Theta series and generalized special cycles on Hermitian locally symmetric manifolds
aa r X i v : . [ m a t h . G T ] S e p THETA SERIES AND GENERALIZED SPECIAL CYCLES ONHERMITIAN LOCALLY SYMMETRIC MANIFOLDS
YOUSHENG SHI*
Abstract.
We study generalized special cycles on Hermitian locally symmet-ric spaces Γ \ D associated to the groups G = U( p, q ), Sp(2 n, R ) and O ∗ (2 n ).These cycles are algebraic and covered by symmetric spaces associated to sub-groups of G which are of the same type. We show that Poincar´e duals of thesegeneralized special cycles can be viewed as Fourier coefficients of a theta series.This gives new cases of theta lifts from the cohomology of Hermitian locallysymmetric manifolds associated to G to vector-valued automorphic functionson the groups G ′ = U( m, m ), O( m, m ) or Sp( m, m ) which forms a reductivedual pair with G . Contents
1. Introduction 1
Part 1. Generalized special cycles on Hermitian locally symmetricspaces
72. Hermitian locally symmetric spaces 73. Generalized special cycles 11
Part 2. Relative Lie Algebra cohomology of the Weil representation
Part 3. Proof of the main theorems
Introduction
Generalized special cycles.
There are four classes of irreducible reductivedual pairs over R of type I in the sense of Howe [How79] (c.f. [Ada94]):(1) (O( p, q ) , Sp(2 n, R )), Date : September 15, 2020.* Partially supported by NSF grant DMS-1518657. (2) (U( p, q ) , U( r, s )),(3) (Sp( p, q ) , O ∗ (2 n )),(4) (O( m, C ) , Sp(2 n, C )).Each group G belonging to any of the seven families of groups in the above tableis the group that preserves a non-degenerate Hermitian or skew-Hermitian form( , ) over a real, complex or quaternionic vector space V . Let Γ be a torsion freecongruence subgroup of G such that Γ \ G is compact. In this introduction we assumethat G = G ( R ) is the set of real points of an algebraic group G and Γ is a congruencesubgroup of G ( Z ). In general we will need to assume G ( R ) = G × G c where G c iscompact, but we omit this technicality in the introduction. Furthermore we assumewe have chosen a lattice L in V which is invariant under Γ. In each of the casesthat we are interested in, the symmetric space D = G/K associated to G , where K is a fixed maximal compact subgroup, has a realization as an open subspace of aLagrangian Grassmannian associated to V . In what follows let M = Γ \ D . Once wehave chosen an orientation of D , after passing to a (possibility deeper) congruencesubgroup of Γ we may assume M is a compact oriented manifold.We define and study cycles which are called “generalized special cycles”, tobe denoted C x ,z ′ (see below for the explanation of the notation), in the locallysymmetric spaces M . In this paper, we restrict our attention to the cases G =U( p, q ), Sp(2 n, R ) and O ∗ (2 n ) which we denote by case A, B and C respectivelythroughout the paper. The members of the above three families of groups are allthe groups that show up in real reductive dual pairs of type I whose symmetricspaces are of Hermitian type with the exception of O( p, M isa compact K¨ahler manifold which is in fact a connected complex algebraic variety([BB66]), and the cycles C x ,z ′ are algebraic cycles (see Theorem 3.4).We now briefly introduce the definition of C x ,z ′ (see Section 3 for more details).Roughly speaking these cycles come from embeddings of smaller groups of the sametype as G into G . In what follows we will let V = C p + q for case A, V = R n forcase B, and V = H n for case C, where H , the Hamilton quaternions, acts by rightmultiplication. We assume that ( , ) is a Hermitian form, skew symmetric form anda skew-Hermitian form on V respectively and G is the linear isometry group of( , ). Let x = ( x , x , · · · , x m ) ∈ V m . We assume that the vectors x , x , · · · , x m are linearly independent and the restriction of the form ( , ) on U = span { x } isnon-degenerate. In particular, in case B this implies that m = 2 r for some positiveinteger r . In case A, let ( r, s ) be the signature of ( , ) | U . We then have the orthogonalsplitting(1.1) V = U ⊕ U ⊥ . For any non-degenerate subspace U ⊆ V define G ( U ) = { g ∈ G | gv = v, ∀ v ∈ U ⊥ } , and D ( U ) to be the symmetric space of G ( U ). We would like to embed D ( U ⊥ ) into D . However such an embedding in general can only be defined after the choice ofa point z ′ ∈ D ( U ): ρ U,z ′ : D ( U ⊥ ) → D. The image of ρ U,z ′ is a complex analytic subvariety of D which we denote by D U,z ′ or D x ,z ′ . We then pass to the locally symmetric space level and still denote by ρ U,z ′ HETA SERIES AND CYCLES 3 (by abuse of notation) the induced map ρ U,z ′ : Γ U \ D ( U ⊥ ) → M = Γ \ D, where Γ U = G ( U ⊥ ) ∩ Γ. The image C U,z ′ or C x ,z ′ of ρ U,z ′ is then an algebraicsubvariety of M which we call a generalized special cycle. In general, ρ U,z ′ is notan embedding and C x ,z ′ is singular. However by passing to a deeper congruencesubgroup of Γ we can resolve the singularity (see Lemma 3.3). We can think of C x ,z ′ as an element in the Chow group Ch ∗ ( M ) or an element in the Singular homologygroup H ∗ ( M ). Remark . It is an important fact that the homology class of C x ,z ′ does not depends on the choice z ′ (Proposition 3.6.1). Hence when only the homology classis considered, we will often write [ C x ] instead of [ C x ,z ′ ]. Remark . When s or r is equal to 0 in cases A, the cycle C x ,z ′ is called a specialcycle by Kudla and Millson (see [KM82], [KM86], [KM87], [KM90]). In these cases G ( U ) is compact and its symmetric space is a single point. In other words, thechoice of z ′ in the definition of C x ,z ′ is not necessary.One of the main goals of the paper is to construct Poincar´e dual of C x ,z ′ interms of differential forms by studying differential forms on D with values in theWeil representation. Let S ( V m ) be the Schwartz functions on V m . Then there isa reductive dual pair ( G, G ′ ) such that G × ˜ G ′ acts on S ( V m ) smoothly and by aunitary representation ω , where ˜ G ′ is a double cover of G ′ . In cases of our interests,we have the following data(1) Case A: G = U( p, q ), G ′ = U( m, m ) with m = r + s , 0 ≤ r ≤ p , 0 ≤ s ≤ q ,(2) Case B: G = Sp(2 n, R ), G ′ = O( m, m ) with m = 2 r , 0 ≤ r ≤ n ,(3) Case C: G = O ∗ (2 n ), G ′ = Sp( m, m ), 0 ≤ m ≤ n .The action of G via ω is just the induced action on functions( ω ( g ) f )( x ) = f ( g − x ) . for f ∈ S ( V m ), where x ∈ V m is viewed as a ( p + q ) by m matrix.Let Ω • ( D, S ( V m )) G be the set of G -invariant differential forms on D with valuesin S ( V m ), it is a chain complex graded by the Hodge bi-degree. Our first main resultis the following theorem which summarizes the results of Section 5, in particularTheorem 5.13 and its corollary. Theorem 1.3.
Assume that we have x = ( x , x , · · · , x m ) ∈ V m with x , x , · · · , x m linearly independent and the restriction of the form ( , ) on U = span { x } is non-degenerate. There exists (a canonical choice of ) a closed differential form ϕ ∈ Ω ( d ′ ,d ′ ) ( D, S ( V m )) G . Here d ′ is the complex codimension of the generalized specialcycle C x ,z ′ . In case A, the construction of ϕ depends on a pair of integers ( r, s ) that matches the signature of U .Remark . In case A, we call the pair of integers ( r, s ) in the theorem the signatureof ϕ . Remark . In case A when s = 0, the cocycle ϕ is actually the special formdefined by Kudla and Millson in [KM90]. YOUSHENG SHI
Recall that L is a lattice in V fixed by Γ. By [Wei64], we can choose an arithmeticsubgroup Γ ′ ⊂ G ′ such that the theta distribution θ L θ L ( ψ ) = X x ∈L m ψ ( x ) , ψ ∈ Ω • ( D, S ( V m )) G is Γ × ˜Γ ′ -invariant. Hence for the ϕ defined in Theorem 1.3 we can define a function(1.2) θ L ,ϕ ( z, g ′ ) = X x ∈L m ω ( g ′ ) ϕ ( z, x ) ∈ Ω • ( M ) ⊗ A (Γ ′ \ G ′ ) , where A (Γ ′ \ G ′ ) is the space of functions on Γ ′ \ G ′ . We also define(1.3) θ L ,β,ϕ ( z, g ′ ) = X x ∈L m , ( x , x )= β ω ( g ′ ) ϕ ( z, x )for a matrix β which is Hermitian in M m ( C )in case A, skew symmetric in M m ( R )in case B and skew Hermitian in M m ( H ) in case C. We have the following Fourierexpansion of θ L ,ϕ ∞ : θ L ,ϕ ( z, g ′ ) = X β θ L ,β,ϕ ( z, g ′ ) , where β runs over all possible inner product matrices ( x , x ). We call θ L ,β,ϕ the β -th Fourier term of θ L ,ϕ as each θ L ,β,ϕ is a character function under the action ofΓ ′ ∩ N ′ , where N ′ is the unipotent radical of the Siegel parabolic (see Section 4).From now on we assume that β is non-degenerate. The set { x ∈ L m | ( x , x ) = β } consists of finitely many Γ-orbits. We choose Γ-orbit representatives { x , . . . , x o } and define U i = span { x i } , ≤ i ≤ o. For each 1 ≤ i ≤ o choose a base point z i ∈ D ( U i ). Let C x i ,z i be the generalizedspecial cycle. Then all these cycles have the same complex codimension d ′ . Let z = { z , z , . . . , z o } . Then define C β, z = o X i =1 C x i ,z i .C β, z is a cycle in the Chow group of Γ \ D . By Remark 1.1, the homology class[ C β, z ] is independent of the choice of z , so we simply denote by [ C β ] its homologyclass.We want to relate the form ϕ in Theorem 1.3 to generalized special cycles. Wewill prove the following theorem in Theorem 6.3. Theorem 1.6.
Let ϕ be as in Theorem 1.3. For x ∈ L m such that U = span { x } is non-dgenerate and in case A of signature ( r, s ) which matches the signature of ϕ , we have X y ∈ Γ · x [ ϕ ( z, g ′ , y )] = κ ( g ′ , β )PD([ C x ]) , where [ ϕ ] is the cohomology class of ϕ in H ∗ ( M ) , PD([ C x ]) ∈ H ∗ ( M ) is thePoincar´e dual of [ C x ] , and κ is a function that is analytic in G ′ . If β = ( x , x ) , thenwe have [ θ L ,β,ϕ ( z, g ′ )] = κ ( g ′ , β )PD([ C β ]) . HETA SERIES AND CYCLES 5
In case A when s = 0, Theorem 1.6 and 1.7 are proved in [KM90], where the exactvalue of κ ( g ′ , β ) is calculated and shown to be never zero. However in the moregeneral case of this paper the exact value of κ ( g ′ , β ) is hard to obtain. Insteadwe calculate certain asymptotic value of κ ( g ′ , β ) in Section 8 which implies thefollowing theorem (see Theorem 6.4). Theorem 1.7.
Let β satisfies the same assumption as in Theorem 1.6. For ageneric g ′ ∈ G ′ , the function κ ( g ′ , β ) in Theorem 1.6 is not zero. In the appendix, we will show that the canonical special class ϕ transforms underan irreducible representation of a maximal compact group ˜ K ′ ⊂ ˜ G ′ . Moreover θ L ,ϕ can be viewed as a matrix coefficient of an automorphic vector bundle on ˜Γ ′ \ ˜ G ′ / ˜ K ′ .1.2. Related works.
The modularity of the generating series of intersection num-bers of special cycles was first studied in [HZ76] in case of Hillbert modular surfaces.Later in a series of work ([KM82], [KM86], [KM87], [KM90]), Kudla and Millsonproved the modularity of generating series of special cycles for higher rank locallysymmetric spaces associated to G = O( p, q ) (resp. U( p, q ) and Sp( p, q )). To bemore specific they constructed via Weil representation differential forms that arePoincar´e duals to C x when ( , ) | span { x } is positive definite. By applying the thetadistribution to these forms one get an automorphic form in the dual group Sp(2 r, R )(resp. U( r, r ), O ∗ (2 r )) of G . Moreover they prove in [KM90] that these differen-tial forms are holomorphic with respect to the dual group G ′ = Sp(2 r, R ) (resp.U( r, r )) on the cohomology level, thus give rise to holomorphic modular forms afterapplying theta distribution.The theory of Kudla and Millson have some generalizations and applications.We just briefly mention some here.(1) When G = O( p, q ) and Γ is not co-compact, the boundary behavior (afterthe compactification of Γ \ D ) of the special forms constructed by Kudla andMillson has been studied in [FM02], [FM06], [FM13] and [FM + G ′ (c.f.[Mok15]), [BMM17] and [BMM16] are able to prove certain cases ofHodge Conjecture on arithmetic hyperbolic spaces and arithmetic quotientsof complex balls.(3) In some cases one can lift the modularity theorem by Kudla and Mill-son from cohomology groups to Chow groups (c.f. [Bor99], [Zha09] and[BWR15]) or even arithmetic Chow groups (c.f. [BHK +
20] and [HP20]).(4) Garcia [Gar18] views the special forms of Kudla and Millson as characteris-tic classes of super connections and generalizes the construction to certainperiod domains.This paper is an attempt to generalize the work of Kudla and Millson. What isnew in this paper is the definition of generalized special cycles and the discovery ofa new class of special forms in Ω( D, S ( V m )) G which turn out to be Poincar´e dualsof the generalized special cycles and can be used as kernels of geometric theta lifts.When G = Sp(2 n, R ) or O ∗ (2 n ) or when G = U( p, q ) but ( , ) | span { x } is not positivedefinite, there is no corresponding special cycles in the sense of Kudla and Millson.So in order to have a similar theory, one has to define generalized special cycles. YOUSHENG SHI
In a sequel [MS], Millson and the author will extend the results of this paper tothe groups G = O( p, q ), Sp( p, q ), O( m, C ) or Sp(2 n, C ).1.3. Sketch of the proof of the main theorems.
The form ϕ in Theorem1.3 is ultimately constructed from ϕ + which is a holomorphic differential formin Ω ( d ′ , ( D, P ) G discovered by [And83], where P is certain Fock model of theWeil representation of a compact dual pair. Using the fact that any holomorphicdifferential form on a K¨ahler manifold is closed together with a result of [And83],we can prove that ϕ + is closed. This implies that ϕ is closed as well.In order to prove Theorem 1.6, we construct a fiber bundle in Section 3.2: π : Γ U \ D → Γ U \ D U,z ′ , whose fibers are (topologically) Euclidean spaces. We show that ϕ ( z, g ′ , x ) is aconstant multiple of the Thom form for the above fibration. To be more precise,(1.4) Z Γ U \ D η ∧ ϕ ( z, g ′ . x ) = κ ( g ′ , β ) Z Γ U \ D U,z ′ η, where(1.5) κ ( g ′ , β ) = Z F D
U,z ′ ϕ ( z, g ′ , x ) , and F D
U,z ′ is any fiber of the fibration Γ U \ D → Γ U \ D U,z ′ . Notice that the inte-gration in (1.5) only depends on β = ( x , x ). The key to proving (1.4) is to showthat the norm of ϕ ( z, g ′ . x ) is fast decreasing on the fiber F D
U,z ′ , which will bedone in Section 7. Then the standard unfolding lemma tells us that Z Γ \ D η ∧ X y ∈ Γ · x ϕ ( z, g ′ . y ) = Z Γ U \ D η ∧ ϕ ( z, g ′ . x ) = κ ( g ′ , β ) Z Γ U \ D U,z ′ η. This identity is nothing but Theorem 1.6.As we have mentioned, in general κ ( g ′ , β ) is hard to compute. One of the reasonsis that F D
U,z ′ is not a sub symmetric space of D except in the Kudla-Millson cases.So instead we show that the asymptotic value of κ ( m ′ ( a ( t )) , β ) when t → ∞ (seeSection 8 for the meaning of m ′ ( a ( t ))) is nonzero using the method of Laplace. Thiswill imply Theorem 1.7.1.4. Outline of the paper.
Section 2 reviews the definition of D and constructscompact arithmetic quotients of D . Section 3 defines the generalized special cycle C x ,z ′ and show that they are algebraic subvarieties. Section 4 reviews some factabout the Weil representation and set up coordinate functions for later use. Section5 reviews the results of [And83], constructs the special class ϕ and proves that itis closed. In Section 6 we will prove Theorem 1.6 assuming the rapid decrease of ϕ on the fiber F D
U,z ′ . Section 7 proves the rapid decrease of ϕ on the fiber F D
U,z ′ .Section 8 proves Theorem 1.7 by the method of Laplace. Appendix A describedthe ˜ K ′ -type of ϕ in terms of highest weight theory. Readers who are familiar witharithmetic groups and the Weil representation can pick up the definition of thegeneralized special cycles in Section 3 and then proceed to section 5 directly, onlygo back to Section 4 if necessary. For Section 4, Section 5, Section 7 and Section8, one can focus only on the case A for first reading as the other two cases are similar. HETA SERIES AND CYCLES 7
Acknowledgements.
First I would like to thank my thesis advisor John Mill-son for introducing me to the subject, for studying the closed form constructed by[And83] together with me, and for asking valuable questions and checking some ofthe proofs in the paper. I would like to thank Jeffrey Adams for teaching me usefulknowledge of the Weil representation and for carefully reading the paper and pro-vide valuable suggestions. I would also like to thank Michael Rapoport and TonghaiYang for helpful suggestions on the definition of generalized special cycles. I wouldlike to thank Patrick Daniels and Hanlong Fang for helpful discussions. Lastly, Iwould like to thank Greg Anderson. Without his thesis [And83], the author’s thesiswould come from nowhere.1.6.
Notations and conventions.
In Section 2 and Section 6 we will let k be atotally real number field, F be a CM extension of k , B be a division algebra withcenter k , V be a vector space over k and G be an algebraic group over k . In caseA, V will be equipped with a Hermitian form.In Section 4, Section 5, Section 7, Section 8 and the appendix, V will denotea real vector space and G will denote a real group. In order to relate case A, Band C using seesaw dual pairs and have uniform statements of results, we equip V with a skew Hermitian form in case A. The identification between Hermitian andskew Hermitian forms is not canonical. In this paper we multiply a skew Hermitianform by i = √− Part Generalized special cycles on Hermitian locally symmetricspaces Hermitian locally symmetric spaces
In this section we recall the construction of the Hermitian locally symmetricmanifolds that are relevant to us. These manifolds are compact arithmetic quotientsof Hermitian symmetric domains associated to the groups U( p, q ), Sp(2 n, R ) andO ∗ (2 n ) and are projective algebraic varieties.2.1. Let k be a totally real number field with ℓ distinct embeddings σ , . . . , σ ℓ into R . Let S ∞ = { v , . . . , v ℓ } be the set of Archimedean places and k v , . . . , k v ℓ be thecorresponding completions. Let F be a CM field whose maximal real subfield is k .There are ℓ pairs of conjugate embeddings of F into C . We choose one inside eachpair, denote them by σ , . . . , σ ℓ by abusing notation.Let ( B, σ ) be a k -algebra with involution of one of the following types:(2.1) ( B, σ ) = ( (the CM field F , the generator of Gal(F/k)) , (a quaternion algebra with center k , the main involution) . Let V = B n regarded as a right B vector space. Let ( , ) be a non-degenerate skewHermitian form or Hermitian form on V satisfying(2.2) ( vb, ˜ v ˜ b ) = b σ ( v, ˜ v )˜ b for v, ˜ v ∈ V and b, ˜ b ∈ B . Let G be the group defined by the equation(2.3) G = { g ∈ GL B ( V ) | ( gv, g ˜ v ) = ( v, ˜ v ) , ∀ v, ˜ v ∈ V } . YOUSHENG SHI
Define V v = V ⊗ k k v and G v = G ( k v ), where v ∈ S ∞ . Extend ( , ) to V v and denotethe new form by ( , ) v . Also define V ∞ = Y v ∈ S ∞ V v , G ∞ = Y v ∈ S ∞ G v . We require the form ( , ) to be anisotropic, which is to say that there is no nonzerovector v ∈ V such that ( v, v ) = 0. As a set, the symmetric space D is defined by D = G ∞ /K ∞ , where K ∞ is a maximal compact subgroup of G ∞ . Later in this section we willsee case by case that D can be regarded as an open subset of a (Lagrangian)Grassmannian. Hence D is a complex variety.Let O k be the ring of integers of k and O B be the integral closure of O k in B .Let L = O nB ⊂ B n = V, and define G ( O k ) = { g ∈ G | g L = L} . For an ideal I in O B , define G ( I ) to be the congruence subgroup(2.4) G ( I ) = { g ∈ G ( O k ) | g ≡ I n (mod I L ) } . By Theorem 17.4 of [Bor69], we can choose an ideal J of O B such that G ( J ) is neat.In particular, G ( J ) acts simply on the symmetric space D . Fix an ideal I ⊆ J andlet Γ = G ( I ). Since we assume that ( , ) is anisotropic, Γ is a co-compact subgroupin G ∞ . Moreover by a theorem of Baily and Borel ([BB66]), Γ \ D is a complexprojective variety.We need a lemma. Lemma 2.1.
There is a c ∈ k such that √ c / ∈ k and σ i ( c ) ∈ U i for ≤ i ≤ ℓ ,where U i is an open subset of k v i ∼ = R .Proof. Choose a prime ideal p of O k such that O k / p is not a field of characteristictwo. As taking square is a two to one map on O k / p , there exists a b ∈ O k suchthat the equation x ≡ b (mod p ) has no solution in O k / p . Thus x = b has nosolution in O k and k . Now choose ǫ small enough such that x = a has a solutionwhen | a − | p ≤ ǫ . By the weak approximation theorem, there exists a c ∈ k suchthat(1) σ i ( c ) ∈ U i (1 ≤ i ≤ ℓ ),(2) | c − b | p < | b | p · ǫ .Then c satisfies the assumption of the lemma. (cid:3) Now we carry out the above construction and give a more detailed descriptionof D in each case.2.2. Case A.
Choose d , . . . , d p + q ∈ k such that(1) σ ( d α ) > σ ( d µ ) < ≤ α ≤ p and p + 1 ≤ µ ≤ p + q ,(2) σ i ( d j ) > ≤ i ≤ ℓ and 1 ≤ j ≤ p + q .This is possible by the weak approximation theorem. Let V be a p + q dimensionalright F vector space and ( , ) be the Hermitian form defined by the diagonal matrixwith diagonal entries d , . . . , d p + q . If G is defined by equation (2.3), we have(1) G v ∼ = U( p, q ), HETA SERIES AND CYCLES 9 (2) G v i ∼ = U( p + q ) for 2 ≤ i ≤ ℓ .Since ( , ) v i is definite for 2 ≤ i ≤ ℓ , V is anisotropic. The symmetric space D canbe defined by D = { z is a subspace of V v | dim C z = q, ( , ) v | z is negative definite } . Case B.
