Calabi-Yau structure and Bargmann type transformation on the Cayley projective plane
aa r X i v : . [ m a t h . DG ] J a n CALABI-YAU STRUCTURE AND BARGMANN TYPE TRANSFORMATION ONTHE CAYLEY PROJECTIVE PLANE
KURANDO BABA AND KENRO FURUTANI ♦ Abstract.
Our purposes are to show the existence of a Calabi-Yau structure on the punctured cotan-gent bundle T ∗ ( P O ) of the Cayley projective plane P O and to construct a Bargmann type trans-formation between the L -space on P O and a space of holomorphic functions on T ∗ ( P O ), whichcorresponds to the Fock space in the case of the original Bargmann transformation. A K¨ahler struc-ture on T ∗ ( P O ) was shown by identifying it with a quadrics in the complex space C \{ } and thenatural symplectic form of the cotangent bundle T ∗ ( P O ) is expressed as a K¨ahler form. Our methodto construct the transformation is the pairing of polarizations, one is the natural Lagrangian foliationgiven by the projection map q : T ∗ ( P O ) −→ P O and the positive complex polarization defined bythe K¨ahler structure.The transformation gives a quantization of the geodesic flow in terms of one parameter group ofelliptic Fourier integral operators whose canonical relations are defined by the graph of the geodesicflow action at each time. It turn out that for the Cayley projective plane the results are not same withother cases of the original Bargmann transformation for Euclidean space, spheres and other projectivespaces. Contents
1. Introduction 22. Representations of quaternion and octanion algebras by complex matrix algebras 33. Cayley projective plane and its punctured cotangent bundle 54. Complex coordinate neighborhoods and Calabi-Yau structure 75. Symplectic manifolds and polarizations 215.1. Integral symplectic manifold 215.2. Real and complex polarizations 225.3. Hilbert space structure on the spaces of polarized sections 225.4. Pairing of polarizations and a Bargmann type transformation 266. Bargmann type transformation 286.1. Holomorphic trivialization and unitary trivialization 296.2. Fock-like space 317. Pairing with the Riemann volume form 347.1. A local coordinates 347.2. Explicit determination of the pairing with the Riemann volume form 368. Invariant polynomials and harmonic polynomials on the Jordan algebra J (3) 398.1. Correspondence between polynomials and differential operators with constant coefficients 398.2. Trace function and invariant polynomials 419. Inverse of Bargmann type transformation 519.1. Inverse transformation 5110. Some additional results 56 Date : January 20, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Cayley projective plane, F , Calabi-Yau structure, polarization, Bargmann transformation,symplectic manifold, Fock space. ♦ The second author was partially supported by JSPS fund 17K05284. Also this work was partially supported byOsaka City University Advanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics andTheoretical Physics JPMXP0619217849). F ε P O Introduction
The fundamental and historical problem in the quantization theory will be how to assign a functionon a phase space to an operator acting on the space of quantum states and the assignment satisfiessome algebraic condition, like a Lie algebra homomorphism. The phase space appearing in the theoryhas a structure, a symplectic structure. There are many theory relating with this problem. Onemethod is the theory of deformation quantization. Also there is the opposite theory, an assignmentof operators to functions, from an operator to a function. In the (pseudo)differential operator theoryand Fourier integral operator theory, the basic assignment of operators to their principal symbol (andsub-principal symbol) is a fundamental isomorphism between the spaces of operators and functionson the phase space modulo lower order classes.The famous transformation, called Bargmann transformation was introduced in [Ba] and gives oneaspect of the quantization of the unitary representation. The method to construct such a transfor-mation is given by the pairing of two polarizations, real polarization and complex polarization, on C n interpreted as C n ∼ = T ∗ ( R n ) ∼ = R n × R n , complex space and fiber space by Lagrangian fibers π : C n → R n . Under precise treatments of this method it was given a similar operator for the case of the spherein [Ra2], in [FY] for the complex projective space and for the quaternion projective spaces in [Fu1].Among the projective spaces the Cayley projective plane P O is the exceptional one and our purposein this paper is to show that we can also construct such an operator for this manifold in the samemethod. This case will be one of the non-trivial examples to which we can apply this method, ” pairingof two polarizations ” ([Ra2], [Ii1], [Ii2], [Fu1], [FY]).In the paper [Fu2] a K¨ahler structure on its punctured cotangent bundle T ∗ ( P O ) was constructedby embedding it into the complex space C \{ } as an intersection of null sets of several quadricpolynomials, which gives the realization of the natural symplectic form as a K¨ahler form. Here weshow the holomorphic triviality of the canonical line bundle of this complex manifold by giving anowhere vanishing global holomorphic 16-form explicitly.There are several study of the existence of K¨ahler structure on the (punctured) cotangent bundleof a certain class of manifolds, like [Sz1], [Sz2], [Koi], [Li], [FT], also see [Be], [So] in relation with aspecial property of the geodesic flow, SC ℓ -manifolds.The classical Bargmann transformation gives a correspondence between monomials on C n andHermite functions on R n , which are the eigenfunctions of the harmonic oscillator and this facts wereapplied to various problems, especially to T¨oplitz operator theory (there are so many, but here I justcite one book [BS]). Also there are many precise treatments and modifications of this transformation(for examples, [Ii1], and recent works in [Ch1], [Ch2] ).For our case we show the restrictions of monomials defined on C \{ } to the embedded puncturedcotangent bundle T ∗ ( P O ) are mapped to eigenfunctions of the Laplacian on P O .This paper is organized as follows. In §
2, we explain a realization of quaternion and octanion numberfields, H and O , in a complex matrix algebra. Multiplication law in the octanion is interpreted in thetwo 2 × C (2) × C (2).In §
3, we introduce the Jordan algebra J (3) of 3 × P O is realized in this Jordan algebra. Following an earlier result in [Fu2] we explain the embedding ofthe punctured cotangent bundle T ∗ ( P O ) of the Cayley projective plane into the complexified Jordanalgebra C ⊗ R J (3) =: J (3) C of 3 × τ O : T ∗ ( P O ) −→ J (3) C . ARGMANN TYPE TRANSFORMATION 3
We denote the image τ O ( T ∗ ( P O )) = X O . Also we state that the natural symplectic form ω P O is aK¨ahler form.In §
4, using the defining equations of the punctured cotangent bundle of the Cayley projective planeembedded in the complex Jordan algebra J (3) C , we give an open covering by complex coordinatesneighborhoods and show by an elementary way that the canonical line bundle of the complex structureis holomorphically trivial by explicitly constructing a nowhere vanishing holomorphic global section(we put it Ω O ), that is, a 16-degree holomorphic differential form which can be seen as the restrictionof a smooth 16-degree differential form on the whole complexified Jordan algebra J (3) C .In §
5, we resume a basic fact on symplectic manifolds with integral symplectic form and a methodof the geometric quantization. Here we consider two types of typical polarizations (real and positivecomplex). Then we apply the method to our case (= T ∗ ( P O )) and give a Bargmann type transfor-mation in the form of a fiber integration on the punctured cotangent bundle T ∗ ( P O ) to the basespace P O .In §
6, first we show the nowhere vanishing holomorphic global section Ω O constructed in § F -invariant. Incidentally we determine the product Ω O ∧ Ω O in terms of the Liouville volume form dV P O := 116 ! (cid:16) ω P O (cid:17) of the cotangent bundle T ∗ ( P O ).Also we introduce a class of subspaces consisting of holomorphic functions on X O satisfying some L conditions. These will correspond to the Fock space in the Euclidean case.In §
7, we determine the exterior product of the Riemann volume form pull backed to the cotangentbundle T ∗ ( P O ) and the nowhere vanishing global holomorphic section Ω O in terms of the Liouvillevolume form dV P O . For this purpose we fix a local coordinates at a point in P O which is also usedin the section § §
8, we discuss invariant polynomials and a similar feature to harmonic polynomials with respectto the natural representation of the group F to the Jordan algebra J (3) and its extension to thepolynomial algebra. Then, based on a general theorem in [He] (also [HL] and [Ko]) we state theeigenfunction decomposition of L space of P O .In §
9, based on the data obtained until § L ( P O ) which are approximated byclassical phenomena, but can not be observed directly by classical mechanical way.Finally in §
10 we mention that our Fock-like spaces have the reproducing kernel and a relation withthe geodesic flow action.2.
Representations of quaternion and octanion algebras by complex matrix algebras
First we fix a representation of quaternion numbers h = h + h i + h j + h k ( h i ∈ R ) as a 2 × ρ H : H ∋ h (cid:18) h + h √− h + h √− − h + h √− h − h √− (cid:19) ∈ C (2) , that is if we put h = ( h + h i ) + ( h + h i ) j = λ + µ j , then ρ H ( h ) = (cid:18) λ µ − µ λ (cid:19) , where we understand that a quaternion λ = h + h i ∈ H is a complex number h + √− h ∈ C inthe complex matrix and so on. Hence by this representation the complexification C ⊗ R H is identified KURANDO BABA AND KENRO FURUTANI with the algebra of whole 2 × C (2) (we put z i = x i + √− y i ∈ C ): C ⊗ R H ∋ h = z + z i + z j + z k = x + x i + x j + x k + √− ⊗ (cid:0) y + y i + y j + y k (cid:1) = ( x + x i ) + ( x + x i ) j + √− ⊗ (cid:0) ( y + y i ) + ( y + y i ) j (cid:1) = λ + µ j + √− ⊗ ( α + β j ) ←→ λ µ − µ λ + √− α β − β α = λ + √− α µ + √− β − µ − √− β λ + √− α (2.1) = z + √− z z + √− z − z + √− z z − √− z . (2.2)For the later use, we remark the converse expression, that is, when we are given a 2 × A = (cid:18) z z z z (cid:19) , then corresponding complex coefficient quaternion a + a i + a j + a k isgiven as(2.3) a + a i + a j + a k = z + z + z − z √− i + z − z j + z + z √− k . From the form of the matrices appearing in the above expression we know that the complexifiedalgebra C ⊗ R H is isomorphic (over C ) to 2 × C (2).For h = h + h i + h j + h k ∈ H , we denote its conjugation by θ ( h ) = h − h i − h j − h k anduse the same notation for elements in C ⊗ R H , that is C ⊗ R H ∋ h θ ( h ) = z − z i − z j − z k ∈ C ⊗ R H . By the expression of h = z + z i + z j + z k = x + x i + x j + x k + √− ⊗ (cid:0) y + y i + y j + y k (cid:1) = ( x + x i ) + ( x + x i ) j + √− ⊗ ( y + y i + ( y + y i ) j = λ + µ j + √− ⊗ (cid:0) α + β j (cid:1) θ ( h ) = z − z i − z j − z k = ( x + √− y ) − ( x + √− y ) i − ( x + √− y ) j − ( x + √− y ) k = x − x i − ( x + x i ) j + √− ⊗ (cid:0) y − y i − ( y + y i ) j (cid:1) = λ − µ j + √− ⊗ (cid:0) α − β j (cid:1) . Hence, if we put ρ H ( h ) = λ µ − µ λ + √− α β − β α = λ + √− α µ + √− β − µ − √− β λ + √− α = (cid:18) w w w w (cid:19) , then ρ H ( θ ( h )) = λ − µµ λ + √− α −√− β √− β √− α = (cid:18) w − w − w w (cid:19) . For h ∈ C ⊗ R H , the product ρ H ( θ ( h )) ρ H ( h ) = ρ H ( h ) ρ H ( θ ( h )) = ( w w − w w ) · Id = det ρ H ( h ) · Id,where Id is 2 × ARGMANN TYPE TRANSFORMATION 5
In the following we represent octanion numbers by two 2 × { e i } i =0 be the standard basis of the octanion number field O such that e is the basis of thecenter and let i , j and k be the basis of the quaternion number field. We identify e = , e = i , e = j , e = k , e i e = e i +4 , i = 0 , , , x = P x i e i as x = X i =0 x i e i + X i =0 x i +4 e i · e = a + b · e ∈ H ⊕ H e as the sum of two quaternion numbers. So, the complexification C ⊗ R O is seen as C ⊗ R O ∼ = C (2) ⊕ C (2) e through the map ρ H . The conjugation in O is defined by h = X h i e i θ ( h ) = h − X i =1 h i e i with the same notation for the case of the quaternion number field.The conjugation operation θ in C ⊗ R O is expressed in the matrix representation of C ⊗ R O as θ : C (2) ⊕ C (2) e ∋ Z + W e = (cid:18) z z z z (cid:19) + (cid:18) w w w w (cid:19) e (2.4) θ ( Z + W e ) = θ ( Z ) − W e = (cid:18) z − z − z z (cid:19) − (cid:18) w w w w (cid:19) e . Remark 1.
Hereafter, for an octanion z = P i =0 z i e i ∈ O or z ∈ C ⊗ R O , we will indicate the coefficientof the basis e i by z i = { z } i sometimes according to the necessity to avoid a confusion in the expression. Remark 2.
We use the conjugation z only for the complex number z = x + √− y and do not usethe operation θ for the conjugate of complex numbers to avoid confusion. So, for a complex octanionnumber z = P { z } i e i , { z } i ∈ C , we mean z = P { z } i e i and it holds θ ( z ) = θ ( z ) . Also for an octanionmatrix A = (cid:16) z ij (cid:17) we mean A = (cid:16) z ij (cid:17) . Cayley projective plane and its punctured cotangent bundle
In this section, we refer [SV] and [Yo] for all the necessary facts on the group F and the Cayleyprojective plane.Let J (3) be a subspace of the 3 × J (3) = t z θ ( y ) θ ( z ) t xy θ ( x ) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, y, z ∈ O , t i ∈ R . We introduce a product in J (3), called a Jordan product, by J (3) × J (3) ∋ ( A, B ) A ◦ B := AB + BA ∈ J (3) . It is called an exceptional Jordan algebra and of 27-dimensional over R . Then the group of R -linearalgebra automorphisms is the exceptional Lie group F :(3.1) F := { g ∈ GL ( J (3)) ∼ = GL (27 , R ) | g ( A ◦ B ) = g ( A ) ◦ g ( B ) , g ( Id ) = Id, A, B ∈ J (3) } . There are various characterizations for the group F (see for examples, [Yo], [SV]).The complexification C ⊗ R J (3) =: J (3) C consists of 3 × J (3) C = ξ z θ ( y ) θ ( z ) ξ xy θ ( x ) ξ (cid:12)(cid:12)(cid:12) x, y, z ∈ C ⊗ R O , ξ i ∈ C KURANDO BABA AND KENRO FURUTANI and is an exceptional Jordan algebra over C of the complex dimension 27. The complex linear auto-morphisms α : J (3) C −→ J (3) C satisfying the condition that α ( A ◦ B ) = α ( A ) ◦ α ( B ) , α ( Id ) = Id is the complex Lie group F C .The Cayley projective plane P O is defined as P O = (cid:8) X ∈ J (3) | X = X, tr X = ξ + ξ + ξ = 1 (cid:9) . It is known that the group F acts on P O in two point homogeneous way.Let X = ∈ P O ⊂ J (3), then it is known that the stationary subgroup of the point X in F is isomorphic to Spin (9) and F ∋ g g · X gives an isomorphism:(3.2) F /Spin (9) ∼ = P O . For X = ξ x θ ( x ) θ ( x ) ξ x x θ ( x ) ξ and Y = η y θ ( y ) θ ( y ) η y y θ ( y ) η ∈ J (3) , we introduce theirinner product by(3.3) < X, Y > J (3) := tr( X ◦ Y ) = X i =1 ξ i η i + 2 < x i , y i > R , where < · , · > R denotes the standard Euclidean inner product of x i and y i ∈ O ∼ = R .This inner product has a property(3.4) < X ◦ Y, Z > J (3) = < X, Y ◦ Z > J (3) , X, Y, Z ∈ J (3) . In particular, since the trace function J (3) ∋ A tr ( A ) is invariant under the F action, that is(3.5) tr ( g · A ) = tr ( A ) , g ∈ F , A ∈ J (3) , this inner product is invariant under the action by F :(3.6) < g · A , g · B > J (3) = tr( g · A ◦ g · B ) = tr (cid:0) g · ( A ◦ B ) (cid:1) = tr ( A ◦ B ) = < A, B > J (3) . The tangent bundle T ( P O ) can be seen as a subspace in J (3) × J (3) such that T ( P O ) = (cid:26) ( X, Y ) ∈ J (3) × J (3) (cid:12)(cid:12)(cid:12) X ∈ P O , X ◦ Y = 12 Y (cid:27) . We equip a Riemann metric g P O on the manifold P O by g P O X ( Y , Y ) := < Y , Y > J (3) , Y i ∈ T X ( P O ) , that is, it is the induced metric from the inner product on J (3). Using this metric on T ( P O ) , hereafter we identify the tangent bundle T ( P O ) and the cotangentbundle T ∗ ( P O ) . Let Y = √ √ ∈ J (3), then Y ∈ T X ( P O ) and || Y || = 1. The stationary subgroup atthe point ( X , Y ) ∈ T ( P O ) is known as being isomorphic to Spin (7) and the two point homogeneityof the action by F gives us the isomorphism F /Spin (7) ∼ = S ( P O ), the unit (co)tangent spherebundle of P O .The inner product on J (3), < · , · > J (3) , is extended to the complexification J (3) C as a complexbi-linear form in a natural way, which we denote by < · , · > J (3) C . Then the extension as the Hermitianinner product on the complexification J (3) C is given by < A, B > J (3) C , A, B ∈ J (3) C . ARGMANN TYPE TRANSFORMATION 7
Note here that the matrix B for B = ξ c θ ( b ) θ ( c ) ξ ab θ ( a ) ξ ∈ J (3) C means B = ξ c θ ( b ) θ ( c ) ξ ab θ ( a ) ξ ∈ J (3) C . We will denote the norm of a ∈ O by | a | = p < a, a > R and by || X || = p < X, X > J (3) the norm of X ∈ J (3) , respectively. Also with the same way for elements a ∈ C ⊗ R O and A ∈ J (3) C , we denotetheir norms. The punctured cotangent bundle T ∗ ( P O ) \{ } =: T ∗ ( P O ) is realized as a subspace in J (3) C withthe following form: Theorem 3.1. ([Fu2]) X O = A = ξ z θ ( y ) θ ( z ) ξ xy θ ( x ) ξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, y, z ∈ C ⊗ R O , ξ i ∈ C , A = 0 , A = 0 . (3.7) The correspondence between T ∗ ( P O ) and X O is given by (3.8) τ O : T ∗ ( P O ) ∋ ( X, Y ) τ O ( X, Y ) = 1 ⊗ (cid:0) || Y || X − Y (cid:1) + √− ⊗ || Y || Y √ . Then
Theorem 3.2. ([Fu2])(3.9) τ O ∗ (cid:16) √− ∂∂ || A || / (cid:17) = ω P O , where we denote by ω P O the natural symplectic form on the cotangent bundle T ∗ ( P O ) . The inverse map σ O of τ O is given by σ O : X O ∋ A ( X, Y ) = ( X ( A ) , Y ( A )) ∈ J (3) × J (3) , (3.10) X ( A ) = || A || · (cid:0) A + A (cid:1) + A ◦ A || A || ,Y ( A ) = − √− √ · || A || − / (cid:0) A − A (cid:1) . We denote by T ∗ ′ ( X O ) C the holomorphic part of the complexified cotangent bundle T ∗ ( X O ) ⊗ R C =: T ∗ ( X O ) C , or it denotes the real cotangent bundle T ∗ ( X O ) being considered as a complex vector bundleby the complex structure on X O .In particular, in the next section we show that the canonical line bundle V T ∗ ′ ( X O ) C is holomor-phically trivial by explicitly constructing a nowhere vanishing global holomorphic section.4. Complex coordinate neighborhoods and Calabi-Yau structure
We consider an open covering by explicit coordinates neighborhoods of the space X O . From theproduct structure of the Jacobians we show that the canonical line bundle has a nowhere vanishingglobal holomorphic section (Theorem 4.7). KURANDO BABA AND KENRO FURUTANI
The condition in (3.7) is expressed in the following six equations in terms of octanions:( ξ + ξ ) x + θ ( yz ) = 0 , (4.1) ( ξ + ξ ) y + θ ( zx ) = 0 , (4.2) ( ξ + ξ ) z + θ ( xy ) = 0 , (4.3) ξ + zθ ( z ) + θ ( y ) y = 0 , (4.4) ξ + θ ( z ) z + xθ ( x ) = 0 , (4.5) ξ + θ ( x ) x + yθ ( y ) = 0 . (4.6)The condition 0 = A ∈ X O is equivalent to one of the components x, y , or z being non zero. Thenthis implies Proposition 4.1. ξ + ξ + ξ = 0 . This property does not appear in an explicit form in (3.7) and plays an important role in § Proof.
