Laplace eigenfunctions on Riemannian symmetric spaces and Borel-Weil Theorem
aa r X i v : . [ m a t h . DG ] F e b LAPLACE EIGENFUNCTIONS ON RIEMANNIAN SYMMETRICSPACES AND BOREL-WEIL THEOREM
DIMITAR GRANTCHAROV, GUEO GRANTCHAROV AND CAMILO MONTOYA
Abstract.
We indicate a geometric relation between Laplace-Beltrami spectraand eigenfunctions on compact Riemannian symmetric spaces and the Borel-Weiltheory using ideas from symplectic geometry and geometric quantization. This isdone by associating via Marsden-Weinstein reduction a generalized flag manifoldparametrizing all maximal totally geodesic tori to each compact Riemannian sym-metric space. In the process we notice a direct relation between the Satake diagramof the symmetric space and the painted Dynkin diagram of its associated flag mani-fold. We consider in detail the examples of the classical simply-connected spaces ofrank one and the space SU (3) /SO (3). In the second part of the paper we providea construction of harmonic polynomials inducing Laplace-Beltrami eigenfunctionson the symmetric space from holomorphic sections of the associated line bundleon the generalized flag manifold. We show that in the examples we consider theconstruction provides all of the eigenfunctions. Introduction
There are two classical geometric interpretations of the representation theory ofthe compact Lie groups. On one side is the Borel-Weil Theorem and its subsequentgeneralization to the Borel-Weil-Bott theory. In particular, every complex represen-tation of a compact Lie group is realized on the space of holomorphic sections ofsome line bundle over a flag manifold. On the other side, the harmonic analysis ona Riemannian symmetric spaces provides irreducible representations of a compactsimple Lie group: the natural action of a transitive simple Lie group of isometries G on the common eigenspaces of the (commutative) algebra of the invariant differen-tial operators on the respective compact symmetric space M = G/K is irreducible.This algebra contains the Laplace-Beltrami operator and is generated by k genera-tors, where k = rk ( M ) is the rank of M . One of the goals of this paper, of whichsome results were reported in [19], is to propose a direct geometric relation betweenthe two theories. The most explicit illustration of the relation is through geometricquantization of the geodesic flow in the case rk ( M ) = 1 and we briefly explain it first.We assume for simplicity that such compact Riemannian symmetric space of rankone (CROSS for short) is simply-connected and irreducible, this leaves the well-knownexamples of the spheres, classical projective spaces (complex and quaternionic) and Date : February 15, 2021. the Cayley plane. These are also the known simply-connected examples in dimensionhigher than two of Riemannian manifolds all of whose geodesics are closed. On aRiemannian manifold (
M, g ) all of whose geodesics are closed there is a natural S -action on its tangent bundle T M and the geodesic flow on the cotangent bundle T ∗ M can be realized as solution to an S -invariant Hamiltonian system. For such systems,under mild conditions, there is a moment map and a symplectic reduction process,called also Marsden-Weinstein reduction. This reduction produces a reduced space T ∗ M//S that can be identified with the space parametrizing all geodesics and thatis equipped with an induced symplectic form. The induced symplectic form dependson a level set of the corresponding moment map which we call the energy level of thegeodesic flow. In many examples the cotangent bundle has a ”complex polarization”- a complex structure compatible with the symplectic form which becomes K¨ahlerform. We use this form to apply a twisted version of Kostant-Souriau geometricquantization scheme (originally due to [11, 25], )and assign a holomorphic line bundlewith first Chern class given by the induced K¨ahler form with an added extra term.This new term is half of the first Chern class of the canonical bundle of the manifold T ∗ M//S . Sometimes this is called prequantum bundle and we use this terminology.The quantum condition is the integrality of that corrected form, while the analogof the Hilbert space of quantum observables is the space of holomorphic sectionsof the prequantum bundle. Our first general result is Theorem 4.1 in Section 4 inwhich we show that the quantized energy levels of the geodesic flow on a simplyconnected rank one symmetric space are, up to a constant, equal to the eigenvalues ofthe Laplace-Beltrami operator on M , and the corresponding complexified eigenspacesare isomorphic to the spaces of the holomorphic sections of the prequantum bundleover the reduced space which is a generalized flag manifold. Although we provide therepresentation theory background for a unified proof, we proceed with a case by caseproof since it illustrates the explicit nature of the calculations and provides a basisfor the next parts of the paper.In the next Sections we consider the case of general rank. We observe that wecan substitute the space parametrizing all geodesics with the space of all maximaltotally geodesic flat submanifolds, which are tori in this case. This space is again ageneralized flag manifold and carries a natural ”polarization” which could be usedfor the quantization - a K¨ahler complex structure. Since our aim is to underline thegeometric approach through the Marsden-Weinstein reduction, we also need a K¨ahlerspace with a (multi-dimensional) Hamiltonian that, after the symplectic reduction,will become the generalized flag manifold with an appropriate reduced symplecticform, a form which is also integral and K¨ahler. This is done in [19] via constructionof a K¨ahler structure on some open subset of the manifold of all tangent spaces of themaximal totally geodesic flat submanifold. Then we prove the main result in Section6 - Theorem 6.1 which was announced in [19]. When the symmetric space has amaximal rank rk ( M ) = rk ( G ), then the corresponding generalized flag manifold isactually the full flag manifold G/T , where T is a maximal torus in G . From the APLACE EIGENFUNCTIONS AND BOREL-WEIL THEOREM 3
Borel-Weyl theorem follows that every irreducible representation of G appears as aspace of holomorphic sections of some line bundle over G/T . This corresponds tothe fact that the symmetric spaces of maximal rank provide the largest variety ofthe irreducible representations of G appearing as subspaces of the eigenspaces of theLaplace-Beltrami operator by the results in [23]. As an example we consider the space SU (3) /SO (3) which is of rank two, and also the simplest example of Riemanniansymmetric space of maximal rank. We note that a general correspondence similar tothe one in Theorem 6.1 has appeared in [15] and [24] Chapter 6, but with a focus onvarious integral transforms.In the remaining part of the paper we indicate a construction which relates theholomorphic sections of the prequantum bundle to the eigenfunctions of the corre-sponding eigenvalue on the symmetric space. We use the standard description ofthe holomorphic sections as holomorphic functions on the total space of the asso-ciated principal C ∗ − bundle with appropriate equivariant condition. Using the basicproperties of the Laplace-Beltrami operator under Riemannian submersions, the rela-tion between holomorphic and harmonic functions on a special non-K¨ahler manifolds,and an extension of the standard relation between harmonic polynomials and eigen-functions on the spheres, in Theorem 7.1 we propose a method of description of theLaplace-Beltrami eigenfunctions out of the holomorphic sections on the associatedquantization space. We apply the method to complex and quaternionic projectivespaces as well as the space SU (3) /SO (3). In these examples we describe a spanningset of all eigenfunctions, which consists of algebraic functions. The cases of quater-nionic spaces and SU (3) /SO (3) are new and extend the known representations forthe spheres and complex projective spaces [32]. We expect that many other symmet-ric spaces will have similar complete description. Note that the known descriptionsof the eigenfunctions so far are based on the integral geometry and various types ofRadon transform which are implicit.The structure of the paper is as follows: in the preliminary Sections we collect thenecessary background facts about symmetric spaces and flag manifolds and noticea simple connection between the Satake diagram of the symmetric space and thepainted Dynkin diagram of the generalized flag manifold. We use it do describe thesecond cohomology group of the quantization space in terms of the Satake diagramof the initial symmetric space. Then in Section 4 we treat the rank one case and inSections 5 and 6 we consider the spaces of any rank. The last two Sections - 7 and 8provide the construction of harmonic polynomials from holomorphic sections of theprequantum line bundle over the corresponding generalized flag manifold.2. Riemannian symmetric spaces and generalized flag manifolds
Let G be a compact semisimple Lie group and K a Lie subgroup given by a fixedpoint set of an automorphic involution θ . Then M = G/K carries a Remannianmetric which makes it a Riemannian symmetric space. Denote by g and k the Lie DIMITAR GRANTCHAROV, GUEO GRANTCHAROV AND CAMILO MONTOYA algebras of G and K . There is an eigenspace decomposition of θ g = k + p wherewe identify p with T o M the tangent space at the o = eK of M . It is known that g n i p defines the non-compact dual Lie algebra of g with respect to θ . Denote by G n the corresponding simply-connected Lie group. It is known that [ k , p ] = p and[ p , p ] = k . Denote by a the maximal abelian subalgebra in p . Then dimension of a iscalled a rank of M . Denote also by g C , k C , p C etc. the complexifications of g , k , p etc.respectively. Note that g C is a complexification of g n and i a is a maximal abelian in i p .The non-compact group G n admits an Iwasawa decomposition G n = KAN where A is the simply-connected Lie group with algebra i a and N is unipotent. Thereis a complex Iwasawa decomposition (see e.g. [9]) given by G C = K C A C N C where K C , A C , N C , and G C are the complexifications of K, A, N, G and G C is some Zariskyopen and dense subset of G C .It is known that K acts transitively on the set of all maximal abelian subalgebrasin p . Denote by M the stabilizer of a in K and by m its Lie algebra. It followsthat it may be expressed as m = { X ∈ k : [ X, a ] = 0 } and since a is maximal, thecentralizer of a in g is l = { X ∈ g : [ X, a ] = 0 } = m + a . If L is the correspondingsubgroup in G , then it contains a maximal torus of G (as a centralizer of an abeliansubgroup) so the space G/L is a generalized flag manifold (also referred to in theliterature as a K¨ahlerian C-space and rational homogeneous manifold) and carries anatural complex structure, as well as a K¨ahler metric. Its geometric interpretation iselucidated in the following comment.
Remark 2.1.
