On the Spectrum of geometric operators on Kähler manifolds
aa r X i v : . [ m a t h . DG ] M a y ON THE SPECTRUM OF GEOMETRIC OPERATORSON K ¨AHLER MANIFOLDS
DMITRY JAKOBSON, ALEXANDER STROHMAIER,AND STEVE ZELDITCH
Abstract.
On a compact K¨ahler manifold there is a canonicalaction of a Lie-superalgebra on the space of differential forms. Itis generated by the differentials, the Lefschetz operator and theadjoints of these operators. We determine the asymptotic distri-bution of irreducible representations of this Lie-superalgebra onthe eigenspaces of the Laplace-Beltrami operator. Because of thehigh degree of symmetry the Laplace-Beltrami operator on formscan not be quantum ergodic. We show that after taking thesesymmetries into account quantum ergodicity holds for the Laplace-Beltrami operator and for the Spin C -Dirac operators if the unitaryframe flow is ergodic. The assumptions for our theorem are knownto be satisfied for instance for negatively curved K¨ahler manifoldsof odd complex dimension. Introduction
Properties of the spectrum of the Laplace-Beltrami operator on amanifold are closely related to the properties of the underlying classicaldynamical system. For example ergodicity of the geodesic flow on theunit tangent bundle T X of a compact Riemannian manifold X impliesquantum ergodicity. Namely, for any complete orthonormal sequence ofeigenfunctions φ j ∈ L ( X ) to the Laplace operator ∆ with eigenvalues λ j ր ∞ one has (see [Shn74, Shn93, Zel87, CV85, HMR])lim N →∞ N X j ≤ N |h φ j , Aφ j i − Z T ∗ X σ A ( ξ ) dµ L ( ξ ) | = 0 , (1) Date : November 21, 2018.2000
Mathematics Subject Classification.
Primary: 81Q50 Secondary: 35P20,37D30, 58J50, 81Q005.
Key words and phrases.
Dirac operator, eigenfunction, frame flow, quantum er-godicity, K¨ahler manifold.The first author was supported by NSERC, FQRNT and Dawson fellowship. for any zero order pseudodifferential operator A , where integrationis with respect to the normalized Liouville measure µ L on the unit-cotangent bundle T ∗ X , and σ A is the principal symbol of A . Quantumergodicity is equivalent to the existence of a subsequence φ j k of countingdensity one such thatlim k →∞ h φ j k , Aφ j k i = Z T ∗ X σ A ( ξ ) dµ L ( ξ ) . (2)In particular, A might be a smooth function on X and the above impliesthat the sequence | φ j k ( x ) | dV g (3)converges to the normalized Riemannian measure dV g in the weaktopology of measures. For bundle-valued geometric operators like theDirac operator acting on sections of a spinor bundle or the Laplace-Beltrami operator the corresponding Quantum ergodicity for eigensec-tions is known in a precise way to relate to the ergodicity of the frameflow on the corresponding manifold [JS]; see also [BoG04, BoG04.2,BO06].This paper deals with a situation in which the frame flow is notergodic, namely the case of K¨ahler manifolds. In this case the con-clusions in [JS] do not hold since there is a huge symmetry algebraacting on the space of differential forms. This algebra is the universalenveloping algebra of a certain Lie superalgebra that is generated bythe Lefschetz operator, the complex differentials and their adjoints. Onthe level of harmonic forms this symmetry is responsible for the richstructure of the cohomology of K¨ahler manifolds and can be seen asthe main ingredient for the Lefschetz theorems. Here we are interestedin eigensections with non-zero eigenvalues, that is in the spectrum ofthe Laplace-Beltrami operator acting on the orthogonal complement ofthe space of harmonic forms. The action of the Lie superalgebra on theorthogonal complement of the space of harmonic forms is much morecomplicated than on the space of harmonic forms where it basicallybecomes the action of sl ( C ). In this paper we classify all finite dimen-sional unitary representations of this algebra and determine the asymp-totic distribution of these representations in the eigenspaces. Since thetypical irreducible representation of the algebra decomposes into fourirreducible representation for sl ( C ) this shows that eigenspaces to theLaplace-Beltrami operator have multiplicities. An important observa-tion in our treatment is that the universal enveloping algebra of thisLie superalgebra is generated by two commuting subalgebras, one ofwhich is isomorphic to the universal enveloping algebra of sl ( C ). This PECTRAL PROPERTIES OF K ¨AHLER MANIFOLDS 3 sl ( C )-action is generated by an operator L t and its adjoint L ∗ t whichis going to be defined in section 3.1. This operator can be interpretedas the Lefschetz operator in the directions of the frame bundle whichare orthogonal to the frame flow. However, L t is not an endomorphismof vector bundles, but it acts as a pseudodifferential operator of orderzero.Guided by this result we tackle the question of quantum ergodicityfor the Laplace-Beltrami operator on ( p, q )-forms. Unlike in the case ofergodic frame flow it turns out that there might be different quantumlimits of eigensections on the space of co-closed ( p, q )-forms because ofthe presence of the Lefschetz operator. Our main results establishesquantum ergodicity for the Dirac operator and the Laplace Beltramioperator if one takes the Lefschetz symmetry into account and underthe assumption that the U ( m )-frame flow is ergodic. For example ouranalysis shows that in case of an ergodic U ( m )-frame flow for any com-plete sequence of co-closed primitive ( p, q )-forms there is a density onesubsequence which converges to a state which is an extension of theLiouville measure and can be explicitly given. For the Spin C -Diracoperators we show that quantum ergodicity does not hold for K¨ahlermanifolds of complex dimension greater than one. Thus, negativelycurved Spin-K¨ahler manifolds provide examples of manifolds with er-godic geodesic flow where quantum ergodicity does not hold for theDirac operator. Our analysis shows that there are certain invariantsubspaces for the Dirac operator in this case and we prove quantumergodicity for the Dirac operator restricted to these subspaces providedthat the U ( m )-frame flow is ergodic.2. K¨ahler manifolds
