit isomorphism. Indeed,it follows from the definitions above thatmp( n ) = {− i b H } , sp( n ) = {− ( H p ∂/∂x − H x ∂/∂p ) } with H as above and Lie brackets equal to the operator commutators. These Lie algebrasare isomorphic and they are also isomorphic to the Lie algebra of homogeneous real quadraticHamiltonians H ( x, p ) ∈ R n + n mp( n ) ≃ sp( n ) ≃ R n + n − i b H ↔ − (cid:16) H p ∂∂x − H x ∂∂p (cid:17) ↔ H ( x, p ) , (3)where we consider the Poisson bracket on the space of Hamiltonians { H ′ , H ′′ } = H ′ x H ′′ p − H ′ p H ′′ x . The fact that the isomorphisms in (3) preserve the Lie algebra structures is proved by a directcomputation.
Isometric linear canonical transformations and their quantization.
In what follows,we consider the maximal compact subgroup Sp( n ) ∩ O (2 n ) of isometric linear canonical trans-formations in Sp( n ). It is well known that this intersection is isomorphic to the unitary group U ( n ) if we introduce the complex structure on T ∗ R n ≃ C n via ( x, p ) z = p + ix , see [30].If we realize sp( n ) in terms of Hamiltonians H ( x, p ) as in (3), then one can show that theLie algebra of the subgroup Sp( n ) ∩ O (2 n ) ⊂ Sp( n ) consists of the Hamiltonians H ( x, p ) = 12 ( x, p ) (cid:18) A − BB A (cid:19) (cid:18) xp (cid:19) , (4)where A and B are real n × n matrices with A symmetric and B skew-symmetric. Moreover,we have the isomorphism of Lie algebrasLie algebra of Sp( n ) ∩ O (2 n ) ⊂ Sp( n ) −→ u ( n ) H ( x, p ) = 12 ( x, p ) (cid:18) A − BB A (cid:19) (cid:18) xp (cid:19) B + iA. (5)Here u ( n ) stands for the Lie algebra of U ( n ); its elements are the skew-Hermitian matrices.Let us illustrate the isomorphism (5) in examples. Example 1. If n = 1, then B = 0 and A = ϕ , ϕ ∈ R , and (4) gives Hamiltonians H ( x, p ) = 12 ( x + p ) ϕ. The solution of the corresponding Hamiltonian system of equations˙ x = ϕp, ˙ p = − ϕx ; x (0) = x , p (0) = p p ( t ) + ix ( t ) = e iϕt ( p + ix ) . For t = 1 we obtain the element e iϕ ∈ U (1), obviously equal to the exponential mapping of iϕ ∈ u (1). On the other hand, (5) gives the same element B + iA = iϕ ∈ u (1). Example 2. If n = 2, then A = (cid:18) k mm l (cid:19) , B = (cid:18) − tt (cid:19) and there are four linearlyindependent Hamiltonians: x + p , x + p , x x + p p , x p − x p . Let us consider for instance the Hamiltonian H ( x, p ) = ( x p − x p ) ϕ = 12 ( x, p ) (cid:18) A − BB A (cid:19) (cid:18) xp (cid:19) ϕ, ϕ ∈ R , where A = 0, B = (cid:18) −
11 0 (cid:19) . On the one hand, the Hamiltonian system of equations˙ x = − ϕx , ˙ x = ϕx , ˙ p = − ϕp , ˙ p = ϕp ; x (0) = x , p (0) = p , has the solution equal to x ( t ) = e Bϕt x , p ( t ) = e Bϕt p , where e Bϕt = (cid:18) cos( ϕt ) − sin( ϕt )sin( ϕt ) cos( ϕt ) (cid:19) For t = 1 we therefore obtain p + ix = e Bϕ ( p + ix ) . Then the element e Bϕ ∈ U (2) is obviously equal to the exponential mapping of Bϕ ∈ u (2). Onthe other hand, (5) gives the same element ( B + iA ) ϕ = Bϕ ∈ o (2) ⊂ u (2).The following lemma will be useful below. Lemma 1. U ( n ) is generated by the orthogonal subgroup O ( n ) and the subgroup U (1) = { diag( z, , . . . , | | z | = 1 } .Proof. It suffices to prove that the Lie algebra of U ( n ) is generated as a vector space by theLie algebra of O ( n ) and the action of the adjoint representation Ad O ( n ) on the Lie algebra of U (1).Indeed, u ( n ) is the set of all matrices B + iA , where A is symmetric and B is skew-symmetric.Since o ( n ) consists of all skew-symmetric matrices, it suffices to show that the set of all iA isgenerated by Ad O ( n ) of the Lie algebra of U (1). This is straightforward: We first generatediagonal matrices using permutation matrices and then generate nondiagonal matrices usingrotations by π/ he homomorphism R : U ( n ) → Mp c ( n ) . It is known that π : Mp( n ) → Sp( n ) is anontrivial double covering. Thus, one can not represent unambigously elements of Sp( n ) bymetaplectic operators. However, it turns out that one can define a representation of the unitarysubgroup U ( n ) ⊂ Sp( n ) by operators in the complex metaplectic group. Proposition 1.
Consider the mapping R : U ( n ) −→ Mp c ( n ) s π − ( s ) √ det s, (6) defined in a neighborhood of the unit element I in U ( n ) , where π − is the section for π :Mp( n ) → Sp( n ) such that π − ( I ) = I and the branch of the square root is chosen such that √ . Then the mapping (6) extends to the entire group U ( n ) as a monomorphism of groups.In terms of Hamiltonians, the homomorphism (6) is defined explicitly as follows. Given H ( x, p ) = 12 ( x, p ) (cid:18) A − BB A (cid:19) (cid:18) xp (cid:19) , where A is symmetric and B is skew-symmetric, we have R (exp( B + iA )) = exp( − i b H ) p det(exp( B + iA )) = exp( − i b H ) exp( i Tr A/ , (7) where b H is the Weyl quantization of H ( x, p ) .Proof. Clearly, this mapping is well defined in a neighborhood of the identity and admits aunique continuation along any continuous path s ( t ) in U ( n ), s (0) = I (since this is true forboth π − ( s ) and √ det s ). Moreover, the result is the same for two homotopic paths withendpoints fixed. Therefore, to prove that the mapping (6) is well defined globally, it sufficesto check that the continuation along the generators of π ( U ( n )) gives the same result as thecontinuation along the constant path.It is well known that π ( U ( n )) ≃ π ( U (1)) = Z and a generator is given by the path s ( t )equal to rotations in the ( x , p )-plane by angles t ∈ [0 , π ]. Then we have π − ( s ( t )) = e − it b H , b H = 12 (cid:18) − ∂ ∂x + x (cid:19) . Since the spectrum of the harmonic oscillator b H is { / k | k ∈ N } , we see that π − ( s (0)) = I, π − ( s (2 π )) = − I. On the other hand, s ( t ) is the diagonal matrix with entries e it , , ...,
1. Hence, we have(det( s ( t ))) / = e it/ , and we see that R s (2 π ) = π − ( s (2 π ))(det( s (2 π ))) / = − I · ( −
1) = I = R s (0) . This implies the desired continuity and also smoothness.Finally, (7) follows from (6) and the fact that the section π − is equal to π − (exp( B + iA )) = exp( − i b H ) , where b H is the Weyl quantization of (5). 6 Elliptic Operators
Shubin type pseudodifferential operators.