From geometric viewpoint a maximal abelian subalgebra of p is tangentto maximal totally geodesic flat torus and every such torus is at a point gK is tangentto a left translate of some a from o = eK to gK . In this terms the generalizedflag manifold G/L parametrizes the set of all such tori. We call the space
G/L aquantization space of
G/K . It is closely related to the horospherical manifold in [24] and [15] . An important invariant related to the quantization space is the dimension of itssecond cohomology. We demonstrate here how this dimension could be identified interms of the data provided by the Satake diagram associated to the noncompact dualRiemannian symmetric space of
G/K . Denote by m = [ m , m ]. Since m is compact, m is the semisimple part of m . We can consider a maximal torus of g which is θ invariant and contains a . Such torus is known to exists and since a is maximal abelianin p , then this torus is a Cartan subalgebra and has the form h = t + a with t ⊂ k .Its complexification h C = t C + a C is a Cartan subalgebra of g C . Let ∆ = ∆ + ∪ ∆ − bea root system with an ordering defining the positive and negative roots of g C . Thereis a set Σ of the so called restricted roots ∆ ⊃ Σ = Σ( g C , a C ) ⊂ ( a C ) ∗ and we canchoose a basis h , ..., h k , h k +1 , ..., h n of ( h C ) ∗ , of basic roots, such that h , ..., h k (afterrestricting them via the projection h C → a C ) are basis for ( a C ) ∗ . We continue to usethe same notation h , ..., h k for the restricted roots. After we choose an ordering of APLACE EIGENFUNCTIONS AND BOREL-WEIL THEOREM 5 the basic roots, or equivalenlty a positive Weyl chamber, every element of ∆ resp. Σis an integer linear combinations with all non-negative or all non-positive coefficientsof h , ..., h n ( h , ..., h k resp.). Similarly we can choose a positive Weyl chamber inthe restricted roots. Then m C has a root decomposition with root spaces which areamong the root spaces g α of g C with respect to h C .3. Satake and painted Dynkin diagrams
The non-compact real form g n of g C which is dual of g have k as a maximal compactsubalgebra. We want to describe the relation between the Satake diagram of g n andthe painted Dynkin diagram associated to the generalized flag manifold G/L (recallthat l = a + m ). We first describe the Satake diagram of g n . For this we paint thesimple roots defining the root spaces of m C in black. The Satake diagram of g n is thenthe Dynkin diagram of g C with the black and white dots as described, but with anadditional arrows between the white roots, when there is an involutive automorphismof g such that the difference between the initial and the endpoint of an arrow is aroot in t ( see [29] Sect. 4.4 or [8] Sect. 2.3, for the facts about Satake diagrams).If we consider the normalizer q of l C = m C + a C , then q is a parabolic Lie algebrain g C , which defines the complex structure of the generalized flag manifold G/L . Asdescribed in [6], q contains a Borel subalgebra and is described by a subset of simpleroots of ∆ + and the complex structure is defined via the ordering. The paintedDynkin diagram of G/L in an analog of the Satake diagram and is defined - see [2, 3]as a diagram with vertices corresponding to the semisimple part of l painted in black.Since l = a + m , the semisimple (ss) part of l is precisely l ss = m . The descriptionleads to the following observation: Theorem 3.1.
Let M = G/K is an irreducible Riemannian symmetric space and
G/L is its quantization space. If g n is the non-compact dual form of g with respectto K , then the Satake diagram of g n with the arrows deleted corresponds precisely tothe painted Dynkin diagram defining the complex structure of G/L . From here we obtain:
Corollary 3.1.
The group H ( G/L, Z ) has no torsion and is generated by the ele-ments corresponding to the white vertices of the Satake diagram of g n .Proof: The fact that H has no torsion is well known. According to [4], thereis an isomorphism Z ( l C ) ∗ ≡ H ( G/L, R ) sometimes called transgression , given by α → i π dα . It has the property, that the fundamental weights in Z ( l C ) ∗ correspondto elements of H ( G/L, Z ) ∈ H ( G/L, R ). Then the first k elements w , ..., w k of thebasis w , ..., w n of the fundamental weights dual w.r.t Killing form on g C to h , ...h n ,define an integral basis of H ( G/L, Z ). It is known [2, 3], that this basis is generatedvia transgression by simple roots corresponding to the white vertices in the paintedDynkin diagram describing the complex structure of G/L which are in bijection with
DIMITAR GRANTCHAROV, GUEO GRANTCHAROV AND CAMILO MONTOYA the basis of the center of l . As explained above these are precisely the white verticesin the corresponding Satake diagram. Q.E.D.
Remark 3.1.
The results clarifies a misstatement in Lemma 12 of [19] . In what follows we use the notations h , ..., h n and w , ..., w n for the simple rootsand dual positive weights throughout the paper, as well as the notion of transgressionfrom the proof above. Example
Consider the complex Grassmanian SU ( p + q ) /S ( U ( p ) × U ( q )) with p ≤ q . Then s ( u ( p ) + u ( q )) = su ( p ) + su ( q ) + R is block-diagonally embedded in su ( p + q ). If su ( p + q ) is represented as the matrices of the form (cid:18) A B − B t C (cid:19) with B a complex p × q matrix and A, C skew-Hermitian, then k = s ( u ( p ) + u ( q )) correspondsto B = 0, p corresponds to A = C = 0 which provides the decomposition g = k + p .As is known (cf. Helgason or others) and could be checked directly, the space a could be identified with the matrices (cid:18) D − D t (cid:19) , where D is a real-valued upper-diagonal matrix d ... ... d ... ... . . ... . ... d p ... with a real diagonal matrix on thefirst p rows and p columns, only 0 entries on the remaining p-q columns. Now we cancheck directly that m is the spaces of the tracefree matrices in k (i.e. B = 0), where A is diagonal matrix with imaginery entries, and C = (cid:18) E F (cid:19) , with E being a p × p diagonal matrix with imaginery entries and F a skew-Hermitian q − p × q − p matrix. frrom here we can see that the center of l = m + a consists of a as wellas the matrices D D
00 0 αId , with D - diagonal with imaginary entries and α = − tr ( D ) - an imaginary complex number. In particular m = su ( q − p ) is thespace with skew-Hermitian matrix in the lower-right corner. Then the generalized flagmanifold is G/L ∼ = SU ( p + q ) /S ( T p × U ( q − p )) and the dimension of H ( G/L, Z )is 2 p . Note that the Cartan subalgebra h in this case is different from the standardone consisting of all diagonal imaginary matrices in su ( p + q ). If { e , ..., e p + q } is thecanonical basis in C n , then the algebra h is represented by the diagonal matrices inthe basis { f i = e i + ie i + p , f i , e j } where 1 ≤ i ≤ p and p + 1 ≤ j ≤ p + q . APLACE EIGENFUNCTIONS AND BOREL-WEIL THEOREM 7 Symplectic reduction and quantization of the geodesic flow ofthe symmetric spaces of rank one
We first recall Marsden-Weinstein (or symplectic) reduction and then the modifiedKonstant - Souriau geometric quantization scheme (twisted half of the canonical bun-dle - see Czyz [11] and Hess [25]). In [27] the authors related the energy spectrum ofthe quantized geodesic flow on a sphere with the eigenvalues of the Laplace-Beltramioperator. In what follows we first notice a genralization of this result from a rep-resentation theory viewpoint to all compact rank-one Riemannian symmetric spaces(CROSSes). Then we provide a similar detailed computations for two examples - C P n and H P n . The exposition follows [19] and is used later to determine a generating setof eigenfunctions defined by harmonic polynomials in an ambient space.For more details on the Marsden-Weinstein (or symplectic) reduction we refer thereader, for example, to [1], here we provide a partial review of the results we needlater. . If ( M, ω ) is a symplectic manifold and H is a function on M , then the vectorfield X H defined as dH ( Y ) = ω ( X H , Y ) is called Hamiltonian vector field. We will call H a Hamiltonian function and the triple ( M, ω, H ) - a Hamiltonian system. If G is agroup of symplectomorphisms then under mild conditions there is a map µ : M → g ∗ ,defined by dµ ( X ) = i X ω, where g is the Lie algebra of G and X ∈ g is identified with the induced vector field on M . When such µ exists, the action is called Hamiltonian and the space N = µ − ( c ) /G is called the Marsden-Weinstein reduction or the symplectic reduction, where c is afixed element of the adjoint action of G on g ∗ . We denote N by M//G . The space
M//G inherits a natural symplectic form ω red such that i ∗ ( ω ) = π ∗ ( ω red ) where i : µ − ( c ) → M is the inclusion and π : µ − ( c ) → N = µ − ( c ) /G is the naturalprojection. The following results will be use repeatedly in the paper (see e.g. [1] forthe proof). Proposition 4.1. If ( N, ω red ) is the symplectic reduction of ( M, ω ) under the actionof a Lie group G and H is a G-invariant function on M , then there is a unique func-tion H red on N such that π ∗ ( H red ) = i ∗ ( H ) . Moreover the flow of the vector field X H preserves µ − ( c ) and projects on N to the flow of the vector fields X H red . Moreover,if we have a second Hamiltonian action of a Lie group G on M which commuteswith the action of G , then the level sets of its moment map µ are G − invariant and µ | µ − ( c ) = π ∗ ( µ ) where µ is the moment map associated to the action of G on N . Proposition 4.2. If G is a compact group of isometries acting freely on the Rie-mannian manifold ( M, g ) and N = M/G is the orbit space, then for the canonicalsymplecitc forms Ω M , Ω N on T ∗ M, T ∗ N respectively we have T ∗ N = T ∗ M//G with Ω N being the reduced form from Ω M . The geodesic flow on a Riemannian manifold is represented as a Hamiltonian flowon its cotangent bundle. The cotangent bundle of each Riemannian manifold (