Let (
X, ω, J ) be a K¨ahler manifold of real dimension n = 2 m . Let g be the metric, h = g + i ω be the hermitian metric, and ω the symplecticform. As usual let J be the complex structure. A k -frame ( e , . . . , e k )for the cotangent space at some point x ∈ X is called unitary if itis unitary with respect to the hermitian inner product induced by h .Hence, a k -frame ( e , . . . , e k ) is unitary iff ( e , J e , e , J e , . . . , e k , J e k )is orthonormal with respect to g . A unitary m -frame at a point x ∈ X is an ordered orthonormal basis for T ∗ x X viewed as a complex vectorspace.Clearly, the group U ( m ) acts freely and transitively on the set ofunitary m -frames. The bundle U m X of unitary m -frames is thereforea U ( m )-principal fiber bundle. Let T ∗ X be the unit cotangent bundlewith bundle projection π . Then projection onto the first vector makes D. JAKOBSON, A. STROHMAIER, AND S. ZELDITCH U m X a principal U ( m − T ∗ X . U m X XT ∗ X ................................................................................................................................................................................................................ ............ U ( m − .......................................................................................................................................................................................................................... ............ S m − ............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... U ( m ) Transporting covectors parallel with respect to the Levi-Civita con-nection extends the Hamiltonian flow on T ∗ X to a flow on U m X whichwe call the U ( m )-frame flow (in the literature it is also referred to asthe restricted frame flow). This is indeed a flow on U m X since J iscovariantly constant and therefore unitary frames are parallel trans-ported into unitary ones. This flow is the appropriate replacement forthe SO (2 m )-frame flow for K¨ahler manifolds as it can be shown to beergodic in some cases, whereas the SO (2 m )-frame flow never is ergodicfor K¨ahler manifoldsSuppose that X is a negatively-curved K¨ahler manifold [Bor]. Wesummarize results that can be found in [Br82, BrG80, BrP74]. We referthe reader to [BuP03, JS, Br82] and references therein for discussionof frame flows on general negatively-curved manifolds. Note that theframe flow is not ergodic on negatively-curved K¨ahler manifolds, sincethe almost complex structure J is preserved. This is the only knownexample in negative curvature when the geodesic flow is ergodic, but theframe flow is not. In fact, given an orthonormal k -frame ( e , . . . , e k ),the functions ( e i , J e j ) , ≤ i, j ≤ k are first integrals of the frame flow.However, the following proposition was proved in [BrG80]: Proposition 2.1.
Let X be a compact negatively-curved K¨ahler man-ifold of complex dimension m . Then the U ( m ) -frame flow is ergodicon U m X when m = 2 , or when m is odd. The Hodge Laplacian and the Lefschetz decomposition
Let ∧ ∗ X be the complex vector bundle ∧ ∗ T ∗ C X , where T ∗ C X is thecomplexification of the co-tangent bundle. Then the Lefschetz operator L : C ∞ ( X ; ∧ ∗ X ) → C ∞ ( X ; ∧ ∗ X ) is defined by exterior multiplicationwith the K¨ahler form ω , i.e. L = ω ∧ . Its adjoint L ∗ is then given byinterior multiplication with ω . Is is well known that[ L ∗ , L ] := H = X k ( m − r ) P r , (4)where P r is the orthoprojection onto C ∞ ( ∧ r X ), and L, L ∗ , H definea representation of sl ( C ) which commutes with the Laplace operator∆ = 2∆ ¯ ∂ = 2∆ ∂ . The decomposition into irreducible representations PECTRAL PROPERTIES OF K ¨AHLER MANIFOLDS 5 on the level of harmonic forms is called the Lefschetz decomposition.We will refer to this decomposition as the Lefschetz decomposition ingeneral. Note that since the Lefschetz operator commutes with ∆ eacheigenspace decomposes into a direct sum of irreducible subspaces forthe sl ( C ) action.The operators L, L ∗ , H, ∂, ¯ ∂, ∂ ∗ , ¯ ∂ ∗ , ∆ satisfy the following relations(see e.g. [B])[ L, ¯ ∂ ∗ ] = − i∂, [ L ∗ , ∂ ] = i ¯ ∂ ∗ , [ L ∗ , ¯ ∂ ] = − i∂ ∗ , [ L, ∂ ∗ ] = i ¯ ∂, [ L ∗ , L ] = H, [ H, L ] = − L, [ H, L ∗ ] = 2 L ∗ , { ∂, ∂ } = { ¯ ∂, ¯ ∂ } = { ∂ ∗ , ∂ ∗ } = { ¯ ∂ ∗ , ¯ ∂ ∗ } = 0 , [ L, ¯ ∂ ] = [ L, ∂ ] = [ L ∗ , ¯ ∂ ∗ ] = [ L ∗ , ∂ ∗ ] = 0 , (5) { ∂, ¯ ∂ } = { ∂, ¯ ∂ ∗ } = { ¯ ∂, ∂ ∗ } = 0 , { ∂, ∂ ∗ } = { ¯ ∂, ¯ ∂ ∗ } = 12 ∆ . Thus, the operators form a Lie superalgebra with central element∆ (see also [FrGrRe99]). Let ∆ − | ker∆ ⊥ the inverse of the Laplaceoperator on the orthocomplement of the kernel of ∆. We view this asan operator defined in L ( X, ∧ ∗ X ) by defining it to be zero on ker∆and write ∆ − slightly abusing notation.3.1. The transversal Lefschetz decomposition.