Since the associativity a · θ ( a ) b = aθ ( a ) · b holds, by multiplying z from the left to the equality ( ξ + ξ ) x + θ ( yz ) = 0 it holds the equality: z · ( ξ + ξ ) x + z · θ ( z ) θ ( y ) = ( ξ + ξ ) zx + zθ ( z ) · θ ( y )= − ( ξ + ξ )( ξ + ξ ) θ ( y ) + zθ ( z ) · θ ( y ) = 0 . Hence if we assume y = 0 ( ξ + ξ )( ξ + ξ ) = zθ ( z )and by the same way ( ξ + ξ )( ξ + ξ ) = θ ( x ) x. These imply that ( ξ + ξ )( ξ + ξ ) + ( ξ + ξ )( ξ + ξ ) + ξ = ( ξ + ξ + ξ ) = 0 . and we have ξ + ξ + ξ = 0 . From the arguments above the same holds for other cases of x = 0 or z = 0. (cid:3) Remark 3.
Although we showed that tr ( A ) = 0 for A ∈ X O within the arguments permitted in theoctanion number field, if we based on the two point homogeneity of the space S ( X O ) by the group F ,we get the result easily. Since we consider the point X := ∈ P O and take a vector Y = ∈ T X ( P O ) , then τ O ( X , Y ) = √− √− − ∈ X O and it is clear tr ( τ O ( X , Y )) = 0 and since the function A tr ( A ) is invariant under the action ofthe group F , we have the result. In relation with the above property we mention the following fact.
ARGMANN TYPE TRANSFORMATION 9
Lemma 4.2.
Let A = ξ z θ ( y ) θ ( z ) ξ xy θ ( x ) ξ ∈ J (3) C , where we put z = P z i e i , y = P y i e i and x = P x i e i with z i , y i , x i , ξ j ∈ C , i = 0 , · · · , , j = 1 , , , and assume f ( A ) = 2 X i =0 ( a i z i + b i y i + c i x i ) + X i =1 α i ξ i is a linear function on J (3) C vanishing on X O . Then f is a constant multiple of the trace function A tr ( A ) , A ∈ J (3) C .Proof. Put a = P a i e i , b = P b i e i , c = P c i e i ∈ C ⊗ R O and B = α a θ ( b ) θ ( a ) α cb θ ( c ) α ∈ J (3) C , where α j ∈ C . Then we can express this function f = f B as f B ( A ) = tr ( B ◦ A ) . Let Y = z θ ( y ) θ ( z ) 0 0 y ∈ T X ( P O ), where z = P z i e i , y = P y i e i ∈ O (that is, z i , y i ∈ R ), then τ O ( X , Y )= | z | + | y | −| z | − θ ( yz )0 − yz −| y | + p | z | + | y | P √− z i e i √− θ ( y ) √− θ ( z ) 0 0 √− y ∈ X O . Here we put y = 0, and we may assume for such A = τ O ( X , Y ) f B ( A ) = tr ( B ◦ A ) = ( | z | )( α − α ) + 2 X √− | z | z i a i = 0 , for any ± z i ∈ R . Then α = α and also a i = 0 for i = 0 , · · · ,
7. Likewise we have α = α and b i = 0for i = 0 , · · · , f B ( τ O ( X , Y )) = < c i e i , θ ( yz ) > R = 0 for any y, z ∈ O . Hence c i = 0 for i = 0 , · · · ,
7, which shows our assertion, that is a = b = c = 0, α := α = α = α and f B ( A ) = α ξ + α ξ + α ξ = α · tr ( A ) . (cid:3) Recall that we express a complexified octanion number z = P a i e i = P i =0 a i e i + (cid:0) P i =0 a i e i (cid:1) e by two 2 × ρ H ( z ) = Z + W e = (cid:18) z z z z (cid:19) + (cid:18) w w w w (cid:19) e , where z i , w i ∈ C . The correspondence between a i and z i , w i are given in the expression (2.2). Likewisewe put ρ H ( y ) = Y + V e = (cid:18) y y y y (cid:19) + (cid:18) v v v v (cid:19) e (4.8) ρ H ( x ) = X + U e = (cid:18) x x x x (cid:19) + (cid:18) u u u u (cid:19) e . (4.9)(4.10) The conditions (4.1), (4.2), (4.3), (4.4), (4.5) and (4.6) are rewritten in terms of the matrices Z , W , Y , V , X , U as(4.11) ξ ( θ ( X ) − U e ) = ( Y + V e )( Z + W e ) = Y Z − θ ( W ) V + ( W Y + V θ ( Z )) e ,ξ ( θ ( Y ) − V e ) = ( Z + W e )( X + U e ) = ZX − θ ( U ) W + ( U Z + W θ ( X )) e ,ξ ( θ ( Z ) − W e ) = ( X + U e )( Y + V e ) = XY − θ ( V ) U + ( V X + U θ ( Y )) e ,ξ + det Z + det W + det Y + det V = 0 ,ξ + det Z + det W + det X + det U = 0 ,ξ + det Y + det V + det X + det U = 0 , that is, these express 27 degree two complex polynomial equations of 27 complex variables.The conditions for matrices in X O require that at least one of the off-diagonal components in thematrix A = ξ z θ ( y ) θ ( z ) ξ xy θ ( x ) ξ is non-zero, that is x = 0, or y = 0, or z = 0.Hence, for example, we assume that there is at least one component in the matrix ρ H ( z ) = Z + W e = (cid:18) z , z z z (cid:19) + (cid:18) w w w w (cid:19) e , say z = 0.Let O z = { A ∈ X O | z = 0 } and we define other open subsets { O z i , O w i , O y i , O v i , O x i , O u i } i =1 ina same way like O z . Then we have Proposition 4.3.
The subsets { O z i , O w i , O y i , O v i , O x i , O u i } i =1 are all coordinate neighborhoodsand totally is an open covering of X O . We denote this open covering by(4.12) U = { O z i , O w i , O y i , O v i , O x i , O u i } i =1 . Proof.
We show only for the case O z . Other cases will be shown by the same way.From the six equations (4.1),(4.2),(4.3),(4.4),(4.5) and (4.6) expressed by octanions, we select 5equations expressed by 2 by 2 complex matrices and two additional equations including the complexvariable z : ξ (cid:18) x − x − x x (cid:19) = (cid:18) y y y y (cid:19) (cid:18) z z z z (cid:19) − (cid:18) w − w − w w (cid:19) (cid:18) v v v v (cid:19) − ξ (cid:18) u u u u (cid:19) = (cid:18) w w w w (cid:19) (cid:18) y y y y (cid:19) + (cid:18) v v v v (cid:19) (cid:18) z − z − z z (cid:19) ξ (cid:18) y − y − y y (cid:19) = (cid:18) z z z z (cid:19) (cid:18) x x x x (cid:19) − (cid:18) u − u − u u (cid:19) (cid:18) w w w w (cid:19) − ξ (cid:18) v v v v (cid:19) = (cid:18) u u u u (cid:19) (cid:18) z z z z (cid:19) + (cid:18) w w w w (cid:19) (cid:18) x − x − x x (cid:19) ξ (cid:18) z − z − z z (cid:19) = (cid:18) x x x x (cid:19) (cid:18) y y y y (cid:19) − (cid:18) v − v − v v (cid:19) (cid:18) u u u u (cid:19) ξ + z z − z z + w w − w w + x x − x x + u u − u u = 0 , and one common equation for all cases in the set U ξ + ξ + ξ = 0 . ARGMANN TYPE TRANSFORMATION 11
From these we can select 11 equations:(4.13) f = − ξ y + z x + z x − ( u w − u w ) = 0 ,f = ξ y + z x + z x − ( u w − u w ) = 0 ,f = ξ v + u z + u z + ( w x − w x ) = 0 ,f = ξ v + u z + u z + ( w x − w x ) = 0 ,f = − ξ x + y z + y z − ( w v − w v ) = 0 ,f = ξ x + y z + y z − ( − w v + w v ) = 0 ,f = ξ u − v z + v z + w y + w y = 0 ,f = ξ u − v z + v z + w y + w y = 0 ,f = − ξ z + x y + x y − ( − v u + v u ) = 0 ,f = ξ + z z − z z + w w − w w + x x − x x + u u − u u = 0 ,f = ξ + ξ + ξ = 0 . It is apparent that the 10 variables(4.14) x , x , u , u , y , y , v , v , ξ , z are coefficients of the variable z , and can be solved easily.In fact, we can solve the 11 variables(4.15) { x , x , u , u , y , y , v , v , z , ξ , ξ } in terms of the remaining 16 variables(4.16) { x , x , u , u , y , y , v , v , z , z , z , w , w , w , w , ξ } , in which except z = 0 other variables { x , x , u , u , y , y , v , v , , z , z , w , w , w , w , ξ } . Here, there are two possibilities to choose ξ or ξ as an independent variable. The last equation f requires that the variable ξ must be a dependent variable, if we choose ξ is independent variable.Explicit solutions are x = ξ y − z x + u w − u w z (4.17) := ϕ ( x , x , u , u , y , y , v , v , z , z , w , w , w , w , ξ ) ,x = − ξ y − z x + u w − u w z := ϕ , (4.18) u = − ξ v − u z − w x + w x z := ϕ , (4.19) u = − ξ v − u z − w x + w x z := ϕ , (4.20) y = ξ x − y z + w v − w v z (4.21) = − ( ξ z + x y + x y + v u − v u ) x − ( y z − w v + w v ) z z := ϕ ,y = − ξ x − y z − w v + w v z := ϕ , (4.22) v = − ξ u + v z − w y − w y z := ϕ , (4.23) v = − ξ u + v z − w y − w y z := ϕ , (4.24) z = − ξ + z z − w w + w w − x x + x x − u u + u u z (4.25) = 1 z · ( − ξ + z z − w w + w w − ϕ x + ϕ x − ϕ u + u ϕ ) := ϕ , ξ = x y + x y + v u − v u z := ϕ , (4.26) ξ = − ξ − x y + x y + v u − v u z (4.27) = − ξ z + x y + x y + v u − v u z := ϕ . Hence the correspondence given as the graph map G z : C × C ∗ × C ∋ ( x , x , u , u , y , y , v v , z , z , z , w , w , w , w , ξ ) ( x , x , u , u , y , y , v , v , z , z , z , w , w , w , w , ξ ; x , x , u , u , y , y , v , v , z , ξ , ξ ) ∈ X O ⊂ C (2) × C defines a holomorphic local coordinates of X O . (cid:3) Corollary 4.4.
Each coordinate neighborhood O ∗ in U is dense and codim X O \ O ∗ = 2 , where ∗ ∈{ x , · · · , w } .Proof. We only show the case O ∗ = O z , since other cases are proved by the similar way.Let A ∈ X O \ O z . Assume, say A ∈ O x , then the subset z = 0 is defined by a quadric polynomialequation z = ξ y − z x + u w − u w x = 0 (see (4.17)). Hence the subset z = 0 must be codimension 2in X O . If A is an element in an open set U ∗ , where we can take z as an independent variable, thenagain the equation z = 0 defines a codimension 2 subset.Hence in any case the complement of each coordinate neighborhood O ∗ ( ∗ ∈ { x , · · · , w } ) is denseand the complement is of codimension 2 in X O . (cid:3) Let P z be the projection map( x , · · · , x , u , · · · , u , y , · · · , y , v , · · · , v ,z , · · · , z , w , · · · , w , ξ , ξ , ξ ) ( x , x , u , u , y , y , v , v , z , z , z , w , w , w , w , ξ ) , then P z ◦ G z = Id .Next we consider, as a case, another open subset O x = { A ∈ X O | x = 0 } . By the similar way asabove we select 10 equations { g i } among 27 equations in (4.11): g = − ξ z + x y + x y − v u + v u = 0 ,g = ξ z + x y + x y − v u + v u = 0 ,g = ξ w + v x + v x + u y − u y = 0 ,g = ξ w + v x + v x + u y − u y = 0 ,g = − ξ y + z x + z x − u w + u w = 0 ,g = ξ y + z x + z x + u w − u w = 0 ,g = ξ v + u z + u z − w x + w x = 0 ,g = ξ v + u z + u z − w x + w x = 0 ,g = − ξ x + y z + y z + w v − w v = 0 ,g = ξ + z z − z z + w w − w w + x x − x x + u u − u u = 0 , and one more which is common equation as above g = f = ξ + ξ + ξ = 0 . In this case the variables { x , x , x , u , u , u , u , y , y , v , v , z , z , w , w , ξ } are independent and the remaining 11 variables { x , y , y , v , v , z , z , w , w , ξ , ξ } ARGMANN TYPE TRANSFORMATION 13 are solved by those 16 variables.In fact they are given explicitly as follows: x = x x − u u + u u − z z + z z − w w + w w − ξ x = ψ ( x , x , x , u , u , u , u , y , y , v , v , z , z , w , w , ξ ) ,y = ξ z − x y + v u − v u x := ψ ,y = − ξ z − x y + v u − v u x := ψ ,v = − ξ w − v x − u y + u y x := ψ ,v = − ξ w − v x − u y + u y x := ψ ,z = ξ y − z x + u w − u w x := ψ ,z = − ξ y − z x − u w + u w x := ψ ,w = − ξ v − u z − u z + w x x := ψ ,w = − ξ v − u z − u z + w x x := ψ ,ξ = y z + y z + w v − w v x := ψ ,ξ = − ξ − ξ = − y z + y z + w v − w v x − ξ := − ψ − ξ . We determine the Jacobian of the coordinate transformation P x ◦ G z . By the observation of the independent variables for O z and O x we know that P x ◦ G z ( x , x , u , u , y , y , v , v , z , z , z , w , w , w , w , ξ )= ( x , x , x , u , u , u , u , y , y , v , v , z , z , w , w , ξ ) . The maps G z ( x , x , u , u , y , y , v , v , z , z , z , w , w , w , w , ξ )= ( x , x , x , x ,u , u , u , u , y , y , y , y , v , v , v , v , z , z , z , z , w , w , w , w , ξ , ξ , ξ )= ( ϕ , ϕ , x , x ,ϕ , u , ϕ , u , ϕ , y , ϕ , y , v , ϕ , v , ϕ , z , z , z , ϕ , w , w , w , w , ϕ , ξ , ϕ )and G x ( x , x , x ,u , u , u , u , y , y , v , v , z , z , w , w , ξ )= ( x , x , x , ψ ,u , u , u , u , ψ , ψ , y , y , ψ , v , ψ , v , ψ , z , ψ , z , w , ψ , w , ψ , ψ , ξ , ψ )imply that P x ◦ G z ( x , x , u , u , y , y , v , v , z , z , z , w , w , w , w , ξ )= ( x , x , x , u , u , u , u , y , y , v , v , z , z , w , w , ξ )= ( ϕ z , ϕ z , x , ϕ z , u , ϕ z , u , ϕ z , y , ϕ z , ϕ z , z , ϕ z , w , w , ξ ) . We reorder the variables( x , x , u , u , y , y , v , v , z , z , z , w , w , w , w , ξ ) as ( x , u , u , y , z , w , w , ξ ; x , y , v , v , z , z , w , w )and put ( x , u , u , y , z , w , w , ξ ; x , y , v , v , z , z , w , w )=( a , a , a , a , a , a , a , a ; a , a , a , a , a , a , a , a ) := a. Also we reorder the variables x , x , x , u , u , u , u , y , y , v , v , z , z , w , w , ξ as ( x , x , x , u , u , u , u , y , y , v , v , z , z , w , w , ξ )= ( x , u , u , y , z , w , w , ξ ; x , x , u , u , y , v , v , z )and put ( b , b , b , b , b , b , b , b , b , b , b , b , b , b , b , b ) . Then b = a , . . . . . . , b = a and b = ϕ ( a ) z = ξ y − z x + u w − u w z ,b = ϕ ( a ) z = − ξ y − z x + u w − u w z ,b = ϕ ( a ) z = − ξ v − u z − w x + w x z ,b = ϕ ( a ) z = − ξ v − u z − w x + w x z ,b = ϕ ( a ) z = − ξ x − y z − w v + w v z ,b = ϕ ( a ) z = − ξ u + v z − w y − w y z ,b = ϕ ( a ) z = − ξ u + v z − w y − w y z ,b = ϕ ( a ) z = − ξ + z z − w w + w w − x x + x x − u u + u u z . The Jacobi matrix ∂b i ∂a j is of the form of (cid:18) Id A B (cid:19)
ARGMANN TYPE TRANSFORMATION 15 where Id is 8 × B = (cid:16) ∂b i ∂a j (cid:17) is a 8 × B = ∂ϕ ∂x ∂ϕ ∂y ∂ϕ ∂v ∂ϕ ∂v ∂ϕ ∂z ∂ϕ ∂z ∂ϕ ∂w ∂ϕ ∂w ∂ϕ ∂x ∂ϕ ∂y ∂ϕ ∂v ∂ϕ ∂v ∂ϕ ∂z ∂ϕ ∂z ∂ϕ ∂w ∂ϕ ∂w ∂ϕ ∂x ∂ϕ ∂y ∂ϕ ∂v ∂ϕ ∂v ∂ϕ ∂z ∂ϕ ∂z ∂ϕ ∂w ∂ϕ ∂w ∂ϕ ∂x ∂ϕ ∂y ∂ϕ ∂v ∂ϕ ∂v ∂ϕ ∂z ∂ϕ ∂z ∂ϕ ∂w ∂ϕ ∂w ∂ϕ ∂x ∂ϕ ∂y ∂ϕ ∂v ∂ϕ ∂v ∂ϕ ∂z ∂ϕ ∂z ∂ϕ ∂w ∂ϕ ∂w ∂ϕ ∂x ∂ϕ ∂y ∂ϕ ∂v ∂ϕ ∂v ∂ϕ ∂z ∂ϕ ∂z ∂ϕ ∂w ∂ϕ ∂w ∂ϕ ∂x ∂ϕ ∂y ∂ϕ ∂v ∂ϕ ∂v ∂ϕ ∂z ∂ϕ ∂z ∂ϕ ∂w ∂ϕ ∂w ∂ϕ ∂x ∂ϕ ∂y ∂ϕ ∂v ∂ϕ ∂v ∂ϕ ∂z ∂ϕ ∂z ∂ϕ ∂w ∂ϕ ∂w = 1 z · − ϕ − z − ξ − ϕ u − u − w − ξ − ϕ − u x − w − ξ − ϕ − u x y x z x z − u x z − w u x z + w − ϕ · x z − ϕ − y y u z x u z − w − u u z + z u z − ϕ · u z − ϕ − y y u z x u z − w − u z u u z + z − ϕ · u z − ϕ − y z ∂ϕ ∂x z ∂ϕ ∂y z ∂ϕ ∂v z ∂ϕ ∂v z ∂ϕ ∂z z ∂ϕ ∂z z ∂ϕ ∂w z ∂ϕ ∂w where the components of the last row are ∂ϕ ∂x = 1 z ( − x + − x z + u y − u w z ) = − ξ y z ,∂ϕ ∂y = − x ξ z , ∂ϕ ∂v = u ξ z , ∂ϕ ∂v = − u ξ z ,∂ϕ ∂z = − ϕ z + x x − x x + u u − u u z ,∂ϕ ∂z = z z , ∂ϕ ∂w = w z , ∂ϕ ∂w = − w z . Hence det B = − x z ×× det − z − ξ u − u − w − ξ − u x − w − ξ − u x y x x − u x − w z u x + z w − y z y u x u − z w − u u + z z u − z y y u x u − z w − u u u + z z − z y − ξ y − ξ x ξ u − ξ u z z z w − z w . (4.28) Proposition 4.5. det B = x z . Proof.