By Lemma 2.1, we can choose c , c such that(1) σ ( c j ) > √ c j / ∈ k for j = 1 or 2,(2) σ i ( c j ) < j = 1 , ≤ i ≤ ℓ .Let B = H k ( c , c ) be the quaternion algebra over k generated by ǫ , ǫ with rela-tions(2.5) ǫ = c , ǫ = c , ǫ ǫ = − ǫ ǫ . We put ǫ = ǫ ǫ . Then an element ξ ∈ B can be written as ξ = ξ + ξ ǫ + ξ ǫ + ξ ǫ , where ξ j ∈ k for 0 ≤ j ≤
3. We define an anti-involution σ on B by σ ( ξ ) = ξ − ξ ǫ − ξ ǫ − ξ ǫ . With the given assumption we know that B ⊗ k k v ∼ = M ( R ) , B ⊗ k k v i ∼ = H for 2 ≤ i ≤ ℓ , where H is the classical Hamiltonian quaternions. Now let V be a n -dimensional right B vector space and ( , ) be a Hermitian form on V satisfying(2.2). Let G be the group defined by (2.3).Let E be any field that contains k [ √ c ]. We define an anti-involution σ ′ on M ( E ) by σ ′ ( x ) = Jx t J − , where J = (cid:18) −
11 0 (cid:19) . We can embed B into M ( E ) as follows i ( ξ + ξ ǫ + ξ ǫ + ξ ǫ ) = (cid:18) ξ + ξ √ c c ( ξ + ξ √ c ) ξ − ξ √ c ξ − ξ √ c (cid:19) . It is easy to check that σ ′ ◦ i = i ◦ σ , so from now on we abuse notation and denoteboth involutions by σ . Let e ij be the matrix with the ( i, j )-th entry 1 and all theother entries zero. Let e = e . As B ⊗ k E ∼ = M ( E ), we get a decomposition V E = V E e + V E e σ as a E vector space, where V E = V ⊗ k E . Let S E be the E -bilinear form on V E e defined by(2.6) S E ( xe, ye ) e = ( xe, ye ) . Following an argument like that of Page 368 of [LM + S E is skewsymmetric and G E ∼ = Sp(( V E e, S E )) ∼ = Sp(2 n, E ) . In particular since k v ∼ = R and σ ( c ) >
0, we know that G v ∼ = Sp(( V v e, S v )) ∼ = Sp(2 n, R ) . And we also have G v i ∼ = Sp( p v i , q v i )for 2 ≤ i ≤ ℓ , where p v i + q v i = n . Moreover we choose the form ( , ) to be definedby a diagonal matrix with diagonal entries d , . . . , d n ∈ k satisfying σ i ( d j ) > ≤ i ≤ ℓ and 1 ≤ j ≤ n . We then have G v i ∼ = Sp( n ) for 2 ≤ i ≤ n . With thischoice G is anisotropic as Sp( n ) is. We have G ∞ ∼ = Sp(2 n, R ) × Sp( n ) ℓ − . The symmetric space D is then the set of n dimensional complex subspace z of the2 n dimensional complex vector space ( V v e ) ⊗ R C such that(1) S k v ( , ) | z is zero ( z is Lagrangian for S k v ( , ).(2) The Hermitian form iS k v (¯ , ) is negative definite on z , where ¯ is the com-plex conjugation on ( V v e ) ⊗ R C .2.4. Case C.
The construction of G in this case is similar to that in case B. Let k be the same totally real number field. By Lemma 2.1, we can choose c , c suchthat(1) σ ( c ) < σ ( c ) < √ c , √ c / ∈ k ,(2) σ i ( c ) > σ i ( c ) < ≤ i ≤ ℓ .Let B = H k ( c , c ) as in equation (2.5). This time we have B ⊗ k k v ∼ = H , B ⊗ k k v i ∼ = M ( R )for 2 ≤ i ≤ ℓ . Now let V be a n -dimensional right B vector space and ( , ) be askew Hermitian form on V satisfying (2.2). Let G be the group defined as in (2.3).Then we see that G v ∼ = O ∗ (2 n ) . Following a result on Page 368 of [LM + G v i ∼ = O( p v i , q v i )for 2 ≤ i ≤ ℓ , where p v i + q v i = 2 n . Moreover we choose d , . . . , d n ∈ k such that σ i ( d j ) > ≤ i ≤ ℓ , 1 ≤ j ≤ n , and let ( , ) be the skew Hermitian formdefined by the diagonal matrix with diagonal entries d ǫ , . . . , d n ǫ . By Lemma2.1 of [LM + G v i (2 ≤ i ≤ ℓ ) is the orthogonal group defined bythe block diagonal matrix S v i with 2 by 2 diagonal blocks − J ( d ǫ ) , . . . , − J ( d n ǫ ).Since − J ( d j ǫ ) = (cid:18) − (cid:19) · (cid:18) c d j d j (cid:19) = (cid:18) d j − c d j (cid:19) , by our assumptions S v i will be positive definite for 2 ≤ i ≤ ℓ . Thus G v i ∼ = O(2 n )for 2 ≤ i ≤ ℓ . This implies that G is anisotropic and G ∞ ∼ = O ∗ (2 n, R ) × O(2 n ) ℓ − . We define a skew Hermitian form H ( , ) on V v regarded as a 2 n dimensional complexvector space by the equation(2.7) H ( v, w ) = a + bi if ( v, w ) = a + bi + cj + dk. Also define a symmetric form S ( , ) on V v regarded as a 2 n dimensional complexvector space by the equation(2.8) S ( v, w ) = H ( vj, w ) . Then the symmetric space D is the set of n dimensional complex subspace z of V v such that(1) S ( , ) | z is zero ( z is Lagrangian for S ( , )),(2) the Hermitian form iH ( , ) | z is negative definite. HETA SERIES AND CYCLES 11 Generalized special cycles
In this section we define generalized special cycles in Γ \ D , where D is thesymmetric space associated to G ∞ = Q v ∈ S ∞ G v or equivalently G v as all theother factors in G ∞ are compact. To ease notations, in this section we write G for G v and assume that(1) In case A: B = C , V ∼ = C p + q , ( , ) is Hermitian of signature ( p, q ) on V,(2) In case B: B = R , V ∼ = R n . ( , ) is symplectic on V ,(3) In case C: B = H , V ∼ = H n . ( , ) is non-degenerate skew Hermitian on V .In particular G = U( p, q ) in case A, Sp(2 n, R ) in case B and O ∗ (2 n ) in case C. Onthe symmetric space level, a generalized special cycle is the fixed point set of a pairof involutions while a special cycle in the sense of Kudla and Millson is the fixedpoint set of a single involution. The definition of a generalized special cycle dependson a choice of a non-degenerate U ⊆ V together with a point in the symmetric spaceassociated to U .As we have seen in the previous section, in each case of our interests, the symmet-ric space D can be described as an open subset of a (Lagrangian) Grassmannian.Meanwhile D can also be described as the set of Cartan involutions of G . Therelation between these two descriptions is as follows. For any z ∈ D , we can findan element r z ∈ G such that(3.1) D r z := { x ∈ D | r z x = x } = { z } . In case A we can simply take r z = Id | z ⊕ ( − Id ) | z ⊥ . In the other two cases we cantake r z to be an element in the center of G z = { g ∈ G | gz = z } not equal to ± σ z = Ad( r z ) ∈ Inn( G ). Then σ z is the Cartan involution associated to z and we have G z = G σ z . Let x = ( x , . . . , x m ) ∈ L m . We require U = span B ( x ) to be non-degenerate with respect to ( , ). We then havethe decomposition V = U + U ⊥ . For any non-degenerate U we define(3.2) G ( U ) = { g ∈ G | gv = v, ∀ v ∈ U ⊥ } . Let D ( U ) be the symmetric space associated to G ( U ). Remark . In order to be consistent with the notation of the work of Kudla andMillson, we will also use the notations G U and D U :(3.3) G U = G ( U ⊥ ) , D U = D ( U ⊥ ) . Define Γ U = Γ ∩ G U . By our construction Γ U \ D U is a compact locally symmetricmanifold. We want to map Γ U \ D U to Γ \ D to get an algebraic cycle. This requiresthe choice of a point in D ( U ) as we will see.For any σ ∈ Aut( G ) and H ≤ G such that σ ( H ) = H , define H σ = { g ∈ H | σ ( g ) = g } . Let r U = Id | U ⊕ ( − Id ) | U ⊥ and σ U = Ad( r U ) ∈ Inn( G ). It is easy to see that G σ U = G ( U ) × G ( U ⊥ ) . Consequently the symmetric space of G σ U is the product D ( U ) × D ( U ⊥ ). There isa natural embedding ρ U defined by ρ U : D ( U ) × D ( U ⊥ ) ֒ → D : ( z , z ) z ⊕ z . For a vector v ∈ z , we have r U ( v ) ∈ z if and only if both proj U ( v ) and proj U ⊥ ( v )are in z . Hence we have(3.4) z ∈ D r U := { z ∈ D | r U ( z ) = z } ⇔ z = z ∩ U ⊕ z ∩ U ⊥ . Hence ρ U defines an isomorphism from D ( U ) × D ( U ⊥ ) onto D r U . From now on wedenote D r U by D ( U, U ⊥ ).Now let z ′ ∈ D ( U ). We define an embedding i z ′ : i z ′ : D U ֒ → D ( U ) × D ( U ⊥ ) , z ( z ′ , z ) . Let ρ U,z ′ be the composition ρ U ◦ i z ′ . In other words(3.5) ρ U,z ′ ( z ) = z ′ ⊕ z. Definition 3.2.
Denote by D U,z ′ (or D x ,z ′ ) the image of D U under the map ρ U,z ′ .We call it a generalized special sub symmetric space of D .As we have explained, it is possible to choose a r z ′ ∈ G ( U ) such that D ( U ) r z ′ = { z ′ } , G ( U ) Ad( r z ′ ) = { g ∈ G ( U ) | gz ′ = z ′ } . We now define σ z ′ = Ad( r ( z ′ )) ∈ Inn( G ), where r ( z ′ ) = r z ′ ⊕ Id U ⊥ . Apparently σ U σ z ′ = σ z ′ σ U , hence we know that G σ U is σ z ′ -stable. Then we define G σ U ,σ z ′ := ( G σ U ) σ z ′ = ( G σ z ′ ) σ U . Since G σ U = G ( U ) × G ( U ⊥ ), we know that G σ U ,σ z ′ = G ( U ) σ z ′ × G ( U ⊥ ) . Notice that G ( U ) σ z ′ is the maximal compact subgroup of G ( U ) fixed by σ z ′ . Thesymmetric space of G σ U ,σ z ′ is D U which we identify with D U,z ′ . Since D r U = D ( U, U ⊥ ) and D ( U, U ⊥ ) r ( z ′ ) = D U,z ′ , we have Proposition 3.2.1. D U,z ′ = D r U ,r ( z ′ ) := ( D r U ) r ( z ′ ) . In particular D U,z ′ is totally geodesic. i z ′ , ρ U and ρ U,z ′ induce maps (still denoted as i z ′ , ρ U and ρ U,z ′ ) of locallysymmetric spacesΓ U \ D U → Γ σ U \ D ( U, U ⊥ ) , Γ σ U \ D ( U, U ⊥ ) → Γ \ D, Γ U \ D U → Γ \ D respectively. Apparently the induced map i z ′ is always an embedding. But ingeneral ρ U (hence ρ U,z ′ ) will not be injective and if this is the case the image of ρ U,z ′ will have self intersections and not be a manifold. The following lemma isLemma 2.1 of [KM90] which ”resolves the singularity” of the image. Lemma 3.3.
There is an arithmetic subgroup Γ ′ ⊆ Γ of finite index such that thefollowing diagram commutes, and ρ ′ U is an embedding. (Γ ′ ) σ U \ D ( U, U ⊥ ) Γ ′ \ D Γ σ U \ D ( U, U ⊥ ) Γ \ D. ρ ′ U ρ U HETA SERIES AND CYCLES 13
In particular, ρ ′ U,z ′ : Γ ′ U \ D U → Γ ′ \ D is also an embedding. We denote the image of ρ U,z ′ by C U,z ′ . We sometimes also use C x ,z ′ to denote C U,z ′ for convenience but it really just depends on U . Theorem 3.4. C U,z ′ is an algebraic subvariety in M = Γ \ D .Proof. First let us assume that the map ρ U,z ′ : Γ U \ D U → Γ \ D is an embedding.The subsymmetric space D U,z ′ is a complex analytic subvariety of D ( U, U ⊥ ) sinceit is a factor. We know that r U is an isometry of D , hence automatically anholomorphic map (Lemma 4.3 of [Hel79]). Hence D ( U, U ⊥ ) = D r U is a complexanalytic subvariety of D . So D U,z ′ is a complex analytic subvariety of D . Since ρ U,z ′ is an embedding, locally D U,z ′ and C U,z ′ are defined by the same analyticequations. In particular, C U,z ′ is complex analytic. So it is a complex algebraicsubvariety of Γ \ D as well by the main theorem of [Cho49].In the general case, we apply Lemma 3.3. By the previous argument we see that C ′ U,z ′ (the image of Γ ′ U \ D U under ρ ′ U,z ′ ) is a complex algebraic subvariety of Γ ′ \ D .Since the map f : Γ ′ \ D → Γ \ D is an analytic covering between complex projectivevarieties, it is automatically a regular map of complex projective algebriac varietiesby [Ser56]. Hence f is projective by Lemma 28.41.15 of Stack Project. Being afinite covering map, f is automatically quasi-finite, hence a finite morphism. Hence f is proper and in particular closed. Then the image C U,z ′ = f ( C ′ U,z ′ ) is a closedsubvariety of Γ \ D . (cid:3) Definition 3.5.
We call C U,z ′ (or C x ,z ′ ) a generalized special cycle.We will need the following lemma later. Lemma 3.6.
The map Γ U \ D U ρ
U,z ′ −→ Γ \ D is a finite birational morphism onto itsimage.Proof. Since gD U,z ′ = D gU,gz ′ , we know that the stabilizer of D U,z ′ is exactly G σ U ,σ z ′ = G ( U ) σ z ′ × G U . ∀ γ ∈ Γ − Γ σ U ,σ z ′ , define D γ = D U,z ′ ∩ γD U,z ′ . Then D γ is an analytic subset of D U,z ′ . We claim that it is a proper subset.Otherwise γ is in the stabilizer of D U,z ′ . Hence γ ∈ Γ σ U ,σ z ′ , a contradiction.The image V γ of D γ under the natural quotient map D U → Γ U \ D U is a properanalytic sub variety of Γ U \ D U . Define V = [ γ ∈ Γ − Γ σU,σz ′ V γ . Then ρ U,z ′ : Γ U \ D U → Γ \ D is injective outside V .The map ρ U,z ′ : Γ U \ D U → Γ \ D factors through φ : Γ U \ D U → C U,z ′ . As in thesituation of Lemma 3.3, we have the following commutative diagramΓ ′ U \ D U C ′ U,z ′ Γ U \ D U C U,z ′ . φ ′ φ By the above commutative diagram and the fact that φ ′ is an isomorphism φ isquasifinite. It is a projective morphism by Lemma 28.41.15 of Stack Project, hence is a finite morphism. By the argument in the previous paragraph, it is injectiveoutside a set of measure 0 with respect to the measure defined by the K¨ahler metricon D U . Hence the degree of the finite morphism ρ U,z ′ must be 1. It must be abirational morphism. (cid:3) Proposition 3.6.1.
The homology class [ C U,z ′ ] ∈ H ∗ (Γ \ D ) does not depends onthe choice of z ′ ∈ D ( U ) .Proof. For any two z ′ , z ′′ ∈ D ( U ), there is a continuous path c : [0 , → D ( U )such that c (0) = z ′ , c (1) = z ′′ . Thus we can define a map D U × [0 , → D by( z, t ) c ( t ) ⊕ z. Since Γ U fixes U , this map defines a map Γ U \ D U × [0 , → Γ \ D which is a homotopyequivalence between two different embeddings of Γ U \ D U . (cid:3) Remark . From now on we specify the choice of embedding if necessary, otherwisewe use the notation [ C U ] or [ C x ] to refer the homology class of C U,z ′ .We now illustrate the above abstract construction in case A.3.1. Example: Case A.
Recall that the symmetric space D can be identified withthe set of negative q -planes in VD = { z ∈ Gr q ( V ) | ( , ) | z < } . Let U be a F -subspace of V . If ( , ) | U has signature ( r, s ), then we have G ( U ⊥ ) ∼ =U( p − r, q − s ) and D U = { z ∈ Gr q − s ( U ⊥ ) | ( , ) | z < } . The choice of a point z ′ ∈ D ( U ) is the same as a choice of an orthogonal decompo-sition of U U = U + ⊕ U − with respect to ( , ) such that ( , ) | U + is positive definite and ( , ) | U − is negativedefinite. Given such a decomposition of U we have z ′ = U − and is s dimensional.Under the embedding ρ U,z ′ defined in (3.5) we have(3.6) D U,z ′ = { z ∈ D | U − ⊆ z ⊆ ( U + ) ⊥ } . Proposition 3.7.1.
When U is positive or negative definite, the above embeddingis canonically defined. To be more precise, the choice of z ′ is unnecessary and wehave D U,z ′ = D ( U, U ⊥ ) . Moreover we have (1) When U is positive definite, D U = { z ∈ D | z ⊆ U ⊥ } . (2) When U is negative definite, D U = { z ∈ D | U ⊆ z } .Proof. When U is positive (negative resp.) definite, the group G ( U ) is compactand the symmetric space D ( U ) consists of a single point z ′ . Since D ( U, U ⊥ ) = D ( U ) × D U , the first statement is proved. The statements in the enumerationfollow from equation (3.6). (cid:3) Remark . When U is positive (resp. negative) definite, we simply denote C U,z ′ by C U . This is the situation in the work of Kudla and Millson ([KM82] [KM86],[KM87]and [KM90]). C U is called a special cycle there. HETA SERIES AND CYCLES 15 π : D → D U,z ′ . For each z ∈ D U,z ′ , we define N z D U,z ′ = { v ∈ T z D | v ⊥ T z D U,z ′ } . Then
N D
U,z ′ = S z ∈ D U,z ′ N z D U,z ′ is the normal bundle of D U,z ′ in D . Then theRiemannian exponential map induces a map F : N D
U,z ′ → D by the formula(3.7) F ( z , v ) = exp z ( v ) . The image of the line through v ∈ N z D U,z ′ under the exponential map is a geodesicthrough z orthogonal to D U,z ′ . Since D U,z ′ is totally geodesic in D which isnegatively curved, Theorem 14.6 of [Hel79] tells us that D is the disjoint union ofthe geodesics which are perpendicular to D U,z ′ . Moreover by a standard Jacobifield calculation we can show that Lemma 3.9. F : N D
U,z ′ → D is a diffeomorphism. For g ∈ G U one can check that the two geodesics gF ( z , vt ) and F ( gz , ( L g ) ∗ vt )are the same by checking that they have the same starting point (at t = 0) andhave the same derivative there. Hence we know that(3.8) gF ( z , v ) = F ( gz , ( L g ) ∗ v ) , ∀ g ∈ G U . By Lemma 3.9 we can define π : D → D U,z ′ by the equation(3.9) π ( F ( z , v )) = z .π : D → D U,z ′ is isomorphic to N D
U,z ′ → D U,z ′ as a fiber bundle. We denote by(3.10) F z D U,z ′ := π − ( z ) , the fiber of π for any z ∈ D U,z ′ . By equation (3.8) and the definition of π , we seethat if z = F ( z , v ) then(3.11) π ( gz ) = π ( gF ( z , v )) = π ( F ( gz , ( L g ) ∗ v )) = gz = gπ ( z )for any g ∈ G U . Thus π induces a fibration π : Γ U \ D → Γ U \ D U,z ′ which we stilldenote by π . Remark . In case A when U is positive or negative definite, F z D U,z ′ as in (3.10)is a sub symmetric space of D . We refer the readers to [KM90] for an explanationof this fact as we do not need it in this paper. Except for these cases, F z D U,z ′ isnot a sub symmetric space. Part Relative Lie Algebra cohomology of the Weil representation The Weil representation and dual pairs
Let G be one of the algebraic group over k introduced in Section 2.1. Then wecan define another algebraic group G ′ over k such that ( G, G ′ ) forms a dual pair inthe sense of [How79]. Let A be the ring of Adeles of k , then the Weil representationis a certain function space on which (a double cover of) ( G × G ′ )( A ) acts. Themain tool of this paper is relative Lie algebra cohomology with values in the Weilrepresentation. However we need two different models of the Weil representation.One is called the Schr¨odinger model, where G acts ”geometrically”, which is thenatural model when we do differential geometry on D . The other model is theFock model, where the maximal compact group of ( G × G ′ ) ∞ acts in a nice way.The Fock model is indispensable for the construction of Anderson’s holomorphicforms ([And83]). In this section we briefly recall knowledge of the two models and write down formulas of the intertwining operators between the two models. Wealso review seesaw dual pairs (4.13) and (4.14).4.1. Dual reductive pairs and the Schr¨odinger model.