We call a smooth function d = d ( x, p ) on T ∗ R n a pseudodifferential symbol (of Shubin type) of order m ∈ R , provided its derivativessatisfy the estimates | D αp D βx d ( x, p ) | ≤ c α,β (1 + | x | + | p | ) m −| α |−| β | for all multi-indices α , β , with suitable constants c α,β . We moreover assume d to be classical,i.e. d admits an asymptotic expansion d ∼ P ∞ j =0 d m − j , where each d m − j is a symbol of order m − j , which is (positively) homogeneous in ( x, p ) for | x, p | ≥ R n ) the norm closure of the algebra of pseudodifferential operators withShubin type symbols of order zero acting on L ( R n ), see also [24]. This closure is a C ∗ -subalgebra in B ( L ( R n )). The symbol mapping in this situation is the homomorphism σ : Ψ( R n ) −→ C ( S n − ) D σ ( D )( x, p )of C ∗ -algebras, induced by the map which associates to a zero order pseudodifferential operator D with symbol d ∼ P ∞ j =0 d − j the restriction of d to S n − . Denoting by K ( L ( R n )) the compactoperators in B ( L ( R n )) we have a short exact sequence0 −→ K ( L ( R n )) −→ Ψ( R n ) σ −→ C ( S n − ) −→ . (8)The unitary group U ( n ) acts on Ψ( R n ) by conjugation with metaplectic transformations: D ∈ Ψ( R n ) , g ∈ U ( n ) R g DR − g ∈ Ψ( R n ) . Moreover, we have an analogue of Egorov’s theorem: σ ( R g DR − g ) = g − ∗ σ ( D ) . Given a discrete group G ⊂ U ( n ), we consider the maximal crossed product Ψ( R n ) ⋊ G (for the theory of crossed products, see e.g. [31, 32]). In the sequel, elements of the crossedproduct are treated as collections { D g } g ∈ G of pseudodifferential operators D g . We have anatural representation Ψ( R n ) ⋊ G −→ B ( L ( R n )) { D g } 7−→ P g ∈ G D g R g . (9)This representation is well defined by the universal property of the maximal C ∗ -crossed productsand the fact that all operators R g are unitary. Operators acting between ranges of projections.
We next introduce a class of operatorsthat is an analogue of operators acting in sections of vector bundles, cf. [3, Sec. 2.2]. Namely,we consider triples (
D, P , P ), where P , P are N × N matrix projections over the maximalgroup C ∗ -algebra denoted by C ∗ ( G ) and D is an N × N matrix operator over Ψ( R n ) ⋊ G . Letus also suppose that D and P , P are compatible in the sense of the following equality: D = P DP .
7f this equality is not satisfied, then we replace D by P DP . To any such triple, we assign theoperator D : Im P −→ Im P , Im P , Im P ⊂ L ( R n , C N ) , (10)called G -operator , where D, P , P are represented as operators on L ( R n , C N ) using formula(9), while Im P , Im P are the ranges of the projections. Ellipticity and Fredholm property.
Let us recall the notion of ellipticity in this situation(see [3, Sec. 2.2]). The symbol homomorphism σ : Ψ( R n ) → C ( S n − ) induces the symbolhomomorphism of the maximal crossed products: σ : Ψ( R n ) ⋊ G −→ C ( S n − ) ⋊ G { D g } 7−→ { σ ( D g ) } . Definition 1.
A triple D = ( D, P , P ) is elliptic if there exists an element r ∈ Mat N ( C ( S n − ) ⋊ G ) such that the following equalities hold P rσ ( D ) = P , σ ( D ) rP = P . (11) Lemma 2.
Elliptic elements have the Fredholm property.Proof.
The crossed product is an exact functor by [32, Proposition 3.19]. Hence the exactnessof the short exact sequence (8) implies the exactness of the corresponding sequence of crossedproducts by G . In particular, the symbol map σ : Ψ( R n ) ⋊ G → C ( S n − ) ⋊ G is surjective.Given r as in Definition 1, we therefore find R ∈ Mat N (Ψ( R n ) ⋊ G ) with symbol equal to r . Then (11) implies that P R : Im P −→ Im P is a two-sided inverse for (10) modulo compact operators. Remark 1. If G is amenable, then the ellipticity condition can be written more explicitly interms of the so called trajectory symbol by the results of Antonevich and Lebedev [1]. Theirresults apply since the action of G on S n − is topologically free. Moreover, it turns out thatellipticity is a necessary and sufficient condition for the Fredhom property. Difference construction.
Given a subgroup G ⊂ U ( n ) and an elliptic G -operator D =( D, P , P ), we define the difference construction for its symbol[ σ ( D )] ∈ K ( C ( T ∗ R n ) ⋊ G ) = K ( C ( C n ) ⋊ G ) (12)following [3, Sec. 4.2].Let us recall the construction of the element (12). We define the matrix projections p = 12 (cid:18) (1 − sin ψ ) P σ − ( D ) cos ψσ ( D ) cos ψ (1 + sin ψ ) P (cid:19) , p = (cid:18) P (cid:19) (13)8ver the C ∗ -crossed product C ( C n ) ⋊ G with adjoint unit, where σ − ( D ) = r (see Definition 1), ψ = ψ ( | z | ) ∈ C ∞ ( C n ) is a real G -invariant function, which for | z | small is identically − π/
2, for | z | large is + π/
2, and is nondecreasing. We set[ σ ( D )] = [ p ] − [ p ] . Remark 2.