The operator Q :=2∆ − ¯ ∂∂ is a partial isometry with initial space Rg( ¯ ∂ ∗ ) ∩ Rg( ∂ ∗ ) andfinal space Rg( ¯ ∂ ) ∩ Rg( ∂ ). Hence, Q ∗ Q is the orthoprojection ontoRg( ¯ ∂ ∗ ) ∩ Rg( ∂ ∗ ) and QQ ∗ is the orthoprojection onto Rg( ¯ ∂ ) ∩ Rg( ∂ ).From the above relations one gets[ L, Q ] = 0 , (6) [ L, Q ∗ ] = 2 i ∆ − ( ¯ ∂ ¯ ∂ ∗ − ∂ ∗ ∂ ) , (7) [ Q ∗ , Q ] = − − ( ¯ ∂ ¯ ∂ ∗ − ∂ ∗ ∂ ) , (8)from which one finds that[ L − iQ, Q ∗ ] = [ L − iQ, Q ] = 0 . (9)We define the transversal Lefschetz operator L t by L t := L − iQ. (10)Then clearly L ∗ t = L ∗ + iQ ∗ and one gets that[ L ∗ t , L t ] = H t = H + [ Q ∗ , Q ] , (11) [ H t , L t ] = − L t , [ H t , L ∗ t ] = − L ∗ t , (12) [ ∂, L t ] = [ ∂ ∗ , L t ] = [ ¯ ∂, L t ] = [ ¯ ∂ ∗ , L t ] = 0 , (13) D. JAKOBSON, A. STROHMAIER, AND S. ZELDITCH and hence, also the transversal Lefschetz operators defines an action ofsl ( C ) on L ( X, ∧ ∗ X ). Unlike the Lefschetz operator the transversalLefschetz operator commutes with the holomorphic and antiholomor-phic codifferentials.Denote by g the Lie-superalgebra generated by a, ¯ a, L, H, a ∗ , ¯ a ∗ , L ∗ and relations[ L, ¯ a ∗ ] = − ia, [ L ∗ , a ] = i ¯ a ∗ , [ L ∗ , ¯ a ] = − ia ∗ , [ L, a ∗ ] = i ¯ a, [ L ∗ , L ] = H, [ H, L ] = − L, [ H, L ∗ ] = 2 L ∗ , { a, a } = { ¯ a, ¯ a } = { a ∗ , a ∗ } = { ¯ a ∗ , ¯ a ∗ } = 0 , [ L, ¯ a ] = [ L, a ] = [ L ∗ , ¯ a ∗ ] = [ L ∗ , a ∗ ] = 0 , (14) { ¯ a, a } = { ¯ a, a ∗ } = { a, ¯ a ∗ } = 0 , { a, a ∗ } = { ¯ a, ¯ a ∗ } = 1 . The subspace of odd elements is spanned by a, a ∗ , ¯ a, ¯ a ∗ , the subspace ofeven elements is spanned by L, L ∗ and H . In the following we will de-note by U ( g ) the universal enveloping algebra of this Lie-superalgebraviewed as a unital ∗ -algebra, i.e. the unital ∗ -algebra generated by thesymbols { L, L ∗ , H, a, a ∗ , ¯ a, ¯ a ∗ } and the above relations.The relations (3.1) are obtained from the relations (5) by sending a to √ − / ∂ and ¯ a to √ − / ¯ ∂ . Therefore, we obtain a ∗ -representationof the Lie-superalgebra g on the orthogonal complement of the kernelof ∆.3.2. The representation theory of U ( g ) . The calculations in theprevious section used the relations in U ( g ) only. Hence, they remainvalid if we regard Q = ¯ aa and L t := L − iQ as elements in the abstract ∗ -algebra U ( g ). Hence, L t , L ∗ t generate a subalgebra in U ( g ) which iscanonically isomorphic to the universal enveloping algebra of sl ( C )and which we therefore denote by U (sl ( C )). Note that U (sl ( C )) com-mutes with a, ¯ a, a ∗ and ¯ a ∗ . Since U ( g ) is generated by two commutingsubalgebras the representation theory for U ( g ) is very simple. The ∗ -subalgebra A generated by a and ¯ a has the following canonical rep-resentation on ∧ ∗ C ∼ = C . For an orthonormal basis { e, ¯ e } of C definethe action of a by exterior multiplication by i e , and the action of ¯ a byexterior multiplication by i ¯ e . It is easy to see that all non-trivial finitedimensional irreducible ∗ -representations of A are unitarily equivalentto this representation.Note that the equivalence classes of finite dimensional irreducible ∗ -representations of U (sl ( C )) are labeled by the non-negative integers.Denote the Verma-module for the Spin- n representation by V n and thedistinguished highest weight vector in V n by h . Remember that V n is PECTRAL PROPERTIES OF K ¨AHLER MANIFOLDS 7 spanned by vectors of the form L kt h with k = 0 , . . . , n and we have L ∗ t h = 0 and H t h = nh .Now define an action of U ( g ) on H n := V n ⊗ ∧ ∗ , ∗ C by L t ( v ⊗ w ) = ( L t v ) ⊗ w,L ∗ t ( v ⊗ w ) = ( L ∗ t v ) ⊗ w,H t ( v ⊗ w ) = ( H t v ) ⊗ w,a ( v ⊗ w ) = v ⊗ aw, (15) ¯ a ( v ⊗ w ) = v ⊗ ¯ aw,a ∗ ( v ⊗ w ) = v ⊗ a ∗ w, ¯ a ∗ ( v ⊗ w ) = v ⊗ ¯ a ∗ w, Clearly, this defines a ∗ -representation of U ( g ) on H n . Theorem 3.1.