Let (
B, σ ) be definedas in (2.1). Let ǫ be − ǫ -Hermitian form ( , ) on a right B vector space V . We can also define a non-dgenerate − ǫ -Hermitian form <, > on a left B vector space W which satisfy < b w , b w > = b < w , w > b σ , < w , w > = − ǫ < w , w > σ , for w , w ∈ W and b , b ∈ B . Let G be defined as in (2.3) and(4.1) G ′ = { g ′ ∈ GL B ( W ) | < w g ′ , w g ′ > = < w , w > ∀ w , w ∈ W } . We view G and G ′ as algebraic groups over k . Let n = dim B V, m = dim B W. Also let W = V ⊗ B W and hh , ii = tr B/k (( , ) ⊗ <, > σ ) . So that W is a k -vector space with the non-degenerate symplectic hh , ii . Then( G, G ′ ) is a dual reductive pair in the sense of [How89].Now we assume that ( W, <, > ) is split over B . i.e. there is a decomposition W = X + Y with B subspaces X and Y which are isotropic for <, > . We can choose a standardsymplectic basis with respect to this decomposition. This choice of basis gives riseto an isomorphism G ′ ∼ = (cid:26)(cid:18) a bc d (cid:19) ∈ GL n ( B ) | ad ∗ − bc ∗ = 1 , ab ∗ = ba ∗ , cd ∗ = dc ∗ (cid:27) , where a ∗ = ( a σ ) t and to isomorphisms W ∼ = V m , X := V ⊗ B X ∼ = V m , Y := V ⊗ B Y ∼ = V m . The parabolic subgroup P ′ ⊂ G ′ which stabilizes Y then has the form P ′ = (cid:26)(cid:18) a b d (cid:19) ∈ G ′ (cid:27) and has unipotent radical(4.2) N ′ = (cid:26) n ′ ( b ) = (cid:18) b (cid:19) | b ∈ M n ( B ) with b ∗ = ǫb (cid:27) and Levi factor(4.3) M ′ = (cid:26) m ′ ( a ) = (cid:18) a
00 ˆ a (cid:19) | a ∈ GL n ( B ) and ˆ a = ( a − ) ∗ (cid:27) . Such a parabolic subgroup is called a Siegel parabolic in the literature.Fix a non-trivial additive chracter ψ of k A trivial on k and let( ω, L ( X )) ∼ = ( ω, L ( V ( A ) m ))be the Schr¨odinger model of the global Weil representation of ^ Sp( W ( A )), the twofold metaplectic cover of Sp( W ( A )), corresponding to ψ and the polarization (1.7)in [Wei64] and [How89]. Let ˜ G ′ ( A ) denote the inverse image of G ′ ( A ) in ^ Sp( W ( A )). HETA SERIES AND CYCLES 17
Then the action of ˜ G ′ ( A ) in L ( V ( A ) n ) defined by the restriction of ω to ˜ G ′ ( A )commutes with the natural action of G ( A ) defined by ω ( g ) ϕ ( x ) = ϕ ( g − x ) , where g ∈ G ( A ) , ϕ ∈ L ( V ( A ) n ).The action of the parabolic subgroup ˜ P ′ ( A ) of ˜ G ′ ( A ) is easy to describe. Fix asection of the covering ^ Sp( W ( A )) → Sp( W ( A )) and hence an identification: ^ Sp( W ( A )) ∼ = Sp( W ( A )) × µ . Then(4.4) ω ( m ′ ( a ) , ζ ) ϕ ( x ) = ζ | a | m ϕ ( x a ) , where | a | is the modulus of multiplying by a on B ( A ) n . Also(4.5) ω ( n ′ ( b ) , ζ ) ϕ ( x ) = ζψ ( 12 tr ( b ( x , x ))) ϕ ( x ) . For the rest of the section, we study real groups. So we switch notation and let B = C or H , V be a real vector space and G be a real Lie group, etc.4.2. In this subsection we recall the construction of the infinitesimal Fock modeltogether with the intertwining operator from the infinitesimal Fock model to theSchr¨odinger model by Section 6 of [KM90]. The key to the construction of the in-tertwining operator is the action of Weyl algebra. In later subsections, we specializeto different dual pairs.Let W be a be a vector space over R with a non-degenerate skew-symmetric form hh , ii and J be a positive definite almost complex structure (i.e. the form hh , J ii is a positive definite symmetric form) on W . We may decompose W ⊗ C accordingto W ⊗ C = W ′ + W ′′ , where W ′ is the + i eigenspace of J and W ′′ is the − i eigenspace of J . Notice thatboth W ′ and W ′′ are isotropic for hh , ii .Choose a nonzero complex parameter λ . Define W λ to be the quotient of thetensor algebra T • ( W ⊗ C ) of the complexification of W by the ideal generated bythe elements x ⊗ y − y ⊗ x − λ hh x, y ii
1, where x, y ∈ W . Then W λ is called the Weylalgebra in the literature. Let p : T • ( W ⊗ C ) → W λ be the quotient map. Clearly p ( T • ( W ′ )) = Sym • ( W ′ ) and p ( T • ( W ′′ )) = Sym • ( W ′′ ). Let I be the left ideal in W λ generated by W ′ . Then P λ := W λ / I is a W λ -module by left multiplication.The natural projection Sym • ( W ′′ ) → W λ induces an isomorphism onto P λ and weobtain an action of W λ by left multiplication. The infinitesmal Fock model is(4.6) P λ ∼ = Sym • ( W ”) ∼ = Pol( W ′ ) , where the second isomorphism is induced by the non-dgenerate pairing hh , ii be-tween W ” and W ′ .There is an embedding of Lie algebra j : sp ( W ⊗ C ) → W λ , so P λ is a sp ( W ⊗ C )module (see page 150 of [KM90]). This induces an action ω λ of sp ( W ⊗ C ) on P λ .Explicitly let { e , . . . , e N , f , . . . f N } be a symplectic basis for W such that(4.7) J e j = f j and J f j = − e j for 1 ≤ j ≤ N . Define w ′ j = e j − f j i and w ′′ j = e j + f j i for 1 ≤ j ≤ N . Then { w ′ , . . . , w ′ N } (resp. { w ′′ , . . . , w ′′ N } ) is a basis for W ′ (resp. W ′′ ). Let u j be the linear functional on W ′ given by u j ( w ′ ) = hh w ′ , w ′′ j ii . Then P λ = Sym • ( W ′′ ) can be identified with Pol( W ′ ) ∼ = Pol( C N ) = C [ u , . . . , u N ].Denote by ρ λ the action of W λ on Pol( C N ). We have (Lemma 6.1 of [KM90]) Lemma 4.1. (1) ρ λ ( w ′′ j ) = u j . (2) ρ λ ( w ′ j ) = 2 iλ ∂∂u j . From now on we specialize to the case λ = 2 πi and let P = P πi , ω = ω πi and ρ = ρ λ . If we decompose W as W = X ⊕ Y , where X and Y are Lagrangian subspaces of W . The Schr¨odinger model can beviewed as the set of Schwartz functions S ( X ) on X . Explicitly if we assume X = span { e , . . . , e N } , Y = span { f , . . . , f N } , then we have(4.8) ρ ( e j ) = ∂∂x j , ρ ( f j ) = 2 πix j , where { x , . . . , x N } are coordinate functions with respect to the basis { e , . . . , e N } .We define(4.9) ϕ = exp( − π N X i =1 x i ) .ϕ is the unique vector in S ( X ) that is annihilated by ρ ( w ′ j ) for all 1 ≤ j ≤ N .Then under the Weil representation ϕ is fixed by ^ U( N ) which is a maximal compactsubgroup of Mp(2 N, R ). There a unique W λ -intertwining (thus sp ( W )-intertwining)operator(4.10) ι : P → S ( X ) . The image ι ( P ) is exactly the ^ U( N )-finite vectors in the Schrodinger model whichconsists of functions on X of the form p ( x ) ϕ ( x ), where p ( x ) is a polynomial functionon X . More specifically, we know that(4.11) ι (1) = ϕ . Using Lemma 4.1 and equation (4.8) one can see immediately that (Lemma 6.3 of[KM90])
Lemma 4.2. ι ( u j ) ι − = ∂∂x j − πx j , ι ( − π ∂∂u j ) ι − = ∂∂x j + 2 πx j , where u j and ∂∂u j are regarded as operators in W λ . HETA SERIES AND CYCLES 19
Since ι intertwines the W λ action, the above lemma and (4.11) determine themap ι completely (see Lemma 4.3).In the rest of the section by using the above framework we are going to writedown coordinate functions of the Fock and the Schr¨odinger model for the dual pairs(4.12) (U( n, n ) , U( m, m ′ )) , (Sp(2 n, R ) , O(2 m, m ′ )) and (O ∗ (2 n ) , Sp( m, m ′ )) . The Fock or Schr¨odinger model of the three dual pairs are the same since they thesame ( W , hh , ii ). In fact the dual pairs can be put into two seesaw dual pairs:(4.13) U( n, n ) O(2 m, m ′ )Sp(2 n, R ) O O ♦♦♦♦♦♦♦♦♦♦ U( m, m ′ ) , O O g g ❖❖❖❖❖❖❖❖❖❖ and(4.14) U( n, n ) Sp( m, m ′ )O ∗ (2 n ) O O qqqqqqqqq U( m, m ′ ) . O O f f ▼▼▼▼▼▼▼▼▼ Case A: the (U( p, q ) , U( m, m ′ )) dual pair. Let V be a p + q dimensional rightcomplex vector space and ( , ) be a non-degenerate skew Hermitian form on V withsignature ( p, q ) satisfying (2.2). Choose an orthogonal basis { v , . . . , v p , v p +1 , . . . , v p + q } of V such that(4.15) ( v α , v α ) = − i and ( v µ , v µ ) = i for 1 ≤ α ≤ p, p + 1 ≤ µ ≤ p + q (in this subsection we keep this convention ofindex).Let W be a m + m ′ dimensional complex vector space with a non-degenerateHermitian form <, > of signature ( m, m ′ ) satisfying < hw, h ′ w ′ > = h < w , w > h ′ for h, h ′ ∈ C and w, w ′ ∈ W . We assume that choose an orthogonal basis { w , . . . , w m + m ′ } of W such that(4.16) < w a , w a > = 1 , < w k , w k > = − ≤ a ≤ m, m + 1 ≤ k ≤ m + m ′ (in this subsection we keep this convention ofindex).Define W = V ⊗ C W and hh , ii h on W by(4.17) hh v ⊗ w, ˜ v ⊗ ˜ w ii h = ( v, ˜ v ) < ˜ w, w > . One checks easily that hh , ii h is a skew Hermitian form that is anti-linear in thefirst variable and linear in the second variable. Define hh , ii = Re hh , ii h , then hh , ii is a symplectic form on the underlying real vector space of W .Define J = iI p,q ⊗ I m,m ′ , where I a,b is the matrix (cid:18) I a − I b (cid:19) . Then J is a positive definite complex structure for the symplectic form hh , ii . Now define W C = W ⊗ R C ∼ = V ⊗ C ( W ⊗ R C ). Denote the new complex structureby right multiplication by i . Define w ′ a = w a − iw a i, w ′′ a = w a + iw a i, (4.18) w ′ k = w k + iw k i, w ′′ k = w k − iw k i. W C = W ′ ⊕ W ′′ , where W ′ ( W ′′ resp.) is the + i ( − i resp.) eigenspace of J . Thenwe have W ′ = span C { v α ⊗ w ′ a , v µ ⊗ w ′′ a , v α ⊗ w ′ k , v µ ⊗ w ′′ k } , W ′′ = span C { v α ⊗ w ′′ a , v µ ⊗ w ′ a , v α ⊗ w ′′ k , v µ ⊗ w ′ k } . Define linear functionals on W ′ : u + αa ( x ) = hh x, v α ⊗ w ′′ a ii , u + µa ( x ) = hh x, v µ ⊗ w ′ a ii , (4.19) u − αk ( x ) = hh x, v α ⊗ w ′′ k ii , u − µk ( x ) = hh x, v µ ⊗ w ′ k ii for x ∈ W ′ . We can now identify Sym • ( W ′′ ) ∼ = Pol( W ′ ) with the space of polynomi-als in complex variables { u + ia , u − ik | ≤ i ≤ p + q, ≤ a ≤ m, m + 1 ≤ k ≤ m + m ′ } and this will be the Fock model P λ .Now we assume m = m ′ . Define e a = 1 √ w a − w a + m ) , f a = 1 √ w a + w a + m )for 1 ≤ a ≤ m . Then E = span { e , . . . , e m } is a Lagrangian subspace of W . TheSchr¨odinger model of the Weil representation is given by the space of Schwartzfunctions on S ( V ⊗ C E ) ∼ = S ( V m ) on V m . We use complex coordinates z =( z , . . . , z m ) with z j = ( z j , . . . , z p + q,j ) t , where z kj = x kj + iy kj (1 ≤ k ≤ p + q )is the coordinate function of the j -th copy of V with respect to the basis { v ⊗ e j , . . . , v p + q ⊗ e j } .The Weil representation of sp ( W , hh , ii ) now arises from the action of Weyl alge-bra W λ . Using Lemma 2.2 of [Kud96], it is easy to derive the following formulas. ρ λ ( v α ⊗ w ′ a ) = 1 √ − λi ¯ z αa + 2 ∂∂z αa ) , ρ λ ( v µ ⊗ w ′′ a ) = 1 √ − λiz µa + 2 ∂∂ ¯ z µa ) , (4.20) ρ λ ( v α ⊗ w ′ a + m ) = 1 √ λiz αa − ∂∂ ¯ z αa ) , ρ λ ( v µ ⊗ w ′′ a + m ) = 1 √ λi ¯ z µa − ∂∂z µa ) ,ρ λ ( v α ⊗ w ′′ a ) = 1 √ λiz αa + 2 ∂∂ ¯ z αa ) , ρ λ ( v µ ⊗ w ′ a ) = 1 √ λi ¯ z µa + 2 ∂∂z µa ) ,ρ λ ( v α ⊗ w ′′ a + m ) = 1 √ − λi ¯ z αa − ∂∂z αa ) , ρ λ ( v µ ⊗ w ′ a + m ) = 1 √ − λiz µa − ∂∂ ¯ z µa ) , where 1 ≤ α ≤ p, p + 1 ≤ µ ≤ p + q , 1 ≤ a ≤ m and ∂∂z jk = 12 ( ∂∂x jk − ∂∂y jk i ) , ∂∂ ¯ z jk = 12 ( ∂∂x jk + ∂∂y jk i ) . If we fix the parameter λ = 2 πi , then we have ϕ ( z ) = exp( − π p + q X k =1 m X a =1 | z ka | ) .ι : P λ → S ( V m ) maps 1 ∈ P λ to ϕ . From (4.20) we have the following lemma. HETA SERIES AND CYCLES 21
Lemma 4.3. ι ( u + αa ) ι − = 1 √ − πz αa + 2 ∂∂ ¯ z αa ) , ι ( u + µa ) ι − = 1 √ − π ¯ z µa + 2 ∂∂z µa ) ,ι ( u − αk ) ι − = 1 √ π ¯ z αa − ∂∂z αa ) , ι ( u − µk ) ι − = 1 √ πz µa − ∂∂ ¯ z µa ) for ≤ α ≤ p, p + 1 ≤ µ ≤ p + q , ≤ a ≤ m , k = a + m . In particular, if p is amonomial in the variables { u + αa , u + µa , u − αk , u − µk | ≤ α ≤ p, p + 1 ≤ µ ≤ p + q, ≤ a ≤ m, m + 1 ≤ k ≤ m } : p = p Y α =1 p + q Y µ = p +1 m Y a =1 2 m Y k = m +1 ( u + αa ) d αa ( u + µa ) d µa ( u − αk ) d αk ( u − µk ) d µk , then we have ι ( p ) = p Y α =1 p + q Y µ = p +1 m Y a =1 2 m Y k = m +1 ( √ d αa + d µa ( −√ d αk + d µk · ( ∂∂ ¯ z αa − πz αa ) d αa ( ∂∂z µa − π ¯ z µa ) d µa ( ∂∂z αa − π ¯ z αa ) d αk ( ∂∂ ¯ z µa − πz µa ) d µa ϕ . We will need the following lemma.
Lemma 4.4.
Suppose p is the same as in the previous lemma. ι ( p ) = ˜ pϕ , where ˜ p is a polynomials of the variables { z αa , z µa , ¯ z αa , ¯ z µa | ≤ α ≤ p, p + 1 ≤ µ ≤ p + q, ≤ a ≤ m } whose unique highest degree term (in every variable) is p Y α =1 p + q Y µ = p +1 m Y a =1 2 m Y k = m +1 ( − √ πz αa ) d αa ( − √ π ¯ z µa ) d µa (2 √ π ¯ z α,k − m ) d αk (2 √ πz µ,k − m ) d µk . Proof.
Since ϕ = p + q Y α =1 m Y a =1 exp( − π | z αa | )and the operators { ι ( u + αa ) , ι ( u + µa ) , ι ( u − αk ) , ι ( u − µk ) | ≤ α ≤ p, p + 1 ≤ µ ≤ p + q, ≤ a ≤ m, m + 1 ≤ k ≤ m } commute with each other, it suffices to prove the lemmafor the case of one variable. That is( ∂∂ ¯ z − πz ) d ( ∂∂z − π ¯ z ) d · exp( − π | z | ) = ˜ p exp( − π | z | ) , where the highest degree term of ˜ p is ( − πz ) d ( − π ¯ z ) d . But this follows from aneasy induction on the bi-dgree ( d , d ). (cid:3) The Dual Pair (Sp(2 n, R ) , O(2 r, s )) . The purpose of this subsection is toexplain the relation between two different constructions of the fundamental mod-ule W of the dual pair (Sp(2 n, R ) , O(2 r, s )). One of them comes directly fromour global construction of the algebraic group in Section 2. The other is whatwe actually use when studying (the relative Lie algebra cohomology of) the Weilrepresentation. In this subsection let B be M ( R ) and V B be a free right B module of rank n .Let σ be the anti-involution of M ( R ) defined by σ ( x ) = Jx t J − , where J = (cid:18) −
11 0 (cid:19) . Let ( , ) B be a Hermitian form on V B satisfying (2.2).Define G to be the isometry group of ( V B , ( , ) B ). Let e = e , then V B e is a2 n dimensional real vector space. Recall from Section 2.3 that we can define asymplectic form ( , ) on V B e by( ve, v ′ e ) e = ( ve, v ′ e ) B . This implies that G ∼ = Sp(2 n, R ). Let W B be a free left B module of rank m and <, > B be a skew Hermitian form on W B satisfying < bw, ˜ b ˜ w > B = b < w, ˜ w > B ˜ b σ . Define G ′ to be the isometry group of ( W B , <, > B ). eW B is a 2 m dimensional realvector space. We can define a symmetric form <, > R on eW B by < ew, ew ′ > R e = < ew, ew ′ > B . This implies that G ′ ∼ = O(2 r, s ).Now let W = V B ⊗ B W B . Then we have W ∼ = V B e ⊗ R eW B as a real vector spaceand as a GL B ( V B ) × GL B ( W B ) module. In fact if one think of v ∈ V B as a 2 n by 2 matrix and w ∈ W B as a 2 by 2 m matrix, the tensor product v ⊗ B w is justthe matrix multiplication vw . Obviously one get the same space by tensoring V B e (the set of 2 n by 1 matrices) with eW B (the set of 1 by 2 m matrices). Moreover,GL B ( V B ) ∼ = GL n ( R ) (resp. GL B ( W B ) ∼ = GL m ( R )) acts by left (resp. right)multiplication on W .One can define hh , ii on W by hh v ⊗ w, ˜ v ⊗ ˜ w ii = tr R [( v, ˜ v ) B < ˜ w, w > B ] . Then hh , ii R is symplectic. One can also define hh , ii R on W (regarded as V B e ⊗ R eW B ) by hh ve ⊗ ew, ˜ ve ⊗ e ˜ w ii R = ( ve, ˜ ve ) < e ˜ w, ew > R . We have the following interesting fact
Lemma 4.5.
There exists a nonzero constant c ∈ R such that hh , ii = c hh , ii R Proof.
We have the following fact (for example see equation (19), page 121 of [FF97]or verify directly) ∧ ( V B e ⊗ eW B ) ∼ = [ ∧ ( V B e ) ⊗ Sym ( eW B )] ⊕ [Sym ( V B e ) ⊗ ∧ ( eW B )] . Both hh , ii and hh , ii R are non-degenerate, skew symmetric and invariant under G × G ′ . By classical invariant theory the invariant subspace of ∧ ( V B e ) ⊗ Sym ( eW B )under G × G ′ is one dimensional and the invariant subspace in Sym ( V B e ) ⊗∧ ( eW B )is trivial. This proves that the two forms are the same up to a constant multiple. (cid:3) HETA SERIES AND CYCLES 23
Case B: the seesaw dual pairs 4.13.
Let V be a 2 n dimensional real vectorspace with a non-degenerate skew symmetric form ( , ) . Let V = V ⊗ C and weextend ( , ) from V to V anti-linearly in the first variable and linearly in the secondvariable. Denote the resulting skew Hermitian form by ( , ). The Hermitian form i () , has signature ( n, n ). In fact let E , . . . , E n , F , . . . , F n be a symplectic basis of( V , ( , ) ). Define(1) v α = √ ( E α − iF α ) for 1 ≤ α ≤ n ,(2) v µ = √ ( E µ − n + iF µ − n ) for n + 1 ≤ µ ≤ n .Then { v , . . . , v n } is an orthogonal basis of ( V, ( , )) such that( v α , v α ) = − i, ( v µ , v µ ) = i for 1 ≤ α ≤ n, n + 1 ≤ µ ≤ n .Let W be a m + m ′ dimensional complex vector space with a Hermitian form <, > of signature ( m, m ′ ) which is linear in the first variable and anti-linear in thesecond variable. And define <, > R = Re <, > . Then <, > R is a symmetric form ofsignature (2 m, m ′ ). We have(4.21) V m + m ′ ∼ = V ⊗ C W = ( V ⊗ R C ) ⊗ C W ∼ = V ⊗ R W ∼ = V m +2 m ′ . Recall that we can define W = V ⊗ C W and a skew Hermitian form hh , ii h on W by (4.17). Also define a symplectic form hh , ii R on W (regarded as V ⊗ R W ) by hh v ⊗ w, ˜ v ⊗ ˜ w ii = ( v, ˜ v ) < ˜ w, w > R . It is easy to check directly that
Lemma 4.6. Re hh , ii h = hh , ii . Remark . The above lemma shows that the seesaw dual pairs in 4.13 share thesame underlying symplectic module ( W , hh , ii ). Thus they give rise to the sameWeil representation. Thus the Fock or Schr¨odinger model of (U( n, n ) , U( m, m ′ ))can serve as the Fock or Schr¨odinger model of (Sp(2 n, R ) , O(2 m, m ′ )) as well. Remark . Combine the above remark with the discussion in Section 4.4, theSchr¨odinger model of (Sp(2 n, R ) , O(2 m, m )) is S ( V m ) ∼ = S ( V m ) ∼ = S (( M ( R ) n ) m ) . Case C: the seesaw dual pair 4.14.
Let V be a n -dimensional right H vector space with skew Hermitian form ( , ) satisfying(4.22) ( vb, ˜ v ˜ b ) = b σ ( v, ˜ v ) ˜ b for v, ˜ v ∈ V and b, ˜ b ∈ H . Define a complex skew Hermitian form ( , ) on V by( v, ˜ v ) = a + bi if ( v, ˜ v ) = a + bi + cj + dk. Then the Hermitian form i ( , ) has signature ( n, n ). In fact choose an H -basis { v , . . . , v n } of V such that( v α , v β ) = − iδ αβ , ≤ α, β ≤ n. Define v µ := v µ − n j, n + 1 ≤ µ ≤ n. Then { v , . . . , v n , v n +1 , . . . , v n } is an orthogonal basis of ( V, ( , )) such that( v α , v α ) = − i and ( v µ , v µ ) = i for 1 ≤ α ≤ n, n + 1 ≤ µ ≤ n .Let W be a m + m ′ dimensional left C vector space with non-degenerate Hermit-ian form <, > of signature ( m, m ′ ) that is complex linear in the first variable andanti-linear in the second variable. Define W H = H ⊗ C W , extend <, > to a form on W H denoted as <, > H satisfying < hv, ˜ h ˜ v > H = h < v, ˜ v > H ˜ h σ for h, ˜ h ∈ H . Then we have a canonical isomorphism V ⊗ H W H ∼ = V ⊗ C W. Let W = V ⊗ H W H . Define hh , ii on W by hh v ⊗ w, ˜ v ⊗ ˜ w ii = Re[( v, ˜ v ) < ˜ w, w > H ] . hh , ii is well-defined on W and is a symplectic form. Also define hh , ii h on W (regarded as V ⊗ C W ) by (4.17). Then we have Lemma 4.9. hh , ii = Re hh , ii h . Remark . The above lemma shows that the seesaw dual pairs in 4.13 share thesame underlying symplectic module ( W , hh , ii ). Thus they give rise to the sameWeil representation. Thus the Fock or Schr¨odinger model of (U( n, n ) , U( m, m ′ ))can serve as the Fock or Schr¨odinger model of (O ∗ (2 n ) , Sp( m, m ′ )) as well.5. Special Schwartz classes in the relative Lie Algebra cohomologyof the Weil representation
In this section we review the construction of holomorphic differential forms in[And83]. We use this result to construct the special canonical class ϕ as in Theorem1.3. We will prove that ϕ is closed. In this section, G denotes a real Lie group.Let g be the Lie algebra of G and g be its complexification. Fix a maximalcompact subgroup z = K of G and the corresponding Cartan decomposition g = k + p . Identify T z D with p , where D = G/K . By parallel translating thetrace form on p by the group G , we endow D with a Riemannian metric denoteedby τ . We assume that D is Hermitian symmetric. Decompose p = p ⊗ C intoholomorphic and anti-holomorphic tangent vectors p = p + + p − . For a ( g , K ) module M , define C • ( g , K ; M ) = Hom K ( ∧ • ( g / k ) , M ) ∼ = Hom K ( ∧ • p , M ) ∼ = ( ∧ • p ∗ ⊗ M ) K . It is a cochain complex and gives rise to the relative Lie algebra cohomology H • ( g , K ; M ) (see [BW13]) . In his thesis [And83], Anderson constructed cochains φ + in Hom K ( ∧ • p + , P p − − ), where P − is the Fock model of a certain Weil represen-tation. Here, the notation P p − − denotes the subspace of P − annihilated by p − . Wecan construct a mirror element φ − ∈ Hom K ( ∧ • p − , P p + + ). Then we take φ to bethe out wedge product of φ + and φ − and ϕ to be ι ( φ ). We now carry out thisconstruction case by case and prove some lemmas (Lemma 5.4, Lemma 5.7 andLemma 5.10) along the way. HETA SERIES AND CYCLES 25
Case A.
We follow the assumptions and notations of Section 4.3. Recall that V is a p + q dimensional complex vector space with a skew Hermitian form ( , ) suchthat i ( , ) has signature ( p, q ). Let ¯ V be the conjugate complex vector space of V :¯ V has the same underlying abelian group as V but the complex multiplication of¯ V is the conjugate of V .Let V ∗ = Hom C ( V, C ). There is a complex linear isomorphism ¯ V → V ∗ givenby v ( v, · ) . One can also identify V ⊗ C ¯ V with Hom C ( V, V ) by the map v ⊗ ˜ v v (˜ v, · ) . Let Sym( V ⊗ C ¯ V ) be the symmetric tensor inside V ⊗ C ¯ V (this makes sense since V and ¯ V have the same underlying Abelian group). By the above identification,Sym( V ⊗ C ¯ V ) acts on V by( v ◦ ˜ v )( x ) = v (˜ v, x ) + ˜ v ( v, x ) , ∀ x ∈ V, where v ◦ ˜ v = v ⊗ ˜ v + ˜ v ⊗ v . One can check that this action satisfies(( v ◦ ˜ v )( x ) , y ) + ( x, ( v ◦ ˜ v )( y )) = 0 . In fact we have
Lemma 5.1.