One can see that (13) defines a projection also in a more general situation (whichis an analogue of the Atiyah–Singer difference construction, see [33] or [3, Sec. 4.2] in thenoncommutative setting). Namely, consider triples( a, P , P ) , where a, P , P ∈ Mat N ( C ( C n ) ⋊ G ) , where P and P are projections, a = P aP , and the triple is elliptic in the sense of Definition 1for | x | + | p | large. More precisely, we require that for the restriction of the triple ( a, P , P )to a subset of the form { ( x, p ) ∈ C n | | x | + | p | ≥ R } for some R > r, P , P ) such that ra = P and ar = P (cf. (11)). Then, if we replace the triple ( σ ( D ) , P , P )in (13) by the triple ( a, P , P ), then the difference of projections (13) gives a well-defined classin K -theory. Of course, such triples are not in general symbols of G -operators. Homotopy classification.
Two elliptic G -operators ( D , P , Q ) and ( D , P , Q ) as in (10)are called homotopic if there exists a continuous homotopy of elliptic operators ( D t , P t , Q t ), t ∈ [0 , t = 0 and t = 1. Two elliptic operators arecalled stably homotopic if their direct sums with some trivial operators are homotopic. Heretrivial operators are operators of the form (1 , P, P ), where P is a projection. It turns out thatstable homotopy is an equivalence relation on the set of elliptic operators. The set of equivalenceclasses of elliptic G -operators is denoted by Ell( R n , G ). This set is an Abelian group, wherethe sum corresponds to the direct sum of operators and the zero of the group is equal to theequivalence class of trivial operators.The difference construction (12) induces the mappingEll( R n , G ) −→ K ( C ( T ∗ R n ) ⋊ G ) , D = ( D, P , P ) [ σ ( D )] . (14) Proposition 2.
The mapping (14) is an isomorphism of Abelian groups.
The proof is standard, see [3, Sec. 4.3] or [34].
Smooth symbols.
In this paper, we obtain a cohomological index formula. To this end, weuse methods of noncommutative geometry and have to assume that our symbol is smooth in acertain sense. More precisely, we make the following assumption. From now on we suppose that G ⊂ U ( n ) is a discrete group of polynomial growth [25]. Under this assumption, one can definesmooth crossed products by actions of G , which are spectrally invariant in the corresponding C ∗ -crossed products (see [35]).Recall that the smooth crossed product A ⋊ G of a Fr´echet algebra A with the seminorms k · k m , m ∈ N , and a group G of polynomial growth acting on A by automorphisms a g ( a )9or all a ∈ A and g ∈ G is equal to the vector space of collections { a g } g ∈ G of elements in A thatdecay rapidly at infinity in the sense that the following estimates are valid: k a g k m ≤ C N (1 + | g | ) − N for all N, m ∈ N , and g ∈ G, where the constant C N does not depend on g . Here | g | is the length of g in the word metricon G . Finally, the action of G on A is required to be tempered: for any m there exists k and apolynomial P ( z ) with positive coefficients such that k g ( a ) k m ≤ P ( | g | ) k a k k for all a and g . Theproduct in A ⋊ G is defined by the formula: { a g } · { b g } = ( X g g = g a g g ( b g ) ) . It follows from the results in [35] that the group K ( C ( C n ) ⋊ G ) is isomorphic to the groupof stable homotopy classes of elliptic symbols ( σ ( D ) , P , P ) that are smooth in the followingsense: their components lie in the smooth crossed products σ ( D ) ∈ Mat N ( C ∞ ( S n − ) ⋊ G ) , P , P ∈ Mat N ( C ∞ ( G )) . (15)Here the smooth group algebra C ∞ ( G ) is interpreted as the smooth crossed product C ⋊ G .Our aim is to define the topological index for smooth elliptic symbols. Algebraic preliminaries.
Suppose that a group G acts by automorphisms on a differentialgraded algebra A with the differential denoted by d . Definition 2 (cf. [9, 26]) . Given s ∈ G , a closed twisted trace is a linear functional τ s : A −→ C such that • τ s ( ab ) = τ s ( bg ( a ))( − deg a deg b for all a, b ∈ A . • τ s ( da ) = 0 for all a ∈ A .Two twisted traces τ s and τ gsg − are compatible if τ gsg − ( a ) = τ s ( g − a ) for all a ∈ A . Example 3.
Let the elements of A and G be represented by operators a and U g on someHilbert space. Then we can set τ s ( a ) = Tr( U s a ) for all a ∈ A, provided that the operator trace Tr exists. Then this collection of functionals is a compatiblecollection of twisted traces. 10iven a compatible collection of twisted traces and a conjugacy class h s i ⊂ G , we definethe functional τ h s i : A ⋊ G −→ C on the algebraic crossed product of A and G by the formula τ h s i { a g } = X g ∈h s i τ g ( a g ) . (16)We claim that this functional is a trace, i.e., we have τ h s i ( ab ) = τ h s i ( ba )( − deg a deg b for all a, b ∈ A ⋊ G .Indeed, if both a and b have a single nonzero component denoted by a g and b h respectively,then ab and ba also have a single nonzero component equal to a g g ( b h ) and b h h ( a g ), and we have τ h gh i ( ab ) = τ gh ( a g g ( b h )) = τ g ( hg ) g − ( a g g ( b h )) == τ hg ( g − ( a g ) b h ) = τ hg ( b h h ( a g ))( − deg a deg b = τ h gh i ( ba )( − deg a deg b . (17) Twisted traces on differential forms.
Let G = U ( n ) act on C n ≃ T ∗ R n and consider theinduced action on differential forms C ∞ c ( C n , Λ( C n )) considered as a differential graded algebra.We now construct a compatible collection of closed twisted traces for all elements of the unitarygroup. To this end, given s ∈ U ( n ), we define the orthogonal decomposition C n = L = L s ⊕ L ⊥ s , where L s is the fixed point subspace of s (equivalently, it is the eigensubspace associated witheigenvalue 1), while L ⊥ s is its orthogonal complement. Then we define the functional τ s : C ∞ c ( C n , Λ( C n )) −→ C ω τ s ( ω ) = Z L s ω | L s . Here, we use the complex orientation on L s (if z j = p j − ix j are the complex coordinates, then Q j dp j ∧ dx j is assumed to be positive). Clearly, this definition does not depend on the choiceof coordinates z . Moreover, these functionals define a compatible collection of twisted tracesin the sense of Definition 2.Thus, for each s ∈ G we get (see (16)) a closed graded trace τ h s i : C ∞ c ( C n , Λ( C n )) ⋊ U ( n ) −→ C on the algebraic crossed product. 11 he definition of the topological index. Let us define the topological index as the func-tional ind t : K ( C ( C n ) ⋊ G ) −→ C . To this end, we represent classes in the latter K -group as formal differences [ P ] − [ P ] ofprojections in the smooth crossed product Mat N ( C ∞ ( C n ) ⋊ G ) such that P = P at infinity in C n . Then we setind t ([ P ] − [ P ]) = X h s i⊂ G − s | L ⊥ s ) tr τ h s i (cid:18) P exp (cid:18) − dP dP πi (cid:19) − P exp (cid:18) − dP dP πi (cid:19)(cid:19) . (18)(cf. [36]). Here the summation is over the set of all conjugacy classes h s i ⊂ G and tr standsfor the matrix trace. Note that each summand in (18) is homotopy invariant. We refer tothis invariant as the topological index localized at the conjugacy class h s i ⊂ G and denote it byind t ([ P ] − [ P ])( s ). The index theorem.Theorem 1.