We explain step by step the calculation of the determinant by the standard way.(1) We make the (7 , j )-components of the above matrix (4.28) for j = 2 , . . . − z − ξ − z x u − z w x − u + w z x − w − ξ − u − z w x x − w w x w x − w − ξ − u − z w x − w x x + w w x y x x − u x − w z u x + z w − y z + z y x x w y x x − w y x x y u x u − z w − u u + z z u z y u x − z y + w y u x − w y u x y u x u − z w − u u u + z z z y u x w y u x − z y − w y u x − x . Then it is enough to consider the matrix − ξ − z x u − z w x − u + w z x − ξ − u − z w x x − w w x w x − ξ − u − z w x − w x x + w w x x − u x − w z u x + z w − y z + z y x x w y x x − w y x x x u − z w − u u + z z u z y u x − z y + w y u x − w y u x x u − z w − u u u + z z z y u x w y u x − z y − w y u x . We put this matrix by (cid:0) µ i,j (cid:1) .(2) We make the components µ ,i , i = 1 , ,
3, to be zero, then the component µ , is µ , = − ( x z ) ξ x . (3) Likewise when we make the components µ ,i , i = 1 , ,
3, to be zero, then fortunately, the(5 , z y u x − ( x u − z w ) − ξ · − z x − ( − u u + z z ) − ξ · − ( u x + z w ) x − u − ξ · ( − u x + z y x ) x = 0 . Hence the (5 , µ , + ( x u − z w ) ξ · ( u x − z w ) x − ( − u u + z z ) − ξ · ( x x − w w ) x − u − ξ · − w x = 1 x ξ {− z y x ξ + w y u ξ + ( x u − z w )( u x − z w )+( − u u + z z )( x x − w w ) − u w (cid:9) = − ( z x ) x ξ . (4) Again fortunately the (6 , µ , + ( x u − z w ) ξ · ( x u − z w ) x − − u − ξ · ( x x − w w ) x − ( u u + z z ) − ξ · − w x = 1 ξ x (cid:8) w y u ξ + ( x u − z w )( x u − z w ) − u ( x x − w w ) − ( u u + z z ) w (cid:9) = 0 . (5) Then finally the (6 , µ , + ( x u − z w ) ξ · ( − x u + w z ) x − − u − ξ · w x − ( u u + z z ) − ξ · ( x x + w w ) x = − ( z x ) x ξ . These data gives us the result(4.29) det B = x z on O z T O x . (cid:3) From this we have (cid:0) d ( P x ◦ G z ) (cid:1) ∗ (cid:16) x dx ∧ dx ∧ dx ∧ du ∧ du ∧ du ∧ du (4.30) ∧ dy ∧ dy ∧ dv ∧ dv ∧ dz ∧ dz ∧ dw ∧ dw ∧ dξ (cid:17) = 1 z dx ∧ dx ∧ du ∧ du ∧ dy ∧ dy ∧ dv ∧ dv ∧ dz ∧ dz ∧ dz ∧ dw ∧ dw ∧ dw ∧ dw ∧ dξ . ARGMANN TYPE TRANSFORMATION 17
The result (4.29) above suggests that the Jacobian of any coordinate transformation P a j ◦ G a i ( a i and a j are one of the coordinates in ( z , z , · · · , x , · · · , u )) will be given in a similar form, that is Proposition 4.6.
Let O a i and O a j be any of two open coordinate neighborhoods in U . Then theJacobian J a j ,a i = det d (cid:0) P a j ◦ G a i (cid:1) of the coordinate transformation P a j ◦ G a i is given by (4.31) J a j ,a i = (cid:18) a j a i (cid:19) on O a j ∩ O a i where a i , a j are the coordinates in { z , · · · , x , · · · , u } indicating the open coordinate neighborhoods O a j and O a i , respectively.Proof. Let σ = (cid:18) (cid:19) . Then the transformation σ : SL (2 , C ) → SL (2 , C ) , S → σ S σ induces an automorphism on J (3) C leaving X O invariant. Also the operation T ( ∗ ) taking transposedmatrix S → T S in SL (2 , C ) induces an automorphism of J (3) C leaving X O .Owing these two automorphisms of X O , it is enough to calculate the Jacobian for the three casesof O z ∩ O z , O w ∩ O z and O x ∩ O z , since other cases are determined by replacing the coordinatesthrough these two transformations and their compositions. The last assertion is guaranteed by thefact that all the intersections, like O z i ∩ O x j are connected, so that by the analytic continuation theJacobian can be determined from the intersection of three open coordinate neighborhood, like the caseof the example: O z i ∩ O x j ∩ O w k ⊂ O z i ∩ O x j .Hence it is enough to determine the Jacobian for the remaining two cases: [J1]-case: J z ,z and[J2]-case: J w ,z .[J1]-case: J z ,z . For this case we consider the coordinate transformation P z ◦ G z , which isgiven by the correspondence:( x , x , u , u , y , y , v , v , z , z , z , w , w , w , w , ξ ) ( x , x , u , u , y , y , v , v , z , z , z , w , w , w , w , ξ ) , where the coordinates ( x , x , u , u , y , y , v , v , z , z , z , w , w , w , w , ξ ) are given by the rationalfunctions: x = ξ y − z x + ( u w − u w ) z ,x = − ξ y − z x + ( u w − u w ) z u = u ,u = u ,y = y ,y = y ,v = − ξ u + v z − w y − w y z ,v = − ξ u + v z − w y − w y z ,z = z ,z = z ,z = − ξ + z z − w w + w w − x x + x x − u u + u u z ,w = w , w = w ,w = w ,w = w ,ξ = ξ . We change the orderings of the coordinates on O z as( x , x , u , u , y , y , v , v , z , z , z , w , w , w , w , ξ ) ( u , u , y , y , z , z , w , w , w , w , ξ , x , x , v , v , z )and O z as ( x , x , u , u , y , y , v , v , z , z , z , w , w , w , w , ξ ) ( u , u , y , y , z , z , w , w , w , w , ξ , x , x , v , v , z ) . Then the Jacobi matrix is of the form that(4.32) (cid:18) Id , C D (cid:19) , where Id is 11 ×
11 identity matrix, 0 , is 11 × D is given by(4.33) D = − z z − z z ∗ ∗ z z − u u z u z ∗ ∗ − u z z z + u u z ∗ ∗ z z (the 5 ×
11 matrix C and components ∗ are given by some functions). Hence the Jacobian J z ,z is J z ,z = det D = (cid:18) z z (cid:19) . [J2]-case: J w ,z . For this case we consider the coordinate transformation P w ◦ G z , which isgiven by the correspondence:( x , x , u , u , y , y , v , v , z , z , z , w , w , w , w , ξ ) ( x , x , u , u , y , y , v , v , z , z , z , z , w , w , w , ξ ) , where the coordinates ( x , x , u , u , y , y , v , v , z , z , z , z , w , w , w , ξ ) are given by the rationalfunctions: x = + ξ y − z x + ( u w − u w z ,x = x ,u = − ξ v − u z − ( w x − w x z ,u = u ,y = − ξ x − y z + ( − w v + w v z ,y = y ,v = v , ARGMANN TYPE TRANSFORMATION 19 v = − ξ u + v z − w y − w y z ,z = z ,z = z ,z = z ,z = − ξ + z z − w w + w w − x x + x x − u u + u u z ,w = w ,w = w ,w = w ,ξ = ξ . We change the orderings of the coordinates on O z as( x , x , u , u , y , y , v , v , z , z , z , w , w , w , w , ξ ) ( x , u , y , v , z , z , z , w , w , w , ξ , x , u , y , v , w )and O w as ( x , x , u , u , y , y , v , v , z , z , z , z , w , w , w , ξ ) ( x , u , y , v , z , z , z , w , w , w , ξ , x , u , y , v , z ) . Then the Jacobi matrix is of the form that(4.34) (cid:18) Id , C ′ D ′ (cid:19) , where the matrix D ′ is given by(4.35) D ′ = w z − w z ∗ ∗ ∗ x z u x z + w z ∗ ∗ u x z − w z u z ∗ ∗ − w w z . (the matrix C ′ and components ∗ are given by some functions) Hence the Jacobian J w ,z is J w ,z = det D ′ = (cid:18) w z (cid:19) . (cid:3) From the above Propositions 4.3, 4.5 and 4.6 we can see that the 1-cocycle defined by { J a j ,a i } a i ,a j ∈{ z , ··· , ··· ,u } is the form of a coboundary of 0-cochain { h i = a i } , we have Theorem 4.7.
The set of holomorphic sections (cid:26) h z i = 1 z i , h w i = 1 w i , h y i = 1 y i , h v i = 1 v i , h x i = 1 x i , h u i = 1 u i (cid:27) , each function is defined on the open coordinate neighborhood O z i , O w i and so on, together define anowhere vanishing global holomorphic section Ω O of the canonical line bundle V T ∗ ′ ( X O ) C .Here the above local sections, for example h z defined on O z , should be understood as the coefficientof a -degree (= highest degree) holomorphic differential form: h z ( x , x , u , u , y ,y , v , v , z , z , z , w , w , w , w , ξ ) = 1 z dx ∧ dx ∧ du ∧ du ∧ dy ∧ dy ∧ dv ∧ dv ∧ dz ∧ dz ∧ dz ∧ dw ∧ dw ∧ dw ∧ dw ∧ dξ . Next, we show that the nowhere vanishing global holomorphic section Ω O coincides with the re-striction of a smooth 16-form e Ω O defined on J (3) C .For each component z i , w i , y i , v i , x i , u i (totally 24 components) there is an open coordinate neighbor-hood O z i , O w i , etc. Among 27 components we choose 16 coordinates on each open set above by whichthe set X O is described as a graph and the Jacobians are given in (4.31). Let { z α i , w β j , y γ k , v δ ℓ , x ǫ m , u τ n , ξ o , ξ o } where the number of each components depends on the point z i , w i , etc.We always line up in this order of the coordinates and denote a smooth 16-form by(4.36) σ z i := z i dz α ∧ · · · dξ o ∧ ξ o . Put(4.37) e Ω O := 1 w ( A ) · X ( − i ∗ σ z i + etc, where w ( A ) is defined by w ( A ) = X | z i | + | w i | + | y i | + · · · . Then by (4.31) it will be apparent that on each open coordinate neighborhood f Ω O | O zi = 1 z i dz α i ∧ · · · ∧ dx i . Let n = X z i dz γ ′ i ∧ · · · ∧ dξ o ′ where the coordinates appearing are the complement for defining the forms σ z i , then Proposition 4.8. e Ω O ∧ n = dz ∧ · · · ∧ dξ . We mention that since the transition function of the canonical line bundle on X O is invariant underthe multiplication by non-zero complex numbers, it is a pull-back of a complex line bundle on thequotient space C ∗ \ X O . More precisely Proposition 4.9. (1)
Interpreting the calculations above in terms of the homogeneous coordinates wesee that the canonical line bundle K C ∗ \ X O of the quotient space C ∗ \ X O , K C ∗ \ X O := V T ∗ ′ ( C ∗ \ X O ) ⊗ C ,is isomorphic to ⊗ L ∗ (cid:12)(cid:12) C ∗ \ X O , where L is the tautological line bundle on the projective space P C , L ⊂ P C × C . (2) Also let V be the kernel of the projection map π : X O −→ C ∗ \ X O , V := ker dπ ⊂ T ( X O ) , which can be seen naturally as a complex line bundle trivialized by the holomorphic vector field corre-sponding to the dilation action X O ∋ A t · A ∈ X O . In this sense we denote it by V C . Then by the exact sequence { } −→ π ∗ ( T ∗ ′ ( C ∗ \ X O ) ⊗ C ) −→ T ∗ ′ ( X O ) ⊗ C −→ V C ∗ −→ { } we know that the canonical line bundle K X O ∼ = π ∗ ( K C ∗ \ X O ) ⊗ V C ∗ is holomorphically trivial, since π ∗ ( L ) is holomorphically trivial. ARGMANN TYPE TRANSFORMATION 21 Symplectic manifolds and polarizations
In this section we review an aspect of a geometric quantization theory in a restricted frameworkfitting our purpose and do not aim a development of a general theory. In the subsections § . § . § Integral symplectic manifold.
Let (
M, ω M ) be a symplectic manifold with the symplecticform ω M . In this paper we assume that[In1] the map H ( M, Z ) → H dR ( M, R ) is injective, or the group H ( M, Z ) has no torsion and,[In2] the de Rham cohomology class [ ω M ] of the symplectic form ω M is in this image.Then the complex line bundle L ∈ H ( M, C ∗ ) ∼ = H ( M, Z ) corresponding to the cohomology class[ ω M ] is unique(of course, up to isomorphism). The first condition is satisfied, for example if M is simplyconnected and our case M = X O satisfies both of these conditions trivially, since H ( X O , Z ) = { } .Under these assumptions, that is, [ ω M ] is integral and the map H ( X O , Z ) → H dR ( X O , R ) is in-jective, the unique complex line bundle L = L ω M has a canonically defined connection ∇ , which isdefined as follows:Let { U i } be an open covering of M consisting of contractive open subsets U i such that the intersec-tions U i ∩ U j are also contractive. Then there are one-forms { f i } , each of which is defined on U i andis satisfying df i = ω M . Then we have collections of smooth functions { c ij } defined on U i ∩ U j suchthat dc ij = f j − f i on U i ∩ U j . By the assumption that c jk − c ik + c ij takes integers on U i ∩ U j ∩ U k (if it is non-empty), the line bundle π : L → M is constructed from the collection of the sets { U i × C } by patching them by the transition functions g ij = e π √− c ij .The connection ∇ is defined as ∇ X ( s i ) = 2 π √− < f i , X > s i on U i , ( X is a vector field)where s i is the nowhere vanishing section on U i identifying U i × C and π − ( U i ) ⊂ L such that U i × C ∋ ( x, z ) z · s i ( x ) ∈ π − ( U i ) , here < f i , X > denotes the pairing of an one-form f i and a tangent vector X . In fact, since s j = g ij s i , by the definition ∇ X ( s j ) = 2 π √− < f j , X > s j = 2 π √− < f j , X > g ij s i and also it must satisfy ∇ X ( s j ) = X ( g ij ) s i + 2 π √− g ij < f i , X > s i . Then by the relation dc ij = f j − f i we have2 π √− (cid:0) < dc ij + f i , X > (cid:1) = 2 π √− < f j , X > for arbitrary vector field X, which shows that the connection ∇ is well-defined.If we choose all the functions c ij being real valued, we may regard that the line bundle L = L ω M is equipped with an Hermitian inner product, which we denote by < · , · > L x at x ∈ M . Hereafter weassume that the line bundle L is equipped with such an Hermitian inner product.We regard that the space C ∞ ( M ) is a Lie algebra by the Poisson bracket { f, g } = ω M ( H f , H g ),where H f denotes the Hamilton vector field with the Hamiltonian f defined by < df, • > = ω M ( H f , • ).The space Γ( L , M ) is a central object in the quantization theory. There is a basic fact that thecorrespondence from g ∈ C ∞ ( M ) to the operator T g , assignment of a function to an operator, T g : Γ( L , M ) ∋ s H g ( s ) + 2 π √− g · s is a Lie algebra homomorphism, [ T g , T h ] = T { g , h } , and it is the main theme in the quantization theoryhow to assign a function on a phase space to an operator. Real and complex polarizations.
Let (
M, ω M ) be a symplectic manifold with the symplecticform ω M (dim M = 2 n ). The skew-symmetric bi-linear form ω Mp : T p ( M ) × T p ( M ) → R at eachpoint p ∈ M is naturally extended to the complexification ω Mp : T p ( M ) ⊗ C × T p ( M ) ⊗ C → C as theskew-symmetric complex bi-linear form which we denote with the same notation.Let F be a subbundle of the complex fiber dimension n in the complexified tangent bundle T ( M ) ⊗ C := T ( M ) C satisfying the properties that(1) F is maximal isotropic with respect to the skew-symmetric bi-linear form ω M , (2) F is integral, that is F ∩ F has constant rank and F, F + F is closedunder bracket operation of vector fields taking values in these subbundles . In this paper we only treat two extreme cases,( P F = F , and( P F + F = T ( M ) C . First one is the complexification of a Lagrangian foliation L ⊂ T ( M ), F = L ⊗ C and we call it a realpolarization. The second case is called a complex polarization.If there is a polarization satisfying the second condition F + F = T ( M ) C , then M has a almostcomplex structure J and the subbundle F is identified with (0 , T ( M ) C (anti-complexsubbundle). The integrability condition implies that M becomes a complex manifold. When we put g ( α, β ) := ω M ( J ( α ) , β ) , α, β vector fields on M , then g is a non-singular symmetric bi-linear form on T ( M ) and moreover it defines an Hermitian formon T ( M ) C . Under the condition that the form g is positive definite, then it is equivalent that M hasa K¨ahler structure. This is equivalent to the condition that −√− ω M ( ξ, ξ ) ≥ ξ ∈ Γ( F ) . We call such a polarization a positive polarization .Hence it is equivalent that if there is a positive complex polarization on the symplectic manifold M , then M is a K¨ahler manifold and the symplectic form ω M is a K¨ahler form.Also real polarization is always positive. In this paper, we consider two polarizations on the space X O , one is the real polarization F definedas the complexification of the kernel of the natural projection map q ◦ ( τ O ) − : X O → P O and aK¨ahler polarization (= positive complex polarization) G described in (3.7) and Theorem (3.2) . Hilbert space structure on the spaces of polarized sections.
Now let M be a symplecticmanifold satisfying the conditions [In1] and [In2] in the subsection § L = L ω M corresponding to the cohomology class [ ω M ] with the connection ∇ and the Hermitian inner productexplained in the above subsections and assume that there is a polarization F on M .Let U be an open subset in M . We introduce a space C F ( U ) ⊂ C ∞ ( U ) by C F ( U ) = { h ∈ C ∞ ( U ) | X ( h ) = 0 , ∀ X ∈ Γ( F, U ) , vector fields taking values in F } and a subspace Γ F ( L , U ) of smooth sections in Γ( L , U ) byΓ F ( L , U ) = { s ∈ Γ( L , U ) | ∇ X ( s ) = 0 , ∀ X ∈ Γ( F, U ) } . Let U be a contractive open subset which is small enough such that there is an one-form θ on U satisfying dθ = ω M , and < θ, X > = 0 for vectors X ∈ F .