M, g ) DIMITAR GRANTCHAROV, GUEO GRANTCHAROV AND CAMILO MONTOYA has a canonical symplectic form given in local coordinates as Ω = P dx i ∧ dy i where( x , ..., x n ) are local coordinates of M and ( x , ..., x n , y , ..., y n ) are the associated localcoordinates of T ∗ M . Then the function H ( x, v ) = g ( v, v ) for x ∈ M and v ∈ T ∗ x M has a Hamiltonian vector field X H and its flow lines project on M to give the geodesics.In particular, i X H Ω = dH . If all the geodesics of M are closed, then they define an S -action on T ∗ M with orbits ( c ( t ) , g ( c ′ ( t )) for a geodesic c ( t ) and the dual 1-form g ( c ′ ( t )) of its tangent vector c ′ ( t ). Which means that the moment map µ ( x, v ) at( x, v ) ∈ T ∗ M, v ∈ T ∗ x M for this S -action is precisely µ = H . When we fix the level c of the moment map, the points in the reduced space H − ( c ) /S represent geodesicson M with tangent vectors of length c . Which explains that Geod ( M ) = T ∗ M//S as sets. We note that the reduced form Ω c from the canonical form on T ∗ M dependson the choice of the level set µ − ( c ) for the moment map of the action µ (which iscalled the energy of the geodesic flow ).Now recall some facts about the quantization scheme of Konstant and Souriau withthe amends of Czyz and Hess [11, 25]. Let X be a compact K¨ahler manifold withK¨ahler form λ . We say that the holomorphic line bundle L is a quantum line bundle if its first Chern class satisfies c ( L ) = 12 π [ λ ] − c ( X ) . Thus X will be quantizable if and only if π [ λ ] − c ( X ) ∈ H ( X, Z ) . The corre-sponding quantum Hilbert space is the (finite dimensional) linear space H ( X, O ( L )).We want to apply the scheme to the space of geodesics of a Riemannian manifold allof whose geodesics are closed.Main examples of such manifolds are the compact Riemannian symmetric spacesof rank one (CROSS for short). Recall that for a compact irreducible Riemanniansymmetric space G/K with simple Lie group G we associated a quantization space G/L parametrizing all maximal totally geodesic flat tori in
G/K . The space
G/L is a generalized flag manifold, so smooth projective variety and from the descriptionof its second cohomology we know that its Picard group is generated by the centerof l = a + m . In particular it contains the fundamental weights w , ..., w k whichcorrespond to the restricted roots for a . Now denote by L = L i ,...,i k the holomor-phic line bundle on G/L determined by w = i w + ...i k w k , where i j ≥
0. By Bottvanishing the higher cohomology of L are zero. The space H ( G/L, O ( L )) is a (uni-tary) representation of G with highest weight w . The Borel-Weil theorem shows thatthe representation is irreducible if w is dominant, and corresponds to the (unique)irreducible representation with highest weight w [31].On the other side, the general theory for the Laplace spectrum on symmetric spaces([7, 32]) tells us that the eigenvalues are given by λ = || ρ (( a C ) ∗ ) + w || − || ρ (( a C ) ∗ ) || where w is as before and ρ (( a C ) ∗ ) is the half sum of positive restricted roots of a C .Then ρ represents one half of the first Chern class of G/L , so ρ (( a C ) ∗ ) + w is the firstChern class of L ⊗ K . APLACE EIGENFUNCTIONS AND BOREL-WEIL THEOREM 9
In this Section we focus on the case of Riemannian symmetric spaces of rank one,since the correspondence in this case is most studied and related to the classicalquantization of the geodesic flow. In the next Sections we’ll generalize the schemeto the symmetric spaces of higher rank. When the rank of M is one, the space ofthe restricted roots Σ is 1-dimensional as is the Weyl chamber in it. The set offundamental weights in it is (see [22]):Λ + = (cid:26) λ ∈ a C | < λ, ψ >< ψ, ψ > ∈ Z + , for all ψ ∈ Σ (cid:27) and in the rank one case is generated by a single element θ . The considerations abovegive the following result, which we’ll generalize in the next Section. Theorem 4.1. let M = G/K be an irreducible simply-connected compact Riemanniansymmetric space of rank one (CROSS). Then up to re-scaling of the metric on M thefollowing are true:i) Under the transgression the reduced symplectic form Ω c on Geod ( M ) = G/L = T ∗ M//S corresponds to π √ cθ and with the choice of the positive Weyl chamber andcomplex structure as above, c ( G/L ) corresponds to N M θ for a positive integer N M .ii) The quantum condition on ( Geod ( M ) , Ω c ) (i.e. [Ω c ] ∈ H ( G/L, Z ) ) providesthe following energy spectrum: c k = 1 / N M + 2 k ) .iii) The spectrum of the (semi)-Laplacian ∆ M on M is given by λ k = || kθ + ρ ( a C ) || − || ρ ( a C ) || and c k = || kθ + ρ ( a C ) || where ρ ( a c ) is the half-sum of the positiverestricted roots of a C .iv) The multiplicities of c k and λ k coincide with the dimension of the (finite - dimen-sional) representation L ( kθ ) of g with highest weight kθ relative to ( h , ∆) . Moreoverthe representation L ( kθ ) is isomorphic to both the (complex) eigenspace L ( M ) λ k of ∆ M corresponding to λ k and the quantization space H ( Geod ( M ) , O ( L k )) .Proof: The spaces in the Theorem are classified and are S n , C P n , H P n , CaP . Inthe two examples below we give a proof in case of C P n and H P n . The case of S n is considered in [27]. In [27] the space of oriented geodesics of M = S n is explicitlyidentified with the complex quadric in C P n via the Marsden-Weinstein reduction.It was noted that the energy levels of the moment map that satisfy a quantizationcondition coincide, up to an additive constant, with the eigenvalues of the Laplace-Beltrami operator and the their multiplicity are the same as the (complex) dimensionof the holomorphic sections of the corresponding quantum bundle L ( kθ ) ( see also[30] for related results).Finaly consider the case CaP = F /Spin (9). Since M has rank one, then thereduction identifies the level set of the moment map with a spheree bundle over M .From [14] Propsition 3.3 follows that it is diffeomorphic to F /Spin (7). This givesan identification of the quantization space with F /Spin (7) × S . Its painted Dynkindiagram from [3], Table 4 and Corollary 3.1 follows that H ( F /Spin (7) × S , Z ) = Z , so the reduced form Ω c is proportional to the generator. Since the generator corresponds to θ under the transgression, and the proportionality constant dependson c , and i ) follows. Then ii ) follows by the quantization condition, iii ) and iv ) bycombining the results from [7] and Borel-Weil Theorem. Q.E.D.
We note that the simply-connected requirement could be lifted and similar state-ment could be stated for R P n . Remark 4.1.
The re-scaling factor mentioned in the Theorem could be different forthe different spaces. It is known that the eigenvalues of the Laplace-Beltrami operatorfor S n and C P n are k ( n + k − and k ( n + k ) in the round metric on S n and theFubini-Studi metric on C P n respectively. So the two metrics are rescaled differently,one by a factor 4 times the other. We continue with the explicit calculations of the two classical simply-connectedprojective spaces.4.1.
Complex projective space.
We’ll use and extend here the results of [12].Throughout this subsection for complex vectors z, w we denote by h z, w i = Re P z i w i their hermitian scalar product and by z.w = P z i w i the complex scalar product so || z || = p h z, z i . For a point [ z ] = [ z , z , ..., z n ] in the complex projective space C P n ,we identify the holomorphic cotangent space T ∗ [ u ] C P n ∼ = { ([ u ] , v ) ∈ { [ u ] } × C n +1 | u.v = 0 } where we used the Fubini-Study metric to identify the tangent and cotangent bundles.To achieve a global description of the cotangent bundle, we use the Hopf map π : S n +1 → C P n which is induced by the standard action of S on S n +1 . This map isdefined by u [ u ], where u ∈ R n +2 = C n +1 with || u || = 1. After identifying thetangent and cotangent bundles of the sphere via the canonical metric, we can identifythe cotangent bundle as T ∗ S n +1 = { ( u, v ) ∈ C n +1 × C n +1 | || u || = 1 , h u, v i = 0 } Then the S -action ρ for the Hopf projection π extends to T ∗ S n +1 as ρ ( e iθ )( u, v ) = ( e iθ u, e iθ v )This action preserves the canonical symplectic form on T ∗ S n +1 , which is given by i ∗ Re ( du ∧ dv ). The moment map for the action ρ can be used to show the followingtheorem. This theorem is first proven in [12], but for reader’s convenience a shortproof is presented. We consider the cotangent bundle with its zero section deleted T ∗ C P n (and T ∗ S n +1 ) in order to avoid the singularity issues since they are irrelevantin the paper. Lemma 4.1.
The space T ∗ C P n is diffeomorphic to both X C and f X C where X C ∼ = { [ u, v ] | || u || = 1 , u.v = 0 , v = 0 } APLACE EIGENFUNCTIONS AND BOREL-WEIL THEOREM 11 with [ u, v ] representing the class of ( u, v ) under ( u, v ) ∼ ( e iθ u, e iθ v ) and f X C ∼ = { [[ u, v ]] | h u, u i = h v, v i 6 = 0 , u.v = 0 } with [[ u, v ]] defined by the relation ( u, v ) ∼ ( e iθ u, e − iθ v ) . Moreover T ∗ C P n is bi-holomorphic to f X C when it is identified with T ∗ S n +1 //S and the reduced complexstructure.Proof: It is well-known that under the action ρ , T ∗ C P n = T ∗ S n +1 //S . Themoment map Φ associated to the action ρ is simply Φ( u, v ) = Im h u, v i . Hence, T ∗ S n +1 //S = Φ − ( µ ) /S , for a generic µ ∈ R = iu (1), is identified with X C whichgives the diffeomorphism T ∗ C P m ∼ = X C . The diffeomorphism between X C and ˜ X C isgiven by the formulas: ˜ u k = 1 √ || v || u k + iv k ) , ˜ v k = 1 √ v k − i || v || u k ) . The biholomorphism follows from the fact that the reduction is K¨ahler, when weconsider the canonical form on T ∗ S n +1 as a K¨ahler form for the complex structureinduced from the embedding in C n +2 as in [30] for example. Q.E.D.