Given an elliptic G -operator D = ( D, P , P ) associated with a discrete group G ⊂ U ( n ) of polynomial growth, the following index formula holds ind D = ind t [ σ ( D )] . (19)The idea of our proof is to use the homotopy invariance of both sides of the index formulaand to use K -theory to reduce the operator to a very special operator, for which one cancompute both sides of the index formula independently and check that they are equal. The aim of this section is to define an isomorphism of Abelian groups β : K ( C ∗ ( G )) −→ K ( C ( T ∗ R n ) ⋊ G ) . This isomorphism will be defined in terms of the Euler operator on R n . If G is trivial, then thisisomorphism coincides with the classical Bott periodicity isomorphism. For nontrivial groups,this isomorphism is a variant of equivariant Bott periodicity. Note also that if G ⊂ O ( n ), thenthis isomorphism was constructed in [3]. Euler operator.
Recall that the classical Euler operator on a Riemannian manifold M isdefined by the formula d + d ∗ : C ∞ ( M, Λ ev ( M )) −→ C ∞ ( M, Λ odd ( M )) . (20)It takes differential forms of even degree to differential forms of odd degree. Here d is theexterior derivative and d ∗ is its adjoint with respect to the Riemannian volume form and the12nner product on forms defined by the Hodge star operator. Let us modify this operator andobtain the following elliptic operator in R n (e.g., see [27]) E = d + d ∗ + xdx ∧ +( xdx ∧ ) ∗ : S ( R n , Λ ev ( C n )) −→ S ( R n , Λ odd ( C n )) . (21)Here xdx = dr / P j x j dx j , where r = | x | . Its symbol is invertible for | x | + | p | = 0. Weconsider this operator in the Schwartz spaces of complex valued differential forms.The following lemma is well known.
Lemma 3.
The kernel ker E can be identified with C e −| x | / , while coker E = 0 . Example 4. If n = 1, then E = ∂∂x + x : S ( R ) −→ S ( R ) (22)is just the annihilation operator modulo √ dx in the differential forms in thetarget space).It follows from the definition that E is O ( n )-equivariant with respect to the natural actionof O ( n ) on differential forms. Let us show that E is equivariant with respect to U ( n ). A unitary representation ρ : U ( n ) → B ( L ( R n , Λ( C n ))) . The identification Λ( R n ) ⊗ C ≃ Λ( C n ) yields a unitary representation U ( n ) −→ Aut(Λ( R n ) ⊗ C ) , namely the natural representation on the algebraic forms: g ∈ U ( n ) , ω ∈ Λ( C n ) g ∗− ω, which is well defined since we consider complex valued forms.Complementing the unitary representation R : U ( n ) −→ B ( L ( R n )) introduced in (6) wenext define the representation ρ : U ( n ) −→ B ( L ( R n , Λ( C n ))) (23)as the diagonal representation: ρ g (cid:16)X I ω I ( x ) dx I (cid:17) = X I R g ( ω I ) g ∗− ( dx I ) , g ∈ U ( n ) , where we represent differential forms as sums P I ω I ( x ) dx I over multi-indices with L coeffi-cients ω I ( x ).As a tensor product of unitary representations, ρ is a unitary representation. Indeed, σ ( E )( x, p ) = ( ip + xdx ) ∧ +(( ip + xdx ) ∧ ) ∗ . Hence, σ ( E ) ( x, p ) = ( | x | + | p | ) Id. ( n ) -equivariance of the Euler operator. Note that for g ∈ O ( n ) ⊂ U ( n ) we have R g = g ∗− , hence in this case ρ g = g ∗− is just the natural action of g on differential forms. Lemma 4. E is U ( n ) -equivariant, i.e., we have ρ g E ρ − g = E for all g ∈ U ( n ) . (24) Proof.
By Lemma 1, U ( n ) is generated by O ( n ) and U (1). Thus, it suffices to prove (24) for g in one of these two subgroups. For g ∈ O ( n ), this equality follows from the definition (since d, d ∗ , dr commute with the action of the orthogonal group by shifts). Thus, it remains to provethe statement for g ∈ U (1). For simplicity, we consider the one-dimensional case (the generalcase is treated similarly).Let n = 1. Then we know (see (6)) R g = e it (1 / − b H ) , where g = e it ∈ U (1) . It is easy to see that g ∗− | Λ ( R ) = 1 , g ∗− | Λ ( R ) = e − it . Hence, the desired equivariance amounts to proving that the operator ∂∂x + x has the property e it ( − / − b H ) (cid:18) ∂∂x + x (cid:19) e − it (1 / − b H ) = ∂∂x + x. Let us prove this identity by Dirac’s method. We define creation and annihilation operators A ∗ = 1 √ (cid:18) − ∂∂x + x (cid:19) , A = 1 √ (cid:18) ∂∂x + x (cid:19) One also has b H = AA ∗ − / A ∗ A + 1 /
2. This enables us to show that e − it b H A = e it/ e − itAA ∗ A = e it/ Ae − itA ∗ A = e it/ Ae − it ( b H − / = e it Ae − it b H . Hence, we get e it ( − / − b H ) Ae − it (1 / − b H ) = e − it e − it b H Ae it b H = e − it e it A = A. This completes the proof of equivariance for n = 1. Twisting by a projection.