Sym( V ⊗ C ¯ V ) ∼ = u ( V, ( , )) = u ( p, q ) . Define V + = span C { v , . . . , v p } and V − = span C { v p +1 , . . . , v p + q } . The splitting V = V + + V − corresponds to a point z = V − in the symmetric space D of G . Itsstabilizer is K ∼ = U( p ) × U( q ). We have a corresponding Cartan decomposition of g = u ( V, ( , )): g = k + p , where k = (Hom( V − , V − ) ⊕ Hom( V + , V + )) ∩ g , p = (Hom( V − , V + ) ⊕ Hom( V + , V − )) ∩ g . More explicitly define(5.1) E mn = v m ◦ v n and F mn = iv m ◦ v n for 1 ≤ m, n ≤ p + q . Then we have(1) k = span R { E αβ , F αβ , E µν , F µν } ,(2) p = span R { E αµ , F αµ } ,where 1 ≤ α, β ≤ p and p + 1 ≤ µ, ν ≤ p + q (in this subsection we keep thisconvention of index). In terms of matrices we have k = (cid:26)(cid:18) A B (cid:19) | A + A ∗ = 0 , B + B ∗ = 0 (cid:27) , p = (cid:26)(cid:18) AA ∗ (cid:19) | A ∈ M p × q ( C ) (cid:27) . We now describe the Ad( K )-invariant almost complex structure J p acting on p that induces the structure of Hermitian symmetric domain on D (c.f. page 14 of[BMM16]). Let ζ = e πi . Define a ( ζ ) by(5.2) a ( ζ )( v α ) = v α ζ and a ( ζ ) v µ = v µ ζ − . Now we define(5.3) J p = Ad( a ( ζ )) . It is easy to check under the identification of V ⊗ C ¯ V ∼ = Hom C ( V, V ) we haveAd( a ( ζ ))( v ⊗ ˜ v ) = ( a ( ζ ) v ) ⊗ ( a ( ζ )˜ v ) . This implies that J p ( E αµ ) = F αµ , J p ( F αµ ) = − E αµ . Define X αµ = E αµ − iF αµ = 2 e αµ , Y αµ = E αµ + iF αµ = 2 e µα , where e ab is the matrix whose ( a, b )-th entry is 1 and all other entries are zero. Let p + (resp. p − ) be the + i (resp. − i ) eigenspace of J p . We then have p + = span C { X αµ | ≤ α ≤ p, p +1 ≤ µ ≤ p + q } , p − = span C { Y αµ | ≤ α ≤ p, p +1 ≤ µ ≤ p + q } . We also let { ξ ′ αµ | ≤ α ≤ p, p + 1 ≤ µ ≤ p + q } (resp. { ξ ′′ αµ | ≤ α ≤ p, p + 1 ≤ µ ≤ p + q } ) be the basis of p ∗ + (resp. p ∗− ) that is dual to { X αµ } (resp. { Y αµ } ).Now fix 1 ≤ r ≤ p, ≤ s ≤ q . Define U + = span C { v , . . . , v r } , U − = span { v p +1 , . . . , v p + s } , and U = U + + U − . We fix z ′ = U − ∈ D ( U ). Recall from equation (3.6) that D U,z ′ = { z ∈ D | U − ⊂ z ⊂ U + } ∼ = D ( U ⊥ ) , where D U,z ′ is the generalized special sub symmetric space in Definition 3.2. Fix abase point z = span { v p +1 , . . . , v p + q } ∈ D U,z ′ . Identify T z D with p , then T z D U,z ′ = span R { E αµ , F αµ | r + 1 ≤ α ≤ p, p + s + 1 ≤ µ ≤ p + q } . Recall that we define a fiber bundle π : D → D U,z ′ in Subsection 3.2 as follows.At each point z ∈ D U,z ′ , the fiber is the union of all geodesics that are perpendicularto D U,z ′ at z . The tangent space of the fiber F z D U,z ′ = π − ( z ) at z can bedescribed as(5.4) N z D U,z ′ = T z D ⊥ U,z ′ = span R { E αµ , F αµ | ( α, µ ) ∈ I } , where I is the index set(5.5) I = { ( α, µ ) | ≤ α ≤ r, p +1 ≤ µ ≤ p + q }∪{ ( α, µ ) | r +1 ≤ α ≤ p, p +1 ≤ µ ≤ p + s } . We also have N + z D U,z ′ = { X αµ | ( α, µ ) ∈ I } . Next let P − be the infinitesimal Fock model defined in Section 4.3 for the dualpair (U( p, q ) , U(0 , r + s )). Recall that P − is the polynomial space in the variables { u − ik | ≤ i ≤ p + q, ≤ k ≤ r + s } . We now define polynomials f − , f − ∈ P − by Definition 5.2. f − = det u − u − . . . u − r . . . . . . . . . . . .u − r u − r . . . u − rr , f − = det u − p +1 r +1 u − p +1 r +2 . . . u − p +1 r + s . . . . . . . . . . . .u − p + s r +1 u − p + s r +2 . . . u − p + s r + s . HETA SERIES AND CYCLES 27
We define an element of ( ∧ • p + ⊗P p − − ) SU( p,q ) following the construction of [And83].To be more precise, let ˜ K be the preimage of K under the map Mp( W ) → Sp( W )and ˜ K be the identity component of ˜ K . Then ˜ K is the det − r + s -cover of K (c.f.[Pau98] or [Kud94]): ˜ K ∼ = { ( g, z ) ∈ K × C × | z = det( g ) − r + s } . Define e D U,z ′ = ^ ( α,µ ) ∈ I X αµ . Also define(5.6) f D U,z ′ = ( f − ) q − s ( f − ) p − r . The polynomials f − , f − and f D U,z ′ are special cases of the harmonic polynomialsstudied in [KV78]. It can be shown that k acts on P − by (c.f. equation (3.5) of[And83]) ω ( e αβ ) = r + s X k =1 u − αk ∂∂u − βk + 12 δ αβ ( r + s ) , ω ( e µν ) = − r + s X k =1 u − νk ∂∂u − µk − δ µν ( r + s ) . The adjoint action of k on p + induces an action on ∧ • p + . Define b = span C { e αβ | ≤ α ≤ β ≤ p } ⊕ span C { e µν | p + 1 ≤ ν ≤ µ ≤ p + q } . Then b is a Borel sub-algebra of k . One can verify that both e D U,z ′ and f D U,z ′ arehighest weight vectors with respect to b . The weight of e D U,z ′ with respect to b is( q, . . . , q | {z } r , s, . . . , s | {z } p − r , − p, . . . , − p | {z } s , − r, . . . , − r | {z } q − s ) . The weight of f D U,z ′ with respect to b is( q + 12 ( r − s ) , ., q + 12 ( r − s ) | {z } r ,
12 ( r + s ) , .,
12 ( r + s ) | {z } p − r ,
12 ( r − s ) − p, .,
12 ( r − s ) − p | {z } s , −
12 ( r + s ) , ., −
12 ( r + s ) | {z } q − s ) . It is easy to observe that these two weights differ by − ( 12 ( r − s ) , . . . ,
12 ( r − s ) | {z } p + q ) . Now denote the irreducible representation of ˜ K generated by e D U,z ′ as V ( U ),the irreducible representation of ˜ K generated by f D U,z ′ as A ( U ), where ˜ K is theidentity component ˜ K . By the theory of highest weight we have a ˜ K -equivariantmap ψ + r,s : V ( U ) → A ( U ) ⊗ det − ( r − s ) such that ψ + r,s ( e D U,z ′ ) = f D U,z ′ ⊗ . Then Theorem A of [And83] can be rephrased as
Theorem 5.3. ψ + r,s ∈ Hom ˜ K ( ∧ rq + ps − rs p + , P p − − ⊗ det − ( r − s ) ) . Let { ǫ , . . . , ǫ d } be a basis of V ( U ) ⊂ ∧ • p + such that each ǫ i is a weight vec-tor of t . Extend { ǫ , . . . , ǫ d } to a basis of ∧ rq + ps − rs p + , take the dual basis inside ∧ rq + ps − rs p ∗ + and denote the first d basis vectors by Ω , . . . , Ω d . We have an iso-morphismHom ˜ K ( ∧ • p + , P p − − ⊗ det − ( r − s ) ) ∼ = ( ∧ • p ∗ + ⊗ P p − − ⊗ det − ( r − s ) ) ˜ K . Under this isomorphism ψ + r,s maps to an element φ + r,s ∈ ( ∧ rq + ps − rs p ∗ + ⊗ P p − − ⊗ det − ( r − s ) ) ˜ K :(5.7) φ + r,s = d X i =1 ψ + r,s ( ǫ i )Ω i . The element thus defined is independent of the choice of the basis { ǫ , . . . , ǫ d } andis actually in ( ∧ • p + ⊗ P p − − ) SU( p,q ) .Let i : F z D U,z ′ → D be the natural embedding. We have the following crucial lemma which states thatwhen restricted to the fiber F z D U,z ′ at z there is only one term left in φ + r,s . Lemma 5.4. i ∗ ( φ + r,s ( x )) | z = ( f − ) q − s ( f − ) p − r i ∗ ( ^ ( α,µ ) ∈ I ξ ′ α,µ ) | z . Proof.
Recall that { X αµ | ( α, µ ) ∈ I } ( I is defined in Equation (5.5)) span theholomorphic tangent space N + z D U,z ′ of F z D U,z ′ at z and X αµ is perpendicularto N + z D U,z ′ if ( α, µ ) / ∈ I . Hence { ξ ′ αµ | ( α, µ ) ∈ I } span the holomorphic cotangentspace of F z D U,z ′ at z and i ∗ ( ξ ′ αµ ) | z = 0 if ( α, µ ) / ∈ I .Now let S = { ( α, µ ) | ≤ α ≤ p, p + 1 ≤ µ ≤ p + q } . Then { ^ ( α,µ ) ∈ T ξ ′ αµ | T ⊆ S, | T | = rq + ps − rs } is a basis of ∧ rq + ps − rs p ∗ + . If T = I , by the argument in the last paragraph we have i ∗ ( ^ ( α,µ ) ∈ T ξ ′ αµ ) | z = 0 . So the only term left in i ∗ ( φ + r,s ( x )) | z is the first term in (5.7) which is the righthand side of the lemma by the definition of φ + r,s . (cid:3) Similarly let P + be the Fock model for the dual pair (U( p, q ) , U( r + s, P + is the polynomial space in the variables { u + ia | ≤ i ≤ p + q, ≤ a ≤ r + s } .We now define polynomials f +1 , f +2 ∈ P + by f +1 = det u +11 u +12 . . . u +1 r . . . . . . . . . . . .u + r u + r . . . u + rr , f +2 = det u + p +1 r +1 u + p +1 r +2 . . . u + p +1 r + s . . . . . . . . . . . .u + p + s r +1 u + p + s r +2 . . . u + p + s r + s . Then ( f +1 ) q − s ( f +2 ) p − r is a lowest weight vector with respect to b of weight( 12 ( s − r ) − q, .,
12 ( s − r ) − q | {z } r , −
12 ( r + s ) , ., −
12 ( r + s ) | {z } p − r , p + 12 ( s − r ) , ., p + 12 ( s − r ) | {z } s ,
12 ( r + s ) , .,
12 ( r + s ) | {z } q − s ) . HETA SERIES AND CYCLES 29 V ( α,µ ) ∈ I Y αµ is a lowest weight vector with respect to b of weight( − q, . . . , − q | {z } r , − s, . . . , − s | {z } p − r , p, . . . , p | {z } s , r, . . . , r | {z } q − s ) . Then there is a unique element ψ − r,s ∈ Hom ˜ K ( ∧ rq + ps − rs p − , P p + + ⊗ det ( r − s ) ) , which maps V ( α,µ ) ∈ I Y αµ to ( f +1 ) q − s ( f +2 ) p − r and a corresponding element φ − r,s ∈ ( ∧ rq + ps − rs p ∗− ⊗ P p + + ⊗ det ( r − s ) ) ˜ K . Now let P be the infinitesimal Fock model for the dual pair (U( p, q ) , U( r + s, r + s )). We have P = P − ⊗ P + ∼ = P − ⊗ det − ( r − s ) ⊗ P + ⊗ det ( r − s ) . We define the following outer wedge product φ r,s = φ + r,s ∧ φ − r,s . It is immediate that φ r,s ∈ ( ∧ rq +2 ps − rs p ∗ ⊗ P ) K . We say ( r, s ) is the signature of φ r,s . Remark . The form φ + r, and φ − r, are also constructed in [BMM16]. The form φ r, is constructed in both [KM90] and [BMM16] and is called the Kudla-Millsonform in the literature.5.2. Case B.
We use the fact that G = Sp(2 n, R ) ∼ = Sp(2 n, C ) ∩ U( n, n ) ([And83]).More precisely let V be a 2 n -dimensional real vector space with a skew symmetricform ( , ) . Then we can extend ( , ) linearly to a skew symmetric form S ( , ) on V = V ⊗ R C . We can also extend ( , ) to a skew Hermitian form ( , ) on V satisfying(2.2). Then Sp( V , ( , ) ) = Sp( V, S ( , )) ∩ U( V, ( , )) . Moreover let g = sp ( V , ( , ) ). Then we have (Section 7 of [KM90]) g ∼ = Sym ( V ) , and g = g ⊗ C ∼ = sp ( V, S ( , )) ∼ = Sym ( V ) , where Sym ( V ) acts on V by( v ⊗ ˜ v + ˜ v ⊗ v )( x ) = vS (˜ v, x ) + ˜ vS ( v, x ) , ∀ x ∈ V. We will denote ( v ⊗ ˜ v + ˜ v ⊗ v ) ∈ Sym ( V ) by v ⋄ ˜ v . The linear transformation a ( ζ )introduced in equation (5.2) sits inside Sp( V, S ( , )) ∩ U( V, ( , )). Thus the almostcomplex structure J p introduced in (5.3) stabilize g and induced an almost complexstructure on g .The map Sp( V , ( , ) ) ֒ → U( V, ( , )) induces an embedding of symmetric spaces D ֒ → ˜ D , where D is the symmetric space of Sp( V , ( , ) ) and ˜ D is the symmetricspace of U( V, ( , )). To be more precise, recall that ˜ D is the set of n -dimensionalsubspaces of V such that z ∈ ˜ D if and only if the Hermitian form i ( , ) is negativedefinite on z . Then we have D = { z ∈ ˜ D | S ( , ) | z is zero } . Choose a symplectic basis { E , . . . , E n , F , . . . , F n } of V and let v α = 1 √ E α − F α i ) , v µ = 1 √ E µ − n + F µ − n i )for 1 ≤ α ≤ n, n + 1 ≤ µ ≤ n (in this subsection we keep this convention of index).Let z = span C { v n +1 , . . . , v n } ∈ D. Its stablizer is K ∼ = U( n ). We have the corresponding Cartan decomposition g = k + p and p = p + + p − , where k is the 0 eigenspace of J p and p + (resp. p − ) is the + i (resp. − i ) eigenspaceof J p . In terms of matrices we have(5.8) k = (cid:26)(cid:18) A − A t (cid:19) | A ∈ M n ( C ) , A ∗ = − A (cid:27) , p = (cid:26)(cid:18) AA ∗ (cid:19) | A ∈ M n ( C ) , A t = A (cid:27) . If we define V + = span C { v , . . . , v n } and V − = span C { v n +1 , . . . , v n } as in thelast subsection, we have the identifications k = span C { v ⋄ ˜ v | v ∈ V + , ˜ v ∈ V − } . Define X αβ = 1 i v α ⋄ v β = e β,α + n + e α,β + n , Y αβ = iv α + n ⋄ v β + n = e α + n,β + e β + n,α for 1 ≤ α ≤ β ≤ n . Then p + = span C { X αβ | ≤ α ≤ β ≤ n } , p − = span C { Y αβ | ≤ α ≤ β ≤ n } . We also let { ξ ′ αβ | ≤ α ≤ β ≤ n } (resp. { ξ ′′ αβ | ≤ α ≤ β ≤ n } ) be the basis of p ∗ + (resp. p ∗− ) dual to the above basis.Now fix 1 ≤ r ≤ n . Define U = span R { E , . . . , E r , F , . . . , F r } . Let z ′ = span C { v n +1 , . . . , v n + r } ∈ D ( U ) . Recall that in Definition 3.2 we define a sub symmetric space D U,z ′ of D . Theholomorphic tangent space of D U,z ′ at z is T + z D U,z ′ = span C { X αβ | r + 1 ≤ α ≤ β ≤ n } . The holomorphic tangent space of the fiber F z D U,z ′ (see Subsection 3.2) at z is(5.9) N + z D U,z ′ = span C { X αβ | ( α, β ) ∈ I } , where I is the index set(5.10) I = { ( α, β ) | ≤ α ≤ r, α ≤ β ≤ n } . Define e D U,z ′ ∈ ∧ • p + by e D U,z ′ = ^ ( α,β ) ∈ I X αβ . Let P − be the infinitesimal Fock model for the dual pair (Sp(2 n, R ) , O(0 , r ))defined in Section 4.5. Recall that P − is the same as the Fock model for the dual HETA SERIES AND CYCLES 31 pair (U( n, n ) , U(0 , r )) and is the polynomial space in the variables { u − ik | ≤ i ≤ n, ≤ k ≤ r } . Define f − ∈ P − by f − = det u − u − . . . u − r . . . . . . . . . . . .u − r u − r . . . u − rr . Also define(5.11) f D U,z ′ = ( f − ) n − r +1 . It can be shown that k acts on W − by (c.f. formula (4.2) of [And83])(5.12) ω ( v α ⋄ v β + n ) = ω ( − ie α,β + ie β + n,α + n ) = − i r X k =1 ( u α + n,k ∂∂u β + n,k + u α,k ∂∂u β,k ) − irδ αβ . The adjoint action of k on p + induces an action on ∧ • p + . Define b = span C { v α ⋄ v β + n | ≤ α ≤ β ≤ n } . b is a Borel sub-algebra of k . Both e D U,z ′ and f D U,z ′ are highest weight vectors of b . Moreover they have the same weight( n + 1 , . . . , n + 1 | {z } r , r, . . . , r | {z } n − r ) . Let ˜ K the preimage of K under the map Mp( W ) → Sp( W ). Using the seesaw pair(5.13) U( n, n ) O(0 , r )Sp(2 n, R ) O O qqqqqqqqq U(0 , r ) O O f f ▼▼▼▼▼▼▼▼▼ and facts about ^ U( n, n ) (see the last subsection), we can see that˜ K = K × {± } . Now denote the irreducible representation of K generated by e D U,z ′ as V ( U ),the irreducible representation of K generated by f D U,z ′ as A ( U ). By the theory ofhighest weight we have a K -equivariant map ψ +2 r : V ( U ) → A ( U ) such that ψ + r ( e D U,z ′ ) = f D U,z ′ . Theorem B of [And83] is
Theorem 5.6. ψ + r ∈ Hom K ( ∧ n ( n +1) − ( n − r )( n − r +1) p + , P p − − ) . Let { ǫ , . . . , ǫ d } be a basis of V ( U ) ⊂ ∧ • p + such that each ǫ i is a weight vector of t . Extend { ǫ , . . . , ǫ d } to a basis of ∧ n ( n +1) − ( n − r )( n − r +1) p + , take the dual basisinside ∧ n ( n +1) − ( n − r )( n − r +1) p ∗ + and denote the first d basis vectors by Ω , . . . , Ω d .We have an isomorphismHom K ( ∧ • p + , P p − − ) ∼ = ( ∧ • p ∗ + ⊗ P p − − ) K . Under this isomorphism ψ + r maps to an element φ + r ∈ ( ∧ n ( n +1) − ( n − r )( n − r +1) p ∗ + ⊗P p − − ) K : φ + r = d X i =1 ψ + ( ǫ i )Ω i . The element thus defined is independent of the choice of the basis { ǫ , . . . , ǫ d } .Let i : F z D U,z ′ → D be the natural embedding. Using the definition of φ + r , we can prove the followinglemma in a similar way as Lemma 5.4. Lemma 5.7. i ∗ ( φ + r ( x )) | z = ( f − ) n − r +1 i ∗ ( ^ ( α,µ ) ∈ I ξ ′ α,µ ) | z . Let P + be the infinitesimal Fock model for the dual pair (Sp(2 n, R ) , O(2 r, P + is the polynomial space in the variables { u + ia | ≤ i ≤ n, ≤ a ≤ r } . Define f + ∈ P + by f + = det u +11 u +12 . . . u +1 r . . . . . . . . . . . .u + r u + r . . . u + rr . Then both ( f + ) n − r +1 and V ( α,β ) ∈ I Y αβ are lowest weight vector with respect to b of weight ( − n − , . . . , − n − | {z } r , − r, . . . , − r | {z } n − r ) . There is a unique element ψ − r ∈ Hom K ( ∧ n ( n +1) − ( n − r )( n − r +1) p − , P p + + ) , which maps ( f + ) n − r +1 to V ( α,β ) ∈ I Y αβ and a corresponding element φ − r ∈ ( ∧ n ( n +1) − ( n − r )( n − r +1) p ∗− ⊗ P p + + ) K . Now let P be the infinitesimal Fock model for the dual pair (Sp(2 n, R ) , O(2 r, r )).We have P = P − ⊗ P + . Define φ r = φ + r ∧ φ − r . Then it is immediate that φ r ∈ ( ∧ n ( n +1) − ( n − r )( n − r +1) p ∗ ⊗ P ) K . Case C.
Let V be a n -dimensional right H vector space with skew Hermitianform ( , ) satisfying (4.22). Let V C denote the underlying complex vector space of V . Define V σ the conjugate left H vector space of V as follows. The underlyingAbelian group of V σ is the same with that of V . The scalar multiplication of V σ is defined by hv = vh σ , ∀ h ∈ H , HETA SERIES AND CYCLES 33 where the left hand side is scalar multiplication in V σ while the right hand side isscalar multiplication in V . There is an isomorphism V σ → Hom H ( V, H ) as left H module given by the form ( , ) v ( v, · ) . One can also identify V ⊗ H V σ with Hom H ( V, V ) by the map v ⊗ ˜ v v (˜ v, · ) . Define Sym( V ⊗ H V σ ) = span R { v ⊗ ˜ v + ˜ v ⊗ v | v, ˜ v ∈ V } ⊂ V ⊗ H V σ . By the above identification, Sym( V ⊗ H V σ ) acts on V by( v ⋄ ˜ v )( x ) = v (˜ v, x ) + ˜ v ( v, x ) , ∀ x ∈ V, where v ⋄ ˜ v = v ⊗ ˜ v + ˜ v ⊗ v . One can check that this action is H -linear and satisfies(( v ⋄ ˜ v )( x ) , y ) + ( x, ( v ⋄ ˜ v )( y )) = 0 . Moreover we have
Lemma 5.8.