By lemma 4.1, if a Lie group G of isometries acts on M , this action induces aHamiltonian action on T ∗ M and the reduced space T ∗ M//G becomes a (reduced)Hamiltonian system. Whenever T ∗ M//G = T ∗ N for some Riemannian manifold N then the solutions of the new system is precisely the geodesic flow on N . In theparticular case of T ∗ S n +1 we obtain the following. Proposition 4.3.
The canonical symplectic form Ω C on T ∗ C P n ∼ = X C is Ω C = 12 ( du ∧ dv + du ∧ dv ) and the Hamiltonian system H C P n = ( X c , Ω C , H C = || v || ) induces the geodesic flowon C P n . The system is equivalent to ( e X C , e Ω C , e H C ) in view of the diffeomorphism inLemma 4.1. Since the orbits of H C P n correspond precisely to the geodesics of C P n , we firstidentify the space parametrizing the geodesics. For this we first consider the geodesicflow on the sphere S n +1 . Since all of the geodesics on the sphere are closed, theflow of X H in the cotangent space has also only closed trajectories. They define an S -action which is given by ( u, v ) → ( e iθ u, e − iθ v ). This action commutes with theaction inducing the Hopf projection and is Hamiltonian. So it defines an action on T ∗ C P n which has orbits - the flow lines of the Hamiltonian vector field defining thegeodesics on C P n . We can identify a geodesic c ( t ) in C P n with the line ( c ( t ) , c ′ ( t ))in T C P n ≃ T ∗ C P n when t is a parameter such that c ′ has constant norm. From here we see that the space parametrizing the geodesics can be identified with theMarsden-Weisntein quotient. Let N c = e H − C ( c ) /S be the reduced space. To identify N c with a flag manifold, we use the Hamiltonian system ( e X C , e Ω C , f H C ). Let F = { ([ z ] , [ w ]) ∈ C P n × C P n : z.w = 0 } = { ([ z ] , [ w ]) ∈ C P n × C P n : h z, z i = h w, w i , z.w = 0 } One can see that F is biholomorphic to the (1,2)-flag in C n +1 with homogeneousrepresentation F = U ( n + 1) /U (1) × U (1) × U ( n − p and p the twoprojections on the corresponding factors of C P n × C P n . Let α be the generator (theFubini-Study form) of H ( C P n , Z ). Then ω = p ∗ α and ω = p ∗ α are generators of H ( F , Z ). With this notation we have the following. Proposition 4.4. If c = 0 then the reduced manifold N c is biholomorphic to the flag F and the reduced K¨ahler form is e ω c = π √ c ( ω + ω ) .Proof: The S -action of the geodesic flow on T ∗ C P n is induced from the one on T ∗ S n +1 . Hence this action is: λ ( z, w ) = ( λz, λw ) , for ( z, w ) ∈ e H − C ( c ). For the sphere S n +1 R of radius R the Hopf projection fits inthe diagram C n +1 S n +1 Ri o o h / / C P n with h ∗ α = πR i ∗ Ω (see [27]). If ˜ π c is theprojection e H − C ( c ) → N c = F then we have the following commutative diagram: e H − C ( c ) (cid:15) (cid:15) ˜ π c / / N c ˜ i c (cid:15) (cid:15) S n +1 × S n +1 h × h / / C P n × C P n , where the vertical arrows correspond to the natural embeddings. Therefore,˜ π ∗ c ( √ cπ ( ω + ω )) = π √ c √− π ( dz ∧ dz || z || + dw ∧ dw || w || )= √ c √− ( dz ∧ dz + dw ∧ dw )= ( du ∧ dv + du ∧ dv )= ˜ i ∗ c ( ˜Ω c )In the above calculation we used that e H C ( z, w ) = c , so || z || = || w || = 2 c . Wewant to use the modified Kostant - Souriau scheme to ”quantize” the geodesic flowof C P n . Proposition 4.5.
We have c ( F ) = n ( ω + ω ) APLACE EIGENFUNCTIONS AND BOREL-WEIL THEOREM 13
Proof:
We apply the adjunction formula for a hypersurface of degree (1,1) in C P n × C P n to obtrain c ( F ) = − ( c ( K C P n × C P n | F ) + c ([ F ] | F ))= c ( C P n × C P n ) | F − c ([ F ] | F )= ( n + 1)( ω + ω ) − ( ω + ω )= n ( ω + ω )Q.E.D. Theorem 4.2.
The energy spectrum of the geodesic flow on C P n is: E k = 12 ( n + 2 k ) , k ∈ N , with corresponding multiplicities m k = (cid:18) n + kk (cid:19) − (cid:18) n + k − k (cid:19) . Proof:
For the exact cohomology sequence: H ( F , O ) → H ( F , O ∗ ) → H ( F , Z ) → H ( F , O )and the identities H ( F , O ) = H ( F , O ) = 0 follows that: c : H ( F , O ∗ ) ∼ = −→ H ( F , Z ) ∼ = Z ⊕ Z . Therefore every holomorphic line bundle L on F is equivalent to L k ,k = k π ∗ ( H ) + k π ∗ ( H ), where H is the hyperplane section on C P n .The quantum condition on c is :12 π [ ω c ] − c ( F ) = c ( L k ,k )which implies √ c − n k where k = k = k is a positive integer. In particular c = 12 (2 k + n ) To count the multiplicities (i.e. dim H ( F , O ( L )) we consider the exact sequenceof sheaves:(1) 0 → O C P n × C P n ( L k − ,k − ⊗ L , ) α −→ O C P n × C P n ( L k,k ) r −→ O| F ( L k,k ) → , where α is the multiplication of sections of L k,k by the polynomial P n z i w i whichdefines F in C P n × C P n and r is the restriction. The corresponding exact cohomologysequence gives:0 → H ( C P n × C P n , O ( L k − ,k − )) → H ( C P n × C P n , O ( L k,k )) → H ( F , O ( L k,k )) → H ( C P n × C P n , O ( L k − ,k − )) = 0 where the last term is zero by the Kodaira vanishing theorem. Thus we have: m k = dim( H ( F , O ( L k,k ))= dim( H ( C P n × C P n , O ( L k,k )) − dim( H ( C P n × C P n , O ( L k − ,k − )))= (cid:0) n + kk (cid:1) − (cid:0) n + k − k (cid:1) . Q.E.D.4.2.
Quaternionic projective space.
We first note that the results in this sub-section were independently obtained in [26] and some of them appear in [12]. Forreaders convenience, in this section we use a slightly different notations to distinguishbetween real complex and quaternionic scalar products. In particular we use h x, y i R h x, y i C and h x, y i H for x.y = P x i y i when x i , y i are in R , C , and H respectively. Thecorresponding norms arising from their rea parts are denoted by || . || R , || . || C , || . || H re-spectively. The geodesic flow on H P n can be described in a similar way as the one for C P n but with the aid of the quaternionic Hopf map. For that we use three equivalentrepresentations of T ∗ S n +3 : T ∗ S n +3 = { ( x, y ) ∈ R n +3 × R n +3 : || x || R = 1 , h x, y i R = 0 } = { ( u, v ) ∈ C n +2 × C n +2 : || u || C = 1 , Re h u, v i C = 0 } = { ( p, q ) ∈ H n +1 × H n +1 : || p || H = 1 , h p, q i R = 0 } , where p k := u k + u k +1 j, q k := v k + v k +1 j and h p, q i R = Re h p, q i H = m Re P p k q k .The quaternionic Hopf map in this case is χ : S n +3 → H P n , p → [ p ] where [ p ] =[ p , p , ..., p n ] is the class of p for the relation p ∼ σp, σ ∈ Sp (1). The next lemma isagain from [12]. Lemma 4.2.
The cotangent space T ∗ H P n is diffeomorphic to both X H and e X H de-fined as follows: X H := {⌊ p, q ⌋ ∈ H n +1 × H n +1 : || p || H = 1 , h p, q i H = 0 } , e X H := {⌊ z, w ⌋ ∈ C n +2 × C n +2 ; || z || C = || w || C , h z, w i C = 0 , I ( z, w ) = 0 } where I ( z, w ) = z w − z w + ... + z n w n +1 − z n +1 w n and ⌊ p, q ⌋ and ⌊ z, w ⌋ denotethe equivalence classes of ( p, q ) and ( z, w ) under ( p, q ) ∼ ( σp, σq ) and ( z, w ) ∼ ( z, w ) g for σ ∈ Sp (1) and g ∈ SU (2) ∼ = Sp (1) .Proof: Consider the action of SU (2) on S n +3 defined by(2) Ψ g ( p, q ) := ( p, q ) g, g ∈ SU (2) . This action has a moment map G : T ∗ S n +3 → su (2) ∗ , given by the formulas G ( p, q ) =( A ( p, q ) , B ( p, q ) , C ( p, q )), where h p, q i H = Re( h p, q i H ) + A ( p, q ) i + B ( p, q ) j + C ( p, q ) k, and the imaginary quaternions are identified with su (2) ∗ . Hence, T ∗ S n +3 //SU (2) = X H ∼ = T ∗ H P n . APLACE EIGENFUNCTIONS AND BOREL-WEIL THEOREM 15
To prove that X H and e X H are diffeomorphic, consider the map t H : X H → ˜ X H , ( z, w ) = t H ( p, q ), where z k := 1 √ || v || C u k + √− v k ) z k +1 := 1 √ −|| v || C u k +1 − √− v k +1 ) w k := 1 √ v k +1 − √− || v || C u k +1 ) w k +1 := 1 √ v k +1 − √− || v || C u k +1 )Q.E.D.The action Ψ defined in (2) commutes with the geodesic flow of S n +3 . Recall thediffeomorphism t H : X H → e X H defined at the end of the last proof. Like in theprevious subsection, we have the following. Proposition 4.6.