Let P = ( P g ) ∈ Mat N ( C ∗ ( G )) be a projection over the group C ∗ -algebra of G ⊂ U ( n ). Then we define a projection1 ⊗ P : L ( R n , Λ( C n ) ⊗ C N ) −→ L ( R n , Λ( C n ) ⊗ C N )by the formula 1 ⊗ P = X g ∈ G (1 ⊗ P g )( ρ g ⊗ N ) . P ⊗ P is defined by a covariant representation, hence, it gives a homomorphismof C ∗ -algebras. This implies that 1 ⊗ P is a projection.Since 1 ⊗ P is a projection, its range is a closed subspace denoted by Im(1 ⊗ P ). Thus, wecan define the twisted operator as E ⊗ N : Im(1 ⊗ P ) −→ Im(1 ⊗ P ) , (25)where we made a reduction to the zero-order operator E = ( E E ∗ + 1) − / E . Since E is equivariant, it follows that ( E ⊗ N )(1 ⊗ P ) = (1 ⊗ P )( E ⊗ N ). Thus, E ⊗ N preserves Im(1 ⊗ P ). This twisted operator is Fredholm with an almost inverse operator equalto E − ⊗ N . Equivariant Bott periodicity.Theorem 2.
The mapping β : K ( C ∗ ( G )) −→ K ( C ( T ∗ R n ) ⋊ G ) P [( σ ( E ⊗ N ) , ⊗ P, ⊗ P )] (26) is an isomorphism of Abelian groups.Proof. The idea (going back to Atiyah [37]) is to include the mapping β in the diagram: K ( C ∗ ( G )) β / / K ( C ( T ∗ R n ) ⋊ G ) β ′ / / ind d d K ( C ( R n ) ⋊ G ) ind ′ e e (27)of Abelian groups and homomorphisms with the following propertiesind ◦ β = I, (28)ind ′ ◦ β ′ = I, (29)ind ′ ◦ β ′ = β ◦ ind . (30)Clearly, if we construct the diagram with these properties, then β and ind are mutually inversehomomorphisms and the theorem is proved.It remains to construct the diagram with these properties. The mappings β, β ′ will bedefined by taking exterior products with the Euler operator, while ind , ind ′ will be analyticindex mappings. Hence, properties (28) and (29) follow from the multiplicative property ofthe index and the fact that the index of the Euler operator is equal to one. It turns out thatthe remaining property (30) also follows from the multiplicative property of the index and anexplicit homotopy of symbols (the so-called Atiyah rotation trick [37]). Let us now give thedetailed proof. 15 . Definition of the mapping β ′ . Consider the doubled space R n = R n × R n , ( x, p, y, q ) ∈ R n , with the diagonal action of G on it. Let us define the mapping β ′ : K ( C ( T ∗ R n ) ⋊ G ) −→ K ( C ( R n ) ⋊ G ) , [ σ ] [ σ σ ( E ) ]in terms of the exterior product of symbols, see [3, Sec. 6.2]. We recall the definition ofthe exterior product. To this end, let a = ( a, P , P ) and b = ( b, Q , Q ) be triples over C ( R n ) ⋊ G and C ( R n ) respectively, see Remark 2. Suppose in addition that the projections P j , Q j are self-adjoint, and b is equivariant. This means that we have two homomorphisms ρ j : G → End(Im Q j ) from G to the group of unitary automorphisms of the vector bundlesequal to the ranges of Q and Q , and b intertwines these homomorphisms: bρ ( g ) = ρ ( g ) b forall g . Definition 3.
The exterior product of the triples a and b is the triple a b = (cid:18)(cid:18) a ⊗ − ⊗ b ∗ ⊗ b a ∗ ⊗ (cid:19) , (cid:18) P ⊗ Q P ⊗ Q (cid:19) , (cid:18) P ⊗ Q P ⊗ Q (cid:19)(cid:19) over C ( R n ) ⋊ G . Here the elements of these matrices are in matrix algebras over the crossedproduct C ( R n ) ⋊ G and they are defined as( a ⊗ g = a g ⊗ ρ ( g ) , ( a ∗ ⊗ g = ( a ∗ ) g ⊗ ρ ( g ) , ( P j ⊗ Q k ) g = P jg ⊗ ρ k ( g ) Q k . We shall frequently abridge this notation and simply write a b = (cid:18) a − b ∗ b a ∗ (cid:19) , omitting the projections and tensor products by identity operators. Lemma 5.
Suppose that the triples a = ( a, P , P ) and b = ( b, Q , Q ) are elliptic. Then theirexterior product a b is elliptic.Proof.
1. Let us state the ellipticity condition for triples in C ∗ -algebraic terms. Consider atriple ( a, P , P ) with components in a C ∗ -algebra A , P j = P ∗ j = P j , and a = P aP . Sucha triple is elliptic if there exists r ∈ A such that ar = P and ra = P . We claim that theellipticity is equivalent to the following two conditions aa ∗ is invertible in the C ∗ -algebra P AP ,a ∗ a is invertible in the C ∗ -algebra P AP . (31)The proof is standard. Namely. ellipticity of ( a, P , P ) is equivalent to that of ( a ∗ , P , P ) and,hence, to that of (cid:18)(cid:18) a ∗ a (cid:19) , (cid:18) P P (cid:19) , (cid:18) P P (cid:19)(cid:19) . (cid:18) a ∗ a (cid:19) in the algebra P AP ⊕ P AP . Finally the invertibility of this self-adjoint matrix isequivalent to the invertibility of its square, which gives the desired result.2. Thus, to prove the lemma, it suffices to prove the invertibility of ( a b ) ∗ ( a b ) and( a b )( a b ) ∗ in the corresponding C ∗ -algebras. Let us prove that the first element is invertible.The verification for the second element is similar. Using the equivariance of b , we obtain thatthe off-diagonal elements in a b and ( a b ) ∗ commute with the elements on the diagonal. Thisimplies that the composition( a b ) ∗ ( a b ) = diag( a ∗ a ⊗ ⊗ b ∗ b, aa ∗ ⊗ ⊗ bb ∗ ) (32)is a diagonal matrix. Let us prove that the upper left corner of this matrix is invertible (theinvertibility of the lower right corner is proved similarly) in the algebra a ∗ a ⊗ ⊗ b ∗ b ∈ ( P ⊗ Q ) Mat N ( C ( R n ) ⋊ G )( P ⊗ Q ) . (33)Let us denote this element and the algebra in (33) as u and A ( R n ) respectively. Moreover,given a U ( n )-invariant closed subset U ⊂ R n , we denote the corresponding algebra by A ( U ).Since ( a, P , P ) is elliptic on the set {| x | + | p | ≥ R } and 1 ⊗ b ∗ b is nonnegative, it followsthat the element (33) is invertible in the algebra A ( R n ∩ {| x | + | p | ≥ R } ) as a sum ofnonnegative elements, one of which is invertible. Denote by r ∈ A ( R n ∩ {| x | + | p | ≥ R } )the inverse element and by e r ∈ A ( R n ) a lift under the projection mapping A ( R n ) −→ A ( R n ∩ {| x | + | p | ≥ R } ) . Such a lift exists by the exactness of the maximal crossed product functor [32, Proposition3.19]. Then the differences u e r − , e r u − {| x | + | p | ≥ R } . Similarly, using the ellipticity of ( b, Q , Q ), we obtainan element e r ∈ A ( R n ) such that the differences u e r − , e r u − {| y | + | q | ≥ R } . Let us now consider the element r = e r χ + e r χ ∈ A ( R n ∩ {| x | + | p | + | y | + | q | ≥ R } ) , where χ , χ ∈ C ∞ ( R n ∩ {| x | + | p | + | y | + | q | ≥ R } ) is a U ( n )-invariant partition of unityassociated with the covering of the set {| x | + | p | + | y | + | q | ≥ R } by the domains R n ∩ {| x | + | p | + | y | + | q | ≥ R } ∩ {| x | + | p | ≥ R } , R n ∩ {| x | + | p | + | y | + | q | ≥ R } ∩ {| y | + | q | ≥ R } . We claim that r is the inverse of u over the domain {| x | + | p | + | y | + | q | ≥ R } . Indeed,we have ur = u e r χ + u e r χ = ( u e r − χ + ( u e r − χ + χ + χ = 0 + 0 + 1 = 1 .