Sym( V ⊗ H V σ ) ∼ = o ∗ ( V, ( , ) ) = o ∗ (2 n ) . Define ( , ) on V C by( v, ˜ v ) = a + bi if ( v, ˜ v ) = a + bi + cj + dk. Then i ( , ) is (complex) Hermitian of signature ( n, n ). We also define S ( , ) on V C by S ( v, ˜ v ) = ( vj, ˜ v ) . One can check that S ( , ) is symmetric complex bilinear. We have the following fact([And83]) O ∗ ( V, ( , ) ) = U( V, ( , )) ∩ O( V, S ( , )) . It can also be shown that g = g ⊗ C ∼ = o ( V, S ( , )) ∼ = ^ ( V C ) , where g = o ∗ ( V, ( , )) and V ( V C ) acts on V by v ∧ ˜ v ( x ) = − vS (˜ v, x ) + ˜ vS ( v, x ) , ∀ x ∈ V. The map O ∗ ( V, ( , ) ) ֒ → U( V, ( , )) induces an embedding of symmetric spaces D ֒ → ˜ D , where D is the symmetric space of O ∗ ( V, ( , ) ) and ˜ D is the symmetricspace of U( V, ( , )). To be more precise, recall that ˜ D is the set of n -dimensionalsubspaces of V C such that z ∈ ˜ D if and only if the Hermitian form i ( , ) is negativedefinite on z . Then we have D = { z ∈ ˜ D | S ( , ) | z is zero } . Explicitly choose an orthogonal basis { v , . . . , v n } of V such that( v α , v β ) = − iδ αβ for 1 ≤ α, β ≤ n . Then { v , . . . , v n , v n +1 , . . . , v n } is a basis of V C , where v µ = v µ − n j for n + 1 ≤ µ ≤ n . The C -linear transformation a ( ζ ) introduced in equation(5.2) sits inside O( V, S ( , )) ∩ U( V, ( , )). Thus the almost complex structure (5.3)stabilize g and induces an almost complex structure on g . Let z = span C { v n +1 , . . . , v n } ∈ D. Its stablizer is K ∼ = U( n ). We have the corresponding Cartan decomposition g = k + p and p = p + + p − , where k is the 0 eigenspace of J p and p + (resp. p − ) is the + i (resp. − i ) eigenspaceof J p . In terms of matrices we have(5.14) k = (cid:8) (cid:18) A − A t (cid:19) | A ∈ M n ( C ) , A ∗ = − A (cid:9) , p = (cid:8) (cid:18) AA ∗ (cid:19) | A ∈ M n ( C ) , A t = − A (cid:9) . If we define V + = span C { v , . . . , v n } and V − = span C { v n +1 , . . . , v n } , we have theidentifications k = span C { v ∧ ˜ v | v ∈ V + , ˜ v ∈ V − } . Define X αβ = iv α ∧ v β = e α,β + n − e β,α + n , Y αβ = 1 i v α + n ∧ v β + n = e β + n,α − e α + n,β for 1 ≤ α < β ≤ n . Then p + = span C { X αβ | ≤ α < β ≤ n } , p − = span C { Y αβ | ≤ α < β ≤ n } . We also let { ξ ′ αβ | ≤ α < β ≤ n } (resp. { ξ ′′ αβ | ≤ α < β ≤ n } ) be the basis of p ∗ + (resp. p ∗− ) dual to the above basis.Now fix 1 ≤ r ≤ n . Define U = span H { v , . . . , v r } . Define z ′ = span C { v n +1 , . . . , v n + r } ∈ D ( U ) . In Definition 3.2 we define a sub symmetric space D U,z ′ of D . The holomorphictangent space of D U,z ′ at z is T + z D U,z ′ = span C { X αβ | r + 1 ≤ α < β ≤ n } . The holomorphic tangent space of the fiber F z D U,z ′ (see Subsection 3.2 ) at z is(5.15) N + z D U,z ′ = span C { X αβ | ( α, β ) ∈ I } , where I is the index set(5.16) I = { ( α, β ) | ≤ α ≤ r, α < β ≤ n } . Define e D U,z ′ ∈ ∧ • p + by e D U,z ′ = ^ ( α,β ) ∈ I X αβ . Let P − be the Fock model defined in the last section for the dual pair (O ∗ (2 n ) , Sp(0 , r ))in Section 4.6. Recall that P − is the same as the Fock model for the dual pair(U( n, n ) , U(0 , r )) and is the polynomial space in the variables { u − ik | ≤ i ≤ n, ≤ k ≤ r } . Define f − ∈ P − by f − = det u − u − . . . u − r . . . . . . . . . . . .u − r u − r . . . u − rr . Also define f D U,z ′ ∈ W − by(5.17) f D U,z ′ = ( f − ) n − r − . HETA SERIES AND CYCLES 35
It can be shown that k acts on W − by (c.f. equation (5.2) of [And83])(5.18) ω ( v α ∧ v β + n ) = ω ( − ie αβ + ie β + n,α + n ) = − i r X k =1 ( u α + n,k ∂∂u β + n,k + u α,k ∂∂u β,k ) − irδ αβ . The adjoint action of k on p + induces an action on ∧ • p + . Define b = span C { v α ∧ v β + n | ≤ α ≤ β ≤ n } . b is a Borel sub-algebra of k . Both e D U,z ′ and f D U,z ′ are highest weight vectors of b . Moreover they have the same weight( n − , . . . , n − | {z } r , r, . . . , r | {z } n − r ) . Let ˜ K be the preimage of K under the map Mp( W ) → Sp( W ). Using the seesawpair(5.19) U( n, n ) Sp(0 , r )O ∗ (2 n ) O O sssssssss U(0 , r ) O O e e ❑❑❑❑❑❑❑❑❑ and facts about ^ U( n, n ) (see subsection 5.7 or [Pau98]), we can see that˜ K = K × {± } . Now denote the irreducible representation of K generated by e D U,z ′ as V ( U ), theirreducible representation of K generated by f D U,z ′ as A ( U ). By the theory ofhighest weight we have a K -equivariant map ψ + : V ( U ) → A ( U ) such that ψ + r ( e D U,z ′ ) = f D U,z ′ . Then Theorem C of [And83] is
Theorem 5.9. ψ + r ∈ Hom K ( ∧ • p + , P p − − ) . Let { ǫ , . . . , ǫ d } be a basis of V ( U ) ⊂ ∧ • p + such that each ǫ i is a weight vector of t . Extend { ǫ , . . . , ǫ d } to a basis of ∧ n ( n − − ( n − r )( n − r − p + , take the dual basisinside ∧ n ( n − − ( n − r )( n − r − p ∗ + and denote the first d basis vectors by Ω , . . . , Ω d .We have an isomorphismHom K ( ∧ • p + , P p − − ) ∼ = ( ∧ • p ∗ + ⊗ P p − − ) K . Under this isomorphism ψ + r maps to an element φ + r ∈ ( ∧ n ( n − − ( n − r )( n − r − p ∗ + ⊗P p − − ) K : φ + r = d X i =1 ψ + ( ǫ i )Ω i . Let i : F z D U,z ′ → D be the natural embedding (see equation (3.10) for the definition of F z D U,z ′ ). Usingthe definition of φ + r , we can prove the following lemma in a similar way as Lemma5.4: Lemma 5.10. i ∗ ( φ + ( x )) | z = ( f − ) n − r − i ∗ ( ^ ( α,µ ) ∈ I ξ ′ α,µ ) | z . Let P + be the infinitesimal Fock model for the dual pair (O ∗ (2 n ) , Sp( r, P + is the polynomial space in the variables { u + ia | ≤ i ≤ n, ≤ a ≤ r } . Define f + ∈ P + by f + = det u +11 u +12 . . . u +1 r . . . . . . . . . . . .u + r u + r . . . u + rr . Then both ( f + ) n − r − and V ( α,β ) ∈ I Y αβ are lowest weight vector with respect to b of weight ( − n + 1 , . . . , − n + 1 | {z } r , − r, . . . , − r | {z } n − r ) . There is a unique element ψ − r ∈ Hom K ( ∧ n ( n − − ( n − r )( n − r − p − , P p + + ) , which maps ( f + ) n − r +1 to V ( α,β ) ∈ I Y αβ and a corresponding element φ − r ∈ ( ∧ n ( n − − ( n − r )( n − r − p ∗− ⊗ P p + + ) K . Now let P be the infinitesimal Fock model for the dual pair (O ∗ (2 n ) , Sp( r, r )).We have P = P − ⊗ P + . Define φ r = φ + r ∧ φ − r . It is immediate that φ r ∈ ( ∧ n ( n − − ( n − r )( n − r − p ∗ ⊗ P ) K . We say r is the rank of φ r . Remark . We have in each case the following subgroups of the dual group G ′ of G :(1) Case A: G ′ = U( r + s, r + s ), K ′ = U( r + s ) × U( r + s ), K + = U( r + s, K − = U(0 , r + s ).(2) Case B: G ′ = O(2 r, r ), K ′ = O(2 r ) × O(2 r ), K + = O(2 r, K − =O(0 , r ).(3) Case C: G ′ = Sp( r, r ), K ′ = Sp( r ) × Sp( r ), K + = Sp( r, K − = Sp(0 , r ).In all cases K ′ = K − × K + and as a K ′ -representation P = P − ⊠ P + , where K + acts trivially on P − and K − acts trivially on P + .5.4. Closedness of holomorphic differentials.
In this subsection we simplywrite φ + for φ + r,s , φ − for φ − r,s and φ for φ r,s in case A and similarly in other cases.We will prove that the cochains φ + , φ − and φ are closed hence cocycles. First werecall the following well-known fact. Lemma 5.12.
A holomorphic form on a compact K¨ahler manifold M is closed. HETA SERIES AND CYCLES 37
Proof.
On a compact K¨ahler manifold we have the following identity of Laplacians∆ d = 2∆ ∂ = 2∆ ¯ ∂ . A holomorphic form ϕ is ¯ ∂ -closed. It is also ¯ ∂ ∗ -closed because it has Hodge-type( p, ϕ is ∆ ¯ ∂ -harmonic, hence ∆ d -harmonic. Hence ϕ is closed. (cid:3) Theorem 5.13.
The form φ + ( φ − resp.) constructed in the previous subsectionsis closed as an element of C • ( g , K ; P − ) ( C • ( g , K ; P + ) resp).Proof. We prove the holomorphic case, the anti-holomorphic case is similar.Recall that in all three cases φ + takes values in the K -representation generatedby f D U,z ′ (see (5.6), (5.11) and (5.17)), a special harmonic polynomial consideredin [KV78]. By [KV78], f D U,z ′ is in a representation A ⊠ θ ( A ), where A is the anirreducible ( g , K ) module and θ ( A ) is an irreducible representation of the compactgroup dual ˜ K − , where K − is as in Remark 5.11. Hence the Anderson cocycle φ + is a holomorphic cocycle in H ( • , ( g , K ; A ).By the proof of Proposition 2.3 in [And83], there is a cocompact lattice Γ of G and a ( g , K )-map I : P − → C ∞ (Γ \ G ) , such that I ( f ) = 0. Since f ∈ A and A is irreducible, we know that I is injectivewhen restricted on A . Hence the map on the cochains I ∗ : C • ( g , K ; A ) → C • ( g , K ; C ∞ (Γ \ G ))is also injective.Now I ∗ ( φ + ) is a holomorphic form on a compact K¨ahler manifold. So it is closedby Lemma 5.12. Since I ∗ is a map of chain complexes we know that I ∗ ( dφ + ) = dI ∗ ( φ + ) = 0 . Because I ∗ is injective on C • ( g , K ; A ), we know that dφ + = 0 . This finishes the proof of the theorem. (cid:3)
Remark . There is an alternative proof of the above theorem. Let C be theCasimir element of the universal enveloping algebra of g . Then one can show byexplicit computation that C acts trivially on A in all three cases. Then Proposition3.1 in Chapter II of [BW13] guarantees that any element in C • ( g , K ; A ) is closed. Corollary 5.14.1.
The cochain φ is closed.Proof. Recall that φ = φ + ∧ φ − . The differential operator d for the chain complex C • ( g, K ; P ) satisfies d = d − ⊗ ⊗ d + , where d − (resp. d + ) is the differential operator for the chain complex C • ( g, K ; P − )(resp. C • ( g, K ; P + )). The corollary now follows from Theorem 5.13. (cid:3) Cocycles in the Schr¨odinger Model.
Recall that we have defined a map ι : P → S ( V m ), where m = r + s in case A, m = r in case B and C. We define(5.20) ϕ r,s = ι ( φ r,s ) in case A , ϕ r = ι ( φ r ) in case B and Cin ( ∧ • p ∗ ⊗ S ( V m )) K (recall from (4.21) that V r ∼ = V r in case B). Since ι is anisomorphism of ( g , K ) modules between P and its image in S ( V m ), by the corollaryto Theorem 5.13, ϕ r,s (resp. ϕ r in case B and C) is closed. More explicitly we havein the Schrodinger Model (see equation (5.7)):(5.21) ϕ r,s or ϕ r = ϕ d X i,j =1 p ij Ω i ∧ ¯Ω j , where Ω i are of Hodge type ( d ′ ,
0) and p ij is a polynomial in the variables { z αa , ¯ z αa , z µa , ¯ z µa | ≤ α ≤ p, p + 1 ≤ µ ≤ p + q, ≤ a ≤ m } (in case B and C, p = q = n ). Define f ′ + = ( − √ π ) r det z z . . . z r . . . . . . . . . . . .z r z r . . . z rr ,f ′− = ( − √ π ) s det ¯ z p +1 r +1 ¯ z p +1 r +2 . . . ¯ z p +1 r + s . . . . . . . . . . . . ¯ z p + s r +1 ¯ z p + s r +2 . . . ¯ z p + s r + s . Recall that i : F z D U,z ′ → D is the an embedding (Section 3.2). Lemma 5.15.
Let ϕ be ϕ r,s in case A and ϕ r in case B and C. The highest term(in terms of the degree of the polynomial in front of ϕ ) of i ∗ ( ϕ ( x )) | z is ( f ′ + ¯ f ′ + ) q − s ( f ′− ¯ f ′− ) p − r i ∗ ( ^ ( α,µ ) ∈ I ξ ′ α,µ ∧ ξ ′′ α,µ ) | z , ( f ′ + ¯ f ′ + ) n − r +1 i ∗ ( ^ ( α,µ ) ∈ I ξ ′ α,µ ∧ ξ ′′ α,µ ) | z , ( f ′ + ¯ f ′ + ) n − r − i ∗ ( ^ ( α,µ ) ∈ I ξ ′ α,µ ∧ ξ ′′ α,µ ) | z , respectively in case A, B and C, where I is specified in (5.5) , (5.10) and (5.16) respectively. Notice that in each case i ∗ ( V ( α,µ ) ∈ I ξ ′ α,µ ∧ ξ ′′ α,µ ) | z is a constant multipleof the volume form of F z D U,z ′ at z .Proof. It follows from combining Lemma 5.4 (resp. Lemma 5.7 or Lemma 5.10),the analogous result for φ − and Lemma 4.4. (cid:3) Part Proof of the main theorems Poincar´e dual and Thom form
In this section we start to prove Theorem 1.6 and Theorem 1.7 in the intro-duction. We resume the notations of Section 2. We give the symmetric space D = G ∞ /K ∞ the G ∞ -invariant Riemannian metric τ induced by the trace form on p . D is then a negatively curved symmetric K¨ahler manifold whose sectional cur-vatures are automatically bounded as it is homogeneous. We suppose the sectionalcurvature of D is bounded below by − ρ . HETA SERIES AND CYCLES 39
Choose x ∈ V m (1 ≤ m ≤ n ) satisfying the following assumptions in the threecases of our interests respectively:(1) Case A: the Hermitian form ( , ) v restricted to span B { x } ⊗ k v is non-degenerate and has signature ( r, s ). In particular 1 ≤ r ≤ p, ≤ s ≤ q and r + s = m .(2) Case B: the Hermitian form ( , ) restricted to span B { x } is non-degenerate. m = r .(3) Case C: the skew-Hermitian form ( , ) restricted to span B { x } is non-degenerate. m = r .In each case we define a 2 m dimensional B -vector space W , a form <, > on W that is non-degenerate and split. We assume that it is Hermitian, skew Hermitianand Hermtian respectively in the three cases. Let G ′ be the group of B -lineartransformations on W preserving <, > . For each Archimedean place v of k define G ′ v = G ′ ( k v ) and let G ′∞ = Q v ∈ S ∞ G ′ v . We have(1) G ′∞ = U( r + s, r + s ) ℓ in case A,(2) G ′∞ = O(2 r, r ) ℓ in case B,(3) G ′∞ = Sp( r, r ) ℓ in case C.We fix a point z in the symmetric space D of G ∞ , or equivalently a maximalcompact group K ∞ of G ∞ . By our assumptions, K ∞ = K v × Q v ∈ S ∞ ,v = v G v . Let g = p ⊕ k be the corresponding Cartan decomposition on the Lie algebra g of G ∞ (we dropthe subscript 0 to indicate complexification). Let Ω • ( D ) be the space of smoothdifferential forms on D with values in C .Let S ( V m ∞ ) be the set of Schwartz functions on V m ∞ . There is an isomorphismgiven by evaluation at z :(6.1) (Ω • ( D ) ⊗ S ( V m ∞ )) G ∞ → ( ∧ • p ⊗ S ( V m ∞ )) K ∞ . Let ˜ G ′∞ be the metaplectic cover of G ′∞ . ˜ G ′∞ acts on S ( V n ∞ ) by the Weilrepresentation ω and the action commutes with that of G ∞ . Any form ψ ∈ ( ∧ • p ∗ ⊗ S ( V m ∞ )) K ∞ give rise to the form ˜ ψ defined by(6.2) ˜ ψ ( z, g ′ , x ) = L ∗ g − z ( ω ( g z , g ′ ) ψ ( x )) , where g z ∈ G ∞ such that g z z = z , g ′ ∈ ˜ G ′∞ and L g denotes the left action by g on D . Then we have ˜ ψ ∈ (Ω • ( D ) ⊗ S ( V m ∞ )) G ∞ × ˜ G ′∞ . Let ϕ v be the vacuum vector (Gaussian function) of S ( V v ) for any Archimedeanplace v = v . Define(6.3) ϕ ∞ = ϕ ⊗ Y v ∈ S ∞ ,v = v ϕ v ∈ ( ∧ • p ⊗ S ( V m ∞ )) K ∞ , where ϕ is the form defined in Section 5 such that(1) ϕ = ϕ r,s in case A,(2) ϕ = ϕ r in case B and C.Applying equation (6.2) to ϕ ∞ we get a form ˜ ϕ ∞ ∈ (Ω • ( D ) ⊗ S ( V m ∞ )) G ∞ × ˜ G ′∞ . Choose an O B lattice L of V and choose Γ ∈ G as in Section 2, then by [Wei64]we can choose an arithmetic subgroup Γ ′ ⊂ G ′ ( k ) such that the theta distribution θ L ( ψ ) = X x ∈L m ψ ( x ) , ∀ ψ ∈ S ( V m ∞ )is Γ × ˜Γ ′ -invariant where ˜Γ ′ = Γ ′ × {± } is the metaplectic cover of Γ ′ .We now apply θ L to ˜ ϕ ∞ to get θ L , ˜ ϕ ∞ = θ L ( ˜ ϕ ∞ ) ∈ Ω • (Γ \ D ) ⊗ C ∞ (˜Γ ′ \ ˜ G ′∞ ) . We also define(6.4) θ L ,β, ˜ ϕ ∞ ( z, g ′ ) = X x ∈L m , ( x , x )= β ˜ ϕ ∞ ( z, g ′ , x )for a matrix β ∈ M m ( B ). We have the following Fourier expansion of θ L , ˜ ϕ ∞ : θ L , ˜ ϕ ∞ ( z, g ′ ) = X β X x ∈L m , ( x , x )= β ˜ ϕ ∞ ( z, g ′ , x )= X β θ L ,β, ˜ ϕ ∞ ( z, g ′ ) , where β runs over all possible inner product matrices ( x , x ).By the construction of ϕ ∞ we know that it has Hodge bi-degree ( d ′ , d ′ ). Let η be any closed differential form on Γ \ D , define a smooth function θ L , ˜ ϕ ∞ ( η ) on ˜ G ′∞ by θ L , ˜ ϕ ∞ ( η )( g ′ ) = Z Γ \ D η ∧ θ L , ˜ ϕ ∞ ( g ′ ) . We call the above map the geometric theta lift defined by ˜ ϕ ∞ . This geometrictheta lift is a map H ( d − d ′ ,d − d ′ ) (Γ \ D, C ) → A (Γ ′ \ ˜ G ′∞ ) , where A (Γ ′ \ ˜ G ′∞ ) is the space of analytic functions on Γ ′ \ ˜ G ′∞ (see the proof ofLemma 6.10 for the statement of analyticity). Also define a L ,β, ˜ ϕ ∞ ( η ) to be the β -coefficient of θ L , ˜ ϕ ∞ ( η ):(6.5) a L ,β, ˜ ϕ ∞ ( η ) = Z Γ \ D η ∧ θ L ,β, ˜ ϕ ∞ ( z, g ′ ) = Z Γ \ D η ∧ X x ∈ L m , ( x , x )= β ˜ ϕ ∞ ( z, g ′ , x ) . Then we have the Fourier expansion: θ L , ˜ ϕ ∞ ( η ) = X β a L ,β, ˜ ϕ ∞ ( η ) . We assume from now on that β is non-degenerate (a nonsingular matrix) and σ ( β ) has signature ( r, s ) in case A (recall that σ i is the map k → k v i ). By atheorem of Borel ([Bor69], Theorem 9.11), the set { x ∈ L m | ( x , x ) = β } consists of finitely many Γ-orbits. We choose Γ-orbit representatives { x , . . . , x o } and define U i = span { x i } , ≤ i ≤ o. HETA SERIES AND CYCLES 41
For each 1 ≤ i ≤ o choose a base point z i ∈ D ( U i ). Let C x i ,z i be the generalizedspecial cycle (Definition 3.5). Then each C x i ,z i has complex codimension d ′ . Let z = { z , z , . . . , z o } . Define C β, z = o X i =1 C x i ,z i .C β, z is a cycle in the Chow group of Γ \ D . By remark 3.6.1, the homology class[ C β, z ] is independent of the choice of z , so we simply denote by [ C β ] its homologyclass. Whenever we take the period of a closed differential form η on Γ \ D we canwrite R C β η (resp. R C x η ). Theorem 6.1.
Assume that β = ( x , x ) is non-degenerate and in case A also as-sume that σ ( β ) has signature ( r, s ) . Then for any closed differential form η ∈ Ω d − d ′ ,d − d ′ (Γ \ D ) we have Z Γ \ D η ∧ X y ∈ Γ · x ˜ ϕ ∞ ( z, g ′ , y ) = κ ( g ′ , β ) Z C x η, and a L ,β, ˜ ϕ ∞ ( η ) = κ ( g ′ , β ) Z C β η, where κ is an analytic function in g ′ .Remark . The proof of Theorem 6.1 only depends on the following two propertiesof ϕ ∞ :(1) ϕ ∞ ∈ ( ∧ ( d ′ ,d ′ ) p ∗ ⊗ ι ( P )) K ∞ , where P is the polynomial Fock space (seeSection 4).(2) ϕ ∞ is closed.So the conclusion is true for any form ψ that satisfies the above two conditions.Let us also briefly recall Poincar´e duality in terms of differential forms. For aclosed submanifold C inside a compact oriented manifold M , we say that τ is aPoincar´e dual form of C if it is a closed form such that Z M η ∧ τ = Z C η for any closed form η . Poincar´e dual form is unique up to exact forms.The above definition of Poincar´e dual form can be applied when C is a subvarietyof the projective variety M if we interpret R C η as the integration of η over thenonsingular locus of C .With the above theory of Poincar´e duality in mind, Theorem 6.1 is equivalentto the following theorem which is Theorem 1.6 in the introduction. Theorem 6.3.
Assume that β = ( x , x ) is non-degenerate and in case A also as-sume that σ ( β ) has signature ( r, s ) . Then X y ∈ Γ · x [ ϕ ( z, g ′ , y )] = κ ( g ′ , β )PD([ C x ]) , where [ ϕ ] is the cohomology class of ϕ in H ∗ (Γ \ D ) , PD([ C x ]) ∈ H ∗ (Γ \ D ) is thePoincar´e dual of [ C x ] , and κ is a function that is analytic in G ′ . Moreover [ θ L ,β,ϕ ( z, g ′ )] = κ ( g ′ , β )PD([ C β ]) . When the function κ ( g ′ , β ) is nonzero, κ ( g ′ ,β ) [ θ L ,β, ˜ ϕ ∞ ( z, g ′ )] (see equation (6.4))is the Poincar´e dual of C β . We will prove the following theorem in section 8. Theorem 6.4.
There exists m ′ ∈ ˜ M ′∞ (see (4.3) ) such that for sufficiently large λ ∈ R , κ ( λm ′ , β ) = 0 . In particular, κ ( g ′ , β ) is nonzero for a generic g ′ as it is analytic. These theorems mean that [ θ L , ˜ ϕ ∞ ] can be seen as a ”generating” series ofPD([ C β ]). Of course as for now we do not have an explanation for all the ”Fouriercoefficients”. Only for those satisfying the condition of Theorem 6.3 do we have anexplanation.6.1. The rest of the section will be devoted to proving Theorem 6.1 under anassumption that we will verify later.First we need the following ”unfolding” lemma: Lemma 6.5. Z Γ \ D η ∧ X y ∈ Γ · x j ˜ ϕ ∞ ( z, g ′ , y ) = Z Γ x j \ D η ∧ ˜ ϕ ∞ ( z, g ′ , x j ) , In particular Z Γ \ D η ∧ X x ∈L m , ( x , x )= β ˜ ϕ ∞ ( z, g ′ , x ) = o X j =1 Z Γ x j \ D η ∧ ˜ ϕ ∞ ( z, g ′ , x j ) . Proof.
The proof is the same as that of Lemma 4.1 of [KM90]. (cid:3)
For x ∈ V m assume β = ( x, x ) satisfies the condition of Theorem 6.1. Let U x = span B v { x } , G x = ( G v ) U x , Γ x = Γ ∩ G x . Choose z ′ ∈ D ( U x ). The critical topological observation is that Γ x \ D x ,z ′ is atotally geodesic sub manifold of the space E x = Γ x \ D and E x is in a naturalway (topologically) a vector bundle over Γ x \ D x ,z ′ . In fact the fibration π : E x → Γ x \ D x ,z ′ defined in Section 3.2 has fibers F z D x ,z ′ diffeomorphic to the vector space N z D x ,z ′ . The following theorem is a special case of Theorem 2.1 of [KM88]. Theorem 6.6.