Although it is not canonical, we may locally identify the spaces C F ( U ) and Γ F ( L , U ) by fixing anowhere vanishing section s : U → L with the property that ∇ X ( s ) = 0 for X ∈ F in such a way that C F ( U ) ∋ ϕ ϕ · s ∈ Γ F ( L , U ) . ARGMANN TYPE TRANSFORMATION 23
Then under this identification, the connection ∇ is ∇ X ( ϕ · s ) = X ( ϕ ) · s + 2 π √− ϕ < θ, X > · s = X ( ϕ ) · s, for vector field X taking values in F .If F is a real polarization, then the function space C F ( U ) consists of such functions that are constantalong each leaf ∩ U of the Lagrangian foliation, and if F is a complex polarization, then C F ( U ) consistsof holomorphic functions on U .We call these sections ∈ Γ F ( L , U ) “ polarized sections ” (with respect to a polarization F ) and arethe main objects in the geometric quantization theory. We may regard, according to the polarization,that they express quantum states in the real polarization case and that they express good classicalobservables in the complex polarization. The above identification indicates the local nature of thepolarized sections according to the polarization. One basic problem is to introduce an inner product (= a sesqui-linear form) on the space Γ F ( L , M ) of L -valued polarized sections and a related space (which will be explained later) in a reasonable way(or without additional assumptions) to make it a (pre-)Hilbert space and the most interesting problemis to see a transformation from one space of polarized sections Γ G ( L , M ) to another space Γ F ( L , M ) of polarized sections by another polarization. We discuss two cases according to the polarizations (real and positive complex) how we introducean inner product below in [RP] (real polarization) and in [CP] (complex polarization). In the nextsubsection § / F be a real polarization . We understand that it is a Lagrangian foliation in T ( M ). Inthis case for avoiding unnecessary generality, we still put three assumptions on the symplectic manifold M and the Lagrangian foliation F such that(RP1) the space of leaves is a smooth manifold with the half dimension of the symplecticmanifold M , or we assume there is a submersionΦ : M −→ N whose fibers are Lagrangian submanifolds so that N is the space of leaves and dim N = dim M ,(RP2) leaves are always connected and,(RP3) the manifold N of the leaves is orientable.So, the real polarization F is defined as the kernel F = Ker d Φ of a surjective submersion Φ : M −→ N and the functions in C F ( M ) are naturally descended to the base space N , that is Φ ∗ ( C ∞ ( N )) = C F ( M ).Let α, β ∈ Γ F ( L , M ), then by the equality0 = < ∇ X ( α ) , β > L + < α, ∇ X ( β ) > L = X ( < α, β > L ) , for X ∈ F , the function < α , β > L is constant on each fiber. Hence it can be naturally identified with a functionon the base manifold N . For such functions we need not integrate along the leaves and it will beenough to consider the integration to the transversal direction of the leaves. This is realized by theintegration on the base space to make the space Γ F ( L , M ) into a (pre) Hilbert space. There are manyway to choose a measure on N , say a Riemann volume form to integrate it.Instead of the space Γ F ( L , M ), we consider L -valued polarized (or exactly to say, we call horizontaland partial) half-densities ϕ ∈ Γ F (cid:16) L ⊗ (cid:12)(cid:12) max V F (cid:12)(cid:12)(cid:17) and/or horizontal and partial -densities ϕ ∈ Γ F (cid:16) L ⊗ r(cid:12)(cid:12) max V F (cid:12)(cid:12)(cid:17) , where F is the annihilator of F , F = { ξ ∈ T ∗ ( M ) | < ξ, X > = 0 , ∀ X ∈ F } . We introduce a (partial) connection ∇ / X ( ξ ) := i X ( dξ ) on max V F = max V ( d Φ) ∗ (Φ ∗ ( T ∗ ( N ))), where ξ is a differential form ∈ Γ (cid:16) max V F , M (cid:17) , X ∈ F and i X denotes the interior product with a tangentvector X ∈ F .Note that i X ( dξ ) = i X ◦ dξ ∈ Γ (cid:16) max ^ F , M (cid:17) , for X ∈ F and ξ ∈ Γ (cid:16) max ^ F , M (cid:17) . Since i X ( ξ ) = 0 for ξ ∈ Γ (cid:16) max V F , M (cid:17) by X ∈ F ∇ / X ( f · ξ ) = i X ◦ d ( f · ξ )= i X ◦ ( df ∧ ξ + f · dξ ) = X ( f ) · ξ − df ∧ i X ( ξ ) + f · i X ( dξ )= X ( f ) ξ + f · ∇ / X ( f · ξ ) , for X ∈ F and f ∈ C ∞ ( M ) , the vector fields taking values in F work as a differentiation on the space of the differential formsΓ (cid:16) max V F , M (cid:17) . Hence we can consider the differentiation ∇ / X along the polarization F for thesections ∈ Γ (cid:16) max V F , M (cid:17) and also sections ∈ Γ (cid:16)(cid:12)(cid:12) max V F (cid:12)(cid:12) , M (cid:17) too.Then under our assumptions (RP1) ∼ (RP3) and according to the definition of the partial connec-tion, the sections ξ ∈ Γ F (cid:16) max V F , M (cid:17) can be descended to the sections ∈ Γ( max V T ∗ ( N ) , N ), hence itholds(5.1) Φ ∗ Γ (cid:16) max ^ T ∗ ( N ) , N (cid:17)! ∼ = Γ F (cid:16) max ^ F , M (cid:17) . We may regard a differential form in Γ F (cid:16) max V F , M (cid:17) a polarized (or horizontal) “partial” half density(or half degree form) on M . Remark 4.
By our assumptions (RP1) ∼ (RP3) , there is an exact sequence (5.2) { } −→ F −→ T ∗ ( M ) −→ F ∗ −→ { } , and the injective bundle map on M , ( d Φ) ∗ : Φ ∗ ( T ∗ ( N )) → T ∗ ( M ) , which is the dual of the differential d Φ . Since the polarization F coincides with the vertical subbundle of the projection map Φ , the image ( d Φ) ∗ (Φ ∗ ( T ∗ ( N ))) = F .By the assumption (RP3) we regard that max V T ∗ ( N ) ∼ = | max V T ∗ ( N ) | (line bundles of the highestdegree differential form and density (volume form) line bundle) and we consider the square root bundle q max V F .Sections in Γ F ( q max V F , M ) or Γ F ( max V F , M ) are not the half densities or / -densities in thetrue sense of the definition, since max V T ∗ ( M ) is trivial by the Liouville volume form and so the linebundle max V F ∗ is the “inverse” of the line bundle max V F and vise versa. So we should call the sectionsin Γ F ( max V F , M ) or in Γ F ( q max V F , M ) polarized “partial” half density or “partial” / -density. Differential forms µ ∈ Γ F (cid:16) max V F , M (cid:17) is descended to densities µ ∗ ∈ Γ( max V T ∗ ( N ) , N ) (highestdegree differential form) on the base manifold N , that is there is a unique highest degree differentialform µ ∗ ∈ Γ( max V T ∗ ( N ) , N ) such that Φ ∗ ( µ ∗ ) = µ by the isomorphism (5.1), and then we can integrate µ ∗ on N . Hence we have a natural linear formI N : Γ F (cid:16) max ^ F , M (cid:17) ∋ µ I N ( µ ) := Z N µ ∗ ∈ C . ARGMANN TYPE TRANSFORMATION 25
If we denote the inverse map of Φ ∗ of (5.1) by Φ ∗ , then Z N µ ∗ = Z N Φ ∗ ( µ ) . In turn, we consider the square root bundle q max V F , which can be seen as a partial 1 / M . Then we can also introduce a partial connection ∇ / X / on the line bundle q max V F andas well it is defined also on the line bundle L ⊗ q max V F . Hence we consider “ L -valued polarized ( orhorizontal ) partial -densities ” α ⊗ η ∈ Γ F (cid:16) L ⊗ q max V F , M (cid:17) and define their product by makinguse of the Hermitian inner product on L with the formulaΓ F (cid:16) L ⊗ s max ^ F , M (cid:17) × Γ F (cid:16) L ⊗ s max ^ F , M (cid:17) −→ Γ F (cid:16) max ^ F , M (cid:17) ∈ ∈ ( α ⊗ µ, β ⊗ ν ) < α , β > L · µ ⊗ ν ∈ Γ F (cid:16) max ^ F , M (cid:17) . (5.3)The resulting horizontal partial half density < α , β > L · µ ⊗ ν ∈ Γ F ( max ∧ F , M ), is identified witha density on N . Hence we can define a pairing (or an inner product) for the sections in Γ F (cid:16) L ⊗ q max V F , M (cid:17) by the integration of the corresponding density on N in a natural way,Γ F (cid:16) L ⊗ s max ^ F , M (cid:17) × Γ F (cid:16) L ⊗ s max ^ F , M (cid:17) −→ C , ∈ ( a ⊗ µ, b ⊗ ν ) I N (Φ ∗ ( < a, b > L · µ ⊗ ν )) = Z N Φ ∗ ( < a, b > L · µ ⊗ ν ) . For the real polarization F on our space X O , first we trivialize the line bundle L by a nowherevanishing polarized section s ∈ Γ F ( L , X O ) with < s , s > L ≡ . We call this trivialization of the linebundle L a unitary trivialization.Next, we choose the Riemannian volume form dv P O on the Cayley projective plane P O and itspull-back { q ◦ ( τ O ) − } ∗ ( dv P O ) to the space X O . We consider its square root q { q ◦ ( τ O ) − } ∗ ( dv P O ) ∈ Γ F (cid:16)s max ^ F , X O (cid:17) , which coincides with { q ◦ ( τ O ) − } ∗ ( p dv P O ) . Then we identify the L -valued polarized partial / -density ξ ⊗ µ ∈ Γ F (cid:16) L ⊗ q max V F , X O (cid:17) with f µ ξ ⊗ p { q ◦ ( τ O ) − } ∗ ( dv P O ) , where f µ p { q ◦ ( τ O ) − } ∗ ( dv P O ) = µ and the function f µ is constant along the fiber of the map { q ◦ ( τ O ) − } .By the definition of the space Γ F (cid:16) L ⊗ q max V F , X O (cid:17) , it can be identified with a half density on N of the form h · p dv P O . Hence we identify the L -space with respect to the Riemann volume form dv P O (we denote it by L ( P O , dv P O )) and the space of L -valued polarized partial / -densities Γ F (cid:16) L ⊗ q max V F , X O (cid:17) ( after taking completion ).[CP] Let G be a positive complex polarization on M whose symplectic form ω M is expressed asa K¨ahler form: √− ∂ ∂φ = ω M . We install an Hermitian inner product on the line bundle L corresponding to the cohomology class [ ω M ]by fixing real valued functions { c ij } defined as dc ij = f j − f i where df i = ω M on each open subset U i and { U i } is a open covering of M consisting of contractive open sets together the intersections U i ∩ U j being contractive too.The inner product < a, b > L of two sections a, b ∈ Γ G ( L , M ) is a function on M and can be integratedwith respect to the Liouville volume form dV M := ( − n ( n − / n ! (cid:8) ω M (cid:9) n (dim M = 2 n ). Hence we canintroduce an inner product on the space Γ G ( L , M ) intrinsically, since we do not depend on any otheradditional assumptions.We can also introduce an inner product on the space of L -valued “polarized” sections of the canonicalline bundle K G for the complex polarization G .The canonical line bundle K G = max V T ∗ ′ ( M ) C is the line bundle of the highest degree exteriorproduct of the holomorphic part T ∗ ′ ( M ) C of the complexified cotangent bundle T ∗ ( M ) C ( (1 ,
0) typecotangent vectors ), which is the annihilator of the complex polarization G ( (0,1) tangent vectors),like F for the real polarization F . The sections of the canonical line bundle can be thought as halfdensities (or complex valued half density) by the isomorphism K G ⊗ K G = max V T ∗ ( M ) C . We canintroduce a partial connection ∇ / X ( X ∈ G ) along the complex polarization G in the similar way asfor the real polarization. Then we consider the space Γ G ( L ⊗ K G , M ) of “ L -valued polarized sectionsof the canonical line bundle” and using the Hermitian inner product on L we have a highest degreedifferential form < a ⊗ µ, b ⊗ ν > = < a, b > L · µ ∧ ν ∈ Γ (cid:16) max ^ T ∗ ( M ) C , M (cid:17) , where a, b ∈ Γ G ( L , M ) and µ, ν ∈ Γ G ( K G , M ). The quantity µ ∧ ν can be seen as a (complex valued)density on M . Hence we have an intrinsic (pre-)Hilbert space structure on the space Γ G ( L ⊗ K G , M ). For the complex polarization G on our space X O , we use a structure so called Calabi-Yau structureon X O to identify the space Γ G ( L ⊗ K G , X O ) and the space C G ( X O ) of holomorphic functions on X O by the correspondence (5.4) γ : C G ( X O ) ∋ h γ ( h ) = h · t ⊗ Ω O ∈ Γ G ( L ⊗ K G , X O ) . The existence of the nowhere vanishing holomorphic -form Ω O on X O was proved in Proposition (4.7) and t is taken for trivializing the line bundle L satisfying the property ∇ / X ( t ) = 0 , where thepartial covariant differential ∇ / X is taken with respect to the complex polarization G .We call a trivialization of the line bundle L by the section t a holomorphic trivialization. We willdetermine the relation of the sections s and t , t = g s in the subsection § Pairing of polarizations and a Bargmann type transformation.
First, we recall the fiberintegration. Let φ : M → N be a differentiable map between two manifolds.Let σ ∈ Γ (cid:16) p V T ∗ ( M ) , M (cid:17) be a differential form with the degree p ≥ dim M − dim N := d . For g ∈ Γ (cid:16) q V T ∗ ( N ) , N (cid:17) with compact support satisfying q = m − p = dim M − p ≥ ( ∗ , ∗ )). We assume Z M | σ ∧ φ ∗ ( g ) | < + ∞ for any such g ∈ Γ (cid:16) q V T ∗ ( N ) (cid:17) and define a linear functional g Z M σ ∧ φ ∗ ( g ) , which is understood as a distribution on the space Γ (cid:16) q V T ∗ ( N ) (cid:17) . We denote this distribution by φ ∗ ( σ ) and denote as(5.5) φ ∗ ( σ )( g ) := Z N φ ∗ ( σ ) ∧ g = Z M σ ∧ φ ∗ ( g ) . If φ is a submersion, then φ ∗ ( σ ) is a smooth differential form of degree p − d . ARGMANN TYPE TRANSFORMATION 27
In the last subsection we introduced inner products on the space Γ F (cid:18) L ⊗ q max V F , M (cid:19) andΓ G (cid:18) L ⊗ max V T ∗ ′ ( M ) C , M (cid:19) = Γ G (cid:0) L ⊗ K G , M (cid:1) for a real polarization F satisfying the conditions (RP1) ∼ (RP3) and a positive complex polarization G on an integral symplectic manifold M . Our main pur-pose is to construct a transformation(5.6) B : Γ G (cid:0) L ⊗ K G , M (cid:1) −→ Γ F (cid:16) L ⊗ s max ^ F , M (cid:17) or it may be understood as a sesqui-linear form onΓ F (cid:16) L ⊗ s max ^ F , M (cid:17) × Γ G (cid:16) L ⊗ K G , M (cid:17) Id × B −→ (5.7) Γ F (cid:16) L ⊗ s max ^ F , M (cid:17) × Γ F (cid:16) L ⊗ s max ^ F , M (cid:17) −→ C . (5.8)For the sections ( α ⊗ µ , β ⊗ ν ) ∈ Γ F (cid:16) L ⊗ q max V F , M (cid:17) ⊗ Γ G (cid:16) L ⊗ K G , M (cid:17) their product < α, β > L · | µ ⊗ ν | ( | ∗ | means a section of (cid:12)(cid:12) K G ⊗ q max V F (cid:12)(cid:12) ) is understood as a partial 3 / M and so we needsome modification to integrate it, since there are no manifold of the dimension 3 / × dim M .Since we identify the half density space Γ (cid:16)q max V T ∗ ( N ) , N (cid:17) with a L -space by fixing a Riemannvolume form dv N , we define a sequi-linear formΓ F (cid:16) L ⊗ s max ^ F , M (cid:17) × Γ G (cid:16) L ⊗ K G , M (cid:17) −→ C by Γ F (cid:16) L ⊗ s max ^ F , M (cid:17) × Γ G (cid:16) L ⊗ K G , M (cid:17) ∋ ( α ⊗ µ , β ⊗ ν )(5.9) Z M < α, β > L · Φ ∗ ( f µ dv N ) ∧ ν, where we can put µ = Φ ∗ ( f µ ) p Φ ∗ ( dv N ) with a function f µ ∈ C ∞ ( N ), that is we multiply thepartial 1 / p Φ ∗ ( dv N ) to the partial 1 / ν = Φ ∗ ( f ν ) p Φ ∗ ( dv N ), then p Φ ∗ ( dv N ) ⊗ p Φ ∗ ( dv N ) ⊗ µ is a (complex valued) highest degree differential form (or can be thought as a density)on M and we can define a sesqui-linear form.Once we have a sesqui-linear form P : Γ F (cid:16) L ⊗ s max ^ F , M (cid:17) × Γ G (cid:16) L ⊗ K G , M (cid:17) −→ C it is rewritten as P ( α ⊗ µ, β ⊗ ν ) = X I N (Φ ∗ < α, α i > L · µ ⊗ µ i ) , where we put B ( β ⊗ ν ) = P α i ⊗ µ i . In our space T ∗ ( P O ) τ O ∼ = X O we have two polarizations F (real) and G ( complex ) ) and we use aCalabi-Yau structure on X O to identify the space Γ G ( L ⊗ K G , X O ) with the space C G ( X O ) of holomorphicfunctions on X O : γ : C G ( X O ) ∋ h γ ( h ) = h · t ⊗ Ω O ∈ Γ G ( L ⊗ K G , X O ) . (5.10) The inner product on the space C G ( X O ) induced by the map γ will be explicitly described in (6.9) atthe end of § Γ G ( L ⊗ K G , X O ) .We recall the sections s and t and describe our Bargmann type transformation including thequantity < s , t > L .Let θ P O be the canonical one-form on the cotangent bundle T ∗ ( P O ) , then dθ P O = ω P O and forany X ∈ F , < θ P O , X > = 0 . So let s be a nowhere vanishing polarized ( with respect to the realpolarization F ) global section of L defining a trivialization X O × C ∼ = L by the correspondence (5.11) X O × C ∋ ( A, z ) ←→ z · s ( A ) ∈ L , with < s , s > L ≡ .Also by the relation τ O ∗ (cid:16) √− ∂∂ || A || / (cid:17) = ω P O , given in Theorem (3.2) , we take a ( complex ) one-form θ G = √− ∂ || A || / , then dτ O ∗ ( θ G ) = ω P O and θ G ( X ) = 0 for X ∈ G , since X is a (0 , tangent vector.Then we can trivialize the line bundle L by making use of a nowhere vanishing global section t ina similar way to (5.11) .Using the identifications (5.10) and the correspondence C ∞ ( P O ) ∋ g q ◦ τ O − } ∗ ( g ) · s · q { q ◦ τ O − } ∗ ( dv P O ) q ◦ τ O − } ∗ ( g ) · s · { q ◦ τ O − } ∗ ( dv P O ) the integral (5.9) is rewritten as (5.12) Z X O { q ◦ τ O − } ∗ ( g ) · h · < s , t > L ·{ q ◦ τ O − } ∗ ( dv P O ) ∧ Ω O , and it is also expressed in terms of the fiber integration as follows: Z X O { q ◦ τ O − } ∗ ( g ) · h · < s , t > L ·{ q ◦ τ O − } ∗ ( dv P O ) ∧ Ω O = Z P O g · { q ◦ τ O − } ∗ ( < s , t > L · h · Ω O ) dv P O . (5.13) Then the Bargmann type transformation B : C G ( X O ) → C ∞ ( P O ) , C G ∋ h B ( h ) , is defined as B ( h ) = { q ◦ τ O − } ∗ ( h · < t , s > L Ω O ) . (5.14) Hence we can express the integral (5.13) as Z P O g · B ( h ) dv P O . Remark 5.