The representations H n are irreducible and pairwise in-equivalent. Any non-trivial finite dimensional irreducible ∗ -representationof U ( g ) is unitary equivalent to some H n .Proof. Since U ( g ) is generated by two commuting subalgebras U (sl ( C ))and A any irreducible ∗ -representation of is also an irreducible ∗ -representation of U (sl ( C )) ⊗ A . If it is finite dimensional it is thereforea tensor product of two finite dimensional irreducible representationsof U (sl ( C )) and A . (cid:3) Corollary 3.2.
Any non trivial finite dimensional irreducible ∗ - repre-sentation of U ( g ) decomposes into a direct sum of equivalent modulesfor the sl ( C ) action defined by L t , L ∗ t , H t . If h n is a highest weight vector of V n then the kernel of L ∗ t in therepresentation H n is given by h n ⊗ ∧ ∗ C . Using the unitary basis e, ¯ e for C as before we see that the vectors h n ⊗ , h n ⊗ e, h n ⊗ ¯ e are in the kernel of L ∗ . Moreover, H ( h n ⊗
1) = ( n − h n ⊗ , (16) H ( h n ⊗ e ) = n ( h n ⊗ e ) , (17) H ( h n ⊗ ¯ e ) = n ( h n ⊗ ¯ e ) . (18)Therefore, in the decomposition of H n into irreducibles of the sl ( C )action defined by L, L ∗ , H the representations V n occur with multiplic-ity at least 2 and the representation V n − occurs with multiplicity atleast 1. The vector h n ⊗ ( e ∧ ¯ e ) has weight n + 1 and therefore, there D. JAKOBSON, A. STROHMAIER, AND S. ZELDITCH must be another representation of highest weight greater or equal than n + 1 occurring. Since4 dim V n − V n − dim V n − = dim V n +1 this shows that as a module for the sl ( C ) action defined by L, L ∗ , H we have H n = V n +1 ⊕ V n ⊕ V n ⊕ V n − . Corollary 3.3.
Every non-trivial finite dimensional irreducible ∗ - rep-resentation of U ( g ) is as a module for the sl ( C ) action defined by L, L ∗ , H unitarily equivalent to the direct sum V n +1 ⊕ V n ⊕ V n ⊕ V n − .By convention V − = { } . Corollary 3.4.
Let V and W be two finite dimensional U ( g ) modules.Then V and W are unitarily equivalent if and only if they are equivalentas modules for the sl ( C ) action defined by L, L ∗ , H . The model representations.
There is another very natural rep-resentation ρ of the ∗ -algebra U ( g ) which is important for our purposes.This representation will be referred to as the model representation andcan be described as follows. Let us view C m ∼ = R m as a real vectorspace with complex structure J . Let { e i } i =1 ,...,m be the standard uni-tary basis in R m . Then in the complexification R m ⊗ C = C m wedefine w i = e i − i J e i , (19) ¯ w i = e i + i J e i . (20)We define ρ ( L ) to be the operator of exterior multiplication by ω = i2 P mi =1 w i ∧ ¯ w i on the space ∧ ∗ C m = L p,q ∧ p,q C m . Let π ( L ∗ ) be itsadjoint, namely the operator of interior multiplication by ω . Let ρ ( a )be the operator of exterior multiplication by i √ w and ρ (¯ a ) be theoperator of exterior multiplication by i √ ¯ w . The operators ρ ( a ∗ ) and ρ (¯ a ∗ ) are defined as the adjoints of ρ ( a ) and ρ (¯ a ). This defines a repre-sentation ρ of U ( g ) on ∧ ∗ C m . This representation decomposes into asum of irreducibles. Note that ρ ( L t ) = ρ ( L − i ¯ aa ) is given by exteriormultiplication by i2 P mi =2 w i ∧ ¯ w i . The restriction of ρ to the two sub-algebras generated by ρ ( L ) , ρ ( L ∗ ) , ρ ( H ) and ρ ( L t ) , ρ ( L ∗ t ) , ρ ( H t ) definerepresentations of sl ( C ). Since the maximal eigenvalue of H is m , onlyrepresentations of highest weight k with k ≤ m can occur in the decom-position of the model representation with respect to the sl ( C )-actionby ρ ( L ) , ρ ( L ∗ ) , ρ ( H ). Consequently, by Cor 3.3 in the decompositionof the model representation into irreducible representations only therepresentations H k with k ≤ m can occur. PECTRAL PROPERTIES OF K ¨AHLER MANIFOLDS 9 Asymptotic decomposition of Eigenspaces
Since the action of U ( g ) commutes with the Laplace operator ∆ onforms each eigenspace V λ = { φ ∈ ∧ ∗ X : ∆ φ = λφ } with λ = 0 is a U ( g )-module and can be decomposed into a direct sumof irreducible U ( g )-modules. In the previous section we classified allirreducible ∗ -representations of U ( g ) and found that they are isomor-phic to H k for some non-negative integer k . Therefore, we may definethe function m k ( λ ) as m k ( λ ) := { the multiplicity of H k in V λ } , (21)so that V λ ∼ = ∞ M k =0 m k ( λ ) H k (22) Theorem 4.1.