Let Φ be a degree d ′ differential form on E x satisfying (1) Φ is closed. (2) || Φ || ≤ exp( − d · ρ · r ) p ( r ) for some polynomial p , where r is the geodesicdistance to Γ x \ D x ,z ′ and ρ is a positive number such that the sectionalcurvature of D is bounded below by − ρ .If η is a closed bounded d − d ′ ) -form on E x then we have Z E x η ∧ Φ = κ Z Γ x \ D x ,z ′ η, where κ = Z F z D x ,z ′ Φ . for any z ∈ Γ x \ D x ,z ′ .Remark . When κ = 1, Φ is a Thom form of the fiber product E x → Γ x \ D x ,z ′ . HETA SERIES AND CYCLES 43
Proof of Theorem 6.1 assuming rapid decrease of ˜ ϕ ∞ on E x : By Lemma6.5 it suffices to show the following. Proposition 6.7.1.
Assume that x ∈ V m and β = ( x, x ) satisfies the condition ofTheorem 6.1. We have Z Γ x \ D η ∧ ˜ ϕ ∞ ( z, g ′ , x ) = κ ( g ′ , β ) Z C x η, where κ ( g ′ , β ) is the function in Theorem 6.1. In other words, the cohomologyclass [ ˜ ϕ ∞ ( z, g ′ , x )] in H ∗ ( E x ) is κ ( g ′ , β ) times the Thom class of the fiber bundle π : E x → Γ x \ D x ,z ′ .Remark . Some literature such as [BT13] requires a Thom form to be compactlysupported on the fiber. However by the same method as in [KM88], ˜ ϕ ∞ ( z, g ′ , x )can be shown to be cohomologous to a compactly supported form. Proof.
We would like to apply Theorem 6.6 to Φ = ˜ ϕ ∞ . By Theorem 5.13 andits corollary, ˜ ϕ ∞ satisfies condition (1) of Theorem 6.6. We will verify that ˜ ϕ ∞ satisfies condition (2) of Theorem 6.6 in Theorem 7.1 and its corollary. Define(6.6) κ ( g ′ , x , z ′ ) := Z F z D x ,z ′ ˜ ϕ ∞ ( z, g ′ , x )for any z ∈ C x ,z ′ . By Theorem 6.6, we know that Z Γ x \ D η ∧ ˜ ϕ ∞ ( z, g ′ , x ) = κ ( g ′ , x , z ′ ) Z Γ x \ D x ,z ′ η. When the map ρ U x ,z ′ : Γ x \ D U x → Γ \ D (see Section 3) is an embedding, one canimmediately conclude that(6.7) Z Γ x \ D η ∧ ˜ ϕ ∞ ( z, g ′ , x ) = κ ( g ′ , x , z ′ ) Z C x ,z ′ η. In general one can use Lemma 3.6 to see that the mapΓ x \ D x ,z ′ → C x ,z ′ induced by ρ U x ,z ′ is birational, so Z C x ,z ′ η = Z Γ x \ D x ,z ′ η if we interpret R C x ,z ′ η as integration over the nonsingular locus of C x ,z ′ . Henceequation (6.7) holds again.Thus in order to prove Theorem 6.1, it remains to show that(1) κ ( g ′ , x , z ′ ) only depends on g ′ and β , so we can define κ ( g ′ , β ) := κ ( g ′ , x , z ′ ).(2) κ ( g ′ , β ) is an analytic function in g ′ . Lemma 6.9. κ ( g ′ , x , z ′ ) is independent of the choice of z ′ . Moreover it only de-pends on β = ( x , x ) when β satisfies the condition of Theorem 6.1.Proof. Let η be a Γ-invariant form on D such that R C x ,z ′ η = 0. Such a η exists byPoincar´e duality. For different choices of z ′ , C x ,z ′ are homologous. Thus R C x ,z ′ η is independent of z ′ . The left hand side of (6.7) is visibly independent of z ′ . This shows that κ ( g ′ , x , z ′ ) is independent of the choice of z ′ so we we can simply denoteit by κ ( g ′ , x ).Let ˜x ∈ V m ∞ and ˜ z be a point in D ( U ˜x ). Assume that ˜ z ∈ D ˜x , ˜ z . Suppose β = ( x , x ) = ( ˜x , ˜x ). Then by Witt’s Theorem there is a g ∈ G ∞ such that g x = ˜x and we have Z F ˜ z D ˜x , ˜ z ˜ ϕ ∞ ( z, g ′ , ˜x ) = Z F ˜ z D g x , ˜ z ˜ ϕ ∞ ( z, g ′ , g x )= Z F ˜ z D g x , ˜ z L ∗ g − ( ˜ ϕ ∞ ( z, g ′ , x ))= Z L g − ( F ˜ z D g x , ˜ z ) ˜ ϕ ∞ ( z, g ′ , x )= Z F g − z D x ,g − z ˜ ϕ ∞ ( z, g ′ , x ) . This proves that κ ( g ′ , ˜x ) = κ ( g ′ , x ). So the proof is finished. (cid:3) In particular, when β satisfies the condition of Theorem 6.1, we can define κ ( g ′ , β ) = κ ( g ′ , x )for any x ∈ V m such that ( x , x ) = β . Lemma 6.10.
The function κ ( g ′ , β ) is analytic in g ′ .Proof. First we claim that ˜ ϕ ∞ ( z, g ′ , x ) is an analytic function on ˜ G ′∞ . First weknow that ˜ G ′∞ = P ′∞ ˜ K ′∞ , where P ′ is the Siegel parabolic as in Section 4 and ˜ K ′∞ is the maximal compact subgroup of ˜ G ′∞ fixing the Gaussian function. There areanalytic functions s : ˜ G ′∞ → P ′∞ , s : ˜ G ′∞ → ˜ K ′∞ such that g = s ( g ) s ( g ). Now any φ in the polynomial Fock space is ˜ K ′∞ -finiteand ˜ K ′∞ analytic. The action of P ′∞ is given by (4.4) and (4.5), hence analytic.This proves the claim.Now by Theorem 7.1, ˜ ϕ ∞ ( z, g ′ , x ) is fast decreasing on the fiber F z D U,z ′ andthe decrease is locally uniform in g ′ . So κ ( g ′ , β ) as defined in (6.6) is an analyticfunction in g ′ . (cid:3) This finishes the proof of Theorem 6.1 and Proposition 6.7.1 under the assump-tion that ˜ ϕ ∞ is rapid decreasing on E x . (cid:3) Rapid decrease of Schwartz function valued forms on the normalbundle
For a moment, we keep the notations of the previous section. For ψ ∈ ( V • p ⊗S ( V v )) K v define(7.1) ˜ ψ ( z, g ′ , x ) = L ∗ g − z ( ω ( g z , g ′ ) ψ ( x )) , where g z ∈ G v such that g z z = z .Recall that the Riemannian distance d ( D ′ , z ) between a totally geodesic sub-manifold D ′ and z ∈ D is the length of the shortest geodesic joining z to a pointof D ′ . This geodesic is necessarily normal to D ′ . Choose a base point z ∈ D . If HETA SERIES AND CYCLES 45 z = exp z ( tu ) for u ∈ N z D ′ and k u k = 1, where exp z denotes the exponentialmap of D at the base point z , then d ( D ′ , z ) = t. The goal of the section is to prove the following theorem.
Theorem 7.1.
Fix a non-dgenerate x ∈ V m , a point z ′ ∈ D ( U ) , where U =span B v { x } , an element g ′ ∈ ˜ G ′ v and ψ ∈ ( V • p ⊗ S ( V v )) K v . Then for any realnumber ρ > , there is a positive constant C ′ ρ such that || ˜ ψ ( z, g ′ , x ) || ≤ C ′ ρ exp( − ρ · d ( D U,z ′ , z )) , where the norm is taken with respect to the Riemannian metric of the symmetricspace D . The constant C ′ ρ depends continuously on g ′ . Corollary 7.1.1.
The form ˜ ϕ ∞ ( z, g ′ , x ) defined in (6.3) satisfies condition (2) ofTheorem 6.6. In particular it is integrable on F z D U,z ′ for any z ∈ D U,z ′ .Proof. Recall that by our assumption, G ∞ = G v × Q v = v G v and G v is compact for v = v . Hence any ψ ∞ ∈ ( V • p ⊗ S ( V m ∞ )) K ∞ is constant in G v for v = v . Theorem7.1 implies immediately that ˜ ϕ ∞ ( z, g ′ , x ) satisfies condition (2) of Theorem 6.6.The integrablity statement follows from Theorem 6.6. (cid:3) In the rest of this section, we work over real vector spaces and real Lie groups.To simplify notations, we denote G v by G , G ′ v by G ′ and K v by K . In otherwords G = U( p, q ) in case A, Sp(2 n, R ) in case B and O ∗ (2 n ) in case C. Throughoutthe section we assume that V is a complex vector space with a non-degenerate skewHermitian form ( , ) such that i ( , ) is of signature ( p, q ). In case B and C, we assumethat p = q = n in addition. We fix an orthogonal basis { v , . . . , v p + q } such that( v α , v α ) = − i for 1 ≤ α ≤ p, ( v µ , v µ ) = 1 for p + 1 ≤ µ ≤ p + q. In case A we denote by D = ˜ D the symmetric space of U( p, q ). In case B (case Cresp.) we denote by D the symmetric space of Sp(2 n, R ) (O ∗ (2 n ) resp.) while wedenote by ˜ D the symmetric space of U( n, n ).In case A, we let m = r + s , x = ( ~x , . . . , ~x r + s ) and U := span C { x } . with the assumption that U has signature ( r, s ) with respect to ( , ).In case B, recall from Section 5.2 that V is a 2 n dimensional real vector spacewith a skew symmetric form ( , ) and V = V ⊗ R C . Then Sp(2 n, R ) = G ( V , ( , ) )is a subgroup of U( n, n ) and its symmetric space D embedds into ˜ D , the symmetricspace of U( n, n ). In this case let m = 2 r and assume x = ( ~x , . . . , ~x m ) ∈ V m .Define U = span R { x } . We assume that U is non-degenerate with respect to ( , ) .In case C, recall from Section 4.6 and 5.3 that V is an n -dimensional right H -vector space with a skew Hermitian form ( , ) . Then O ∗ (2 n ) = G ( V, ( , ) ) is asubgroup of U( n, n ) and its symmetric space D embedds into ˜ D . In this case let m = r and assume x = ( ~x , . . . , ~x r ) ∈ V r and U = span H { x } . Assume that U is non-degenerate with respect to ( , ) . In all three cases, a point z ∈ D (in the latter two cases view z as in ˜ D insteadof D ) gives rise to a involution(7.2) θ z : V → V, θ z = ( − Id z ) ⊕ Id z ⊥ . Define a positive definite Hermitian form ( , ) z on V by(7.3) ( x, y ) z = i ( x, θ z y ) . We call this form the majorant of ( , ) with respect to z . We denote ( v, v ) z as k v k z . z also induces a Cartan decompostion g = k + p . We identify T z D with p .In all the above cases, for any z ∈ D , define M z : D × V m → R to be thefunction(7.4) M z ( z, x ) = m X ℓ =1 || g − z ~x ℓ || z , where g z ∈ G is any element such that g z z = z . For any k ∈ K (the isotropicgroup of z ) we have θ z k = kθ z . Hence we know that( kx, ky ) z = i ( kx, θ z ky )= i ( kx, kθ z y )= i ( x, θ z y ) = ( x, y ) z . Hence the function M z ( z, x ) is well-defined. Lemma 7.2. M hz ( hz, h x ) = M z ( z, x ) . Proof.
Choose a g such that gz = z , then hgh − hz = hz . Hence M hz ( hz, h x ) = m X ℓ =1 || hg − h − h~x ℓ || hz = i m X ℓ =1 ( hg − h − · h~x ℓ , hθ z h − · ( hg − h − · h~x ℓ ))= i m X ℓ =1 ( hg − ~x ℓ , hθ z g − ~x ℓ )= i m X ℓ =1 ( g − ~x ℓ , θ z g − ~x ℓ )= M z ( z, x ) . (cid:3) The main technical ingredient for proving Theorem 7.1 is the following estimateof M z ( z, x ) for all three cases. Theorem 7.3.
Let x and U be as above in each cases and choose z ′ ∈ D ( U ) together with z ∈ D U,z ′ . There is positive constants b and c depending on x and z ′ such that (7.5) M z ( z, x ) ≥ c exp(2 b · d ( D U,z ′ , z )) . HETA SERIES AND CYCLES 47
It is easy to see that(7.6) d ( D U,z ′ , z ) = d ( D gU,gz ′ , gz ) , ∀ g ∈ G. Since g fixes x , U and z ′ for any g ∈ G U , by Lemma 7.2 we know that M gz ( gz, x ) = M gz ( gz, gx ) = M z ( z, x ) , ∀ g ∈ G U . The above equation together with Lemma 7.2 implies that the constant c in The-orem 7.3 is independent of the choice of z ∈ D U,z ′ since G U acts transitively on D U,z ′ . Moreover (7.6) and Lemma 7.2 implies that in order to prove Theorem 7.3it suffices to assume (by replacing ( x , z ) by ( g x , gz ) for some g ∈ G ) that(1) In case A, span { x } = span C { v , . . . , v r , v p +1 , . . . , v p + s } and z = span C { v p +1 , . . . , v p + q } .(2) In case B, span R { x } = span R { E , . . . , E r , F , . . . , F r } and z = span C { E + iF , . . . , E n + iF n } .(3) In case C, span H { x } = span H { v , . . . , v r } and z = span C { v j, . . . , v n j } .Recall that we define in Section 3.2 a G U -equivariant fibration π : D → D U,z ′ with fiber F w D U,z ′ over w ∈ D U,z ′ . Since G U fixes x , by the definition of M z ( z, x )(Equation (7.4)), we see that M z ( gz, x ) = M z ( z, x ) , ∀ g ∈ G U . Moreover since G U stabilizes D U,z ′ , we have d ( D U,z ′ , gz ) = d ( g − D U,z ′ , z ) = d ( D U,z ′ , z ) , ∀ g ∈ G U . Hence in order to prove Theorem 7.3 we can translate z by elements in G U andassume that z ∈ F z D U,z ′ . In other words z = exp z ( X ) , for some X ∈ N z D U,z ′ . It is well-known that (c.f. Section 3 of Chapter IV of[Hel79]) exp z ( X ) = exp( X ) z , where we identify N z D U,z ′ as a subspace of p and exp is the exponential map ofthe group G . From now on we assume that z = exp( X ) z with X ∈ N z D U,z ′ ⊂ p .7.1. Theorem 7.3 in case A.
We assume n = p + q in this subsection. Thetheorem will be a consequence of Lemma 7.4 through 7.9. Let Herm n be the set of n × n Hermitian matrices with values in C . For a matrix A , its norm is defined by k A k = tr( AA ∗ ) . By our choice of z , { v , . . . , v p + q } is an orthonormal basis for ( V, ( , ) z ). Hence thenorm k · k z is the standard one with respect to the basis. Lemma 7.4.
Let A ∈ Herm n and ǫ > be given. Then there exists δ dependingon ǫ and A such that for any B ∈ Herm n with k A − B k < δ there exist R, S ∈ U( n ) such that RAR − and SBS − are diagonal with k RAR − − SBS − k < ǫ and k R − S k < ǫ . Proof.
For U ∈ U( n ), the statement in the lemma is true for A if and only if it istrue for U AU − (with the same ǫ and δ ). Hence without lost of generality we canassume we can assume A is diagonal of the form(7.7) A = λ I r · · · λ I r · · · · · · λ k I r k ,λ , λ , . . . λ r are distinct eigenvalues of A and r + r + · · · + r k = n .First we assume that λ , λ , . . . λ r are all nonzero.Define a Lie subalgebra u A ⊆ u ( n ) to be the set of matrices of the form(7.8) X = r X · · · X k − X ∗ r · · · X k − X ∗ k − X ∗ k · · · r k . We define a map φ : u A × Herm r × · · · × Herm r k → Herm n by the formula(7.9) φ ( X, m , · · · , m k ) = exp ( X ) A exp( m ) 0 · · ·
00 exp( m ) · · ·
00 0 · · · exp( m k ) exp ( − X ) . Then the differential of φ at (0 , , . . . , dφ ( X, m , · · · , m k ) = A m · · · m · · ·
00 0 · · · m k + [ X, A ] , which in turn is equal to λ m ( λ − λ ) X · · · ( λ k − λ ) X k ( λ − λ ) X ∗ λ m · · · ( λ k − λ ) X k ( λ k − λ ) X ∗ k ( λ k − λ ) X ∗ k · · · λ k m k . Because we assume that λ , λ , . . . λ k are distinct and nonzero, dφ is an isomor-phism from u A × Herm r × · · · × Herm r k to Herm n . Hence by inverse functiontheorem, φ is a diffeomorphism from the product of a ball B (0 , η ) of radius η around the origin in u A with a ball B (0 , η ′ ) of radius η ′ around the origin ofHerm r × · · · × Herm r k to a neighborhood U ( η, η ′ ) of A in Herm n .For a given ǫ , shrink the size of η and η ′ if necessary such that(7.10) X ∈ u A and || X || < η ⇒ || I − exp( X ) || < ǫ, and(7.11) Y ∈ Herm r × · · · × Herm r k and k Y k ≤ η ′ ⇒ k A − A exp ( Y ) k < ǫ. HETA SERIES AND CYCLES 49
Choose δ such that B ( A, δ ) ⊂ U ( η, η ′ ). Suppose B ∈ B ( A, δ ). Since B ∈ U ( η, η ′ )we have a unique expression B = exp ( X ) A exp ( Y ) exp ( − X ) with X ∈ B (0 , η ) and Y ∈ B (0 , η ′ ) . Put R = I and S = exp ( − X ) so R AR − = A is diagonal and S BS − = A exp ( Y ) is block diagonal of the form(7.12) S BS − = B · · · B · · ·
00 0 · · · B kk , where B ii is of size r i × r i .By the above choices of η (equation (7.10)), η ′ (equation (7.11)) and δ , it is clearthat we have(7.13) k A − B k < δ ⇒ k R − S k < ǫ and k R AR − − S BS − k < ǫ. Now there is a block diagonal unitary matrix R = R · · · R · · ·
00 0 · · · R kk , where R ii ∈ U( r i ) such that RS BS − R − is diagonal. Notice that RA = AR ,hence RAR − = A is also diagonal.Let S = RS . Since R is unitary, by (7.13) we have || R − S || = || R ( R − S ) || < ǫ, and || RAR − − SBS − || = || R ( A − S BS − ) R − || < ǫ. The lemma is now proved for A a block diagonal matrix of the form (7.7) and λ , . . . , λ k are nonzero.In general we can choose a λ such that A ′ = A + λI n does not have zero eigenvalue.By the previous argument, there are B ′ ∈ Herm n and R, S ∈ U( n ) such that || R − S || < ǫ and || RA ′ R − − SB ′ S − || < ǫ . Now let B = B ′ − λI n . The lemma isnow proved. (cid:3) Recall (see Section 5.1) that p are Hermitian matrices of the form p = (cid:26)(cid:18) AA ∗ (cid:19) | A ∈ M p × q ( C ) (cid:27) , and the tangent space N z D U,z ′ of the fiber F z D U,z ′ at z can be identified witha subspace of p given by equation (5.4).Let X ∈ N z D U,z ′ so in particular X ∈ p z and is Hermitian with respect to ( , ) .Let ˜ v , · · · ˜ v n be an orthonormal basis for ( , ) z of V consisting of eigenvectors of X . Then X (˜ v k ) = λ k ˜ v k , ≤ k ≤ n. Suppose ~x j = n X k =1 a kj ˜ v k for 1 ≤ k ≤ n . Then k ~x j k z = X λ k k p λ k ( ~x j ) k z = n X k =1 | a kj | , where the summation is over all eigenvalues λ k of X and p λ k is the orthogonalprojection with respect to the metric ( , ) z onto the eigenspace of eigenvalue λ k . Remark . When it is necessary to distinguish to which X ∈ N z D U,z ′ the num-bers a kj and λ k belong, we will write a kj ( X ) and λ k ( X ). Lemma 7.6.
We have (exp ( − tX ) ~x j , exp ( − tX ) ~x j ) z = X λ k k p λ k ( ~x j ) k z exp ( − λ k t ) , where the sum is over all eigenvalues of X .Proof. Since { ˜ v k : 1 ≤ k ≤ n } is an orthonormal basis for V we have(exp ( − tX ) ~x j , exp ( − tX ) ~x j ) z = n X i =1 | (˜ v i , (exp − ( tX ) ~x j ) z | = n X i =1 | (cid:0) ˜ v i , exp ( − tX )( n X k =1 a kj ˜ v k ) (cid:1) z | = n X i =1 | (˜ v i , n X k =1 a kj exp ( − λ k t )˜ v k ) z | = n X i =1 | a ij | exp ( − λ i t ) . (cid:3) We now define(7.14) f ( X ) = − r + s X j =1 n X i =1 λ i ( X ) < | a ij | ( X ) λ i ( X ) = − r + s X j =1 X λ ( X ) < k p λ ( X ) ( ~x j ) k z λ ( X ) . Since all the terms in the sum defining f ( X ) are nonnegative it follows that f ( X ) = 0 if and only all the term in the sum are zero. By Lemma 7.4, we can provethe following. Lemma 7.7. f ( X ) is continuous on p ∼ = T z D .Proof. Let X ∈ p . Then for any X ∈ p we have(7.15) | f ( X ) − f ( X ) | = | r + s X j =1 n X i =1 λ i ( X ) < | a ij | ( X ) λ i ( X ) − r + s X j =1 n X i =1 λ i ( X ) < | a ij | ( X ) λ i ( X ) | . Let ǫ > A = X to find δ such that whenever k X − X k < δ , there exist unitary matrices R, S such that RX R − and SXS − are diagonal and k R − S k < ǫ and k RAR − − SBS − k < ǫ. But k R − S k < ǫ implies that suitably chosen eigenvectors of A and B are close.More precisely, if ˜ v i ( X ) (resp. ˜ v i ( X )), 1 ≤ i ≤ n is the eigenvector of X (resp. HETA SERIES AND CYCLES 51 X ), corresponding to the eigenvalue λ i ( X ) (resp. λ i ( X )) which is the i -th row of R (resp. i -th row of S ), we have ǫ > k R − S k = n X i =1 k ˜ v i ( X ) − ˜ v i ( X ) k z ⇒ k ˜ v i ( X ) − ˜ v i ( X ) k z < ǫ , ≤ i ≤ m. Hence, for all i, j, ≤ i ≤ n and 1 ≤ j ≤ n we have | a ij ( X ) − a ij ( X ) | = | ( ~x j , ˜ v i ( X ) − v i ( X )) z | ≤ k ~x j k z k ˜ v i ( X ) − ˜ v i ( X ) k z < k ~x j k z ǫ . Similarly | a ij ( X ) | < k ~x j k z . Also k RAR − − SBS − k = m X i =1 ( λ i ( X ) − λ i ( X )) , consequently k RAR − − SBS − k < ǫ ⇒ n X i =1 ( λ i ( X ) − λ i ( X )) < ǫ ⇒ ( λ i ( X ) − λ i ( X )) < ǫ , ≤ i ≤ n. Hence, for all i, ≤ i ≤ n we have | λ i ( X ) − λ i ( X ) | < ǫ. Since X is fixed, we assume that λ i ( X ) ≤ M, λ i ( X ) ≤ M for 1 ≤ i ≤ n .Now using the identity | ab − a ′ b ′ | ≤ | b || a − a ′ | + | a ′ || b − b ′ | we obtain || a ij | ( X ) λ i ( X ) −| a ij | ( X ) λ i ( X ) | ≤ | λ i ( X ) || a ij | ( X ) −| a ij | ( X ) | + | a ij | ( X ) || λ i ( X ) − λ i ( X ) | . Since | λ i ( X ) | ≤ M and | a ij | ( X ) ≤ k ~x j k z we have || a ij | ( X ) λ i ( X ) − | a ij | ( X ) λ i ( X ) | ≤ M || a ij | ( X ) − | a ij | ( X ) | + k ~x j k z | λ i ( X ) − λ i ( X ) |≤ M || ~x j || z ǫ + || ~x j || z ǫ = || ~x j || z ( M ǫ + ǫ ) . Suppose that the strictly negative eigenvalues of X are λ ( X ) , · · · , λ k ( X ) andthe strictly negative eigenvalues of X are λ ( X ) , · · · , λ ℓ ( X ). We assume k > ℓ .The case k = ℓ is easier (in this case, we have only the first sum in Eqation (7.16)below) and the case k < ℓ can be treated in a manner symmetrical to that of thecase k > ℓ .We have(7.16) | f ( X ) − f ( X ) | = | r + s X j =1 ℓ X i =1 (cid:18) | a ij | ( X ) λ i ( X ) −| a ij | ( X ) λ i ( X ) (cid:19) − r + s X j =1 k X i = ℓ +1 | a ij | ( X ) λ i ( X ) | The first sum is clearly majorized by ℓ P r + sj =1 k ~x j k z (2 M ǫ + ǫ ) using the inequalityimmediately above. To majorize the second sum we note that ℓ < i ≤ k ⇒ λ i ( X ) < λ i ( X ) ≥ . Hence | λ i ( X ) − λ i ( X ) | = − λ i ( X ) + λ i ( X ). Note that each of the two terms ispositive. But | λ i ( X ) − λ i ( X ) | < ǫ ⇒ − λ i ( X ) + λ ( X ) < ǫ ⇒ − λ i ( X ) < ǫ. Hence the second summand is majorized by ( k − ℓ ) P r + sj =1 k ~x j k z ǫ . Lemma 7.7 follows. (cid:3)
Let S ( N z D U,z ′ ) be the unit sphere of N z D U,z ′ , then we have Lemma 7.8. f ( Y ) does not take the value zero on S ( N z D U,z ′ ) . As S ( N z D U,z ′ ) is compact and f ( Y ) ≥ , there exists C > so that f ( Y ) ≥ C, ∀ Y ∈ S ( N z D U,z ′ ) . Proof.