Let Ω H = Ω T ∗ H P n be the canonical symplectic form on T ∗ H P n .Then Ω H = 12 ( du ∧ dv + du ∧ dv ) Moreover the geodesic flow of H P n is the flow of the equivalent Hamiltonian systems ( X H , Ω H , G H ) ∼ = ( e X H , e Ω H , e G H ) where G H = || q || H || v || C , e Ω H = t ∗ H Ω H and e G H = t ∗ H ( G H ) . Next we compute the energy spectrum of the geodesic flow on H P n in a similarway as in the case of C P n . We consider again the reduced space O c = T ∗ H P n //S =˜ G − ( c ) /S with the induced symplectic form ω c obtained from ˜ i ∗ c Ω H = ˜ π ∗ c ω c , where˜ i c : e G − ( c ) → T ∗ H P n and ˜ π c : e G − ( c ) → O c . Denote by F is the isotropic Grassmannmanifold F is = { Λ ∈ Gr ( C n +2 ); I | Λ = 0 } = { [[ z, w ]] ∈ C n +1 × C n +1 : || z || C = || w || C = 1 , h z, w i C = I ( z, w ) = 0 } where [[ z, w ]] is representative of ( z, w ) for ( z, w ) ∼ = ( λz, λw ) g , λ ∈ S , g ∈ SU (2)or equivalently ( z, w ) ∼ = ( z, w ) g, g ∈ U (2). Alternatively, F is is a hyperplane in Gr ( C n +2 ): F is ∼ = { ( λ ij ) ∈ Gr ( C n +2 ) : λ + λ + ... + λ n +1 , n +2 = 0 } , where ( λ ij ) are the Pl¨ucker coordinates on Gr ( C n +2 ), as well as a homogeneousspace: F is ∼ = Sp ( n + 1) /U (2) Sp ( n − Proposition 4.7. If c = 0 then the reduced space O c is isomorphic to F is equippedwith the K¨ahler form e ω c = π √ cω , where ω is the restriction of the canonical K¨ahlerform on Gr ( C n +2 ) which generates H ( Gr ( C n +2 ) , Z ) .Proof: The S action of the geodesic flow on e G − H ( c ) ⊂ T ∗ H P n ∼ = e X H is: λ ⌊ z, w ⌋ = ⌊ λz, λw ⌋ . which commutes with the action of Sp (1) ∼ = SU (2) defining the quaternionic Hopffibration. Now from e G H ( z, w ) = c we have || z || C = || w || C = 2 c . If λ ij = z i w j − z j w i are the Pl¨ucker coordinates on Gr ( C n +2 ) then:˜ π ∗ c ( π √ cω ) = π √ c √− π dλ ij ∧ dλ ij P i,j || λ ij || = √ c √− ( dz ∧ dz + dw ∧ dw )= ˜ i ∗ c ( e Ω H )Q.E.D. Proposition 4.8.
We have c ( F is ) = (2 n + 1) ω .Proof: We note that c ( Gr ( C n +2 ) | F is = (2 n + 2) ω and then proceed with theadjunction formula as in Proposition 2.4 using the fact that F is is a hypersurface in Gr ( C n +2 ).Q.E.D. Theorem 4.3.
The energy spectrum of the geodesic flow on H P n is E k = 12 (2 n + 1 + 2 k ) , k ∈ N with corresponding multiplicities: m k = (2 n + 2 k − n + k − n − Y j =3 (cid:18) k + jj (cid:19) . Proof:
We only sketch the proof since it is similar to the C P n case. We have c : H ( F is , O ∗ ) → H ( F is , Z ) = Z . Therefore all holomorphic line bundles on F is which arise from the quantization are L k := S ⊗ k , where S = ι ∗ ([ H ]) and ι is theinclusion ι : F is → Gr ( C N +2 )). Hence, √ c − n + 12 = k. The dimension is calculated in [7].
Q.E.D.
This Theorem finishes the case by case proof of Theorem 4.1.
APLACE EIGENFUNCTIONS AND BOREL-WEIL THEOREM 17
Remark 4.2.
Existence of a symplectic form on the cotangent bundle doesn’t havea direct analog to use in the case of higher-rank symmetric spaces. However not allsimple and simply connected Lie group acts transitively on a CROSS. And the rep-resentation theory suggests that the correspondence could be extended to the higherdimensional case. In the next Sections we present one possible extension of the cor-respondence to the higher rank spaces based on geometry of toric bundles over flagmanifolds. Symplectic geometry of complex torus bundles
Let T n = S × S × ... × S be a (real) n-dimensional torus. Then its tangent bundleis trivial and there is a well-known the identification T ( T n ) ≡ T ( S ) × T ( S ) × ... × T ( S ) ≡ ( C ∗ ) n ≡ ( T n ) C which is the complex n-dimensional torus. In particularit is an open and dense subset of C n and has an induced complex structure andK¨ahler metric. If z k = r k e iθ k are the coordinates in ( C ∗ ) n then the K¨ahler form canbe written as ω = P d ( r k ) ∧ dθ k = d ( P r k dθ k ). One can see that the T n action on( C ∗ ) n ( z , ..., z n ) → ( e iα z , ..., e iα n z n ) is Hamiltonian with moment map µ ( z , ..., z n ) =( r , ..., r n ) = ( | z | , ..., | z n | ). Now we want to extend it to torus bundles: Theorem 5.1.
Let π : P → M be a principal T n -bundle over a K¨ahler manifold M with characteristic classes of type (1 , . Let P C = P × T n ( C ∗ ) n be the associatedcomplex torus bundle with the standard right action of T n on ( C ∗ ) n . then P C isopen and dense subset of the vertical tangent bundle V of P and carries a naturalcomplex structure and compatible symplectic (pseudo-K¨ahler) form ω . Moreover the T n action on P C is Hamiltonian and the Marsden-Weinstein reduction P C //T n isdiffeomoerphic to M for a generic level set of the corresponding moment map.Proof: A principal torus bundle is determined, up to an isomorphism, by its char-acteristic classes on the base. Then we have a closed and integral (1 , M , ω , ..., ω n and a connection 1-forms θ , .., θ n on P , such that dθ k = π ∗ ( ω k ). The pro-jection map z : P × ( C ∗ ) n → ( C ∗ ) n defines functions r k = | z k | which are T n -invariantand descend to P C . Now the forms θ k also descend to connection 1-forms on P C andwe can define an almost complex structure on P C as I ( dr k ) = θ k and on the horizon-tal co-vectors is just a pull-back of the complex structure on the base. It defines thestandard complex structure on the fibres ( C ∗ ) n . Its integrability follows from the factthat ω k are of type (1 ,
1) (see [17]). The symplectic form is ω = P d ( r k θ k ) + π ∗ ( ω M ),where ω M is a K¨ahler form which is positive enough to ensure that ω is non-degeneratein the horizontal directions for almost all x i . Now it is clear that for a basis of verti-cal vector fields X k which are defined by the T n action and satisfy θ i ( X j ) = δ ji , themoment map is µ = ( r , ..., r n ) as a R n -valued function on P C . So it is clear that for c ∈ R n where all coordinates are positive, µ − ( c ) ≡ P where we identify P with theset of points in P C with r k = 1 for all k . Then it is clear that P C //T n ≡ M .Q.E.D. We can see that the reduced symplectic form depends on the level c and is integralwhenever c satisfies some integrality condition - which will provide the quantumcondition for the correspondence in the higher rank symmetric spaces.We identify the reduced symplectic form on P C //T n in the following way: Corollary 5.1.
In the notations of the Theorem 5.1 and its proof, the symplecticform on P C is given by ω = d ( P x i θ i ) + π ∗ ω M , and the reduced symplectic form on P C //T n = µ − ( c , ..., c n ) ≡ M for a generic choice of ( c , ..., c n ) is ˜ ω M = P c i dθ i + ω M . Symmetric spaces of general rank
Now we apply the Corollary and the Theorem of the previous Section to the spaceparametrizing the maximal totally geodesic tori of a Riemannian symmetric space.Let as before M = G/K be a symmetric space with G compact and semisimple.Every maximal totally geodesic tori is tangent to a translated maximal commutativesubspace of m . Denote again by a one such fixed subspace. Also L is the connectedsubgroup of G with Lie algebra l = m + a , where m is the centralizer of a in k . We canalso write L = M A where M and A are the corresponding Lie groups (see [15], [16]).Then G/L is a generalized flag manifold parametrizing the maximal totally geodesictori in M . As such it caries a natural complex structure, which depends on the choiceof a Cartan subalgebra of g C and a partial order in it which determines a positiveWeyl chamber and it defines a positive Weyl chamber in ( a C ) ∗ . The later is dual to thecone of restricted dominant weights. Then as a complex manifold G/L is equivalentto G C /M C A C N C and has a principle A C -bundle G C /M C N C → G C /M C A C N C withtotal space - the horospherical manifold Θ. Since A C is the complexification of thereal torus T r = A and can be identified with the cotangent bundle T ∗ T r , then Θ canbe identified with the total space of the vertical (co)tangent bundle of the principalbundle G/M → G/L with fiber A . This is also the set of all tangent planes to allmaximal totally geodesic tori in M . In case the rank of M is r = 1, this is just T ∗ M . Since the characteristic classes of the bundle G/M → G/L are determined viatransgression by the simple roots in a ∗ , then we can apply the constructions of theprevious Section. Theorem 6.1.