Let X be any compact K¨ahler manifold of complexdimension m . Then in the decomposition of the eigenspaces of theLaplace-Beltrami operator ∆ into irreducible representations of U ( g ) the proportion of irreducible summands of type H k in L ( X ; ∧ ∗ X ) isin average the same as the proportion of such irreducibles in the modelrepresentation of U ( g ) on ∧ ∗ C m : (23) 1 N ( λ ) X j : λ j ≤ λ m k ( λ j ) ∼ ∧ ∗ C m ) m k ( ∧ ∗ C m ) , where N ( λ ) = trΠ [0 ,λ ] and Π [0 ,λ ] is the spectral projection of the Laplace-Beltrami operator ∆ . We recall that N ( λ ) ∼ rk ( E ) vol ( X )(4 π ) m Γ( m +1) λ m for the Laplacian on a bundle E → X of rank rk ( E ) over a manifold X of real dimension 2 m . Notethat apart from the fact that we are not dealing with a group butwith a Lie superalgebra the action is neither on X , nor on T ∗ X , butrather on the total space of the vector bundle π ∗ ( ∧ ∗ X ) → T ∗ X . Theaction there leaves the fibers invariant and therefore it is rather differentfrom a group action on the base manifold. The above theorem thusfalls outside the scope of the equivariant Weyl laws of articles such as[BH1, BH2, GU, HR, TU]. In fact its conclusion is rather different fromthe conclusions in these articles as in our case only a fixed number oftypes of irreducible representations may occur. Proof.
For a compact K¨ahler manifold U ( g ) acts by pseudodifferen-tial operators on C ∞ ( X ; ∧ ∗ X ). Therefore, the symbol map defines an action of U ( g ) on each fiber of the bundle π ∗ ( ∧ ∗ X ) → T ∗ X . The rep-resentation of U ( g ) on each fiber is easily seen to be equivalent to themodel representation. Since the maximal eigenvalue of H , acting on L ( X ; ∧ ∗ X ), is m , only representations of highest weight k with k ≤ m can occur in the decomposition of L ( X ; ∧ ∗ X ) into irreducible sub-spaces with respect to the sl ( C )-action by L ∗ , L, H . Again, by Cor 3.3types H k with k > m cannot occur in the decomposition with respectto the U ( g )-action. Let P k be the orthogonal projection onto the type H k in L ( X ; ∧ ∗ X ). Then P k is actually a pseudodifferential operator oforder 0. Namely, the quadratic Casimir operator C of the sl ( C )-actionby L ∗ t , L t , H t given by C = L ∗ t L t + L t L ∗ t + 12 H , (24)is a pseudodifferential operator of order 0. On a subspace of type H k itacts like multiplication by k + k . Therefore, if Q is a real polynomialthat is equal to 1 at k + k and equal to 0 at l + l for any integer l = k between 0 and m it follows that P k = Q ( C ). Thus, P k is apseudodifferential operator of order 0 and its principal symbol at ξ projects onto the subspace in the fiber π ∗ ξ ( ∧ ∗ X ) which is spanned bythe representations of type H k . Therefore, for every ξ :1dim( H k ) tr( σ P k ( ξ )) = m k ( ∧ ∗ C m ) . (25)Applying Karamatas Tauberian theorem to the heat trace expansiontr( P k e − t ∆ ) = (4 π ) − m Vol( X ) Z T ∗ X tr( σ P k ( ξ )) dξ ! t m + O ( t m − ) . (26)gives 1 N ( λ ) X j : λ j ≤ λ tr(Π [0 ,λ ] P k ) ∼ m k ( ∧ ∗ C m )dim( H k ) 1dim( ∧ ∗ C m ) . (27)After dividing by dim( H k ) this reduces to the statement of the theorem. (cid:3) Remark 4.2.
A natural question is whether, for generic K¨ahler met-rics, the eigenspaces of the Laplace-Beltrami operator are irreduciblerepresentations of the Lie superalgebra g and of complex conjugation.Such irreducibility is suggested by the heuristic principle of ‘no acci-dental degeneracies’, i.e. in generic cases, degeneracies of eigenspacesshould be entirely due to symmetries (see [Zel90] for some results and PECTRAL PROPERTIES OF K ¨AHLER MANIFOLDS 11 references). Cor. 5.3 would then suggest that for a generic K¨ahler man-ifold the spectrum of ∆ on the space of primitive co-closed ( p, q ) -formsshould be simple for fixed p and q . Quantum ergodicity for the Laplace-Beltramioperator
We will now investigate the question of quantum ergodicity for theLaplace-Beltrami operator on a compact K¨ahler manifold X and wekeep the notations from the previous sections. As shown in [JS] thisquestion is intimately related to the ergodic decomposition of the tra-cial state on the C ∗ -algebra C ( X ; π ∗ ∧ ∗ X ). The transversal Lefschetzdecomposition plays an important role here.5.1. Ergodic decomposition of the tracial state.