Assume Y ∈ S ( N z D U,z ′ ) and f ( Y ) = 0. Suppose v is an eigenvector of Y corresponding to a strictly negative eigenvalue so Y ( v ) = λv, λ < . Then f ( Y ) = 0 ⇒ k p λ ( ~x j ) k z = 0 ⇒ ( ~x j , v ) z = 0 , ≤ j ≤ r + s. Let U + = span { v , . . . , v r } and U − = span { v p +1 , . . . , v p + s } . Let U ⊥ + = span { v r +1 , . . . , v p } and U ⊥− = span { v p + s +1 , . . . , v p + q } . Then v ⊥ U + ⊕ U − as span { ~x , . . . , ~x r + s } = U + ⊕ U − . For any u ∈ V , we may write u = ( v , w , v , w ) with v ∈ U + , w ∈ U ⊥ + , v ∈ U − , w ∈ U ⊥− . Then in this representation we have v = (0 , w , , w ) with w = 0 or w = 0 , and Y ( v ) = λv = (0 , λw , , λw ) . But since Y ∈ N z D x ,z ′ we have (see equation (5.4)) Y = a b c a ∗ c ∗ b ∗ . Hence Y ( v ) = ( bw , , c ∗ w ,
0) = λv = (0 , λw , , λw ) . Since λ < w and w = 0, a contradic-tion. (cid:3) Lemma 7.9.
Let x = ( ~x , . . . , ~x m ) and U = span { x } and suppose X ∈ S ( N z D U,z ′ ) .Then there exists strictly positive numbers b, c depending only on x and z ′ and anegative eigenvalue λ of X such that for some j, with ≤ j ≤ m , such that | λ | = − λ ≥ b and k p λ ( ~x j ) k z ≥ c. Proof.
Since f ( X ) = − r + s X j =1 n X i =1 λ i ( X ) < | a ij | ( X ) λ i ( X )is bounded below by C , at least one of the terms in the sum is bounded below by c = CN , where N is the number of terms in the sum ( N ≤ n ( r + s )). Suppose thisterm is −| a ij | λ i . Hence | a ij | | λ i | = −| a ij | λ i ≥ c for some i, j. HETA SERIES AND CYCLES 53
But since P ni =1 | a ij | = k ~x j k z it follows that | a ij | ≤ k ~x j k z , ≤ i ≤ n, ≤ j ≤ r + s. Hence | λ i | ≥ c k ~x j k z . We put b = c k ~x j k z . Since k X k = P ni =1 λ i = 1, it follows that | λ i | ≤ , hence k p λ i ( ~x j ) k z ≥ | a ij | ≥ c. (cid:3) Proof of Theorem 7.3 in Case A:
We assume X ∈ S ( N z D x ,z ′ ), z = exp( Xt ) z and g = exp( Xt ). Then M z ( z, x ) = r + s X j =1 (exp ( − tX ) ~x j , exp ( − tX ) ~x j ) z = r + s X j =1 n X i =1 | a ij | exp ( − λ i t ) . But all the terms in the sum immediately above are nonnegative and we have provedin Lemma 7.9 that one of them is minorized by c exp (2 bt ). Hence the entire sumis also minorized by c exp (2 bt ) and we obtain M z ( z, x ) ≥ c exp (2 bt ) . Since d ( D U,z ′ , z ) = t, Theorem 7.3 is proved. (cid:3)
Proof of Theorem 7.3 in case B.
One way to proceed is to use the seesawdual pair:(7.17) U( n, n ) O(2 r, r )Sp(2 n, R ) O O qqqqqqqqq U( r, r ) O O f f ▼▼▼▼▼▼▼▼▼ together with a relation between D U,z ′ and ˜ D U ⊗ C ,z ′ to reduce a substantial part ofthe problem to the unitary dual pair case. However we use a direct approach hereinstead. We proceed quickly by omitting the proofs that are similar to the unitarygroup case.We know that Sp(2 n, R ) ֒ → U( n, n ) and the symmetric space D embedds into ˜ D (see Section 5.2). Recall that we have assumed thatspan R { x } = span R { E , . . . , E r , F , . . . , F r } and z = span C { E + iF , . . . , E n + iF n } . The first condition is equivalent to (recall from Section 4.5 for our convention ofthe basis)(7.18) span C { x } = span C { v , . . . , v r , v n +1 , . . . , v n + r } . Also recall that(7.19) M z ( z, x ) = r X j =1 k g − z ~x j k z . Since T z D ⊂ T z ˜ D , for X ∈ T z D we can define f ( X ) as in equation (7.14) (seethe paragraphs before equation (7.14) for the definition of λ ( X ) and p λ ( X ) ):(7.20) f ( X ) = − r X j =1 n X i =1 λ i ( X ) < k p λ i ( X ) ( ~x j ) k z λ i ( X ) . By Lemma 7.7, we know f is continuous on T z D . Lemma 7.10. f ( Y ) does not take zero value on S ( N z D U,z ′ ) . As S ( N z D U,z ′ ) iscompact there exists C > such that f ( Y ) ≥ C, ∀ Y ∈ S ( N z D U,z ′ ) . Proof.
Assume Y ∈ S ( N z D U,z ′ ) and f ( Y ) = 0. Suppose v ∈ V is an eigenvectorof Y corresponding to a strictly negative eigenvalue so Y ( v ) = λv, λ < . Then f ( Y ) = 0 ⇒ k p λ ( ~x j ) k z = 0 ⇒ ( ~x j , v ) z = 0for 1 ≤ j ≤ r . Under the basis { v , . . . , v n , v n +1 , . . . , v n } and the assumption(7.18), we have v = (0 , w , , w ) with w = 0 or w = 0 , where w ∈ span C { v r +1 , . . . , v n } and w ∈ span C { v n + r +1 , . . . , v n } .Recall that for Y ∈ N z D U,z ′ we have (see equation (5.9) and equation (5.8)) Y = a b b t a ∗ ¯ b b ∗ , where a = a t . Hence Y ( v ) = ( bw , , ¯ bw ,
0) = λv = (0 , λw , , λw ) . Since λ < w = 0 and w = 0, a contra-diction. (cid:3) With Lemma 7.10, the conclusion of Lemma 7.9 holds for Y ∈ N z D U,z ′ as well.Hence the rest of the proof of Theorem 7.3 in case B is the same as that of case A. (cid:3) Proof of Theorem 7.3 in case C.
We know that O ∗ (2 n ) ֒ → U( n, n ) and thesymmetric space D embedds into ˜ D (see section 5.3). Recall that we have assumedthat(7.21) span H { x } = span H { v , . . . , v r } and z = span C { v n +1 , . . . , v n } . Also recall that(7.22) M z ( z, x ) = r X j =1 k g − z ~x j k z . HETA SERIES AND CYCLES 55
Since T z D ⊂ T z ˜ D , for X ∈ T z D we can define f ( X ) as in equation (7.14) (seethe paragraphs before equation (7.14) for the definition of λ ( X ) and p λ ( X ) ):(7.23) f ( X ) = − r X j =1 n X i =1 λ i ( X ) < k p λ i ( X ) ( ~x j ) k z λ i ( X ) . By Lemma 7.7, we know f is continuous on T z D . Lemma 7.11. f ( Y ) does not take zero value on S ( N z D U,z ′ ) . As S ( N z D U,z ′ ) iscompact there exists C > such that f ( Y ) ≥ C, ∀ Y ∈ S ( N z D U,z ′ ) . Proof.
Assume Y ∈ N z D U,z ′ identified with a subspace of p . Since Y ∈ o ∗ (2 n ),it commutes with right multiplication by j on V. In particular if λ (recall that λ must be real since Y is Hermitian) is an eigenvalue of Y and V λ is the λ -eigenspaceof Y , V λ is preserved by right multiplication by j .Suppose V λ is the λ -eigenspace of Y corresponding to a strictly negative eigen-value λ . Assume f ( Y ) = 0, then f ( Y ) = 0 ⇒ k p λ ( ~x α ) k z = 0 ⇒ ~x α ⊥ V λ , for 1 ≤ α ≤ r . As V λ and ( , ) z are preserved by right multiplication by j , theabove implies ~x α j ⊥ V λ for 1 ≤ α ≤ r . Under the basis { v , . . . , v n , v j, . . . , v n j } and the assumption (7.21),we then have v = (0 , w , , w ) with w = 0 or w = 0 , where w ∈ span C { v r +1 , . . . , v n } and w ∈ span C { v r +1 j, . . . , v n j } .Recall that for Y ∈ N z D U ,z ′ we have (see equation (5.15) and equation (5.14)) Y = a b − b t a ∗ − ¯ b b ∗ , where a = − a t . Hence Y ( v ) = ( bw , , − ¯ bw ,
0) = λv = (0 , λw , , λw ) . Since λ < w = 0 and w = 0, a contra-diction. (cid:3) With Lemma 7.10, the conclusion of Lemma 7.9 holds for Y ∈ N z D U,z ′ as well.Hence the rest of the proof of Theorem 7.3 in case C is the same as that of case A. (cid:3) Rapid decrease of the cocycles on the fiber F z D x ,z ′ . In this subsectionwe prove Theorem 7.1.
Theorem 7.12.
All the assumptions are as in Theorem 7.3. For any Schwartzfunction ψ ∈ S ( V m ) ( S ( V m ) in case B) and any constant ρ > , there is a constant C ρ such that (7.24) ψ ( g − x ) ≤ C ρ exp( − ρ d ( D U,z ′ , gz )) , ∀ g ∈ G. Proof.
Since ψ is a Schwartz function, for any positive integer N , there is a positiveconstant C N such that ψ ( x ) ≤ C N ( || x || z ) N . By Theorem 7.3, we know that k g − x k z = m X j =1 k g − x j k z ≥ c · exp(2 b · d ( D U,z ′ , gz )) . Hence ψ ( g − x ) ≤ C N c exp( − N b · d ( D U,z ′ , gz )) . We fix a
N > ρ b and let C ρ = C N c , the theorem is proved. (cid:3) Remark . Suppose ψ g ′ = ω ( g ′ ) ψ for g ′ ∈ ˜ G ′ . Since ˜ G ′ acts smoothly on S ( V m ), the constant C N and C ρ in the above proof for ψ g ′ depends continuouslyon g ′ . Proof of Theorem 7.1:
Fix a g ∈ G such that gz = z . Assume ψ = d X I ψ I Ω I , where ψ I ∈ S ( V m ) ( S ( V m ) in case B) are polynomial, Ω I ∈ V • p ∗ and are mutuallyperpendicular. Then˜ ψ ( z, g ′ , x ) = X I ( ω ( g ′ ) ψ I )( g − x ) L ∗ g − (Ω I ) . Weil representation preserves the space of Schwartz functions, hence ω ( g ′ ) ψ I ∈S ( V m ) ( S ( V m ) in case B).By Theorem 7.12, we know that for any I and ρ >
0, there is a constant C Iρ > ω ( g ′ ) ψ I )( g − x ) ≤ C Iρ exp( − ρ d ( D U,z ′ , z )) . Since the left action of G on D is isometric, we know that || L ∗ g − (Ω I ) || = || Ω I || . Hence define C ′ ρ = sX I ( C Iρ ) || Ω I || . We know that || ˜ ψ ∞ ( z, g ′ , x ) || ≤ C ′ ρ exp( − ρd ( D U,z ′ , z )).By the remark after Theorem 7.12 each constant C Iρ depends continuously on g ′ , hence C ′ ρ also depends continuously on g ′ . (cid:3) HETA SERIES AND CYCLES 57 Asymptotic evaluations of fiber integrals
We go back to the settings of Section 6. The goal of this section is to proveTheorem 6.4. We want to compute the fiber integral κ ( g ′ , β ) defined in (6.6) by κ ( g ′ , β ) = Z F z D U,z ′ ˜ ϕ ∞ ( z, g ′ , x ) , where ( x , x ) = β , U = span B v { x } , z ′ ∈ D ( U ) and z ∈ D U,z ′ . Our goal is to proveTheorem 6.4. Recall that ϕ ∞ = ϕ ⊗ Q v = v ϕ v , where ϕ is the cocycle specifiedafter equation (6.3), ϕ v is the Gaussian function of V mv and v is an Archimedeanplace for the number field k . We know that ϕ v ( g v , g ′ v ) only depends on g ′ v and isnonzero for v = v . We have κ ( g ′ , β ) = Z F z D U,z ′ ˜ ϕ ( z, g ′ v , x ) · Y v ∈ S ∞ ,v = v ϕ v ( g ′ v ) . So in order to prove the integral κ ( g ′ , β ) is nonzero, it suffices to compute thefollowing rescaled integral(8.1) Z F z D U,z ′ ˜ ϕ ( z, g ′ , x )with g ′ ∈ G ′ v , where ˜ ϕ is defined in (7.1). So from now on in this section, wechange our notation and let G = G v , G ′ = G ′ v and κ ( g ′ , β ) = R F z D U,z ′ ˜ ϕ ( z, g ′ , x ).Recall that ( G, G ′ ) can be the following three dual pairs(1) case A: (U( p, q ) , U( r + s, r + s )),(2) case B: (Sp(2 n, R ) , O(2 r, r )),(3) case C: (O ∗ (2 n ) , Sp( r, r )).Let B be C in case A, R in case B and H in case C. Let m be r + s in case A, 2 r in case B and r in case C. Recall that the group M ′ ⊂ G ′ is M ′ = (cid:26) m ′ ( a ) = (cid:18) a
00 ( a ∗ ) − (cid:19) | a ∈ GL m ( B ) (cid:27) . An element ( m ′ ( a ) , ζ ) in its double cover acts by (4.4). From this we know that(8.2) κ ( g ′ ( m ′ ( a ) , ζ ) , β ) = ζ | a | m κ ( g ′ , a ∗ βa ) . Suppose β satisfies the condition of Theorem 6.1, namely, iβ is non-degenerate andis of signature ( r, s ) in case A. By the above formula and Gram-Schmidt process,we can choose m = m ′ ( a ) such that a ∗ βa is of the following form:(1) the r + s by r + s diagonal matrix with diagonal entries {− i, . . . , − i | {z } r , i, . . . , i | {z } s } in case A,(2) the 2 r by 2 r matrix (cid:18) − I r I r (cid:19) in case B,(3) the r by r H -valued diagonal matrix with diagonal entries {− i, . . . , − i | {z } r } incase C. So from now on we assume β is of the above form. By translating by appropriate g ∈ G , we can further assume x = ( v , . . . , v r , v p +1 , . . . , v p + s ) , z = span C { v p +1 , . . . , v p + q } , (8.3) x = ( E , . . . , E r , F , . . . , F r ) , z = span C { v n +1 , . . . , v n } , x = ( v , . . . , v r ) , z = span C { v n +1 , . . . , v n } in the three cases respectively. Let a ( t ) ∈ GL m ( B ) be the scalar matrix t · Id .The exact value of κ ( g ′ , β ) is hard to compute in general, instead we approximate κ ( g ′ , β ) for g ′ = ( m ′ ( a ( t )) ,
1) as t → ∞ . We need the following theorem. Theorem 8.1.
Let f ( x ) , h ( x ) be smooth functions on R n . And let J ( t ) be theintegral J ( t ) = Z R n f ( x ) e − th ( x ) dx. And we assume that (1)
The integral J ( t ) converges absolutely for all t > . (2) For every ǫ > , ρ ( ǫ ) > , where ρ ( ǫ ) = inf { h ( x ) − h (0) : x ∈ R n , | x − | ≥ ǫ } . (3) the Hessian matrix A = ( ∂ h∂x i ∂x j ) | x =0 is positive definite.Then we have J ( t ) ∼ ( 2 πt ) n f (0)det( A ) − exp[ − th (0)] as t → ∞ . The above theorem is one special case of the so-called method of Laplace. Theproof can be found in Section 5 of Chapter IX in [Won01]. To apply Theorem 8.1,we choose a base point z ∈ D U,z ′ . Recall that we identify T z D ∼ = g / k with p . Proof of theorem 6.4 under assumptions that will be checked later :Define ρ : N z D U,z ′ → F z D U,z ′ by ρ : Y exp( − Y ) z , where we regard N z D U,z ′ as a linear subspace of p . Our strategy is to applyTheorem 8.1 to(8.4) J ( t ) = Z F z D U,z ′ ˜ ϕ ( z, g ′ ( t ) , x ) = Z N z D U,z ′ ρ ∗ ( ˜ ϕ )( Y, g ′ ( t ) , x )for g ′ ( t ) = ( m ′ ( a ( t )) , ϕ ∞ ( z, g ′ ) = ( L g − ) ∗ ( ω ( g ′ ) ϕ ) = ( L g − ) ∗ ( ω ( g ′ ) ϕ ) , where gz = z . By equation (5.21), we have ϕ = ϕ d X i,j =1 p ij Ω i ∧ ¯Ω j , HETA SERIES AND CYCLES 59 where p ij are polynomial functions on V m , Ω i ∈ V • p ∗ + and ϕ ( x ) = exp( − π m X i =1 || ~x m || z )with x as in (8.3). Define h : N z D U,z ′ → R by(8.5) h ( Y ) = π · M z (exp( Y ) z , x ) , where M z ( z, x ) is the function defined in (7.4). Then we have ϕ (exp( − Y ) x ) = exp( − h ( Y )) . We also have (recall (4.4))( ρ ∗ )( ˜ ϕ )( Y, g ′ ( t ) , x )(8.6) = t e exp( − h ( Y ) t ) X i,j p ij (exp( − Y ) x · t ) ρ ∗ ( L ∗ exp( − Y ) (Ω i ) ∧ L ∗ exp( − Y ) (Ω j )) , where e is ( p + q )( r + s ) in case A and is 2 nr in case B and C. Since ˜ ϕ is of Hodgedegree ( N, N ), where N is the dimension of N z D U,z ′ , each term ρ ∗ ( L ∗ exp( − Y ) (Ω i ) ∧ L ∗ exp( − Y ) (Ω j )) in the above equation is some function times dvol , the volume formof the Euclidean space N z D U,z ′ . After combining terms according to t -degree, wehave(8.7) ( ρ ∗ )( ˜ ϕ )( Y, g ′ ( t ) , x ) = t e X i t i f i ( Y ) exp( − h ( Y ) t ) dvol. Theorem 8.1 can be applied to compute each Z N z D U,z ′ f i ( Y ) exp( − h ( Y ) t ) dvol. Condition (1) of Theorem 8.1 is checked in the corollary of Theorem 7.1. Condition(2) and (3) will be checked later in this section.As we are interested in asymptotic value when t → ∞ , only the highest degreeterm of t in Equation (8.7) matters if it is nonzero. By Lemma 5.15, we know thatthe highest degree term of ( ρ ∗ )( ˜ ϕ )( Y, g ′ ( t ) , x ) evaluated at x and Y = 0 is nonzero.Hence the asymptotic value of J ( t ) as t → ∞ is nonzero and Theorem 6.4 is proved. (cid:3) The rest of the section will be devoted to verifying condition (2) and (3) ofTheorem 8.1 (see Proposition 8.3.1, Proposition 8.4.1 and Proposition 8.5.1) andcomputing J ( t ) as in equation (8.4) in each case. We will emphasize on case A andproceed the other two cases quickly.8.1. Case A.
As before we assume (8.3). For 1 ≤ j ≤ p + q , define functions M j ( X ) : D → R by M j ( z ) = || g − z v j || z , where z ∈ D and g z · z = z . It is well defined as || · || z is K invariant. Let U j = span C { v j } , D j = D U j , ≤ j ≤ p + q. Recall that the definition of D j does not require a base point (see Remark 3.8). Lemma 8.2. D U,z ′ = ( r \ j =1 D j ) \ ( p + s \ j = p +1 D j ) . Proof.
By Proposition 3.7.1, we know that By definition for 1 ≤ j ≤ p , we have D α = { z ∈ D | z ⊆ U ⊥ α } , D µ = { z ∈ D | U µ ⊆ z } for 1 ≤ α ≤ p and p + 1 ≤ µ ≤ p + q . On the other hand by (3.6) we have D U,z ′ = { z ∈ D | ⊕ p + sj = p +1 U j ⊂ z ⊂ ( ⊕ rj =1 U j ) ⊥ } . The lemma follows. (cid:3)
Lemma 8.3.
For any element z ∈ D , we have M j ( z ) = cosh ( t ) + sinh ( t ) , where t = d ( z, D j ) . In particular, M j ( z ) ≥ M j ( z ) . Equality holds if and only if z ∈ D j .Proof. Let us assume 1 ≤ j ≤ p . The case p + 1 ≤ j ≤ p + q is similar. Withoutloss of generality we can assume j = 1. It is easy to see thatexp( − tE αµ ) v = ( cosh ( t ) · v − isinh ( t ) · v µ if α = 1 ,v otherwise , exp( − tF αµ ) v = ( cosh ( t ) · v − sinh ( t ) · v µ if α = 1 ,v otherwise , where E αµ and F αµ are defined in Section 5.1. Recall that the group G U fixes v .Hence M ( z ) = M ( gz ) , ∀ g ∈ G U . We have a G U -equvariant fibration π : D → D (see Section 3.2). By translating z using an element in G U , we can assume that π ( z ) = z and is of the form z = exp( X ) z , where X ∈ N z D ⊆ p . We have N z D = span R { E µ , F µ | p + 1 ≤ µ ≤ p + q } . We assume that X = p + q X µ = p +1 x µ E µ − p + q X µ = p +1 y µ F µ = p + q X µ = p +1 v ◦ ( x µ + iy µ ) v µ . We define t = vuut p + q X µ = p +1 ( | x µ | + | y µ | ) . and v = 1 t p + q X µ = p +1 ( x µ + iy µ ) v µ . Then ( v, v ) = i and ( v , v ) = 0. We haveexp( − X )( v ) = cosh ( t ) · v − isinh ( t ) · v. So we have (see (7.3) for the definition of k · k z ) M ( z ) = k cosh ( t ) · v − isinh ( t ) · v k z = cosh ( t ) + sinh ( t ) . Since z = exp( tv ◦ v ) z we know that t = d ( z, z ) = d ( z, D ). The claim of thelemma is proved. (cid:3) HETA SERIES AND CYCLES 61
Define M : D → R by (compare with (7.4)) M ( z ) = M z ( z, x ) = r X j =1 M j ( z ) + p + s X j = p +1 M j ( z ) . Define h j : N z D U,z ′ → R by(8.8) h j = M j ◦ exp z ◦ i, where i is the injection i : N z D U,z ′ ֒ → T z D . Then we have (see (8.5) for thedefinition of h ( Y ))(8.9) h ( Y ) = r X j =1 h j ( − Y ) + p + s X j = p +1 h j ( − Y ) . Proposition 8.3.1.