Let M = G/K be a compact Riemannian symmetric space of rank k with G semisimple and let θ , ..., θ k be the basis of fundamental weights that is dualto the simple restricted roots of a C . Let Θ be the associated horospherical manifoldand Θ → G/L be the corresponding principal ( C ∗ ) k -bundle, so G/L parametrizesthe maximal totally geodesic tori in
G/K . Let ω M = ı2 π dρ be the 2-form on G/L representing c ( G/L ) , so ρ is the half sum of the positive roots in g c vanishing on l c (as in [2, 3] ). Then there exists a symplectic form ω on Θ with the followingproperties:i) There are positive numbers n i such that the reduced form ˜ ω on Θ //T k corre-sponding to ω via the Marsden-Weinstein reduction is ˜ ω = P ki =1 n i dθ i + ω M , on G/L . APLACE EIGENFUNCTIONS AND BOREL-WEIL THEOREM 19 ii) When the 2-form dα for α = n θ + n θ + ... + n k θ k is integral (up to afactor of π ) and determines a dominant weight, then the corresponding quantumbundle L defined by ˜ ω ∈ c ( L ) has the property that its space of holomorphic sections H ( G/L, O ( L )) is an irreducible unitary representation of G with highest weight α .iii) The complexified eigenspaces of the Laplace-Beltrami operator ∆ M correspond-ing to the eigenvalue λ α = || α + ρ a || − || ρ a || on M have dimension equal to the sumover all α with || α + ρ a || = λ α + || ρ || of the dimensions of H ( G/L, O ( L )) definedin ii). All eigenvalues of ∆ M are equal to λ α for some α .Proof: Because c ( G/L ) > M is K¨ahler.The form ω is closed and since θ i correspond to a basis of the positive Weyl chamber in a C , wesee that the ω is also K¨ahler as a sum of a positive and non-negative form.The reduced form coincides with of ˜ ω M by Corollary 5.1, which proves i ).The quantum line bundle is well defined since for generalized flag manifolds thePicard group is isomorphic to H ( G/L, Z ). Then ii ) follows from Borel-Weil Theorem.Finally iii ) is valid in view of the fact that the eigenspaces of the Laplace-Beltramioperator are sums of irreducible G -modules.Q.E.D. Example 6.1.
The space M = SU (3) /SO (3) . The space SU (3) /SO (3) has rank 2 - which is the rank of SU (3) (i.e. it is ofmaximal rank). If Z = X + iY ∈ su (3), then X ∈ so (3) and the decomposition of theLie algebra su (2) is determined by su (3) = so (3) + p where p is the space of purelyimaginary matrices. The space a is given by the diagonal ones and the Lie algebra of M is trivial. In particular the generalized flag manifold G/L is the standard manifoldof full flags in C identified with SU (3) /S ( U (1) × U (1) × U (1)). Then a choice ofthe simple roots is given by diagonal matrices , which up to a factor of √− α = (1 , − , , α = (0 , , −
1) and the other positive root is α − α + α . So thehalf sum of the positive roots is ρ = (2 α + 2 α ) = α = (1 , , −
1) for two diagonalmatrices H = ( h , h , h ) , H ′ = ( h ′ , h ′ , h ′ ) the product h H, H ′ i = Tr(ad H ad H ′ ) = P i
In this Section we describe a procedure to obtain an explicit algebraic expressionof the eigenfunctions of the Laplace-Beltrami operator on compact symmetric spacesthrough harmonic polynomials.We shortly describe the idea of the construction first. Consider the space of holo-morphic sections of a line bundle in the Borel-Weil Theorem. It is identified withholomorphic functions f on a principal C ∗ -bundle P over the (generalized) flag man-ifold F = G/L such that f ( xa ) = χ ( a ) f ( x ), where χ is the character of the repre-sentation in H ( F , L χ ) for the associated with P line bundle L χ from the Borel-WeilTheorem. The structure group of P could be reduced to S so P has a structure ofa cone P ∼ = R + × S , for S - the total space of an S -bundle over F . Note that P is different from Θ and sometimes can be represented as its quotient. The S actionon S is induced from the C ∗ -action on P such that for a = re iθ ∈ C ∗ we have theaction R a ( x, t ) = ( e iθ x, rt ). We note that S has a Sasakian metric g S (see [5]) andthere is a cone metric g P on P such that g P = dr + r g S . Then g P is the K¨ahlercone metric - as in Boyer-Galicki approach to Sasakian geometry [5]. In particularevery holomorphic function on P is also harmonic. Now the relation between theLaplace-Beltrami operators on the cone P and the base S is∆ P ( u ) = ∂ u∂r + n r ∂u∂r + r − ∆ S ( u )where u = u ( r, x ) and ∆ S ( u ) is calculated when S is embedded in P as r = constant .If the function u is corresponding to a holomorphic section then the equivariancecondition above gives for x = e iθ yu ( x, r ) = u ( e iθ y, r.
1) = r k e ikθ u ( e − iθ x, k = χ ( re iθ ). Then from the formulas we obtain ∂u∂r = kr u and(3) ∆ S u = λu when u ( x ) = u ( x,
1) and λ depends on k . In particular u determines an eigenfunctionof the Laplace-Beltrami operator on S . Now, in many cases, we can pull-back thefunction to Θ and if this pull-back is K -invariant, then it will define a function on G/K . This function is an eigenfunction if the projection is a Riemannian submersionwith totally geodesic fibers. To make this strategy work we have to resolve twoproblems. First we need to see when a pull-back to Θ is possible. Second, the metricson F which will lead to such projection are not K¨ahler - they arise from the biinvariantmetric on G . So we need a modification of this idea for non-K¨ahler metrics. We startwith the second problem. APLACE EIGENFUNCTIONS AND BOREL-WEIL THEOREM 21
Recall that a Hermitian metric g on a complex manifold M with a fundamentalform ω is called balanced, if dω n − = 0 where n is the complex dimension of M .A result in [20] shows that a holomorphic function on a balanced manifold is againharmonic. To use this property we need a few Lemmas. Lemma 7.1.
Suppose that M is a compact complex manifold of dimension n with abalanced metric g M which has fundamental form ω M , i.e. dω n − M = 0 Let π : P ∼ = R + × S → M be a principal C ∗ -bundle with U (1) -connection 1-form θ on S and acone metric of the form g = dr + r ( θ + π ∗ ( g M )) . Let dθ = π ∗ ( ω ) , where ω is aform of type (1,1), be the curvature of S (and P ). With respect to a natural complexstructure I on P compatible with g P , such g P is balanced iff ( ω − ω M ) ∧ ω n − M = 0 . Proof:
The complex structure on P is given by I ( dr ) = rθ, I ( θ ) = d log r and thepull-back of the complex structure on M on the horizontal spaces ker( θ ) ∩ ker ( dr ). Thefact that it is integrable follows from the condition that ω is (1,1) (see [17]). Supposethat the complex dimension of M is n , so dim C P = n + 1. Then the fundamentalK¨ahler form of g P is given by ω P = rdr ∧ θ + r π ∗ ( ω M ). For convenience we write ω M for π ∗ ( ω M ) where it is not confusing. Now ω nP = nr n +1 dr ∧ θ ∧ ω n − M + ω nM and dω nP = n (2 n + 1) r n dr ∧ ( ω M − ω ) ∧ ω n − M since dω n − M = 0. Q.E.D.
Then we have
Lemma 7.2.
Let P ∼ = R + × S be a principal C ∗ -bundle over a generalized flag manifold M = G/L where G is a compact simply-connected and semisimple Lie group and L isa centralizer of a torus. Assume that S has a connection 1-form θ which is G -invariantand its curvature ω has a cohomology class [ ω ] such that [ ω ] / π ∈ H ( M, Z ) and isnot an integer multiple of another class. Then there is a projection π : G → S whichis a factor-bundle. With respect to the natural complex structure, P admits a balancedcone metric g P = dr + r g S with an induced g S metric on S , such that the projection π is a Riemannian submersion with totally geodesic fibers when G is equipped with itsbiinvariant metric, after possible rescaling.Proof: The fact that there is such a projection follows from [18]. For every in-variant g M , the Hodge-dual ∗ d ( ω M ) n − of dω n − M is an invariant 1-form and the Eulercharacteristic of M is positive, so g M is balanced. From Lemma 7.1 we see that both ω nM and ( ω ∧ ω n − M ) are proportional to the invariant volume form on M . Whichmeans that up to a rescaling of the metric on M we could make them equal, so g P isbalanced. Lemma 7.3.
Every holomorphic function on P is harmonic with respect to the metric g P from Lemma 7.2.Proof: The result follows for example from [20].
Lemma 7.4.