On the space of( p, q )-forms denote by P p,q the projection onto the space of transversally-primitive forms, i.e. onto the kernel of L ∗ t . Let P p,q,k be the projectiononto the range of L kt P p − k,q − k . The operators P = P ∂ ¯ ∂ = 4∆ − ∂ ¯ ∂ ¯ ∂ ∗ ∂ ∗ = QQ ∗ , (28) P = P ∂ ∗ ¯ ∂ ∗ = 4∆ − ∂ ∗ ¯ ∂ ∗ ¯ ∂∂ = Q ∗ Q, (29) P = P ∂ ¯ ∂ ∗ = 4∆ − ∂ ¯ ∂ ∗ ¯ ∂∂ ∗ , (30) P = P ∂ ∗ ¯ ∂ = 4∆ − ∂ ∗ ¯ ∂ ¯ ∂ ∗ ∂ (31)are projections onto the ranges of the corresponding operators. Wehave P H + X i =1 P i = 1(32)where P H is the finite dimensional projection onto the space of har-monic forms. Using the transversal Lefschetz decomposition we obtaina further decomposition min( p,q ) X k =0 P p,q,k P H + min( p,q ) X k =0 4 X i =1 P p,q,k P i = 1(33)where each of the subspaces onto which P p,q,k P i projects is invariantunder the Laplace operator.Note that the principal symbols of these projections are invariantprojections in C ( T ∗ X, π ∗ End( ∧ p,q X )) and the above relation gives riseto a decomposition of the tracial state ω tr on C ( T ∗ X, π ∗ End( ∧ p,q X )) defined by ω tr ( a ) := 1rk( ∧ p,q X ) Z T ∗ X tr( a ( ξ )) dξ (34)into invariant states. Thus, the tracial state is not ergodic. However,if the U ( m )-frame flow is ergodic this decomposition turns out to beergodic. Proposition 5.1.
Suppose that the U ( m ) -frame flow on U m X is er-godic. Let P be one of the projections P p,q,k P i , ≤ i ≤ , ≤ k ≤ min( p, q ) . Then the state ω P on C ( T ∗ X ; π ∗ End( ∧ p,q X )) defined by ω P ( a ) := c P ω tr ( σ P a ) is ergodic. Here c P = ω tr ( σ P ) − .Proof. The bundle ∧ p,q X can be naturally identified with the associatedbundle U m X × ˆ ρ ∧ p,q C m , where ˆ ρ is the representation of U ( m ) on ∧ p,q C m = ∧ p C m ⊗ ∧ q C m . obtained from the canonical representation on C m . The pull back π ∗ ∧ p,q X of ∧ p,q X can analogously be identified with the associatedbundle U m X × ˆ ρ ∧ p,q C m , (35)where ˆ ρ is the restriction of ˆ ρ to the subgroup U ( m − C m is invariant under the action of U ( m −
1) we havethe decomposition ∧ p,q C m = ∧ p,q C m − ⊕ ∧ p − ,q C m − ⊕ ∧ p,q − C m − ⊕ ∧ p − ,q − C m − into invariant subspaces. The projections onto these subspaces in eachfiber is exactly given by the principal symbols σ P i of the projections P i . The representation of U ( m −
1) on ∧ p ′ ,q ′ C m − may still fail to beirreducible. However, it is an easy exercise in representation theory(c.f. [FuHa91], Exercise 15.30, p. 226) to show that the kernel of σ L ∗ t in each fiber is an irreducible representation of U ( m − σ P projects onto a sub-bundle F of π ∗ ∧ p,q X that is associated with anirreducible representation ρ of U ( m − F ∼ = U m X × ρ V ρ . (36)This identification intertwines the U ( m )-frame flow on U m X and theflow β t . To show that the state ω P is ergodic it is enough to show thatany positive β t -invariant element f in σ P L ∞ ( T ∗ X, π ∗ End ∧ p,q X ) σ P is PECTRAL PROPERTIES OF K ¨AHLER MANIFOLDS 13 proportional to σ P (see [JS], Appendix). Under the above identification f gets identified with a function ˆ f ∈ L ∞ ( U m X ; V ρ ) which satisfiesˆ f ( xg ) = ρ ( g ) ˆ f ( x ) ρ ( g ) − , x ∈ U m X, g ∈ U ( m − . (37)If such a function is invariant under the U ( m )-frame flow it follows fromergodicity of the U ( m )-frame flow that it is constant almost everywhere.So almost everywhere ˆ f ( x ) = M , where M is a matrix. By the abovetransformation rule M commutes with ρ ( g ). Since ρ is irreducibleit follows that M is a multiple of the identity matrix. Thus, ˆ f isproportional to the identity and consequently, f is proportional to σ P . (cid:3) Applying the abstract theory developed in [Zel96] the same argumentas in [JS] can be applied to obtain
Theorem 5.2.