The section s is free of U (1) -multiple and t is of free by a constant ∈ C ∗ . Bargmann type transformation
For expressing the Bargmann type transformation explicitly and to determine its L continuity, weneed to know the function < s , t > L = g , and relations of Ω O ∧ Ω O and { q ◦ τ O − } ∗ ( dv P O ) ∧ Ω O interms of Liouville volume form dV P O explicitly.In this section we determine them. ARGMANN TYPE TRANSFORMATION 29
Holomorphic trivialization and unitary trivialization.
The relation of the sections s nd t must be given by a function g = < s , t > L (6.1) t = g · s , and the function g satisfies an equation ∇ X ( t ) = 2 π √− < √− ∂ || A || / , X > g · s = ∇ X ( g s ) = X ( g ) s + 2 π √− g · < θ P O , X > s . Since F + G = F + G = T ( X O ) C , we have an equation for the function g :(6.2) 2 π √− · (cid:16) τ O ∗ (cid:16) √− √ ∂ || A || / (cid:17) − θ P O (cid:17) g = dg . We can put g = e π √− λ , and then the equation (6.2) reduces to the equation(6.3) dλ = τ O ∗ (cid:16) √ √− ∂ || A || / (cid:17) − θ P O . To get a solution λ we need to consider the real and imaginary parts in the formula √ √− ∂ || A || / separately.So, put τ O ∗ (cid:0) √ √− ∂ || A || / (cid:1) := a + √− b with real and imaginary parts of the one-form τ O ∗ (cid:0) √ √− ∂ || A || / ) on J (3) × J (3). Then d dλ = d (cid:0) τ O ∗ (cid:0) √ √− ∂ || A || / (cid:1) − θ P O (cid:1) = 0implies that there are real valued functions λ Re and λ Im such that a − θ P O = dλ Re , and dλ Im = b. The problem to solve the equation (6.3) reduces to find explicitly the functions λ Re and λ Im .Let ( X, Y ) ∈ T ∗ ( P O ) ⊂ J (3) × J (3). Here again we remark that we identify the cotangentspace T ∗ X ( P O ) and the tangent space T X ( P O ) by the Riemannian metric defined by ( Y , Y ) P O X :=tr ( Y ◦ Y ) for Y i ∈ T X ( P O ) ∼ = J (3). If we express Y i ∈ T X ( P O ) ⊂ J (3) , i = 1 , Y = ǫ u θ ( u ) θ ( u ) ǫ u u θ ( u ) ǫ , Y = η v θ ( v ) θ ( v ) η v v θ ( v ) η ,ǫ i , η i ∈ R , u i , v i ∈ O ∼ = R , the inner product in T X ( P O ) (see (3.3)) is expressed as(6.4) ( Y , Y ) P O X := tr ( Y ◦ Y ) = X ǫ i η i + 2 X ( u i , v i ) R . By the notation (
Y, dX ) for X = ξ x θ ( x ) θ ( x ) ξ x x θ ( x ) ξ ∈ J (3) , Y = ǫ u θ ( u ) θ ( u ) ǫ u u θ ( u ) ǫ ∈ J (3) , we mean the canonical one-form( Y, dX ) := X ǫ i dξ i + 2 X { u } i d { x } i + { u } i d { x } i + { u } i d { x } i on T ∗ ( J (3)) ∼ = J (3) × ( J (3)) ∗ ∼ = J (3) × ( J (3)), or its restriction to T ∗ P O , since in the inner productexpression (6.4) it can be understood as η i and { v k } i ( k = 1 , , , i = 0 , · · · ,
7) are replaced by thedifferentials dξ i and d { x k } i of the corresponding components in X ∈ J (3), respectively. Also for
A, B ∈ J (3) C we express the complex one form ( A, dB ) in the same way. Here we regardthat the pairing ( · , · ) J (3) C is the natural bi-linear extension of the inner product ( · , · ) P O X , so thatHermitian inner product < · , · > J (3) C on C ⊗ J (3) is given by < A , B > J (3) C = tr ( A ◦ B ) = ( A, B ) J (3) C . Now we recall the map τ O : τ O : J (3) × J (3) ⊃ T ∗ ( P O ) ∋ ( X, Y ) A := τ O ( X, Y ) ∈ C ⊗ R J (3) = J (3) C τ O ( X, Y ) = 1 ⊗ (cid:0) || Y || X − Y (cid:1) + √− ⊗ || Y || Y √ . Based on this expression of the matrix A = τ O ( X, Y ) ∈ J (3) C we have Proposition 6.1. ( see [Fu2]) 12 || Y || = || a || = || b || , || A || = || a || + || b || = || Y || , and ( da, a ) = || Y || ( Y, dY ) = ( db, b ) . In the expression τ ∗ ( dA, A ) = ( τ ∗ ( dA ) , τ ∗ ( A ) = ( a − √− b, da + √− db ) = d || A || = ( a, da ) + ( b, db ) + √− (cid:0) ( a, db ) − ( b, da ) (cid:1) , ( a, db ) − ( b, da ) = 2( db, a ) = 2 √ · (cid:8) || Y || ( dY, X ) − || Y || ( dY, Y ◦ Y ) (cid:9) and it is proved in [Fu2] (page 179) that ( dY, Y ◦ Y ) = 0 . Hence τ ∗ O ( √ √− ∂ || A || / ) − θ P O = √ √− || Y || {√− · ( dY, X ) − || Y || ( Y, dY ) − θ P O = − ( dY, X ) − ( Y, dX ) + √− √ || Y || ( Y, dY ) = √− √ d || Y || , since d ( X, Y ) = ( dX, Y ) + (
Y, dX ) = d tr( X ◦ Y ) = 0 for ( X, Y ) ∈ T ∗ ( P O ). Hence finally we maychoose the solutions λ Re and λ Im with λ Re ≡ λ Im = 1 √ || Y || . Hence
Proposition 6.2. g = e −√ π || Y || , or it is expressed on X O as g = e −√ π || A || / . Now we have B : C G ( X O ) → C ∞ ( P O ) , B ( h ) = { q ◦ ( τ O ) − } ∗ ( h · < t , s > L Ω O ) = { q ◦ ( τ O ) − } ∗ ( h · e −√ π || A || / Ω O ) . (6.5) Remark 6.
The solution λ Re can be an arbitrary real constant. However the absolute value | g | doesnot depend on the chosen constant λ Re . ARGMANN TYPE TRANSFORMATION 31
Fock-like space.
First, we show
Proposition 6.3.
The nowhere vanishing global holomorphic section Ω O of the canonical line bundle K G is F -invariant.Proof. Let α ∈ F . The action of α on X O is naturally defined from the action on P O and the actionis holomorphic. We denote it with the same notation α : X O → X O .We can put α ∗ (Ω O ) = K α · Ω O with a nowhere vanishing holomorphic function K α = K α ( A ).Then α ∗ (Ω O ) ^ α ∗ (Ω O ) = α ∗ (cid:0) Ω O ^ Ω O (cid:1) = | K α | · Ω O ^ Ω O . We can express Ω O ^ Ω O = D · { τ O − } ∗ (cid:16) ( ω P O ) (cid:17) by the Liouville volume form 116! ( ω P O ) and a function D = D ( A ) on X O . Hence α ∗ (cid:0) Ω O ^ Ω O (cid:1) = α ∗ ( D ) · { τ O − } ∗ (cid:16) ( ω P O ) (cid:17) , since the action by α on X O is symplectic. Hence α ∗ ( D ) = | K α | · D. By comparing the behaviours of Ω O and the Liouville volume form dV P O under the dilation actionby positive numbers: T t : X O → X O , A → t · A, we can see on the coordinate neighborhood O z T ∗ t (cid:0) Ω O ^ Ω O (cid:1) = 1( tz ) d ( tz ) ∧ · · · ∧ d ( tξ ) ^ tz ) d ( tz ) ∧ · · · ∧ d ( tξ )= t z dz ∧ · · · ∧ dξ ^ z dz ∧ · · · ∧ dξ = t · D ( A ) · { τ O − } ∗ (cid:16) ( ω P O ) (cid:17) = D ( t · A ) · t { τ O − } ∗ (cid:16) ( ω P O ) (cid:17) . Hence D ( t · A ) = t · D ( A ) . Note that the action T t on T ∗ ( P O ) defined via the map τ O is(6.6) τ O − ◦ T t ◦ τ O : T ∗ ( P O ) ∋ ( X, Y ) ( X, √ tY ) ∈ T ∗ ( P O ) . Then since || α ( A ) || = || A || α ∗ ( D )( A ) = D ( α ( A )) = D (cid:18) || α ( A ) || · α ( A ) || α ( A ) || (cid:19) = || A || · D (cid:18) α ( A ) || α ( A ) || (cid:19) = || A || · | K α ( A ) | D (cid:18) A || A || (cid:19) , hence(6.7) D (cid:18) α ( A ) || α ( A ) || (cid:19) = | K α ( A ) | D (cid:18) A || A || (cid:19) . This equality implies that the function K α is bounded on X O . Especially, if we consider it on thecoordinate open subset O z ∼ = C ∗ × C ( z = 0), then it can be extended to a holomorphic functionon C × C ⊃ O z and is bounded there. Hence the function K α is a constant function on the wholespace X O .Then by the property K α · β = K α · K β , α, β ∈ F ,F ∋ α K α is a one-dimensional representation of the compact simply connected group F , sothat we have not only | K α | ≡ α ∈ F , but also it must hold always K α ≡
1. This impliesΩ O is F -invariant. (cid:3) Corollary 6.4.
Since the action of F on S ( X O ) = { A ∈ X | || A || = 1 } is transitive, the function isof the form D ( A ) = C × || A || with the constant C = 2 . Especially we have (6.8) τ O ∗ (cid:0) Ω O ∧ Ω O (cid:1) ( X, Y ) = 2 || Y || (cid:16) ω P O (cid:17) . Proof.
It is enough to determine the constant C .The components of the the matrix A = η c ′ + c ′′ e θ ( b ′ + b ′′ e ) θ ( c ′ + c ′′ e ) η a ′ + a ′′ e b ′ + b ′′ e θ ( a ′ + a ′′ e ) η ∈ J (3) C is expressed in terms of the coordinates( x , · · · , x , u , · · · , u , y , · · · , y , v , · · · , v , z , · · · , z , w , · · · , w , ξ , ξ , ξ )= (cid:18)(cid:18) x x x x (cid:19) , (cid:18) u u u u (cid:19) , (cid:18) y y y y (cid:19) , (cid:18) v v v v (cid:19) , (cid:18) z z z z (cid:19) , (cid:18) w w w w (cid:19) , ξ , ξ , ξ (cid:19) by the correspondence ρ H ( c ′ ) + ρ H ( c ′′ ) e = (cid:18) z z z z (cid:19) + (cid:18) w w w w (cid:19) e ,ρ H ( b ′ ) + ρ H ( b ′′ ) e = (cid:18) y y y y (cid:19) + (cid:18) v v v v (cid:19) e ,ρ H ( a ′ ) + ρ H ( a ′′ ) e = (cid:18) x x x x (cid:19) + (cid:18) u u u u (cid:19) e as η i = ξ i , i = 1 , , ,c ′ + c ′′ e = z + z + z − z √− i + z − z j + z + z √− k + (cid:18) w + w + w − w √− i + w − w j + w + w √− k (cid:19) e b ′ + b ′′ e = y + y + y − y √− i + y − y j + y + y √− k + (cid:18) v + v + v − v √− i + v − v j + v + v √− k (cid:19) e a ′ + a ′′ e = x + x + x − x √− i + x − x j + x + x √− k + (cid:18) u + u + u − u √− i + u − u j + u + u √− k (cid:19) e . By a simple calculation we have || A || = X | η i | + 2 X | a ′ | + | a ′′ | + | b ′ | + | b ′′ | + | c ′ | + | c ′′ | = X i =1 | ξ i | + X i =1 | z i | + | w i | + | y i | + | v i | + | x i | + | u i | . So we express an element A = η c ′ + c ′′ e θ ( b ′ + b ′′ e ) θ ( c ′ + c ′′ e ) η a ′ + a ′′ e b ′ + b ′′ e θ ( a ′ + a ′′ e ) η ∈ ˜ O z ⊂ X O in terms ofthe coordinates( x , x , u , u , y , y , v , v , z , z , z , w , w , w , w , ξ ; x , x , u , u , y , y , v , v , z , ξ , ξ )= ( s , · · · · · · , s ; s , · · · , s ) , ARGMANN TYPE TRANSFORMATION 33 then || A || = s P i =1 | s i | , the first 16 coordinates give local coordinates on ˜ O z and the remaining coordi-nates ( s , · · · , s ) are rational functions of the coordinates ( s , · · · , s ), that is s j = s j ( s , · · · , s ), j ≥
17 (especially s = z and the explicit form of each s j for j >
16 is given in (4.17)).In particular, we see from the explicit form of these functions at the point A = A ( z )= A (0 , · · · , z , · · · , , ; 0 , · · · , s j ( A ( z )) = s j (0 , · · · , , z , , · · · ,
0) = s j (0 , · · · , , s , , · · · ,
0) = 0 , ≤ j ≤ , and for i ≤ j ≥ ∂s j ∂s i ( A ( z )) = 0 . On O z it holdsΩ O ^ Ω O = 1 | z | dx ∧ dx ∧ · · · ∧ dw ∧ dξ ^ dx ∧ dx ∧ · · · ∧ dw ∧ dξ = 1 | s | ds ∧ · · · ∧ ds ^ ds ∧ · · · ∧ ds = D ( A ) · { τ O − } ∗ (cid:16) ( ω P O ) (cid:17) . We calculate the right hand side at the point A ( z ) = (0 , · · · , , z , , . . . ; 0 , · · · , ∈ O z using theexpression of ω P O (see Theorem 3.2): ω P O = { τ O } ∗ (cid:16) √− ∂∂ || A || / (cid:17) . Then ∂∂ || A || / = 14 · ∂ X i =1 | s i | ! − / · X s i ds i and ∂ X i =1 | s i | ! − / · X s i ds i = − X i =1 | s i | ! − / X j =1 s j ds j ^ X i =1 s i ds i + 14 X i =1 | s i | ! − / X i =1 ds i ∧ ds i . Here we evaluate it at the point A ( z ), then it is given by − | s | − / · | s | ds ∧ ds + 14 | s | − / X i =1 ds i ∧ ds i = 116 | s | − / ds ∧ ds + 14 | s | − / X ≤ i ≤ ≤ i ≤ ds i ∧ ds . Hence ( ω P O ) = (cid:16) √− ∂∂ || A || / (cid:17) = 16! · · ( √− · | s | / × ds ∧ ds ∧ · · · · · · ∧ ds ∧ ds = 16! · · | s | ds ∧ ds ∧ · · · · · · ∧ ds ∧ ds at the point A ( z ) = A (0 , · · · , , s , , · · · ,
0; 0 , · · · ,
0) ( s = z ). Consequently we haveΩ O ∧ Ω O (cid:12)(cid:12) A ( z ) = Ω O ∧ Ω O (cid:12)(cid:12) A ( z ) = D (cid:18) s | s | (cid:19) · | s | · | s | − ds ∧ ds ∧ · · · · · · ∧ ds ∧ ds
164 KURANDO BABA AND KENRO FURUTANI and the constant C is C = 2 , D ( A ) = 2 || A || , Ω O ^ Ω O = 2 || A || { τ O − } ∗ (cid:16) ( ω P O ) (cid:17) . (cid:3) In the following, we denote the space consisting of the restrictions of polynomials C [ J (3) C ] ∼ = C [ x , · · · , w , ξ , ξ , ξ ] (= P P k [ x , · · · , ξ ] := P P k : sum of homogeneous polynomials) of 27 com-plex variables ( x i , u i , y i , v i , z i , w i , ξ , ξ , ξ ) ( i = 1, · · · ,4, see § X O ) to the subspace X O is denoted by P P k [ X O ] and is identified by the correspon-dence γ : X O ∋ p γ ( p ) = p · t ⊗ Ω O ∈ Γ G ( L ⊗ K G , X O )with a subspace consisting of L -valued polarized sections Γ G ( L ⊗ K G , X O ).We define a parameter family of inner products { ( ∗ , ∗ ) ε } ε ∈ R on the space Γ G ( L ⊗ K G , X O ) in thefollowing way thatΓ G ( L ⊗ K G , X O ) × Γ G ( L ⊗ K G , X O ) ∋ ( h · t ⊗ Ω O , g · t ⊗ Ω O ) Z X O h · g < t , t > L ·|| A || ε · Ω O ^ Ω O = 2 Z X O h · g · e − √ π || A || / · || A || ε · { τ O − } ∗ ( dV P O )(6.9) = ( h , g ) ε . (6.10) Remark 7.
According to the value of ε , the integral (6.9) for functions f, g ∈ P k [ X O ] need not befinite. In fact, for k > − − ε/ the integral (6.9) converges. We denote by F ε the completion ofthe space P k > − − ε/ P k [ X O ] with respect to the inner product (6.10) and the the remaining finitedimensional space P k ≤− − ε/ P [ X O ] with a suitable inner product. Pairing with the Riemann volume form
Let dv P O be the Riemann volume form on P O . The purpose in this section is to determine thepairing { q ◦ τ O − } ∗ ( dv P O )( A ) ^ Ω ( A )= C RC ( A ) · { τ O − } ∗ (cid:18) (cid:16) ω P O (cid:17) (cid:19) ( A ) , A = τ O ( X, Y ) ∈ τ O ( T ∗ ( P O )) = X O (7.1)explicitly. Since both of dv P O and Ω are invariant under the action by F and also the action of F on S ( X O ) is transitive, the function C RC must has the form that C RC ( A ) = C RC ( A/ || A || ) · || A || = C RC ( X, Y ) = C RC ( Y / || Y || ) · || Y || and the function C RC ( A/ || A || ) is equal to a constant 2 (Corollary7.7).The homogeneity order is determined by comparing their orders in the both sides (see the relation(6.6)).7.1. A local coordinates.
For the determination of the constant C RC ( A/ || A || ) we choose a localcoordinates around the point X = ∈ P O .The condition X = X = t c θ ( b ) θ ( b ) t ab θ ( a ) t for X ∈ P O ⊂ J (3) is expressed as(7.2) ( t + t ) a + θ ( bc ) = a, ( t + t ) b + θ ( ca ) = b, ( t + t ) c + θ ( ab ) = c, t + cθ ( c ) + θ ( b ) b = t ,t + θ ( c ) c + aθ ( a ) = t , t + θ ( a ) a + bθ ( b ) = t andtr X = t + t + t = 1 . ARGMANN TYPE TRANSFORMATION 35 where a, b, c ∈ O , t i ∈ R . The first 6 conditions are rewritten in the forms of(7.3) t a = θ ( bc ) , t b = θ ( ca ) , t c = θ ( ab ) , ( t − / + cθ ( c ) + θ ( b ) b = ( t − / + | c | + | b | = 1 / , ( t − / + θ ( c ) c + aθ ( a ) = ( t − / + | c | + | a | = 1 / , ( t − / + θ ( x ) x + bθ ( b ) = ( t − / + | a | + | b | = 1 / . Let(7.4) W = n ( b, c ) ∈ O (cid:12)(cid:12)(cid:12) | c | + | b | < o . Then we can solve the equations (7.3) in the following order:First, we solve the fourth equation in (7.3) with respect to t under the condition | c | + | b | < with the solution t = 12 + r − | b | − | c | > . Next, the component a is given by ( b, c ) by the first equation in (7.3) as a = θ ( bc ) t , and this solution a satisfies the inequality: | a | = | bc | t < · | b | + | c | < . Under these conditions we can solve the fifth equation in (7.3) with respect to t with the solution t = 12 − r − | c | − | a | < , since | c | + | a | <
18 + 164 < . Now, with these solutions expressed in terms of the variables ( b, c ) ∈ W we define a map(7.5) M : W ∋ ( b, c ) X = t c θ ( b ) θ ( c ) t ab θ ( a ) 1 − t − t ∈ P O . Then the matrix M ( b, c ) satisfies the condition (7.3), so that we can choose components ( b, c ) as alocal coordinates around the point X . We denote by f W = M ( W ). The point X corresponds to(0 , ∈ W . Lemma 7.1.