17 similar computation shows that ru = 1.Thus, we proved that ( a b ) ∗ ( a b ) and ( a b )( a b ) ∗ are invertible in the corresponding C ∗ -algebras. Hence, by part 1 of the proof, the exterior product a b is elliptic. Remark 3.
One similarly defines the exterior product if the first factor is equivariant. Moregenerally, whenever we write an expression of the form a b , we implicitly assume that one ofthe factors is equivariant, and depending on which of the factors is equivariant, we apply thecorresponding definition. (If both symbols are equivariant, we can use any of the definitions;both give the same result.)
2. Definition of the mapping ind . Given an elliptic G -operator ( D, P , P ) on R n , where D = X g D g R g , P j = X g P j,g R g , j = 1 , , we now construct a G -operator acting in Hilbert modules over the group C ∗ -algebra C ∗ ( G )following the construction in [3, Sec. 5.2]. To this end, let L g be the operator of left translationby g in the free C ∗ ( G )-module C ∗ ( G ) N . We define operators e D = X g D g R g ⊗ L g , e P j = X g P j,g R g ⊗ L g , j = 1 , , (36)acting in the space L ( R n , C ∗ ( G ) N ). The operators are well defined by the universal propertyof the maximal crossed product. Then we consider the operator e P e D e P : Im e P −→ Im e P (37)over the C ∗ -algebra C ∗ ( G ) acting between the ranges of the projections e P j : L ( R n , C ∗ ( G ) N ) −→ L ( R n , C ∗ ( G ) N )considered as right Hilbert C ∗ ( G )-modules. We claim that the operator (37) is C ∗ ( G )-Fredholmin the sense of Mishchenko and Fomenko [38]. Indeed, its almost-inverse operator is equal to e P g D − e P , where D − is a G -operator with the symbol r , see Definition 1. Thus, the operator(37) has an index ind C ∗ ( G ) ( e P e D e P : Im e P −→ Im e P ) ∈ K ( C ∗ ( G )) . Then we defineind : K ( C ( T ∗ R n ) ⋊ G ) −→ K ( C ∗ ( G ))[( σ ( D ) , P , P )] ind C ∗ ( G ) ( e P e D e P : Im e P −→ Im e P ) . (38)
3. Definition of the mapping ind ′ . We define the index mappingind ′ : K ( C ( R n ) ⋊ G ) −→ K ( C ( T ∗ R n ) ⋊ G )18s follows. Let ( x, p, y, q ) be variables in R n . Then each class in K ( C ( R n ) ⋊ G ) contains arepresentative of the form( a, P , P ) , a ∈ Mat N ( C ( R n ) ⋊ G ) , P , ∈ Mat N ( C ( T ∗ R n ) ⋊ G ) , (39)which is elliptic for large ( x, p, y, q ) and such that(1) a ( x, p, y, q ) = P ( x, p ) = P ( x, p ) = diag(1 , .., , , ...,
0) if | x | + | p | ≥ R for some R > a ( x, p, y, q ) is homogeneous of degree zero in ( y, q ) for ( y, q ) large uniformly in ( x, p ).Such a representative can be obtained if we use stable homotopies of the symbol and theprojections. Note that here we use the realization of the group K ( C ( R n ) ⋊ G ) in terms oftriples (39), where the element a defines the equivalence of projections at infinity (see Remark 2).We treat the triple in (39) as a symbol of a G -operator and associate to it as in (36) thecorresponding operator e a (cid:18) x, p, y, − i ∂∂y (cid:19) : e P L ( R ny , ( C ( T ∗ R n ) ⋊ G ) + ⊗ C N ) −→ e P L ( R ny , ( C ( T ∗ R n ) ⋊ G ) + ⊗ C N )(40)acting in Hilbert ( C ( T ∗ R n ) ⋊ G ) + -modules. This operator is Fredholm with almost inverseoperator defined by the triple ( a − , P , P ). Consider the index of this operatorind C ( T ∗ R n ) ⋊ G e a (cid:18) x, p, y, − i ∂∂y (cid:19) ∈ K ( C ( T ∗ R n ) ⋊ G ) (41)as the ( C ( T ∗ R n ) ⋊ G ) + -index of operator (40). A priori this index lies in K (( C ( T ∗ R n ) ⋊ G ) + ),but one can readily show that the homomorphism ( C ( T ∗ R n ) ⋊ G ) + → C , whose kernel is C ( T ∗ R n ) ⋊ G , takes the operator (40) to the identity operator (by our assumption (1) above),whose index is zero, and henceind ( C ( T ∗ R n ) ⋊ G ) + e a (cid:18) x, p, y, − i ∂∂y (cid:19) ∈ K ( C ( T ∗ R n ) ⋊ G ) ≡ ker (cid:0) K (( C ( T ∗ R n ) ⋊ G ) + ) → K ( C ) (cid:1) . Finally, we define ind ′ [( a, P , P )] as the index (41).