A harmonic function F on P which satisfies f ( x, r ) = r k f ( x, inducesan eigenfunction of the Laplace-Beltrami operator on S .Proof: It follows from (3) and the calculations there.Now we consider the problem of existence of a pull-back of a function to Θ. Denoteby L ss the subgroup of G with Lie algebra l ss = [ l , l ] = [ m . m ] which is the semisimplepart of l . Consider G/L ss as a T k -principal bundle over the flag manifold G/L . Ithas characteristic classes given by γ i = π dω i where ω i , i = 1 , ..k are the fundamen-tal weights. Any principal S -bundle can be characterised topologically by its firstChern class, which is a positive integer combination of these. Let S be determinedby c ( S ) = P n i γ i , where n i are positive integers. According to Lemma 3 in [18],if gcd( n , ..., n k ) = 1, then we can find a basis of generators β = c ( S ) , β , ..., β k of H ( F , Z ) and they will define an equivalent principal bundle to G → F = G/T k . Inparticular, there is a principal T k − -bundle G/L ss → S and we can use the construc-tion above. If we dont have this condition, then c ( S ) = mβ , for some β and m positive integer, which satisfies it. Now we can replace S with another bundle S withcharacteristic class β . By a standard argument (for example comparing the Eulerclasses - see e.g. [10] Ex 3.26.) S = S/ Z m as a finite cover, so we have the projections G/L ss → S → S . Now G/L ss have two fibrations - over S and over the symmetricspace G/K . When we induce the metrics on
G/K, G/L ss , and S from the biinvari-ant metric on G , both fibrations are Riemannian submersions with totally geodesicfibers. For a such a Riemannian submersion π : M → N the relation between theLaplace-Beltrami operators on M and N is:(4) ∆ M ( f ◦ π ) = (∆ N f ) ◦ π for any smooth function f on N - see [33]The above considerations lead to: Theorem 7.1.
Suppose that
G/K is a compact Riemannian symmetric space and F = G/L is the associated generalized flag manifold - the quantization space. Let f ∈ H ( F , L χ ) be a holomorphic section of a positive line bundle L χ over a flag manifold F which is considered as function on the corresponding principle bundle P = S × R + with f ( za ) = χ ( a ) f ( z ) for a ∈ C ∗ . Let π : G/L ss → S and π : G/L ss → G/K be thenatural projections. If the pull-back of f to G/L ss × R + via π is K -invariant, then f satisfies the conditions of Lemma 7.4 and the function u ( x ) = f ( x, on S definesan eigenfunction u of the Laplace-Beltrami operator on the Riemannian symmetricspace G/K with π ∗ ( u ) = π ∗ ( u ) .Proof From the Lemmas above, u is an egienfunction on S . By the property (4) π ∗ ( u ) is an eigenfunction on G , and by the K -invariance it is a pull-back of a functionon G/K . Then again by (4) u is an eigenfunction on G/K . Q.E.D.
APLACE EIGENFUNCTIONS AND BOREL-WEIL THEOREM 23
Remark 7.1.
Using the Cartan embedding i : G/K → G ,we see that we needfunctions on G/K depending on the parameters defining the image of
G/K . We aregoing to use this in the examples below. Moreover, the generalized flag manifold F has more than one invariant complex structures - see [4] for example. Each of themdefines a set of eigenfunctions described in the Theorem. In the examples below thisprocess in fact generates all of the eigenfunctions. It is likely that this happens formost of the irreducible compact symmetric spaces. We mention briefly the relation of the construction to the so-called spherical repre-sentations. A spherical representation π of group G in a vector space V (with respectto the Riemannian symmetric space G/K ) is called spherical, if V contains a vector,fixed by all operators in π ( K ). Any unitary spherical representation of G with a unitvector e fixed by π ( K ), the function G ∋ dx → h e , π ( x ) e i is positive-definite andspherical ([23], Therorem 3.4). A function on a Lie group G is called positive definite,if for every x , .., x n ∈ G and α , ..., α n ∈ C we have X i,j φ ( x − i x j ) α i α j ≥ φ is called spherical if it is K -bi-invariant (left and right) and also a commoneigenfunction of all left-invariant operators on G , which are also right K − invariant.The Cartan-Helgason Theorem [23] characterizes the irreducible spherical represen-tations as the ones for which the highest weight λ : h → C satisfies λ ( i ( h ∩ k ) = 0and h λ, α ih α, α i ∈ Z + , ∀ α ∈ Σ + Note that the irreducible representations for compact G are characterized by thesecond condition, with h λ,α ih α,α i ∈ Z + instead of h λ,α ih α,α i ∈ Z + . In particular, when G/K isa Riemannian symmetric space of maximal rank, the first condition is trivial, so thespherical representations form“half” of all irreducible representations - we call them even .In fact, the relation between the Borel-Weil theorem and the Laplace-Beltramieigenfunctions can potentially reveal more information. The spaces of holomorphicsections in the Borel-Weil theorem are irreducible representations and the eigenspaceson the symmetric space
G/K are not unless it is a CROSS. The irreducible sphericalrepresentations are characterized as the common eigenspaces of the invariant differen-tial operators on
G/K . So we expect that the correspondence in the Theorem couldbe extended to the irreducible spherical representations. Another approach to that(see [24, 15]) is through the integral geometry and variations of Radon transform.But such approach provides only expressions of the spherical functions in terms ofintegral formulas, which are not explicit in general. Examples of harmonic polynomials and eigenfunctions
The classical example we want to generalize is that of the sphere S n . It is knownthat the spherical harmonics (Laplace-Beltrami eigenfunctions), are restrictions ofthe harmonic polynomials on R n +1 . A similar description is known for C P n . Beforewe present it we recall briefly the facts we need from the theory of invariant harmonicpolynomials.Let V be a real or complex vector space and G a group of linear transformationsof V . Then G acts on the ring of polynomials identified as the symmetric algebra S ( V ∗ ). If f ∈ S ( V ∗ ) is a polynomial and X ∈ V , then the directional derivative ∂ ( X )acts on f as ( ∂ ( X ) f )( Y ) = ( ddt f ( Y + tX )) | t =0 This extends to a map L from S ( V ) to the algebra of differential operators on V which is an isomorphism. Take a positive bilinear form B on V and define with itthe isomorphism B : V → V ∗ . The space S ( V ∗ ) has a bilinear form hh , ii defined as hh p, q ii = ( ∂ ( P ) q )(0)where P is the image of p under the isomorphism L ◦ B . This coincides with he usualextension of B to S ( V ), so is a positive scalar product. We have the following propertyfor polynomials p, q, r and their corresponding differential operators P, Q, R ::(5) hh p, qr ii = hh ∂ ( Q ) p, r ii so the multiplication by q is adjoint to the operator ∂ ( Q ). Let I ( V ∗ ) is the ideal gen-erated by the invariant polynomials and I + ( V ∗ ) ⊂ I ( V ∗ ) is the subset of polynomialswithout constant term. For the action of G denote by H ( V ∗ ) the set of G − harmonic polynomials h , i.e. ∂ ( J )( h ) = 0 for every invariant differential operator J = L ◦ B ( j ), j ∈ I + ( V ∗ ). Assuming that G is compact by [23] Ch. 3, Theorem 1.1: S ( V ∗ ) = I ( V ∗ ) H ( V ∗ )and from the proof we see that S k ( V ∗ ) = ( I + ( V ∗ ) S ( V ∗ )) k + H k ( V ∗ )is an orthogonal decomposition with respect to hh , ii . We are going to use a particularcase, when I + ( V ) is generated by one homogeneous polynomial p of degree l . Thenthe multiplication by p gives an embedding P : S k ( V ∗ ) → S k + l ( V ∗ ) such that wehave an identification of the quotient space S k + l ( V ∗ ) /P ( S k ( V ∗ )) = H k + l ( V ∗ )with the harmonic polynomials which in this case are just ker( ∂ ( P )).Now we start with a preliminary example to illustrate the correspondence betweeneigenfunctions and holomorphic sections in the quantization space: APLACE EIGENFUNCTIONS AND BOREL-WEIL THEOREM 25