In terms of the local coordinates ( b, c ) = X i =0 { b } i e i , X i =0 { c } i e i ! introduced above around the point X , the Riemann volume form dv P O at the point X is (7.6) dv P O (0 ,
0) = d { b } ∧ · · · ∧ d { b } ∧ d { c } ∧ · · · ∧ d { c } . Proof.
We can see this by d M (0 , (cid:18) ∂∂ { b } (cid:19) = (cid:18) ∂∂ { b } (cid:19) X + X i =1 ∂t i (0 , ∂ { b } (cid:18) ∂∂t i (cid:19) X + X i =0 ∂ { a } i (0 , ∂ { b } (cid:18) ∂∂ { a } i (cid:19) X = (cid:18) ∂∂ { b } (cid:19) X , where we know ∂t (0 , ∂ { b } = −{ b } p / − | b | − | c | (cid:12)(cid:12) b =0 ,c =0 = 0 , ∂t (0 , ∂ { b } = − b − P i =0 { a } i ∂ { a } i ∂ { b } p / − | b | − | a | (cid:12)(cid:12) b =0 ,c =0 = 0 ,etc., since a (0 ,
0) = P { a } i e i = 0. Other derivatives are also ∂t i ∂ { b } j (cid:12)(cid:12) (0 , = 0 , ∂t i ∂ { c } j (cid:12)(cid:12) (0 , = 0 , ∂ { a } j { a } k ∂ { b } i (cid:12)(cid:12) (0 , = 0 , ∂ { a } j { a } k ∂ { c } i (cid:12)(cid:12) (0 , = 0 . Hence d M (0 , (cid:18) ∂∂ { b } i (cid:19) = (cid:18) ∂∂ { b } i (cid:19) X , d M (0 , (cid:18) ∂∂ { c } i (cid:19) = (cid:18) ∂∂ { c } i (cid:19) X . Then the metric tensor g ij with respect to the coordinates ( b, c ) at the point ( b, c ) = (0 ,
0) is g ij = δ ij . (cid:3) Explicit determination of the pairing with the Riemann volume form.
Let A = ξ z θ ( y ) θ ( z ) ξ xy θ ( x ) ξ ∈ X O , where ξ i ∈ C , z, y, x ∈ C ⊗ R O . Put τ O − ( A ) = ( X ( A ) , Y ( A )), then X ( A ) = A + A || A || + A ◦ A || A || = 12 || A || · ξ z θ ( y ) θ ( z ) ξ xy θ ( x ) ξ + ξ z θ ( y ) θ ( z ) ξ xy θ ( x ) ξ + 12 || A || · | ξ | + zθ ( z ) + θ ( y ) y ξ z + zξ + θ ( y ) θ ( x ) ξ θ ( y ) + zx + θ ( y ) ξ θ ( z ) ξ + ξ θ ( z ) + xy θ ( z ) z + | ξ | + xθ ( x ) θ ( z ) θ ( y ) + ξ x + xξ yξ + θ ( x ) θ ( z ) + ξ y yz + θ ( x ) ξ + ξ θ ( x ) yθ ( y ) + θ ( x ) x + | ξ | + | ξ | + zθ ( z ) + θ ( y ) y ξ z + zξ + θ ( y ) θ ( x ) ξ θ ( y ) + zx + θ ( y ) ξ θ ( z ) ξ + ξ θ ( z ) + xy θ ( z ) z + | ξ | + xθ ( x ) θ ( z ) θ ( y ) + ξ x + xξ yξ + θ ( x ) θ ( z ) + ξ y yz + θ ( x ) ξ + ξ θ ( x ) yθ ( y ) + θ ( x ) x + | ξ | = ξ + ξ || A || + | ξ | + | z | + | y | || A || z + z || A || + − ξ z − ξ z + θ ( xy + xy )2 || A || θ ( y + y )2 || A || + − θ ( ξ y + ξ y )+ zx + zx || A || θ ( z + z )2 || A || + − θ ( ξ z + ξ z )+ xy + xy || A || ξ + ξ || A || + | ξ | + | z | + | x | || A || x + x || A || + − ξ x − ξ x + θ ( yz + yz )2 || A || y + y || A || + − ξ y − ξ y + θ ( zx + zx )2 || A || θ ( x + x )2 || A || + − θ ( ξ x + ξ x )+ yz + yz || A || ξ + ξ || A || + | ξ | + | x | + | y | || A || . From the above expression of τ O − ( A ) = ( X ( A ) , Y ( A )) we consider two components of the matrix X ( A ) ∈ P O c = z + z || A || + − ξ z − ξ z + θ ( xy + xy )2 || A || , b = y + y || A || + − ( ξ y + ξ y ) + θ ( zx + zx )2 || A || , where we assume A ∈ ˜ U z .We consider the map q ◦ τ O − : X O −→ P O , around the point A = √− e √− e − ∈ X O , where q ◦ τ O − ( A ) = X . On the other handthe point A ∈ X O corresponds to the matrices( X, U, Y, V, Z, W, ξ , ξ , ξ ) ARGMANN TYPE TRANSFORMATION 37 = (cid:18)(cid:18) (cid:19) , (cid:18) (cid:19) , (cid:18) (cid:19) , (cid:18) (cid:19) , (cid:18) √− √− (cid:19) , (cid:18) (cid:19) , , − , (cid:19) ∈ C (2) × C (see the matrix representation (4.7) of octanion numbers).So we may consider points in the local coordinates neighborhood ˜ U z and assume( x , x , u , u , y , y , v , v , z , z , z , w , w , w , w , ξ ) ∈ U z around the point (0 , , , , , , , , z = √− , , , , , , , ξ = − . Consequently, by the explicit expression (4.17) of the other dependent variables( x , x , u , u , y , y , v , v , z , ξ , ξ ) in the matrix representation ( X, U, Y, V, Z, W, ξ , ξ , ξ ) of A is( x , x , u , u , y , y , v , v , z , ξ , ξ ) = (0 , , , , , , , , √− , , . To distinguish the coefficients of the (complex) octanions z, y, x appeared in the expression of thematrix A and the matrix components corresponding to A by the complex matrices ( X, U, Y, V, Z, W,ξ , ξ , ξ ), for example Z = (cid:18) z z z z (cid:19) , we use the notation z = P i =0 { z } i e i and do not express z = P z i e i (see Remark 1). Hence for example (see (2.3)) { z } = z + z , { z } = z − z √− , { z } = z − z , { z } = z + z √− , { z } = w + w , { z } = w − w √− , { z } = w − w , { z } = w + w √− , { y } = y + y , { y } = y − y √− , { y } = y − y , { y } = y + y √− , { y } = v + v , { y } = v − v √− , { y } = v − v , { y } = v + v √− , · · · . Now we determine the differentials modulo anti-holomorphic differentials { q ◦ τ − O } ∗ ( dc ) = X i =0 { q ◦ τ − O } ∗ ( d { c } i ) ⊗ e i = X i =0 d (cid:0) { q ◦ τ − O } ∗ ( { c } i ) (cid:1) ⊗ e i , and { q ◦ τ − O } ∗ ( db ) = X i =0 { q ◦ τ − O } ∗ ( d { b } i ) ⊗ e i = X i =0 d (cid:0) { q ◦ τ − O } ∗ ( { b } i ) (cid:1) ⊗ e i , at the point A .Each component of b and c is given by { c } i = { z } i + { z } i || A || + − ξ { z } i − ξ { z } i + { θ ( xy + xy ) } i || A || , and { b } i = { y } i + { y } i || A || + − ξ { y } i − ξ { y } i + { θ ( zx + zx ) } i || A || . The pull-back { q ◦ τ O − } ∗ ( dv P O ) is expressed as { q ◦ τ O − } ∗ ( dv P O ) = X i =0 Σ i , where Σ i ∈ Γ − i ^ T ∗ ′ ( X O ) C ⊗ i ^ T ∗ ′′ ( X O ) C ! . In particular, Σ i ∧ Ω = 0 for i ≥ , and Σ j = Σ − j . Hence for the determination of the constant C RC ( Y / || Y || ), it is enough to consider the terms consistingof holomorphic differentials dx , dx , du , du , dy , dy , dv , dv , dz , dz , dz , dw , dw , dw , dw , dξ and may ignore the anti-holomorphic differentials dx , dx , etc, so that in the expression of equal-ities below we denote them as ∗ ≡ ∗ , which means both sides coincide modulo anti-holomorphicdifferentials .Here we gather up relations of the holomorphic differentials of dependent variables by independentvariables at the point A . See (4.17) for the explicit expression of each variable x , x , · · · , ξ in termsof independent variables x , x , · · · , ξ . All the equalities in the Lemmas blow hold at the point A . Lemma 7.2. || A || = 2 , d || A || | A ≡ { z dz + z dz + ξ dξ + ξ dξ } | A = − dξ ,z ( A ) = √− , dz | A = − dz − √− dξ , ξ ( A ) = 0 , dξ | A = 0 , dξ | A = − dξ ,dx | A = √− dy , dy | A = −√− dx , dy | A = √− dx , dx | A = −√− dy ,dv | A = √− du , dv | A = √− du , du | A = −√− dv , du | A = −√− dv . Lemma 7.3. d { c } i | A ≡ d { z } i || A || − { z } i + { z i }|| A || · dξ , and for each i = 0 , · · · , d { c } | A ≡ −√− dξ , d { c } | A ≡ dξ − √− dz , d { c } | A ≡ dz − dz ,d { c } | A ≡ dz + dz √− , d { c } | A ≡ dw + dw , d { c } | A ≡ dw − dw √− ,d { c } | A ≡ dw − dw , d { c } | A ≡ dw + dw √− , d { b } | A ≡ = dy − √− dx ,d { b } i | A ≡ d { y } i + √− d { x } i , where we can ignore the term { zx } , since { x } i | A = 0 and d { zx } i | A ≡ P e α e β = e i { z } α d { x } β = { z } d { x } i | A = −√− d { x } i and for i = 1 , · · · , , d { b } | A ≡ dx − √− dy , d { b } | A ≡ dy − √− dx , d { b } | A ≡ dx − √− dy ,d { b } | A ≡ dv + √− du , d { b } | A ≡ − du + √− dv ,d { b } | A ≡ − dv + √− du , d { b } | A ≡ du − √− dv . Based on these data
Proposition 7.4.
At the point A , the holomorphic component of the pull-back { q ◦ τ O − } ∗ ( dv P O ) is equal to { q ◦ τ O − } ∗ ( dv P O ) | A = { q ◦ τ O − } ∗ ( d { c } ∧ · · · ∧ d { c } ∧ d { b } ∧ · · · ∧ d { b } ) | A ≡ −√− dξ ∧ dξ − √− dz ∧ dz − dz ∧ dz + dz √− ∧ dw + dw ∧ dw − dw √− ∧ dw − dw ∧ dw + dw √− ∧ dy − √− dx ∧ dx − √− dy ∧ dy − √− dx ∧ dx − √− dy ∧ dv + √− du ∧ − du + √− dv ∧ − dv + √− du ∧ du − √− dv = 12 √− · dx ∧ dx ∧ du ∧ du ∧ dy ∧ dy ∧ dv ∧ dv ∧ dz ∧ dz ∧ dz ∧ dw ∧ dw ∧ dw ∧ dw ∧ dξ . Hence
Corollary 7.5. { q ◦ τ O − } ∗ ( dv P O ) ^ Ω ( A ) = C RC { τ O − } ∗ (cid:18)(cid:16) ω P O (cid:17) (cid:19) ( A )= 2 · || A || · { τ O − } ∗ (cid:18)(cid:16) ω P O (cid:17) (cid:19) ( A ) . (7.7)By this formula (7.7) we have an expression of the Bargmann type transformation (5.14). Corollary 7.6. B ( h )( X ) · dv P O ( X ) = { q ◦ τ O − } ∗ (cid:16) h · g · { q ◦ τ O − } ∗ ( dv P O ) ^ Ω O (cid:17) = { q ◦ τ O − } ∗ (cid:18) h · g · · || A || · { τ O − } ∗ (cid:16)(cid:0) ω P O (cid:1) (cid:17)(cid:19) = 2 · q ∗ (cid:16) h ( τ O ( X, ∗ )) · e −√ π ||∗|| || ∗ || · dV P O ( X, ∗ ) (cid:17) . (7.8)8. Invariant polynomials and harmonic polynomials on the Jordan algebra J (3)In this section we describe invariant polynomials on J (3) and commuting differential operatorswith constant coefficients under the action by the automorphism group F of the Jordan algebra J (3). Mostly we follow the framework given in [HL] and [He] based on the detailed and explicitdeterminations given in [Yo] for various properties of the group F and the Cayley projective plane P O .8.1. Correspondence between polynomials and differential operators with constant coef-ficients.
Let R N × R N ∋ ( x, A ) = ( x , . . . , x N , A . . . , A N ) < x, A > = X x i A i ∈ R , be the standard non-degenerate symmetric bi-linear form. We also use the same notation for itsextension to the complex bi-linear form defined on C N × C N .Differential operators D with constant (complex) coefficients are expressed in the form D = X | α |≤ k a α ∂ | α | ∂x α = X a α D αx where a α ∈ C , α = ( α , . . . , α N ) being multi-indices, | α | = P α i and D αx := ∂ | α | ∂x α = ∂ α + ··· + α N ∂x α · · · ∂x N α N . Let D = P a α D αx be a constant coefficient partial differential operator defined on R N , then by therelation(8.1) e −
Let g ∈ GL( N, R ) and D a linear differential operator with constant coefficients. Then, P g ◦ D = D ◦ P g on the space of the whole polynomial functions, if and only if Q D ( ξ ) = Q D ( t g − ( ξ )) .Proof. If P g ◦ D = D ◦ P g on the space of the whole polynomial functions, then by the Weierstrasspolynomial approximation theorem this commutativity holds on the space C ∞ ( R N ) of all smoothfunctions.Then P g ◦ D ( e < • ,ξ> )( x ) = D ( e < • ,ξ> )( g − ( x )) = ( e
Lemma 8.2.
Let D = P a α D αx be a differential operator with constant coefficients and P the poly-nomial corresponding to D according to the correspondence (8.1) , that is e −
We recall several properties on the action of thegroup F on J (3). Theorem 8.3 ([SV],[Yo]) . . (1) Th group F is a closed subgroup of SO (27) = SO ( J (3)) and closed under the operation F ∋ g t g . The transpose t g is taken with respect to the inner product : tr ( g ( X ) ◦ Y ) = < g ( X ) , Y > J (3) = < g ( X ) , t g ( Y ) > J (3) . (2) For any A ∈ J (3) , there exists an element α ∈ F such that (8.7) α ( A ) = ξ ξ
00 0 ξ , where the triple of quantities { ξ i } depends only on A and does not depend on such an element α ∈ F which send A to a diagonal matrix in J (3) . (3) The trace function J (3) ∋ A tr ( A ) is invariant under the action of F : (8.8) tr ( α ( A )) = tr ( A ) , α ∈ F . Of course this property implies F ⊂ SO (27) . Also, as was remarked in [Yo],
Theorem 8.4.
The representation of F to J (3) is decomposed into two mutually orthogonal irre-ducible subspaces, that is J (3) = J (3) ⊕ R · Id, where J (3) = { A ∈ J (3) | tr A = 0 } and Id is the × identity matrix which is the fixed point in J (3) under the action of F .It holds the same decomposition in the complexified Jordan algebra J (3) C by the action of thecomplex group F C = { algebra isomorpphisms of the complex Jordan algebra : J (3) C } . We denote the (complex valued) polynomial algebra over J (3) by C [ J (3) ] and treat the algebra C [ J (3) ] as the algebra of polynomial functions on the Jordan algebra J (3) too. It is equipped withan Hermitian inner product explained in the preceding subsection § z , · · · , z , y , · · · , y , x , · · · , x , ξ , ξ , ξ ) in J (3) of the components of the matrix A = ξ z θ ( y ) θ ( z ) ξ xy θ ( x ) ξ ∈ J (3) , z = X z i e i = X { z } i e i and so on . Sometimes we will denote these coordinates as( z , · · · , z , y , · · · , y , x , · · · , x , ξ , ξ , ξ ) = ( s , · · · · · · , s )(8.9)or ( z , · · · , z , y , · · · , y , x , · · · , x , ξ , ξ , ξ ) = ( s , · · · · · · , s , ξ , ξ , ξ ) . (8.10)Then, we can identify in the isometric way the space P [ J (3)] ∼ = (cid:0) J (3) C (cid:1) ∗ with the space J (3) C by the correspondence(8.11) J (3) C ∋ A ←→ h A ∈ P [ J (3)] , h A ( X ) = < X , A > J (3) C . The action of the group F is extended to the space C [ J (3)] as denoted in § . C [ J (3)] ∋ Q ( P g ( Q )( X ) := Q ( g − ( X ))and the extended action leaves the degree of the polynomials and the inner product. Definition 8.5.
We denote a subspace in each P k [ J (3)] by I k consisting of invariant polynomialsunder the extended action of the group F and put I = I F = P k ≥ I k , the algebra of invariantpolynomials under the action of the Lie group F on J (3) . Although the power A n in the Jordan algebra J (3) is not defined in a unique way for n ≥
4, wecan define invariant polynomials T k of degree k as(8.12) T k : J (3) ∋ A tr ( A k ) = < A, A ◦ ( A ◦ ( ◦ A ( ◦ · · · ◦ A ) · · · ) > J (3) . The well-definedness of these polynomials is guaranteed by the property (3.4). Also these invariantpolynomials can be seen as being defined on the complexified algebra J (3) C .Then, Proposition 8.6.
All the invariant polynomials in P k [ J (3)] are given by the linear sums of polyno-mials of the products T i · T i · T i under the condition that i + 2 · i + 3 · i = k (0 ≤ i , i , i ≤ k ) . The polynomials of different indices ( i , i .i ) are linearly independent. So dim C I k = { number of the solutions ( i , i , i ) under the condition i + 2 · i + 3 · i = k } = [ k/ X ℓ =0 (cid:20) k − ℓ (cid:21) + 1 . (8.13) Proof.