4. Proof of (28) . Let us prove that ind ◦ β = I . To this end, note that if [ P ] ∈ K ( C ∗ ( G )),where P is a projection over C ∗ ( G ), then the class β [ P ] ∈ K ( C ( T ∗ R n ) ⋊ G ) is represented bythe elliptic symbol ( σ ( E ⊗ N ) , ⊗ P, ⊗ P ) . Hence ind β [ P ] is equal to the C ∗ ( G )-index of the operator e E ⊗ N : 1 ⊗ e P L ( R n , Λ ev ( C n ) ⊗ C ∗ ( G ) N ) −→ ⊗ e P L ( R n , Λ odd ( C n ) ⊗ C ∗ ( G ) N ) . (42)However, the cokernel of E is trivial, and the kernel is one-dimensional and consists of G -invariant elements. Thus, the cokernel of the operator (42) is trivial and the kernel isker 1 ⊗ e P ( e E ⊗ N ) = ker E ⊗ Im e P ≃ Im P ⊂ C ∗ ( G ) N . We obtain the desired equality ind β [ P ] = [ P ] . (43)19 . Proof of (29) . The proof is similar to that in [3]. For the sake of completeness, let usgive a shorter proof here. Given an arbitrary element in K ( C ( T ∗ R n ) ⋊ G ), we choose itsrepresentative of the form a = ( a, P , P ) , a, P , P ∈ C ( T ∗ R n , Mat N ( C )) ⋊ G, where P = P = a = diag(1 , , .., , , , ..,
0) in the domain | x | + | p | ≥ R for some R > β ′ [ a ] we choose the following representative a σ ( E ) = (cid:18) a ( x, p ) ⊗ − χ ( x, p )(1 ⊗ σ ∗ ( E )( y, q )) χ ( x, p )(1 ⊗ σ ( E )( y, q )) a ∗ ( x, p ) ⊗ (cid:19) (44)where χ ( x, p ) is a smooth U ( n )-invariant function with compact support on T ∗ R n such that χ ( x, p ) ≡ | x | + | p | ≤ R . Furthermore, we suppose that here σ ( E )( y, q ) ishomogeneous at infinity and continuous at y = q = 0. Clearly, this representative satisfiesthe properties in the definition of the mapping ind ′ . Hence, we have by the definition of themapping ind ′ the following equalityind ′ β ′ [ a ] = ind ( C ( T ∗ R n ) ⋊ G ) + e A, where e A is an operator in Hilbert ( C ( T ∗ R n ) ⋊ G ) + -modules associated with the symbol (44).We make the following choice of e A : e A = (cid:18) e a ( x, ξ ) ⊗ (1 − Π) − χ ( x, ξ )(1 ⊗ E ∗ )) χ ( x, ξ )(1 ⊗ E ) e a ∗ ( x, ξ ) ⊗ (cid:19) : e P L ( R ny , ( C ( T ∗ R n ) ⋊ G ) + ⊗ C N ⊗ Λ ev ( C n )) ⊕ e P L ( R ny , ( C ( T ∗ R n ) ⋊ G ) + ⊗ C N ⊗ Λ odd ( C n )) −→ e P L ( R ny , ( C ( T ∗ R n ) ⋊ G ) + ⊗ C N ⊗ Λ ev ( C n )) ⊕ e P L ( R ny , ( C ( T ∗ R n ) ⋊ G ) + ⊗ C N ⊗ Λ odd ( C n )) (45)where Π is the orthogonal projection on the subspace ker E = C e −| y | / .Then we haveker e A = ker e A ∗ e A =ker diag (cid:16)e a ∗ e a ( x, p ) ⊗ (1 − Π) + χ ( x, p )(1 ⊗ E ∗ E ) , e a e a ∗ ( x, p ) ⊗ χ ( x, p )(1 ⊗ E E ∗ ) (cid:17) (46)We claim that the operator ( e a e a ∗ )( x, p ) ⊗ χ ( x, p )(1 ⊗ E E ∗ ))is strictly positive and, hence, invertible. Indeed, this operator is a sum of two nonnegativeoperators and for | x | + | p | ≤ R the second summand is strictly positive since ker E ∗′ = 0,20hile for | x | + | p | ≥ R the first term is strictly positive, since e a is invertible here. One showssimilarly that the kernel of operator( e a ∗ e a )( x, p ) ⊗ (1 − Π) + χ ( x, p )(1 ⊗ E ∗ E )is equal to Im P ⊗ ker E ≃ Im P and this operator is strictly positive on the orthogonalcomplement of this subspace. Thus, we haveker e A = (Im P ⊗ ker E ) ⊕ ≃ Im P . The kernel of the adjoint operator is similarly equal toker e A ∗ = ker e A e A ∗ = (Im P ⊗ ker E ) ⊕ ≃ Im P . Hence, we obtainind ( C ( T ∗ R n ) ⋊ G ) + e A = [ker e A ] − [ker e A ∗ ] = [ P ] − [ P ] ∈ K ( C ( T ∗ R n ) ⋊ G ) . This proves (29).
6. Proof of (30) . Given [ a ] ∈ K ( C ( T ∗ R n ) ⋊ G ), we claim that the element a σ ( E ) ishomotopic within elliptic symbols to an element unitarily equivalent to σ ( E ) a . Indeed, thehomotopy σ t = a ( x cos t + y sin t, p cos t + q sin t ) σ ( E )( y cos t − x sin t, q cos t − p sin t )for t ∈ [0 , π/
2] takes a ( x, p ) σ ( E )( y, q ) to a ( y, q ) σ ( E )( − x, − p ), and then the 180 ◦ rotation inthe ( x, p )-plane takes it to the symbol unitarily equivalent to σ ( E ) a . Moreover, this homotopypreserves the ellipticity of the symbol, since the diagonal action of G on R n commutes withthe rotation homotopy( x, p, y, q ) ( x cos t + y sin t, p cos t + q sin t, y cos t − x sin t, q cos t − p sin t ) . Finally, the following equality holdsind ( C ( T ∗ R n ) ⋊ G ) + [ σ ( E ) a ] = β ind[ a ] . (47)The proof of this equality coincides with the proof of Lemma 6.7 in [3]. Both sides of the index formula (19) are homomorphisms of Abelian groupsind , ind t : Ell( R n , G ) −→ C . The group Ell( R n , G ) ≃ K ( C ( T ∗ R n ) ⋊ G ) is generated by the stable homotopy classes oftwisted Euler operators (25) by the equivariant Bott periodicity (see Theorem 2). Hence, itsuffices to prove that the analytic index is equal to the topological index for the twisted Euleroperators. 21 he analytic index of twisted Euler operators. The cokernel is trivial (this follows fromthe fact that E ⊗ N is surjective and commutes with 1 ⊗ P ), while the kernel is equal to P C N exp( − r / E ⊗ N , ⊗ P, ⊗ P ) = rk P | C N exp( − r / = Tr P | C N exp( − r / = X g ∈ G tr P g =: X h g i⊂ G ch g [ P ] . (48)Here Tr stands for the operator trace on L ( R n , C N ), tr is the matrix trace, P g are the com-ponents of P ∈ Mat N ( C ∞ ( G )), and we used the fact that the Gaussian function exp( − r /
2) is U ( n )-invariant. The topological index of twisted Euler operators.