Complex projective space C P n . The representation of the eigenfunctions interms of harmonic polynomials is known in this case - see [32] for example.The quantization space is the generalized flag manifold SU ( n + 1) /S ( U (1) × U (1) × U ( n − C P n is a bundle over theflag SU ( n + 1) /S ( U (1) × U (1) × U ( n − C P n × C P n (Section 3.1). Since the bundle should correspond to L p,q for some non-negative integers p, q , then H ( F , O ( L p,q )) = { F : C n +1 × C n +1 → C : ∂F = 0 , deg( F ) = ( p, q ) } / { z.w = 0 } where the set on the right is spanned by F = f ( z ) g ( w ) , deg( f ) = p, deg( g ) = q, - homogeneous polynomials f, g of degree p, q respectively, up to the identity z.w =0. For the sections of the prequantum bundle L k,k , k > H ( F , O ( L k,k )) can be identified with the polynomials p ( z, w ) forwhich P ∂ ∂z i ∂w i ( p ) = 0. Since the space of holomorphic sections of the budnle O ( k, l )over C P n × C P n is spanned by F = f g as above, it is easy to see that H ( F , O ( L k,k )) isspanned by (the restriction to F of) p ( z, w ) = ( a, z ) k ( b, w ) k for all a , b with ( a, b ) = 0.On the other side - under the Cartan embedding a matrix in SU ( n + 1) with thefirst row z is mapped into a matrix with entries z i z j for i = j and 1 + | z i | on thediagonal. On the other side we see from [32] that the functions ( a, z ) k ( b, z ) k span theharmonic polynomials on C n +1 which induce the eigenfunctions for the k th eigenvalue λ k of C P n , which is known to be 4 k ( n + k ) for the Laplace-Beltrami operator of theFubini-Study metric (and could be seen from Theorem 4.1). So in this case therestriction of p ( z, w ) given by w = z gives after taking the span, the space of alleigenfunctions of λ k .8.2. Quaternionic projective space H P n . Next we can consider the functions of z = ( z , ..., z n ) and w = ( w , ..., w n ) such that q i = z i + w i j are the quaternioniccoordinates of H n +1 . The harmonic polynomials on C n +1 which induce the eigen-functions on C P n are precisely the ones on R n +2 which are invariant under S .And the invariant functions which generate these polynomials are exactly z i z j , whichalso fits the Cartan embedding interpretation. Similarly for H n +1 = R n +4 = C n +2 the harmonic polynomials which determine the eigenfunctions on H P n are the onesright-invariant under Sp (1) = SU (2), and they are functions of the variables q l q k = z l z k + w l w k + ( z k w l − z l w k ) j . In particular we see that the following functions spanthe Sp (1) invariant quadratic harmonic polynomials:( a, z )( b, z ) + ( a, w )( b, w )( a, z )( b, w ) − ( a, w )( b, z )for ( a, b ) = 0, as well as their conjugates. Now we relate the functions to the quantization space which is the symplecticisotropic Grassmannian F is (2 , n + 2) = F is . Using the construction in Section 3.2,it is embedded in the regular Grassmanian by a hyperplane section given by theholomorphic symplectic form denoted by I ( z, w ). Then the Grassmanian is embeddedin P (Λ ( C n +2 ) by Plucker relations. Now the Picard group of F is (2 , n + 2) is Z , soevery line bundle is of type O ( k ) for some power of the hyperplane section bundle O F is (1) which is the restriction of O P (Λ C n +2 ) (1). In particular we can consider thesection as a homogeneous polynomials of degree k on the variables U i V j − U j V i for U, V ∈ C n +2 which are orthogonal in hh , ii to P U i V n + i +1 − U n + i +1 V i = I ( U, V ).They can be characterized as the polynomials which are also in the kernel of (cid:3) = P ( ∂ ∂U i ∂V n + i +1 − ∂ ∂V i ∂U n + i +1 ). By modifying the result for the generalized flag manifold F in the C P n case, we observe that such polynomials can be written as a sum of p ( U, V ) = l ( U, V ) k where l ( U, V ) = (
A, U )( B, V ) − ( B, U )( A, V )for
A, B satisfying certain conditions. To describe them we need to establish somenotations. First consider every element in C n +2 as pairs A = { a, b } and B = { c, d } of elements in C n . Then I ( { a, b } , { c, d } )) = ( a, d ) − ( b, c ) and ( { a, b } , { c, d } )) =( a, c ) + ( b, d ). If U = { z, w } , V = { u, v } , then (cid:3) ( l ( U, V )) = ( a, d ) − ( b, c )To calculate (cid:3) ( l ( U, V ) k ) we use U, V, A, B as above and ∂ ∂U i ∂V n + i +1 ( l ( U, V ) k ) = ∂ ∂z i ∂v i ( l ( U, V ) k ) = ∂ ∂z i ∂v i ([( a, z ) + ( b, w )][( c, u ) + ( d, v )] − [( a, u ) + ( b, v )][( c, z ) + ( d, w )]) k = ∂∂z i k ( l ( U, V ) k − )( d i [( a, z ) + ( b, w )] − b i [( c, z ) + ( d.w )] = k ( k − l ( U, V ) k − )( a i [( c, u )+( d, v )] − c i [( a, u )+( b, v )])( d i [( a, z )+( b, w )] − b i [( c, z )+( d.w )])+ k ( l ( U, V ) k − )( a i d i − b i c i )Similarly ∂ ∂V i ∂U n + i +1 l ( U, V ) k = ∂ ∂w i ∂u i l ( U, V ) k = k ( k − l ( U, V ) k − )( b i [( c, u ) − d [ ( a, u ) + ( b, v )])( c i [( a, z ) + ( b, w )] − a i [( c, z ) + ( d, w )])+ k ( l ( U, V ) k − )( c i b i − a i d i )Now subtracting the two identities gives: (cid:3) ( l ( U, V ) k ) = ( k + k )( l ( U, V ) k − ) X i ( a i d i − b i c i ) = ( k + k )( l ( U, V ) k − )(( a, d ) − ( b, c )) APLACE EIGENFUNCTIONS AND BOREL-WEIL THEOREM 27
So we have the Lemma:
Lemma 8.1.
The holomorphic sections of the quantization bundle O F is ( k ) are thelinear combinations of the polynomials p ( U, V ) satisfying ( a, d ) − ( b, c ) = 0Now for the functions which restrict to harmonic polynomials on H n we take V = jU = { w, − z } This transforms (cid:3) into the Laplacian of H n , so that p ( U, jU ) are pull-backs of theeigenfunctions for the k -th eigenvalue. More precisely we have: Theorem 8.1.
The eigenfunctions of the Laplace-Beltrami operator on H P n corre-sponding the k − th eigenvalue are the linear combinations of the functions with pull-back to H n +1 of the form p ( U, jU ) = ([( a, z ) + ( b, w )][( c, w ) − ( d, z )] − [( c, z ) + ( d, w )][( a, w ) − ( b.z )]) k where ( a, d ) − ( b, c ) = 0 .Proof: It remains to prove that the span H of p ( U, jU ) covers all of the space ofharmonic polynomials. To do this, one can mimic the proof of Theorem 14.2 and 14.4in [32], but in our case we can use the quantization bundle. From the Lemma 8.1,we have a description of the space of its holomorphic sections, and we directly checkthat the spanning sets are inn 1 - 1 correspondence, so they determine spaces of thesame dimension. Since both are irreducible representations of Sp ( n + 1) as well asthe k th eigenspace of H P n , the proof is complete. Q.E.D.
We note that a similar set of harmonic polynomials for H n +1 defining eigenfunctionson H P n was found in [13] Proposition 3.4, but its span was not explicitly discussedthere. Example 8.1.
The space SU (3) /SO (3) . In this example we extend the results from [21] and find a generating set for alleigenfunctions of the Laplace-Beltrami operator on SU (3) /SO (3). Following the no-tations there we denote by z, w etc. matrices in the Lie groups SU ( n ) or SO ( n ) (so z ∈ SU ( n )). If zz T = I with I being the identity matrix) and by Z, W matrices inthe corresponding Lie algebras. Denote by z ij the entries of z . The standard metricon SU ( n ) is given by g ( Z, W ) = Re (Trace( ZW t )). Then the Laplace Beltrami oper-ator ∆ on SU ( n ) (denoted by τ in [21]) satisfies ∆( ϕ.g ) = ∆( f ) g + k ( f, g ) + ∆( g ) f where k ( f, g ) = g ( ∇ f, ∇ g ). If E ij is the matrix entries δ ik δ jl on the ( k, l ) th placethen an orthonormal basis ˆ B of u ( n ) is given by Y ij = ( E ij − E ji ) / √ , X ij = i ( E ij + E ji ) / √ , D ii = iE ii . Using the relation between the operators ∆ and k on U ( n ) and SU ( n ) in ??? (from above) the calculations give that for any orthonormal basis B of su ( n ) one obtains for SU ( n ) ∆( z ij ) = ( 1 n − n ) z ij and k ( z ij , z kl ) = X Z ∈B z ( ZE jl Z T ) z T = − z il z kj + 1 n z ij z kl Now we can directly see in the same way∆( z ij ) = ( 1 n − n ) z ij and k ( z ij , z kl ) = X Z ∈B z ( ZE jl Z T ) z T = − z il z kj + 1 n z ij z kl Take Φ jα = ( zz T ) jα = P r z jr z αr Now using the fact that zz T = I , or P r z jr z rα = δ j,α , and following the calculationsin [21], pp. 7 and 8, we get: k (Φ jα , Φ kβ ) = − δ rα δ jβ − δ αβ δ kj + 4 n Φ α Φ kβ Now from here we take φ = P j,α a j a α Φ jα , ψ = P k,β b k b β Φ kβ . Then the samecalculations as in [21] again give: k ( φ, ψ ) = − a, b ) + 4 n φψ where ( a, b ) = P a i b i . In particular for ( a, b ) = 0 we get that k ( φ, ψ ) = n φψ .Now for multiplication of functions we get k ( f n , h m ) = nmf n − h m − g ( ∇ f, ∇ h ) = nmf n − h m − k ( f, h ).Then using the fact that φ and ψ are both eigenfunctions of ∆ and k (calledeigenfamily in [21] we get that ∆( φ n ψ n ) = λ n,m φ n ψ m for ( a, b ) = 0. Theorem 8.2.
On the Riemannian symmetric space SU (3) /SO (3) the functions φ n ψ m satisfying ( a, b ) = 0 above generate the space of eigenfunctions of the k th eigen-value λ k after varying m and n such that λ n,m = λ k . APLACE EIGENFUNCTIONS AND BOREL-WEIL THEOREM 29
Proof:
First we notice that the functions φ and ψ are SO (3)-invariant and descendto the quotient SU (3) /SO (3). So they define eigenfunctions on SU (3) /SO (3) and wehave to prove that every eigenfunction is a linear combination of them. For this weuse the correspondence and the Borel-Weil theory.Now the horospherical manifold corresponding to SU (3) /SO (3) as explained abovea bundle over the flag SU (3) /T , which embedded as a quadric (1,1)-hypersurface in C P × C P . Since the bundle should correspond to L p,q for some non-negative integers p, q , then H ( F , O ( L p,q )) = { F : C × C → C : ∂F = 0 , deg( F ) = ( p, q ) } / { z.w = 0 } where the set on the right is also F = f ( z ) g ( w ) , deg( f ) = p, deg( g ) = q, for homogeneous polynomials f and g of degree p, q respectively, up to the identity z.w = 0.Such functions are spanned by the set h p,qa,b where, as in [32], h p,qa,b ( z, w ) = h a, z i p h v, w i q , h a, b i = 0and h , i is the complex quadric. Now it is clear that the map above h m,na,b → φ m ψ n is 1-to-1 for fixed m, n , so the spaces they generate are isomorphic. Since by Borel-Weil Theorem h m,na,b generate all irreducible SU (3)-modules, and the eigenspaces ofthe Laplace-Beltrami operators on Riemannian symmetric spaces are finite sums ofsuch modules, we obtain the result. Q.E.D.
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D. Grantcharov: Department of Mathematics, University of Texas Arilington,Arlington, TX 76019-0408, USA
Email address : [email protected] G. Grantcharov and C. Montoya: Department of Mathematics and Statistics,Florida International University, Miami, FL 33199, USA
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