Let f ∈ I k be an invariant polynomial. Then by the property (8.7) in Theorem 8.3 and theinvariance of the trace function (8.8), the values f ( A ) = f ( α ( A )) = f ξ ξ
00 0 ξ depend onlyon the triple { ξ i } which appears when it is expressed as a diagonal matrix given in the above Theorem8.3. ARGMANN TYPE TRANSFORMATION 43
Let σ : J (3) → J (3) be a permutation defined by(8.14) σ : J (3) ∋ A A ∈ J (3) . Likewise we can define other two permutations σ and σ among the quantities { ξ i } by the matrices and , respectively. These are elements in F and the values f ( σ i ( A )) = f ( A ).Hence the invariant polynomial ring I = I F = P k ≥ I k in C [ J (3)] is generated by three elementarysymmetric polynomials ξ + ξ + ξ , ξ ξ + ξ ξ + ξ ξ and ξ ξ ξ . This is equivalent to say that the subalgebra of invariant polynomials of positive degree I + := P k ≥ I k (i.e., without constant terms ) is generated by three invariant polynomials T , T and T ,T ( A ) = tr A = X i =1 ξ i , T ( A ) = tr A = 2 X i =0 (cid:0) z i + y i + x i (cid:1) + X i =1 ξ i = || A || T ( A ) = tr A ◦ ( A ◦ A ) = < A, A ◦ A > J (3) = < A ◦ A , A > J (3) = X i =1 ξ i + 3 (cid:0) | z | ( ξ + ξ ) + | y | ( ξ + ξ ) + | x | ( ξ + ξ ) (cid:1) + zx · y + θ ( zx · y ) + xy · z + θ ( xy · z ) + yz · x + θ ( yz · x )2+ x · yz + θ ( x · yz ) + y · zx + θ ( y · zx ) + z · xy + θ ( z · xy )2= X i =1 ξ i + 3 (cid:0) | z | ( ξ + ξ ) + | y | ( ξ + ξ ) + | x | ( ξ + ξ ) (cid:1) + 6 · Re ( x · yz ) , ( R = { x · yz } is the real part of the octanion x · yz )and we have an algebra isomorphism:(8.15) C [ Y , Y , Y ] ∼ = I = X I k , Y i T i . These three polynomials T i are irreducible.We can see the last formula (8.13) in an elementary way by solving the equation i + 2 · i + 3 · i = k . (cid:3) In the proof above we used a property of the multiplication low in the octanion O : Re ( x · yz ) = Re ( y · zx ) = Re ( z · xy ) . Also note that Re ( zx · y ) = Re ( θ ( y ) · θ ( zx )) = Re ( y · zx ) , and so on . For lower degrees, we list a basis of I k : I = [ { T } ] , dim I = 1 ,I = [ { T , T } ] , dim I = 2 ,I = [ { T , T T , T } ] , dim I = 3 ,I = [ { T T , T , T T , T } ] , dim I = 4 . Next we discuss a relation between the invariant polynomials and commuting differential operatorswith the F action, that is, by the lemma 8.1 and the group F is closed under the transpose operationwith respect to the inner product in J (3) we see Proposition 8.7.
The invariant polynomial ring I = P I k and differential operators with constantcoefficients commuting with the F action are isomorphic. Especially the differential operators corre-spond to the generators T , T and T of the invariant polynomial ring are L = L ( z, y, x, ξ , ξ , ξ ) := ∂∂ξ + ∂∂ξ + ∂∂ξ ←→ T = e −
Let H k be H k := { Q ∈ P k [ J (3)] | L ( Q ) = 0 , ∆( Q ) = 0 , Γ( Q ) = 0 } . Then H k = H k . This characterization says that our space of Cayley harmonic polynomials is a subspace of the spaceof usual harmonic polynomials that hereafter we just call them simply Cayley-harmonic polynomials.
Lemma 8.10.
For each k the space (8.17) I k + H · I k − + · · · + H k − · I = I k + P · I k − + · · · + P k − · I . ARGMANN TYPE TRANSFORMATION 45
The right hand side need not be a direct sum, while the left hand side is a direct sum which will beproved later after several preparations.Proof.
It is apparent k − X i =0 H i · I k − i ⊂ X P i · I k − i . Since I ⊃ I · I , H · I + I ⊃ ( H · I + I · I ) + I = P · I + I . Hence H · I + I = P · I + I .Assume j − X i =0 H i · I j − i = j − X i =0 P i · I j − i , for j ≤ k. Then using the property I j ⊃ I a · I b for a + b = j , we can show inductively j − X i =0 H i · I j − i ⊃ j − X i =0 P i · I j − i , for j ≤ k. For example, I k +1 + H · I k + H · I k − + · · · + H k · I ⊃ I k · I + H · I k − · I + H · I k − · I + · · · + H k · I ⊃ ( I k + H · I k − + · · · + H k ) · I = P k · I . (cid:3) Proof of the Proposition Q ) = 0 , ∆( Q ) = 0 , and L ( Q ) = 0are together equivalent to the condition that a polynomial Q ∈ P k is orthogonal to the subspace I k + H · I k − + · · · + H k − · I . ✷ Lemma 8.11. L ( T ) = 3 , L ( T ) = 2 T , L ( T ) = 3 T , ∆( T ) = 198 , ∆( T ) = 198 T , Γ( T ) = 562 . Remark 8.
Invariant polynomials above need not be orthogonal. For example ≪ T , T ≫ = L ( T )(0) = L ( L ( T ))(0) = 2 L ( T )(0) = 6 . Now we show that the sum (8.16) is direct sum. By definition it is enough to show the sum H k − · I + · · · + H · I k − + I k is a direct sum. For this purpose we prepare several lemmas. Lemma 8.12.
The map L : I k −→ I k − is surjective for all k = 1 , , · · · , Proof.
Let t : I k −→ I k +1 be a map defined by t ( T ) = T · T, then t is injective. In fact, if there is an element T ∈ I k satisfying L ◦ t ( T ) = L ( T · T ) = 3 T + T · L ( T ) = 0 , then again we have 3 L ( T ) + 3 L ( T ) + T · L ( T ) = 0 and 3 T − T · L ( T ) = 0 . By iterating this procedure we have T = 0 . Hence the map L ◦ t is injective, which means that the map L : I k +1 → I k is already surjective (infact isomorphic) on t ( I k ). (cid:3) Based on the equality (8.17) and the Lemma below (8.14), we can construct an orthogonal basis ofthe space I k inductively { ϕ k ( i ) } dim I k i =1 of I k in the following way: Definition 8.13. I =[ { ϕ (1) = T } ] ,I =[ { ϕ (1) = T = T · ϕ (1) , ϕ (2) = T − / T } ] , where ϕ (2) is taken to be orthogonal to ϕ (1) and equivalently L ( ϕ (2)) = 0 ,I =[ { ϕ (1) = T = T ϕ (1) , ϕ (2) = T ϕ (2) , ϕ (3) = T − T T + 2 / T } ] , where ϕ (3) is taken to be orthogonal to ϕ (1) and ϕ (2) , which is also takento satisfy L ( ϕ (3)) = 0 and is determined uniquely up to constant multiples, I =[ { ϕ (1) = T = T ϕ (1) , ϕ (2) = T ϕ (2) , ϕ (3) = T ϕ (3) ϕ (4) = T − / T T + 1 / T } ] , where ϕ (4) is taken to satisfy L ( ϕ (4)) = 0 and is equal to ϕ (4) = ϕ (2) . Likewise we can continue the construction in such a way that if { ϕ k ( i ) } dim I k i =1 is constructed as abovefor k = 1 , , , , then we define for k ≥ ϕ k +1 ( i ) = T ϕ k ( i ) for i = 1 · · · , dim I k and for j = 1 , · · · , dim I k +1 − dim I k ,ϕ k +1 (dim I k + j ) is chosen as being orthogonal to all ϕ k +1 ( i ) , i = 1 , · · · , dim I k + j − . The orthogonality condition ≪ ϕ k +1 (dim I k + j ) , T ϕ k ( i ) ≫ = 0 implies that L ( ϕ k +1 (dim I k + j )) = 0 . Lemma 8.14.
The construction is guaranteed by the property that if f and g ∈ I k is orthogonal and L ( g ) = 0 , then T · f and T · g is orthogonal, since ≪ T f, T g ≫ = ≪ f, L ( T g ) ≫ = 3 ≪ f , g ≫ = 0 . Lemma 8.15.
Put N k = { T ∈ I k | L ( T ) = 0 } , then dim N k +1 = dim I k +1 − dim I k , and is equal to the number of the non-negative integer solutions ( a, b ) of the equation (8.18) 2 a + 3 b = k + 1 . Proof.
Put ϕ (2) = T − / T := ϕ and ϕ (3) = T − T T + 2 / T := ϕ . Then both of theseare irreducible polynomials, since they are not decomposed into lower degree polynomials even on thesubspace z = y = x = 0.By L ( ϕ · ϕ ) = 0, products of any powers of these two polynomials are in the kernel of the map L . So corresponding to the non-negative integer solutions ( a, b ) of (8.18) we have a basis of the kernel N k +1 . (cid:3) Lemma 8.16.
For any j and ℓ dim H j · I ℓ = dim H j · dim I ℓ . ARGMANN TYPE TRANSFORMATION 47
Proof.
It will be apparent if dim I ℓ = 1.We prove the property by induction and we show that the natural map H j ⊗ N ℓ −→ H j · N ℓ isisomorphic. So we assume for ∀ ℓ ≤ k and any j ≥ H j ⊗ N ℓ ∼ = H j · N ℓ . Let(8.20) X ( a,b ) run through the solutions of (8.18) h a,b · ϕ a · ϕ b = 0 , h a,b ∈ H j . Let ( a , b ) , ( a , b ) , · · · , ( a n , b n ) be all the solutions of (8.18):2 a i + 3 b i = k + 1 , Assume a > a > · · · > a n , then b < b < · · · < b n . Then the we can assume the expression (8.20)has one of the following two forms:[1] : if a n > , ϕ a n · p + ϕ b n · h a n ,b n = 0 , or(8.21) [2] : if a n = 0 , ϕ a n − · p + ϕ b n · h a n ,b n = 0 . (8.22)In any case the polynomial ϕ does not divide the polynomial ϕ , so that we may put h a n ,b n = ϕ · Q with a polynomial Q ∈ P j − . Then by the equality (8.17) the polynomial Q · ϕ ∈ H j − · I + H j − · I + · · · + H · I j − + I j . On the other hand Q · ϕ = h a n ,b n ∈ H j . Hence by the definition of the space H j which is orthogonal complement of the space H j − · I + H j − · I + · · · + H · I j − + I j , hence h a n ,b n = 0 and also p = 0. By iterating the arguments we see that in the expression X ( a i ,b i ) run through the solutions of (8.18) h a i ,b i · ϕ a i · ϕ b i all the coefficient polynomials h a i ,b i must be zero.Finally we see from the sequences { } −−−−→ H j ⊗ N k +1 −−−−→ H j ⊗ I k +1 Id ⊗ L −−−−→ H j ⊗ I k −−−−→ { } y y y { } −−−−→ H j · N k +1 inclusion −−−−−−→ H j · I k +1 −−−−→ H j · I k −−−−→ { } two spaces H j ⊗ I k +1 ∼ = H j · I k +1 are isomorphic. (cid:3) Proposition 8.17.
For each k , the sum H k + H k − · I + · · · H · I k − + I k is a direct sum.Proof. First we remark that the sums P = H + I and P = H + H · I + I are orthogonal sums.The first one is included in the definition and the second one is shown as ≪ h T , T ≫ = ≪ h , L ( T ) ≫ = ≪ h , T ≫ = ≪ L ( h ) , T ≫ = 0 , where h ∈ H .Then we assume that the sum H j + H j − · I + · · · H · I j − + I j are direct sums for j ≤ k .We express T ∈ H k · I + H k − · I + · · · + H · I k + I k +1 as T = h k · T + X i =1 h k − ( i ) · ϕ ( i ) + · · · + dim I k X i =1 h ( i ) ϕ k ( i ) + dim I k +1 X i =1 h ( i ) ϕ k +1 ( i ) = 0 , where h j ( i ) ∈ H j and ϕ j ( i ) are the basis polynomials of I j constructed in the Definition 8.13. Thenby the induction hypothesis, L ( T ) = 0 implies h k = 0 , h k − (1) ϕ = 0 , h k − (1) ϕ (1) + h k − (2) ϕ (2) = 0 , · · · , dim I k X i =1 λ i ϕ k ( i ) = 0 , that is, the coefficient polynomials h j ( i ) of the basis included in the orthogonal complement of N j arezero.Hence it will be enough to show(8.23) h k − (2) ϕ + h k − (3) ϕ + · · · + dim I k X i =dim I k − +1 h ( i ) ϕ k ( i ) + dim I k +1 X i =dim I k +1 λ i ϕ k +1 ( i ) = 0implies all the coefficient polynomials h j ( i ) = 0 and constants λ i = 0. As in the proof of the Lemma8.16, the equation (8.23) can be rewritten as(8.24) ϕ · P = − ϕ · Q where the polynomial P = h k − (2) + · · · is the sum of all the terms including some power ( ≥
0) of ϕ and Q = g + g ϕ + · · · (especially g = h k − (3) ∈ H k − ) is a polynomials of the polynomial ϕ withthe coefficient polynomials g i ∈ H j with the degree of g i = k + 1 − i . Since ϕ does not divide ϕ , Q must be divided by ϕ , that is we have ϕ · Q = Q = g + g ϕ + · · · , where Q ∈ P k − . Hence by Lemma 8.10 Q = g + g ϕ + · · · ∈ H k − I + H k − I + · · · + I k − , which implies that g = 0. Hence we can rewrite (8.24) as ϕ · P = − ϕ · Q . By iterating the same arguments as above we see that Q = 0. Hence P = 0.Then we can apply the same argument to the polynomial P by expressing P as P = ϕ P + R = 0 , where P is the sum of terms in P of the form h a,b · ϕ a ϕ b , a > , b ≥ h a,b ∈ H k − − a − b and R isa polynomial of ϕ , R = h ′ + h ′ ϕ + h ϕ + · · · with coefficients h ′ a ∈ H k − − a .Again by the same argument as above we see that P = 0, which proves our assertion. (cid:3) We put H := P k ≥ H k , and denote by I + ( J (3) C ) = P k> I k ( J (3) C ) invariant polynomial functionsextended to the complexification J (3) C in the natural way.Since the function taking the trace A tr ( A ) is linear and A A k is an operation inside theJordan algebra J (3) and its complexification J (3) C , the extensions of the invariant polynomials T i to J (3) C coincide with the trace functions J (3) C ∋ A tr ( A ) , A tr ( A ) and A tr ( A ) , which are also extensions to the complexification.Let N J (3) C be the common null set (other than zero) of the invariant polynomial functions consideredon the complexified space J (3) C : N J (3) C := A = ξ z θ ( y ) θ ( z ) ξ xy θ ( x ) ξ ∈ J (3) C (cid:12)(cid:12)(cid:12) A = 0 , T ( A ) = T ( A ) = T ( A ) = 0 . ARGMANN TYPE TRANSFORMATION 49
Remark 9.
Let A ∈ N J (3) C . Then at least one of the three components z, y, x does not vanish.Since if A ∈ N J (3) C and assume z = y = x = 0 , then T ( A ) = P ξ i = 0 , T ( A ) = P ξ i = 0 and T ( A ) = P ξ i = 0 . Hence these imply that ξ i = 0 too. By the Proposition 4.1
Proposition 8.18. X O = τ O ( T ∗ ( P O )) ⊂ N J (3) C and the non-singular part of the space N J (3) C has dim N J (3) C = 24 .Proof. Let A ∈ X O . Then T ( A ) = η + η + η = 0 (Proposition 4.1), and A = 0 implies T ( A ) = 0and T ( A ) = 0 trivially. Hence X O = τ O ( T ∗ ( P O )) ⊂ N J (3) C .The second assertion is seen by noting that at the points z = y = x = 0 the three differentials dT , dT , dT are linearly independent. (cid:3) Let A ∈ J (3) C and consider the functions of the form(8.25) J (3) ∋ X tr X ◦ A = h A ( X ) . Since X O is F invariant, the nontrivial subspace in H linearly spanned by the functions J (3) ∋ X tr ( X ◦ A ) := h A ( X ) , A ∈ J (3) C , tr ( A ) = 0 , is an invariant subspace in H . Here note that tr ( g ( X ) ◦ A ) = tr ( X ◦ t g ( A )) for g ∈ F and tr ( t g ( A )) =tr ( A ) = 0.However the representation of the group F to H is irreducible (Theorem 8.4), the space H mustbe spanned by these functions. Also the same holds that the subspace in H linearly spanned by thefunctions n tr X ◦ A = h A ( X ) | A ∈ N J (3) C o coincides with H . Incidentally we have Proposition 8.19.
All the point in N J (3) C can be expressed as a linear sum of points in X O . Alsothe space N J (3) C is path-wise-connected.Proof. Since any linear function J (3) ∋ X tr ( X ◦ A ) = h A ( X ) with A ∈ N J (3) C is a linear sumof functions of the form tr ( X ◦ B i ) = h B i ( X ) with B i ∈ X O ,tr ( X ◦ A ) = X c i tr ( X ◦ B i ) on J (3) , where B i ∈ X O ,A = P c i B i with these B i ∈ X O .Let A and A ′ ∈ N J (3) C . Assume A = P c i B i and A ′ = P c ′ i B ′ i where B i , B ′ i ∈ N J (3) C . Then thesecond assertion is proved by connecting points B i and B ′ i suitably in X O . (cid:3) In general, the space H k of “Cayley-harmonic polynomials” is an orthogonal sum of two subspaces H (1) k and H (2) k , H (1) k is the subspace linearly spanned by the powers < X , A > k of linear functionstr ( X ◦ A ) = < X , A > with A ∈ N J (3) C and H (2) k is the orthogonal complement of H (1) k in H k . Theorthogonality is equivalent to the property that Cayley-harmonic functions in H (2) k are vanishing onthe subset N J (3) C .In our case the second subspace H (2) k is always { } , that is, Proposition 8.20. H (2) k = H k \ q I + ( J (3) C ) = H k \ I + ( J (3) C ) = { } . Proof.
The first equality is a consequence of Hilbert Nullstellensatz and the irreducibility of N J (3) C implies the second equality.We see the latter one by the following observation that the equation T ( A ) = 0 is linear so that ifwe replace the variable ξ by ξ = − ξ − ξ , then the space N J (3) C can be seen as a subset defined by T ( A ) = 0 in the quadrics Q = { A ∈ C \{ } | T ( A ) = 0 } and the polynomial T restricted onthe space z = y = x = 0 is irreducible even modulo T , i.e., there are no decomposition such that T ( A ) = ξ ξ + ξ ξ = ( aξ + bξ )( αξ + βξ ξ + γξ ) on ξ + ξ + ξ ξ = 0. Hence the space N J (3) C must be irreducible and we have q I + ( J (3) C ) = I + ( J (3) C ) . (cid:3) In fact, our space N J (3) C is an irreducible algebraic manifold and a complete intersection. Inparticular there are points in N J (3) C at which the differentials dT , dT , dT are linearly independent(see the Lemma 4 on page 345 [Ko] for these aspects).Especially, as a corollary of Proposition 8.19 we have Proposition 8.21.
The representation of F to the space H k = H (1) K is irreducible for each k .Proof. Since X O is connected, if the space H k is decomposed into two invariant subspaces, H k = G ⊕ G , then they are orthogonal, because F action is orthogonal. Consequently, according to thisdecomposition the space X O must be separated into two non intersecting closed subsets X O = F S F , F T F = φ . This can be seen in a following way: we put F i = { A ∈ X O | (tr A ◦ X ) k ∈ G i } . If thereis an element A ∈ F T F such that A = 0, then this implies that { } = F T F ∋ (tr A ◦ X ) k = 0.This is a contradiction. Hence each H k must be irreducible under the action by the group F . (cid:3) Now we sum up a conclusion as
Theorem 8.22.