The function h satisfies condition (2) and (3) of Theorem 8.1.In particular, is the unique minimal point of h .Proof. By Lemma 8.3 and Lemma 8.2, M ( z ) obtains its minimal value at z if andonly if z ∈ D U,z ′ . In particular, z is the unique minimal point of the function M on F z D U,z ′ . Hence 0 is the unique minimal point of the function h on N z D U,z ′ .Suppose that X ∈ S ( N z D U,z ′ ). We define h X : R → R by h X ( t ) = h (exp z ( Xt )) . Then t = 0 is a global minimum for h X ( t ), thus we have(8.10) ddt h X ( t ) | t =0 = 0 . By Lemma 7.6, we have(exp ( − tX ) v j , exp ( − tX ) v j ) z = m X k =1 | a kj | exp ( − λ k t ) . From this we know(8.11) d dt h X ( t ) = n X k =1 m X j =1 | a kj | λ k exp ( − λ k t ) . With Lemma 7.9 in mind, we have a uniform lower bound of d dt h X ( t ) for all X ∈ S ( N z D U,z ′ ) and t ≥
0. A similar argument works for t <
0. So we can assumethat d dt h X ( t ) ≥ C for a positive constant C and all t ∈ R , X ∈ S ( N z D U,z ′ ). Itfollows that h ( X ) ≥ h (0) + 12 C || X || . Hence h satisfies condition (2) of Theorem 8.1. Condition (3) is also satisfiedbecause we know from the above that the Hessian matrix of h is positive definitewith the smallest eigenvalue bigger or equal to C . (cid:3) Any X ∈ N z D U,z ′ can be written as X = X ( α,µ ) ∈ I x αµ E αµ + X ( α,µ ) ∈ I y αµ F αµ , where I is the index set defined in equation (5.5). We need Corollary 8.3.1.
Suppose ( α, µ ) , ( β, ν ) ∈ I . For ≤ j ≤ p , we have ∂ h j ∂x αµ ∂x βν | X =0 = ∂ h j ∂y αµ ∂y βν | X =0 = 4 δ αβ δ αj δ µν , and ∂ h j ∂x αµ ∂y βν | X =0 = 0 . For p + 1 ≤ j ≤ p + q , we have ∂ h j ∂x αµ ∂x βν | X =0 = ∂ h j ∂y αµ ∂y βν | X =0 = 4 δ αβ δ µj δ µν , and ∂ h j ∂x αµ ∂y βν | X =0 = 0 . Proof.
We need to compute the following ∂ ∂s∂t [(exp ( sX + tY ) v, exp ( sX + tY ) v ) z ] | s = t =0 . Using the second order approximation of the exponential mapexp ( sX + tY ) = I +( sX + tY )+ 12 ( s X + t Y + stXY + stY X )+higher order terms , one can check that ∂ ∂s∂t [(exp ( sX + tY ) v, exp ( sX + tY ) v ) z ] | s = t =0 (8.12)= 12 (( XY + Y X ) v, v ) z + 12 ( v, ( XY + Y X ) v ) z + ( Y v, Xv ) z + ( Xv, Y v ) z =2( Xv, Y v ) z + 2( Y v, Xv ) z . The last equality follows from the fact that (
Xv, w ) z = ( v, Xw ) z for any X ∈ p and v, w ∈ V . We also have E αµ ( v α ) = − iv µ , E αµ ( v µ ) = iv α ,F αµ ( v α ) = − v µ , F αµ ( v µ ) = − v α for 1 ≤ α ≤ p, p + 1 ≤ µ ≤ p + q . Also recall that ( , ) z is a positive definiteHermitian form with an orthonormal basis { v , . . . , v p + q } . With these preparations,the formulas in the lemma follow from straightforward calculations. (cid:3) The Hessian matrix A of h at 0 is a 2( rq + ps − rs ) by 2( rq + ps − rs ) matrix. Corollary 8.3.2. A is diagonal with diagonal entries ∂ h∂ x αµ = ∂ h∂ y αµ = 8 for ≤ α ≤ r, p + 1 ≤ µ ≤ p + s . And ∂ h∂ x αµ = ∂ h∂ y αµ = 4 for ≤ α ≤ r, p + s + 1 ≤ µ ≤ p + s or r + 1 ≤ α ≤ p, p + 1 ≤ µ ≤ p + s . Inparticular it is positive definite with determinant det( A ) = 4 rq +2 ps − rs . Proof.
The corollary follows from 8.3.1 and (8.9). (cid:3)
Recall that J ( t ) is defined in (8.4). Theorem 8.4. J ( t ) ∼ ( − i ) ps + rq − rs ps + rq − rs π ps +2 rq − rs t ( p + q )( r + s ) − rs exp( − ( r + s ) πt ) . HETA SERIES AND CYCLES 63
Proof.
Recall the proof of Theorem 6.4 at the beginning of this section. By Lemma5.15, we know that the highest degree term of (8.7) evaluated at x = ( v , . . . , v r , v p +1 , . . . , v p + s )is (2 √ πt ) rq + ps − rs ) i ∗ ( ^ ( α,µ ) ∈ I ξ ′ αµ ∧ ξ ′′ αµ ) | z , where I is specified at (5.5). By Theorem 8.1 we know J ( t ) ∼ t ( p + q )( r + s ) · ( 2 πt ) ps + rq − rs · ( − √ πt ) r ( q − s )+2 s ( p − r ) · det( πA ) − exp( − t πh (0)) · ( − i ) ps + rq − rs =( − i ) ps + rq − rs ps + rq − rs π ps +2 rq − rs t ( p + q )( r + s ) − rs exp( − ( r + s ) πt ) . The theorem is proved. (cid:3)
Method of Laplace for Case B.
We want to apply Theorem 8.1 to compute J ( t ) for the dual pair (Sp(2 n, R ) , O(2 r, r )). We need to check conditions (2) and(3) of Theorem 8.1 again. We proceed quickly by omitting the proofs that aresimilar to those of case A. In this subsection we use the assumptions and notationsof Section 4.5.Recall that we assume that z = span { v n +1 , . . . , v n } , and x = ( ~x , . . . , ~x r ) = ( E , . . . , E r , F , . . . , F r ) . Recall from (8.5) that h : N z D U,z ′ → R is defined by h ( Y ) = r X j =1 k exp( − Y ) ~x j k z = r X j =1 ( k exp( − Y ) E j k z + k exp( − Y ) F j k z ) . Since ( , ) z is Hermitian, we have the following identity(8.13) k x k z + k y k z = 12 k x − iy k z + 12 k x + iy k z , ∀ x, y ∈ V. In our coordinates, it is more natural to write h ( Y ) = 12 r X j =1 ( k exp( − Y )( E j − F j i ) k z + k exp( − Y )( E j + F j i ) k z )= r X j =1 ( k exp( − Y ) v j k z + k exp( − Y ) v n + j k z ) . We then have the following proposition.
Proposition 8.4.1.
The function h satisfies condition (2) and (3) of Theorem 8.1.Proof. For X ∈ S ( N z D U,z ′ ), define h X : R → R by h X ( t ) = h ( Xt ) . Then as in the proof of Proposition 8.3.1, (8.11) still holds and by Lemma 7.9 weknow that d dt h X ( t ) ≥ C for all t ∈ R and X ∈ S ( N z D U,z ′ ). Using the formula ddt (exp( Xt ) v, exp( Xt ) v ) z | t =0 = ( Xv, v ) z + ( v, Xv ) z , one can also show that all first order derivatives of h at Y = 0 vanish. This suggeststhat Y = 0 is the unique minimal point of h on N z D U,z ′ . These facts imply that h ( X ) ≥ h (0) + 12 C || X || . Hence h satisfies condition (2) The Hessian matrix of h is positive definite with thesmallest eigenvalue no smaller than C . Hence condition (3) of Theorem 8.1 is alsosatisfied.. (cid:3) Recall that J ( t ) is defined in (8.4). In this case we have Theorem 8.5. J ( t ) ∼ ( − i ) (2 nr + r − r ) nr + r − r π r ( n − r +1) t nr + r − r exp( − rπt ) . Proof.
Apply Lemma 4.4, Lemma 5.7 and Theorem 8.1. The details are similar tothose of the proof of Theorem 8.4. (cid:3)
Method of Laplace for Case C.
We want to apply theorem 8.1 to compute J ( t ) in case of the dual pair (O ∗ (2 n ) , Sp( r, r )). We need to check conditions (2)and (3) of theorem 8.1. We use the assumptions and notations of Section 4.6.Recall that we assume that z = span { v n +1 , . . . , v n } , and x = ( ~x , . . . , ~x r ) = ( v , . . . , v r ) . Recall from (8.5) that h : N z D U,z ′ → R is defined by h ( Y ) = r X s =1 k exp( − Y ) ~x s k z = r X s =1 k exp( − Y ) v s k z . We then have
Proposition 8.5.1.
The function h satisfies condition (2) and (3) of Theorem 8.1.Proof. Similar to that of Proposition 8.4.1. (cid:3)
Recall that J ( t ) is defined in (8.4). In this case we have Theorem 8.6. J ( t ) ∼ ( − i ) (2 nr − r − r ) nr − r − r π r ( n − r − t nr − r − r exp( − rπt ) . Proof.
Apply Lemma 4.4, Lemma 5.10 and Theorem 8.1. The details are similarto those of the proof of Theorem 8.4. (cid:3)
Appendix A. The associated vector bundle
We go back to the settings of Section 6. Recall that ˜ G ′∞ is the metaplectic coverof G ′∞ , where G ′∞ = Q v G ′ v . Let ˜ K ′∞ be the subgroup of ˜ G ′∞ which fixes theGaussian function in equation (6.3). Then ˜ K ′∞ is a maximal compact subgroup of˜ G ′∞ which is the metaplectic cover of K ′∞ = Q v K ′ v , a maximal compact subgroupof G ′∞ .˜ G ′ acts on S ( V n ∞ ) by the Weil representation ω and the action commutes withthat of G . In this appendix, we show that θ L , ˜ ϕ is a matrix coefficient of an auto-morphic vector bundle E → ˜Γ ′ \ ˜ G ′∞ / ˜ K ′∞ . For each Archimedean place v of the number field k we have HETA SERIES AND CYCLES 65 (1) G ′ v = U( r + s, r + s ), K v = U( r + s ) × U( r + s ),(2) G ′ v = O(2 r, r ), K v = O(2 r ) × O(2 r ),(3) G ′ v = Sp( r, r ), K v = Sp( r ) × Sp( r ),in case A, B and C respectively.In order to determine the ˜ K ′∞ action on ϕ ∞ = ϕ ⊗ Y v = v ϕ v , it suffices to compute the ˜ K ′ v action on ϕ since ˜ K ′ v acts on ϕ v trivially for v = v .It turns out that often times ˜ K ′ v is the trivial two-fold cover of K ′ v and the actiondescends to K ′ v . The following general argument applies to both ˜ K ′ v and K ′ v representations so we just deal with the ˜ K ′ v case for brevity.We will see that ϕ is a highest weight vector of an irreducible representation of˜ K ′ v . We denote this representation by σ . To be more precise there is an irreduciblerepresentation ( E σ , σ ) of ˜ K ′∞ inside, where E σ ⊂ P ⊂ S ( V m ∞ ) such that ϕ ∞ ∈ E σ and(A.1) ω ( g ′ k ′ ) φ = ω ( g ′ )( σ ( k ′ ) φ )for all g ′ ∈ ˜ G ′∞ , k ′ ∈ ˜ K ′∞ and φ ∈ E σ .Let E ∗ σ = Hom C ( E σ , C ) be the dual representation of E . There is a canonicalelement Φ ∈ E σ ⊗ C E ∗ σ which corresponds to the identity element in E σ ⊗ C E ∗ σ ∼ =Hom C ( E σ , E σ ). Explicitly we choose a basis { ϕ , . . . , ϕ d } of E σ and we assume ϕ = ϕ ∞ . Let { e , . . . , e d } be the corresponding dual basis of E ∗ σ . Then we haveΦ = d X i =1 ϕ i ⊗ e i . By definition the diagonal action of ˜ K ′∞ on Φ leaves it invariant:(A.2) ( σ ⊗ σ ∗ )( k ′ )Φ = Φ , k ′ ∈ ˜ K ′∞ . Equivalently(A.3) ( σ ⊗ Id )( k ′ )Φ = ( Id ⊗ σ ∗ )(( k ′ ) − )Φ , ∀ k ′ ∈ ˜ K ′∞ , where Id stands for the trivial action. For each x ∈ V m ∞ ,Φ( g ′ , x ) = (( ω ( g ′ ) ⊗ Id )Φ)( x )is a function on ˜ G ′∞ with values in E ∗ σ . Equation (A.1) and equation (A.3) togetherimply that(A.4) Φ( g ′ k ′ , x ) = ( Id ⊗ σ ∗ )(( k ′ ) − )Φ( g ′ , x ) . In other words, Φ( · , x ) is a section of the vector bundle E → ˜ G ′∞ / ˜ K ′∞ associatedto the representation ( E ∗ σ , σ ∗ ) of ˜ K ′∞ .We now apply θ -distribution to get θ L , Φ ( g ′ ) = X x ∈L m Φ( g ′ , x ) . Then θ L , Φ ( g ′ ) is ˜Γ ′ -invariant: θ L , Φ ( γ ′ g ′ ) = θ L , Φ ( g ′ ) , γ ′ ∈ ˜Γ ′ . By equation (A.4) and the ˜Γ ′ -invariance of θ L , Φ , θ L , Φ is a section of the bundle˜Γ ′ \E → ˜Γ ′ \ ˜ G ′∞ / ˜ K ′∞ . The cocycle θ L ,ϕ ∞ is then a matrix coefficient of θ L , Φ : θ L ,ϕ ∞ ( g ′ ) = h ϕ ∞ θ L , Φ ( g ′ ) i , where h , i is an ˜ K ′∞ -invariant bilinear pairing between E σ and E ∗ σ .In case A, we will decide the ( ˜ K ′ v ) (the identity component of ˜ K ′ v ) action bycomputing highest weights. In case B and C, ˜ K ′ v is the trivial two-fold cover of K ′ v and the action descends to K ′ v , so we will decide the K ′ v action. In case Aand B the calculations are essentially done in [KV78]. In each case ϕ = ι ( φ + ∧ φ − )(see (5.20)) and ( ˜ K ′ v ) = K − × K + such that K − acts trivially on φ − and K + acts trivially on φ − (see Remark 5.11). So it is enough to determine the K − actionon φ + and K + action on φ − .A.1. Case A.
In this case G ′ v ∼ = U( m, m ), where m = r + s . Recall that in Section4.3 we choose a basis { w , . . . , w m , w m +1 , . . . , w m } of W v with the Hermitian form <, > such that(1) < w a , w a > = 1,(2) < w k , w k > = − ≤ a ≤ m, m + 1 ≤ k ≤ m and < w j , w k > = 0 if j = k . Define W + = span C { w , . . . , w m } , W − = span C { w m +1 , . . . , w m } . Denote the group U( W + , <, > ) (U( W − , <, > ) resp.) by U( m,
0) (U(0 , m ) resp.).Then K ′ v = U(0 , m ) × U( m,
0) is a maximal compact subgroup of G ′ v . Let k ′ beits Lie algebra and k ′ be its complexification.Following a calculation like that of Section 7 of [KM90], we can show that k ′ actson the Fock model by ω ( e ab ) = − p X α =1 u − α,b ∂∂u − α,a + p + q X µ = p +1 u − µ,a ∂∂u − µ,b − δ ab ( p − q )for 1 ≤ a, b ≤ m and(A.5) ω ( e kℓ ) = p X α =1 u − α,k ∂∂u − α,ℓ − p + q X µ = p +1 u − µ,ℓ ∂∂u − µ,k + 12 δ kℓ ( p − q )for m + 1 ≤ k, ℓ ≤ m .Recall that in Section 5 we define the element φ + r,s using the special harmonic( f − ) q − s ( f − ) p − r . Here f − and f − are as in Definition 5.2 except that we shift b to b + m , where b is the second index of the variable u − ab . This is because in Definition5.2 our assumption is that W is negative definite.Let t be the diagonal torus of u (0 , m ), n be the strictly upper triangular Liealgebra of u (0 , m ): n = span C { e kℓ | m + 1 ≤ k < ℓ ≤ m } . Using equation (A.5), it is easy to see that ( f − ) q − s ( f − ) p − r has weight( − s + 12 ( p + q ) , . . . , − s + 12 ( p + q ) | {z } r , r −
12 ( p + q ) , . . . , r −
12 ( p + q ) | {z } s ) HETA SERIES AND CYCLES 67 under t . Moreover we can show that ( f − ) q − s ( f − ) p − r is annihilated by n . We havethree cases(1) m + 1 ≤ k < ℓ ≤ m + r ,(2) m + 1 ≤ k ≤ m + r , m + r + 1 ≤ ℓ ≤ m + r + s ,(3) m + r + 1 ≤ k < ℓ ≤ m + r + s .In case (1), e kℓ ( k < ℓ ) replaces a column of f − by an existing column and actstrivially on all the variables in f − . In case (2), e kℓ acts trivially on all the variablesin f − and f − . In case (3), e kℓ acts trivially in all the variables in f − and replaces acolumn of f − by an existing column. In any case e kℓ annihilates ( f − ) q − s ( f − ) p − r .The conclusion is that φ + r,s is a highest weight vector of u (0 , m ). Similarly φ − r,s is a lowest weight vector of u ( m,
0) with weight( s −
12 ( p + q ) , . . . , s −
12 ( p + q ) | {z } r , − r + 12 ( p + q ) , . . . , − r + 12 ( p + q ) | {z } s ) . It generates an irreducible representation of highest weight( − r + 12 ( p + q ) , . . . , − r + 12 ( p + q ) | {z } s , s −
12 ( p + q ) , . . . , s −
12 ( p + q ) | {z } r ) . A.2.
Case B.
In this case G ′ v = O(2 r, r ).Let us use the notation of subsection 4.5. Recall that W is a complex vector spacewith a Hermitian form <, > of signature ( r, r ). We denote by W R the underlyingreal vector space of W and let <, > R = Re <, > . G ′ v is the linear isometry groupof ( W R , <, > R ). We can choose an orthonormal basis { w a , w k | ≤ a ≤ r, r + 1 ≤ k ≤ r } of W such that < w a , w a > = 1 , < w k , w k > = − ≤ a ≤ r, r + 1 ≤ k ≤ r . Define W + = span C { w , . . . , w r } , W − = span C { w r +1 , . . . , w r } . Denote the group O( W + , <, > R ) (O( W − , <, > R ) resp.) by O(2 r,
0) (O(0 , r ) resp.).Then K ′ v = O(0 , r ) × O(2 r,
0) is a maximal compact subgroup of G ′ v . Let k ′ beits Lie algebra and k ′ be its complexification. Then (c.f. Section 7 of [KM90]) k ′ ∼ = ∧ R ( W + ) ⊕ ∧ R ( W − ) = o (2 r, ⊕ o (0 , r ) , k ′ ∼ = ∧ C ( W + ⊗ R C ) ⊕ ∧ C ( W − ⊗ R C ) . First we focus on O(0 , r ). Denote the complex structure 1 ⊗ i on W − ⊗ R C byright multiplication by i . Define w ′ k = w k + iw k i, w ′′ k = w k − iw k i, where r + 1 ≤ k ≤ r . We take { w ′′ r +1 , . . . , w ′′ r , w ′ r +1 , . . . , w ′ r } to be the basis of W − ⊗ R C . Notice that if we extend the form <, > R complex linearly to a symmetricform on W R ⊗ R C . Then < w ′ k , w ′′ ℓ > R = − δ kℓ , < w ′ k , w ′ ℓ > R = < w ′′ k , w ′′ ℓ > R = 0 . Then we have a split torus t of o (0 , r ) ⊗ R C spanned by { w ′ k ∧ w ′′ k | r + 1 ≤ k ≤ r } . We also have a nilpotent algebra n of o (0 , r ) ⊗ R C : n = span { w ′′ k ∧ w ′′ ℓ | r + 1 ≤ k, ℓ ≤ r } ⊕ span { w ′ k ∧ w ′′ ℓ | r + 1 ≤ k < ℓ ≤ r } . The Lie algebra o (0 , r ) ⊗ R C acts on the Fock model in the following way: ω ( w ′ k ∧ w ′′ ℓ ) = ω (2 e r + k,r + ℓ − e ℓ,k ) = 2 n X α =1 ( u − α + n,k ∂∂u − α + n,ℓ − u − α,ℓ ∂∂u − α,k ) ,ω ( w ′ k ∧ w ′ ℓ ) = ω (2 e r + k,ℓ − e r + ℓ,k ) = 2 n X α =1 ( u − α + n,k ∂∂u − α,ℓ − u − α + n,ℓ ∂∂u − α,k ) ,ω ( w ′′ k ∧ w ′′ ℓ ) = ω (2 e k,r + ℓ − e ℓ,r + k ) = 2 n X α =1 ( u − α,k ∂∂u − α + n,ℓ − u − α,ℓ ∂∂u − α + n,k ) . Recall that in Section 5 we define the element φ + r using the special harmonic( f − ) n − r +1 . Here f − is as in Definition 5.2 except that we shift b to b + r , where b is the second index of the variable u − ab . This is because in Definition 5.2 ourassumption is that W is negative definite.Then it is easy to see that ( f − ) n − r +1 is annihlated by n and has weight(( n − r + 1) , . . . , ( n − r + 1) | {z } r )under t . So φ + r is a highest weight vector of so (0 , r ).Under the group O(0 , r ), φ + r generates an irreducible representation that splitsinto two irreducible representations of so (0 , r ) with highest weights(( n − r + 1) , . . . , ( n − r + 1) | {z } r − , ± ( n − r + 1)) . Similarly φ − r is a lowest weight vector of so (2 r,
0) of weight( − ( n − r + 1) , . . . , − ( n − r + 1) | {z } r ) . It generates an irreducible representation of O(2 r,
0) with highest weights(( n − r + 1) , . . . , ( n − r + 1) | {z } r − , ± ( n − r + 1)) . A.3.
Case C.
In this case G ′ v = Sp( r, r ).Let us use the notation of subsection 4.5. Recall that W H is a H -vector spacewith a Hermitian form <, > H of signature ( r, r ). G ′ v ∼ = is the linear isometry groupof ( W H , <, > H ). We can choose an orthonormal basis { w a , w k | ≤ a ≤ r, r + 1 ≤ k ≤ r } of W H such that < w a , w a > H = 1 , < w k , w k > H = − ≤ a ≤ r, r + 1 ≤ k ≤ r . Define W + = span H { w , . . . , w r } , W − = span H { w r +1 , . . . , w r } . Denote the group Sp( W + , <, > H ) (Sp( W − , <, > H ) resp.) by Sp( r,
0) (Sp(0 , r ) resp.).Then K ′ v = Sp(0 , r ) × Sp( r,
0) is a maximal compact subgroup of G ′ v . Let k ′ beits Lie algebra and k ′ be its complexification. Then k ′ ∼ = Sym C ( W + ) ⊕ Sym C ( W − ) ∼ = sp ( r, ⊗ R C ⊕ sp (0 , r ) ⊗ R C ∼ = sp (2 r, C ) ⊕ sp (2 r, C ) . First we focus on Sp(0 , r ). We take the complex basis { w r +1 , . . . , w r , jw r +1 , . . . , jw r } of W − . We have a split torus t of sp (2 r, C ): t = span C { e a,a − e a + r,a + r | ≤ a ≤ r } . HETA SERIES AND CYCLES 69
We also have a nilpotent algebra n of sp (2 r, C ): n = span { e r + ℓ,r + k − e k,ℓ | r +1 ≤ k < ℓ ≤ r }⊕ span { e k,s + ℓ + e ℓ,s + k | r +1 ≤ k, ℓ ≤ r } . The Lie algebra sp (0 , r ) acts by the Weil representation in the following way: ω ( e k,ℓ − e r + ℓ,r + k ) = n X α =1 ( u − α,k ∂∂u − α,ℓ − u − α + n,ℓ ∂∂u − α + n,k ) ,ω ( e k,r + ℓ + e ℓ,r + k ) = n X α =1 ( u − α,k ∂∂u − α + n,ℓ + u − α,ℓ ∂∂u − α + n,k ) ,ω ( e r + k,ℓ + e r + ℓ,k ) = n X α =1 ( u − α + n,ℓ ∂∂u − α,k + u − α + n,k ∂∂u − α,ℓ ) . Recall that in Section 5 we define the element φ + r using the special harmonic( f − ) n − r − . Here ( f − ) is as in Definition 5.2 except that we shift b to b + r , where b is the second index of the variable u − ab . This is because in Definition 5.2 ourassumption is that W is negative definite.Moreover it is easy to see that ( f − ) n − r − is annihlated by n and has weight(( n − r − , . . . , ( n − r − | {z } r )under t . So φ + r is a highest weight vector of Sp(0 , r ).Similarly φ − r is a lowest weight vector of Sp( r,
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Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA
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