Let P be one of the projections P p,q,k P i , ≤ i ≤ , ≤ k ≤ min( p, q ) . and let ( φ j ) be an orthonormal basis in Rg( P ) with ∆ φ j = λ j φ j , (38) λ j ր ∞ . If the U ( m ) -frame flow on U m X is ergodic, then quantum ergodicityholds in the sence that N N X j =1 |h φ j , Aφ j i − ω P ( σ A ) | → , (39) for any A ∈ ΨDO cl ( X, ∧ p,q X ) . Since for co-closed forms primitivity and transversal primitivity areequivalent there is a natural gauge condition that manages without theabove heavy notation.
Corollary 5.3.
Let φ j be a complete sequence of primitive co-closed ( p, q ) -forms such that ∆ φ j = λ j φ j , (40) λ j ր ∞ . Then, if the U ( m ) -frame flow on U m X is ergodic, quantum ergodicityholds in the sence that N N X j =1 |h φ j , Aφ j i − ω P ( σ A ) | → , (41) for any A ∈ ΨDO cl ( X, ∧ p,q X ) , where P = P p,q, P is the orthogonalprojection onto the space of primitive co-closed ( p, q ) -forms. Quantum ergodicity for Spin C -Dirac operators In this section we consider the quantum ergodicity for Dirac typeoperators rather than Laplace operators. The complex structure onK¨ahler manifolds gives rise to the so-called canonical and anti-canonicalSpin C - structures. The spinor bundle of the latter can be canonicallyidentified with the bundle ∧ , ∗ X in such a way that the Dirac opera-tor gets identified with the so-called Dolbeault Dirac operator. OtherSpin C - structures (e.g. the canonical one) can then be obtained bytwisting with a holomorphic line bundle. Let us quickly describe theconstruction of the twisted Dolbeault operator.Let L be a holomorphic line bundle. Then the twisted Dolbeaultcomplex is given by . . . ∧ ,k − X ⊗ L ∧ ,k X ⊗ L . . . .............................................................................. ............ ¯ ∂ ...................................................................................................................... ............ ¯ ∂ .......................................................................................... ............ ¯ ∂ This is an elliptic complex and the twisted Dolbeault Dirac operator isdefined by D = √
2( ¯ ∂ + ¯ ∂ ∗ ) . (42)As mentioned above this operator is the Dirac operator of a Spin C -structure on X where the spinor bundle is identified with S = ∧ , ∗ X ⊗ L . Note that Spin structures on X are in one-one correspondencewith square roots of the canonical bundle K = ∧ n, T X , i.e. withholomorphic line bundles L such that L ⊗ L = K . In this case theDirac operator D is exactly the twisted Dolbeault Dirac operator.The twisted Dolbeault Dirac operator is a first order elliptic formallyself-adjoint differential operator. It is therefore self-adjoint on the do-main H ( X ; ∧ , ∗ X ⊗ L ) of sections in the first Sobolev space. As D isa first order differential operator its spectrum is unbounded from bothsides. PECTRAL PROPERTIES OF K ¨AHLER MANIFOLDS 15
The Dolbeault Laplace operator is given by 2( ¯ ∂ ¯ ∂ ∗ + ¯ ∂ ∗ ¯ ∂ ) = D andwill be denoted by ∆ L . The Hodge decomposition is C ∞ ( X ; ∧ ,k X ⊗ L ) =(43) ker(∆ Lk ) ⊕ ¯ ∂C ∞ ( X ; ∧ ,k − X ⊗ L ) ⊕ ¯ ∂ ∗ C ∞ ( X ; ∧ ,k +1 X ⊗ L ) . Note that the Dirac operator leaves ker(∆ Lk ) invariant since it commuteswith ∆ L . Moreover, D maps ¯ ∂C ∞ ( X ; ∧ ,k − X ⊗ L ) to ¯ ∂ ∗ C ∞ ( X ; ∧ ,k X ⊗ L ) and ¯ ∂ ∗ C ∞ ( X ; ∧ ,k X ⊗ L ) to ¯ ∂C ∞ ( X ; ∧ ,k − X ⊗ L ). Therefore, thesubspaces H k = ¯ ∂C ∞ ( X ; ∧ ,k − X ⊗ L ) ⊕ ¯ ∂ ∗ C ∞ ( X ; ∧ ,k X ⊗ L )(44)are invariant subspaces for the Dirac operator. The orthogonal projec-tions Π k onto the closures of H k are clearly zero order pseudodifferentialoperators which commute with the Dirac operator.Let ΨDO cl ( X ; S ) be the norm closure of the ∗ -algebra of zero orderpseudodifferential operators in B ( L ( X, S )). Then the symbol mapextends to an isomorphismΨDO cl ( X ; S ) / K ∼ = C ( T ∗ X, π ∗ End( S )) . (45)By theorem 1.4 in [JS] ΨDO cl ( X ; S ) is invariant under the automor-phism group α t ( A ) := e − i (∆ L ) / t Ae + i (∆ L ) / t and the induced flow β t on C ( T ∗ X, π ∗ End( S )) is the extension of the geodesic flow defined byparallel translation along the fibers.As in the analysis for the Laplace-Beltrami operator we have to con-sider the tracial state ω tr ( a ) = 1rk( S ) Z T ∗ X tr( a ( ξ )) dξ, (46)As already remarked in [JS] this state is not ergodic with respect to β t since it has a decomposition ω tr ( a ) = 12 ω + ( a ) + 12 ω − ( a ) , (47)where ω ± ( a ) = ω tr ((1 ± σ sign( D ) ) a )(48)On Spin C -manifolds with ergodic frame flows the states ω ± were shownin [JS] to be ergodic. On K¨ahler manifolds of complex dimensiongreater than one they are not ergodic since we have a further decom-position ω ± ( a ) = X k ω ± ( σ Π k a )(49) into invariant states. Proposition 6.1.