Given g ∈ G , let us compute thelocalized topological index ind t [ σ ( E ⊗ N , ⊗ P, ⊗ P )]( g ). Let P and P be matrix projectionsover C ∞ ( C n ) ⋊ G such that[ σ ( E ⊗ N , ⊗ P, ⊗ P )] = [ P ] − [ P ] . By the definition of the localized topological index, we haveind t [ σ ( E ⊗ N , ⊗ P, ⊗ P )]( g ) = 1det(1 − g | L ⊥ g ) X s ∈h g i tr( τ s ( ω s )) = 1det(1 − g | L ⊥ g ) X s ∈h g i Z L s tr( ω s | L s ) , (49)where the functional τ s was defined in (4), L = C n , L s is the fixed-point subspace for s ∈ U ( n ),and we set ω = { ω s } s ∈ G = P exp (cid:18) − dP dP πi (cid:19) − P exp (cid:18) − dP dP πi (cid:19) ∈ Mat N ( C ∞ c ( C n , Λ( C n )) ⋊ G ) . We claim that the following equality holds X s ∈h g i Z L s tr( ω s | L s ) = Z L g ch g [ σ ( E ⊗ N , ⊗ P, ⊗ P )] (50)where ch g [ σ ( E ⊗ N , ⊗ P, ⊗ P )] ∈ H evc ( L g ) is the localized Chern character of the symbolof the twisted Euler operator defined in [3, p.92]. Indeed, it follows from the definitions in thecited monograph thatch g [ σ ( E ⊗ N , ⊗ P, ⊗ P )] = X s ∈h g i Z G g,s tr( h ∗ ω s ) | L g dh, (51)where G g,s = kC g ⊂ U ( n ), C g = { h ∈ U ( n ) | gh = hg } is the centralizer of g in U ( n ) (it isa compact Lie group), and k is an arbitrary element such that kgk − = s . Finally, dh is themeasure on G g,s induced by the element k from the normalized Haar measure on C g . Integrating2251) over L g gives us the desired equality: Z L g ch g [ σ ( E ⊗ N , ⊗ P, ⊗ P )] = X s ∈h g i Z G g,s Z L g tr( h ∗ ω s ) | L g dh = X s ∈h g i Z G g,s Z L s tr( ω s | L s ) dh = X s ∈h g i Z L s tr( ω s | L s ) . Here we used the fact that each h ∈ G g,s defines a diffeomorphism h : L g → L s of the fixed-pointsets of g and s . Thus, Eqs. (49) and (50) give us the following equalityind t [( E ⊗ N , ⊗ P, ⊗ P )]( g ) = 1det(1 − g L ⊥ g ) Z L g ch g [ σ ( E ⊗ N , ⊗ P, ⊗ P )] . (52)The localized Chern character is multiplicative (see [3, Lemma 9.10]) and we havech g [ σ ( E ⊗ N , ⊗ P, ⊗ P )] = ch g [ P ] ch i ∗ ( σ ( E ))( g ) , (53)where ch g [ P ] = P s ∈h g i tr P s ∈ C and i ∗ ( σ ( E )) is the restriction of symbol σ ( E ) to the subspace L g . A direct computation shows that the restriction of the symbol of the Euler operator to thefixed-point set is equal to i ∗ σ ( E ) = (1 Λ ev ( L ⊥ g ) ⊗ σ ( E L g )) ⊕ (1 Λ odd ( L ⊥ g ) ⊗ σ ( E ∗ L g )) , where Λ ev/odd ( L ⊥ g ) are the vector spaces of even/odd algebraic forms of L ⊥ g , and we denote thesymbol of the Euler operator on a vector space L by σ ( E L ). Now note that the action of g isnontrivial only on the exterior algebra of L ⊥ g . Hence, the localized Chern character is equal to ch( i ∗ σ ( E ))( g ) = tr g ([Λ ev ( L ⊥ g )( g )] − [Λ odd ( L ⊥ g )]) · ch( σ ( E L g )) = det(1 − g L ⊥ g ) · ch( σ ( E L g )) . This equality follows from the definition of the localized Chern character and the fact thattr g ([Λ ev ( L ⊥ g )( g )] − [Λ odd ( L ⊥ g )]) = det(1 − g L ⊥ g ), which is easy to see if we diagonalize g L ⊥ g .Substituting the expression for the localized Chern character in (52), we obtainind t [ σ ( E ⊗ N , ⊗ P, ⊗ P )]( g ) = ch g [ P ] det(1 − g L ⊥ g )det(1 − g L ⊥ g ) Z L g ch( σ ( E L g )) . (54) Recall the definition of the localized Chern character for a trivial G -space X :ch( · )( g ) : K G ( X ) ≃ K ( X ) ⊗ R ( G ) ch ⊗ tr g −→ H ∗ ( X ) ⊗ C , where K G ( X ) ≃ K ( X ) ⊗ R ( G ) is the natural isomorphism, R ( G ) is the ring of virtual representations of G , chis the Chern character, while tr g : R ( G ) → C takes a virtual representation to the value of its character at theelement g ∈ G . Z L g ch( σ ( E L g )) = Z C ch( σ ( E C )) dim L g = 1 . This equality is a special case of Riemann–Roch formula for the embedding pt ⊂ L g , see e.g. [39].Hence we obtain the formula for the localized topological index of the twisted Euler operatorind t [ σ ( E ⊗ N , ⊗ P, ⊗ P )]( g ) = ch g ( P ) . Then the topological index itself is equal toind t [ σ ( E ⊗ N , ⊗ P, ⊗ P )] = X h g i⊂ G ch g [ P ] . (55)Comparing the expressions for the analytic index in (48) and the topological index in (55) wesee that they are equal. The proof of the index formula is now complete. References [1] A. Antonevich, M. Belousov, and A. Lebedev.
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