Suppose that the U ( m ) -frame flow on U m X is er-godic. Then the states ω k ± := c k ω ± ( σ Π k a ) are ergodic with respect tothe group β t . Here c k := ω ± ( σ Π k ) − .Proof. Let R be one of the projections ± sign( D )2 Π k and let σ R be itsprincipal symbol. Hence, σ R is a projection in C ( T ∗ X, π ∗ End( S )) ∼ = C ( T ∗ X, π ∗ End( ∧ , ∗ X )) . (50)We need to show that a → ω tr ( σ R ) − ω tr ( σ R a ) is an ergodic state. Asin the proof of Proposition 5.1 this is equivalent to showing that anypositive element in L ∞ ( T ∗ X, π ∗ End( ∧ ,k X )) σ R is proportional to σ R .A positive element in L ∞ ( T ∗ X, π ∗ End( ∧ ,k X )) σ R is also in σ R L ∞ ( T ∗ X, π ∗ End( ∧ ,k X )) σ R = L ∞ ( T ∗ X, End F ) , where F is the sub-bundle of π ∗ ∧ ,k X onto which σ R projects. Since σ R is β t -invariant the flow clearly restricts to a flow on the sub-bundleEnd F of π ∗ End( ∧ ,k X ). We will show that under the stated assump-tions an invariant element in L ∞ ( T ∗ X, End F ) is proportional to theidentity in L ∞ ( T ∗ X, End F ). Note that π ∗ ∧ ,k X is naturally identifiedwith an associated bundle π ∗ ∧ ,k X ∼ = U m X × ∧ k ˜ ρ ∧ k C m , (51)where ˜ ρ is the restriction of the anticanonical representation of U ( m )on ¯ C m to U ( m − U m X as a U ( m − T ∗ X . Note that ∧ k ˜ ρ is not irreducible but splits intoa direct sum of two irreducible representations. This corresponds tothe splitting ∧ k ( ¯ C m − ⊕ ¯ C ) = ∧ k − ¯ C m − ⊕ ∧ k ¯ C m − . Under the abovecorrespondence the projections onto the sub-representations are exactlythe principal symbols of the projections onto Rg( ¯ ∂ ) and Rg( ¯ ∂ ∗ ). Onefinds that F is associated with a representation ρ of U ( m − F ∼ = U m X × ρ V ρ , (52)where ρ is equivalent to ∧ k ˆ ρ and ˆ ρ is the anticanonical representationof U ( m − ρ is irreducible. Hence, elements in f ∈ L ∞ ( T ∗ X, End F ) can be identified with functions ˆ f ∈ L ∞ ( U m X, End V ρ )that satisfy the transformation propertyˆ f ( xg ) = ρ ( g ) ˆ f ( x ) ρ ( g ) − , x ∈ U m X, g ∈ U ( m − . (53)This identification intertwines the pullback of the frame flow with β t .Now exactly in the same way as in the proof of Proposition 5.1 we con-clude that an invariant element in L ∞ ( T ∗ X, End F ) must be a multipleof the identity. Thus, the corresponding state is ergodic. (cid:3) PECTRAL PROPERTIES OF K ¨AHLER MANIFOLDS 17
The above theorem gives rise to an ergodic decomposition of the tra-cial state on the C ∗ -algebra of continuous sections of π ∗ End( S ) whichis different from the decomposition obtained from Prop. 5.1. Theadvantage of this decomposition is that it is more suitable to studyquantum ergodicity for the Dirac operator. Namely, the decomposi-tion (47) corresponds to the splitting into negative energy and positiveenergy subspaces of the Dirac operator. Thus, if we are interested inquantum limits of eigensections with positive energy we need to de-compose the state ω + into ergodic components. This is achieved byProp. 6.1.In the same way as in [JS] one obtains Theorem 6.2.
Let X be a K¨ahler manifold and let L be a holomor-phic line bundle. Let D be the associated Spin C -Dirac operator and let L ( X, S ) be the positive spectral subspace of D . Suppose that ( φ j ) isan orthonormal basis in Π k L ( X, S ) such that Dφ j = λ j φ j , (54) λ j ր ∞ . If the U ( m ) -frame flow on U m X is ergodic, then N N X j =1 |h φ j , Aφ j i − ω k ( σ A ) | → , (55) for any A ∈ ΨDO cl ( X, S ) . Here ω k is the state on C ( T ∗ X, π ∗ End( S )) defined by ω k ( a ) = C Z T ∗ X tr (cid:0) (1 + σ sign( D ) ( ξ )) σ Π k ( ξ ) a ( ξ ) (cid:1) dξ, (56) where integration is with respect to the normalized Liouville measureand C is fixed by the requirement that ω k (1) = 1 . This shows that quantum ergodicity for the Dirac operators holdsonly after taking the symmetry Π k into account. The states ω k differfor different k . Therefore, Dirac operators on a K¨ahler manifolds ofcomplex dimension greater than one are never quantum ergodic in thesense of [JS]. References [B] W. Ballmann,
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Department of Mathematics and Statistics, McGill University, 805Sherbrooke Str. West, Montr´eal QC H3A 2K6, Canada.
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