Since the functions in the invariant polynomials I k are constant on the manifold P O , by restricting polynomial functions in P k [ J (3)] to P O the decompositions P k [ J (3)] = H k + I H k − + · · · + I k for each k give totally a decomposition of a subspace in C ∞ ( P O ) as ∞ X k =0 H k | P O , which is dense in C ∞ ( P O ) .Proof. Based on the preceding arguments it will be enough to remark the last assertion, which is astandard argument.Since any smooth function on P O can be extended to a smooth function on an open neighborhoodof P O and the Weierstrass approximation theorem guarantees that any smooth function can beapproximated in the C ∞ − topology and the approximation holds when it is restricted to P O . Hencethe space P H k | P O is dense in C ∞ ( P O ). (cid:3) Before interpreting the decomposition stated in Theorem 8.22 in the framework of the Peter-Weyltheorem for a symmetric space of our case P O we remark about the Riemannian metric on P O . Proposition 8.23.
The Cayley projective plane P ( O ) ∼ = F /Spin (9) is an irreducible Riemann-ian symmetric space, that is, the stationary subgroup Spin (9) acts irreducibly on the tangent space T X P ( O ) . By Schur’s lemma this implies that P ( O ) has an essentially unique F -invariant Rie-mannian metric. Thus, ( · , · ) P O coincides with the metric on P ( O ) induced from the Killing formof the Lie algebra of F up to a constant factor. Let Φ k : H k ⊗ H k ∗ −→ C ∞ ( F ) be a map defined by H k ⊗ H k ∗ ∋ h ⊗ ϕ Φ k ( h ⊗ ϕ )( g ) = ϕ ( P g − ( h )) , g ∈ F , then the Peter-Weyl theorem says that the image of the map Φ k is a subspace consisting of the dim H k number of the spaces, all of which are isomorphic to H k .Recall we explained the identification ( ?? ) of the quotient space F / Spin(9) with P O through thecorrespondence F ∋ g g ( X ) ∈ P O . ARGMANN TYPE TRANSFORMATION 51
If we consider a subspace H k ∗| Spin(9) consisting of linear forms in H k ∗ which are invariant under theaction by Spin(9), then the functions in Φ k ( H k ⊗ H k ∗| Spin(9) )are Spin(9) invariant, so that it can be descended naturally to functions on F / Spin(9) ∼ = P O ⊂ J (3).For X ∈ J (3) we denote the linear form J X ∈ H k ∗ H k ∋ h J X ( h ) = h ( X ) , that is, this is an evaluation at X ∈ J (3). In particular, we take a linear form J X ∈ H k ∗| Spin(9) , thenit can be written as J X ( P g − ( h )) = P g − ( h )( X ) = h ( g ( X )) . Hence through the identification F / Spin(9) ∼ = P O the function J X ( P g − ( h )) is the restriction of theoriginal polynomial function h ∈ H k to P O . Then we have ∞ X k =0 Φ k ( H k ⊗ { J X } ) = ∞ X k =0 H k | P O . Since dim H k +1 > dim H k (see Appendix) and the space ∞ P k =0 H k | P O is already dense in C ∞ ( P O ),a fundamental theorem on compact symmetric spaces gives us Proposition 8.24.
Each irreducible representation of the group F appears in C ∞ ( P O ) with multi-plicity one as in the above way and incidentally dim H k ∗| Spin (9) = 1 . Moreover by the Proposition 8.23we can see that this decomposition is the eigenspace decomposition of the Laplacian on P O .The dimension of the space H k ∗| Spin (9) is always one and the linear form J X can be seen as a basevector of the space H k ∗| Spin (9) for any k . Inverse of Bargmann type transformation
In this section, based on the data obtained until § B : X P k [ X O ] −→ C ∞ ( P O )with respect to the parameter family of the inner products { ( ∗ , ∗ ) ε } − <ε on the space P P k [ X O ] onits boundedness and invertibility. It has a dense image from P P k [ X O ] always for a possible value ofthe parameter ε , but unlike the cases of spheres and other projective spaces (see [Ra2],[Fu1], [FY]), itneed not be an isomorphism when ε = 0. This means in cases of the values of the parameter ε > − / L ( P O ) which can not be seen by classical observables.9.1. Inverse transformation.
Let A k be a transformation defined by A k : H k ∋ ϕ Z P O ϕ ( X ) · (cid:16) tr ( X ◦ A ) (cid:17) k dv P O ( X ) ∈ P k [ X O ] . (9.1)and A k : H k ∋ ϕ −→ A k ( ϕ ) = γ ◦ A ( ϕ ) = A k ( ϕ ) · t · Ω ∈ Γ G (cid:0) L ⊗ K G , X O (cid:1) . (9.2)The correspondence by γ is defined in (5.4). Proposition 9.1.
For any inner product defined on the space P k [ X O ] according to the value of theparameter ε , the operator A k is a constant times a unitary operator.Proof. For ϕ ∈ H k the inner product(9.3) ( A k ( ϕ ) , A k ( ϕ ) ) ε is expressed as ( A k ( ϕ ) , A k ( ϕ ) ) ε = Z X O (cid:12)(cid:12)(cid:12)(cid:12)Z P O ϕ ( X )( tr ( X ◦ A ) ) k dv P O ( X ) (cid:12)(cid:12)(cid:12)(cid:12) · e − √ π || A || / || A || ε Ω O ∧ Ω O = Z P O Z P O Z X O (cid:0) tr ( ˜ X ◦ A ) (cid:1) k (cid:0) tr ( X ◦ A ) (cid:1) k · e − √ π || A || / · || A || ε · Ω O ∧ Ω O ! ×× ϕ ( ˜ X ) ϕ ( X ) dv P O ( X ) dv P O ( ˜ X ) . Here we consider the operator B k P k [ X O ] ∋ h k ( h ) := Z X O h ( A ) · (tr X ◦ A ) k e − √ π || A || / · || A || ε · Ω O ( A ) ∧ Ω O ( A ) ∈ H k . Since H k consists of linear sums of functions of the form (tr X ◦ A ) k by arbitrary A ∈ X O (seeProposition 8.20), we see that B k ( h ) ∈ H k . Then the inner product (9.3) is understood as ( A k ( ϕ ) , A k ( ϕ ) ) ε = ( B k ◦ A k ( ϕ ) , ϕ ) P O . Then the operator B k ◦ A k commutes with the F action on H k . Hence it must be a constant timesidentity operator (which constant we put b k ) so that the kernel function defined by the integral L k ( X, ˜ X ) := Z X O (cid:0) tr ( ˜ X ◦ A ) (cid:1) k (cid:0) tr ( X ◦ A ) (cid:1) k · e − √ π || A || / · || A || ε · Ω O ∧ Ω O ! must satisfies the invariance:(9.4) L k ( g ( X ) , ˜ X ) = L k ( X, g − ( ˜ X )) , for g ∈ F , X, ˜ X ∈ P O . Then the constant b k is given by(9.5) Trace of the operator B k ◦ A k = Z P O L k ( X, X ) dv P O = b k dim H k . and the integral Z P O L k ( X, X ) dv P O is given by Z P O L k ( X, X ) dv P O ≡ L k ( X, X ) · Vol ( P O ) , since by the invariance (9.4) the function L k ( X, X ) is a constant function and apparently is non-zero.Now we know B k is injective and so dim H k ≤ dim P k [ X O ] . On the other hand, degree k polynomials generated by the invariant polynomials which are naturallyextended to the complexification J (3) C , that is, the polynomials k X i =0 P k − i [ J (3) C ] · I k − i = k X i =0 H k − i · I i (see Lemma 8.10) are all vanishing on the manifold X O so thatdim P [ X O ] ≤ dim P [ J (3) C ] − k X i =0 dim H k · dim I k , (see Proposition 8.17).Hence the operator B k is also surjective to the space P k [ X O ]. Consequently, the operator B k is aconstant times a unitary operator. (cid:3) Next, we determine the concrete value of the constant b k : ARGMANN TYPE TRANSFORMATION 53
Proposition 9.2. L k ( X, X ) = b k · dim H k = 2 · Vol ( S ( P O )) · Γ(4 k + 44 + 2 ε )2 k +66+3 ε π k +44+2 ε , where the constant Vol ( S ( P O )) is the volume of the unit cotangent sphere bundle S ( P O ) of P O with respect to the volume form dσ S ( P O ) := 116! · θ P O ∧ (cid:0) ω P O ) (cid:12)(cid:12) S ( P O ) . Proof.
Since L k ( X, X ) = Z X O (cid:12)(cid:12)(cid:12) tr ( X ◦ A ) (cid:12)(cid:12)(cid:12) k · e − √ π || A || / || A || ε · Ω O ( A ) ∧ Ω O ( A ) does not depend onthe point X ∈ P O , we have L k ( X, X ) = Z X O (cid:12)(cid:12)(cid:12) tr ( X ◦ A ) (cid:12)(cid:12)(cid:12) k · e − √ π || A || / · || A || ε · Ω O ( A ) ∧ Ω O ( A )= Z F Z X O (cid:12)(cid:12)(cid:12) tr ( g − ( X ) ◦ A ) (cid:12)(cid:12)(cid:12) k · e − √ π || A || / · || A || ε · Ω O ( A ) ∧ Ω O ( A ) ! dv F ( g )= Z X O Z F (cid:12)(cid:12)(cid:12)(cid:12) tr (cid:18) X ◦ g (cid:18) A || A || (cid:19) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) k dv F ( g ) ! · || A || k + ε · e − √ π || A || / Ω O ( A ) ∧ Ω O ( A ) , (9.6)where dv F is the normalized Haar measure on F .The function(9.7) Z F (cid:12)(cid:12)(cid:12)(cid:12) tr (cid:18) X ◦ g (cid:18) A || A || (cid:19) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) k dv F ( g )does not depend neither on X ∈ P O nor on A ∈ X O , since the trace function A tr ( A ) is F -invariant, the group F acts both on the spaces P O and the cotangent sphere bundle S ( P O ) τ O ∼ = S ( X O )transitively and the Haar measure dv F is bi-invariant.Let ( X, Y ) ∈ T ∗ ( P O ). Put A g ( X, Y ) := g ( τ O ( X, Y )), then g ( τ O ( X, Y )) = g (cid:18) || Y || X − Y + √− ⊗ || Y ||√ Y (cid:19) = g ( || Y || ) g ( X ) − g ( Y ) + √− ⊗ || g ( Y ) ||√ g ( Y ) . Hence τ O − ( A g ( X, Y )) = ( X ( A g ( X, Y )) , Y ( A g ( X, Y ) ) = ( g ( X ) , g ( Y ) ) ∈ T ∗ ( P O ) . The integral (9.7) is expressed as1 || A g ( X, Y ) || k Z F (cid:12)(cid:12) tr X ( A g ( X, Y )) ◦ A g ( X, Y ) (cid:12)(cid:12) k dv F ( g )= 1 || Y || k Z F (cid:12)(cid:12)(cid:12)(cid:12) tr g ( X ) ◦ (cid:18) || g ( Y ) || g ( X ) − g ( Y ) + √− ⊗ || g ( Y ) ||√ g ( Y ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) k dv F ( g )= 1 || Y || k Z F (cid:18) || g ( Y ) || (cid:19) k dv F = 12 k , since g ( X ) = g ( X ) , tr g ( X ) = 1 , g ( X ) ◦ g ( Y ) = 12 g ( Y )and we used the property tr ( X ◦ Y ) ◦ Z = tr X ◦ ( Y ◦ Z ) . Now the integral (9.6) is Z X O Z F (cid:12)(cid:12)(cid:12) tr X ◦ g ( A ) (cid:12)(cid:12)(cid:12) k dv F ( g ) · e − √ π || A || / || A || ε · Ω O ( A ) ∧ Ω O ( A ) = Z X O k || A || k + ε · e − √ π || A || / · Ω O ( A ) ∧ Ω O ( A )= 2 k Z T ∗ ( P O ) || Y || k +28+2 ε · e − √ π || Y || · dV P O . (9.8)where we used the relation (6.8). Then according to the decomposition of the space T ∗ ( P O ) ∼ = R + × S ( P O ), we can decompose the Liouville volume form dV P O as dV P O = t dt ∧ dσ S ( P O ) , where dσ S ( P O ) is the volume form on the unit cotangent sphere bundle S ( P O ). Finally we have theintegral (9.8) as 2 k · Z T ∗ ( P O ) || Y || k +28+2 ε · e − √ π || Y || dV P O = 2 k · Z S ( P O ) dσ S ( P O ) Z ∞ t k +28+2 ε e − √ πt · t dt = 2 k · Vol ( S ( P O )) · Γ(4 k + 44 + 2 ε )(2 √ π ) k +44+2 ε (9.9) = 12 π · Vol ( S ( P O )) · Γ(4 k + 44 + 2 ε )2 k +3 ε π k +2 ε , and(9.10) b k = 12 ε π ε · Vol ( S ( P O )) · Γ(4 k + 44 + 2 ε )2 k π k dim H k . (cid:3) Proposition 9.3.
Since both of the transformations A k and the restriction of the transformation B to the space P [ X O ] ( for short we denote it by T k := B |P [ X O ] ) commute with F action and therepresentation of F on H k is irreducible (see (8.24)) , the composition T k ◦ A k on H k | P O is a constantmultiple operator T k ◦ A k = a k Id and the constant a k is given by a k = 2 · Vol ( S ) · Vol ( P O ) · Γ(2 k + 22)2 k +11 · π k +22 dim H k = 12 π · Vol ( S ) · Vol ( P O ) · Γ(2 k + 22)2 k · π k dim H k . (9.11) Proof.
Let f ∈ H k then by Corollary 7.8 T k (cid:16) A k ( f ) (cid:17) ( X ) · dv P O ( X )= 2 q ∗ (cid:18)Z P O f ( ˜ X ) · { tr ( ˜ X ◦ τ O ( X, ∗ ) ) } k · dv P ( ˜ X ) · e −√ π ·||∗|| · || ∗ || · dV P O ( X, ∗ ) (cid:19) = 2 Z P O f ( ˜ X ) q ∗ (cid:16) { tr ( ˜ X ◦ τ O ( X, ∗ ) ) } k · e −√ π ·||∗|| · || ∗ || dV P O ( X, ∗ ) (cid:17) dv P O ( ˜ X )= 2 Z P O f ( ˜ X ) K k ( ˜ X, X ) dv P O ( ˜ X ) · dv P O ( X ) , where we put the fiber integral as K k ( ˜ X, X ) · dv P O ( X ) := q ∗ (cid:16) { tr ˜ X ◦ τ O ( X, ∗ ) } k · e −√ π ·||∗|| · || ∗ || dV P O ( X, ∗ ) (cid:17) . The kernel function K k ( ˜ X, X ) satisfies the property similar to the kernel function L k ( ˜ X, X ):(9.12) K k ( g · ˜ X, X ) = K k ( ˜ X, g − ( X )) . Then by this property (9.12) that K k ( X, X ) is constant and we havetrace of the operator T k ◦ A k = a k · dim H k ARGMANN TYPE TRANSFORMATION 55 = 2 · Z P O K k ( X, X ) dv P O ( X ) = 2 · K k ( X, X ) · Vol ( P O ) . Since tr (cid:0) X ◦ τ O ( X, Y ) (cid:1) = 1 / || Y || , q ∗ (cid:16) { tr ( X ◦ τ O ( X, ∗ ) ) } k · e −√ π ·||∗|| · || ∗ || · dV P O ( X, Y ) (cid:17) = (1 / k · q ∗ (cid:16) || ∗ || k +6 e −√ π ·||∗|| · dV P O ( X, ∗ ) (cid:17) . If we choose a point X = X , then the above fiber integral is expressed as(1 / k · q ∗ (cid:16) || ∗ || k +6 e −√ π ·||∗|| · dV P O ( X , ∗ ) (cid:17) = (1 / k · Z q − ( X ) || Y || k +6 e −√ π || Y || dβ ∧ · · · ∧ dβ ∧ dγ ∧ · · · ∧ dγ ∧ db ∧ · · · ∧ db ∧ dc ∧ · · · ∧ dc , = (1 / k · Z q − ( X ) || Y || k +6 e −√ π || Y || dβ ∧ · · · ∧ dβ ∧ dγ ∧ · · · ∧ dγ ∧ dv P O ( X ) , (9.13)where we express the integral using the local coordinates on W (see (7.4)) around the point X andthe dual coordinates ( X, Y ) = ( b, c, β, γ ) ←→ P i β i db i + γ i dc i ∈ T ∗ X ( W ). Then the integral (9.13)over the point X is(1 / k · Z q − ( X ) || Y || k +6 e −√ π || Y || dβ ∧ · · · ∧ dβ ∧ dγ ∧ · · · ∧ dγ = (1 / k · Z R (cid:16)X β i + γ i (cid:17) k +3 e −√ π √ P ( β i + γ i ) dβ · · · dβ dγ · · · dγ = Γ(2 k + 22)2 k +11 · π k +22 · Vol ( S ) . Here
Vol ( S ) is the volume of the standard 15-sphere. (cid:3) Now we have a k dim H k = 2 · Γ(2 k + 22)2 k +11 · π k +22 · Vol ( S ) · Vol ( P O )= 12 π · Vol ( S ) · Vol ( P O ) · Γ(2 k + 22)2 k · π k . Proposition 9.4. B |P k [ X O ] ◦ A k = 12 π Γ(2 k + 22)2 k π k · dim H k · Vol ( S ) · Vol ( P O ) Id.
Corollary 9.5.
The operator norm || B − |P k [ X O ] || is given by || B − |P k [ X O ] || = √ b k a k = q V ol ( S ( P O )Γ(4 k +44+2 ε ))2 ε π ε · k π k dim H k V ol ( S ) · V ol ( P O ) · Γ(2 k +22)2 k π k · dim H k , which we express as = C ( ε ) · N ( k ) , (9.14) where C ( ε ) includes only ε and N ( k ) is a function of k and (9.15) N ( k ) = 2 k · dim H k · Γ(4 k + 44 + 2 ε )2 k Γ(2 k + 22) . It is enough to see (9.15) for the behavior of the norm (9.14) when k −→ ∞ and for this purposewe mention two properties of the Gamma function. Lemma 9.6. lim k →∞ Γ( k + α ) · · · Γ( k + α ℓ )Γ( k + β ) · · · Γ( k + β ℓ ) = + ∞ , if P α i > P β i , , if P α i = P β i , , if P α i < P β i . Lemma 9.7. Γ( nz ) = n nz − / (2 π ) ( n − / · n − Y j =0 Γ (cid:18) z + jn (cid:19) . Then by Lemma 9.7(9.16) N ( k ) = 2 ε √ π · dim H k · Q j =0 Γ( k + 11 + ε/ j/ k + 11) Γ( k + 11 + 1 / . By the relation of the Poincar´e polynomials
P P ( t ) = P H ( t ) · P I ( t ) (see (A.1)), the dimension of H k is given as(9.17) dim H k = k − C k + 2 · k − C k − + 2 · k − C k − + k − C k − . Hence (9.16) is N ( k ) = 2 ε √ π · (cid:18) Γ(24 + k )Γ( k + 1) + 2 Γ(23 + k )Γ( k ) + 2 Γ(22 + k )Γ( k −
1) + Γ(21 + k )Γ( k − (cid:19) ×× Q j =0 Γ( k + 11 + ε/ j/ k + 11) Γ( k + 11 + 1 / . Hence finally by Lemma 9.6 have
Theorem 9.8. (1)
Let ε = − , then the Bargmann type transformation B : F − / −→ L ( P O , dv P O ) is an isomorphism, although it is not unitary. (2) If − < ε < − , then the inverse of the Bargmann type transformation B − : L ( P , dv P O ) −→ F ε is bounded, but and the Bargmann type transformation can not be extended to the whole Fock-like space F ε . (3) If ε > − , then the Bargmann type transformation is bounded with the dense image, but notan isomorphism between the spaces F ε and L ( P , dv P O ) . (4) Let ε ≤ − . Then, for such a k that k + 44 + 2 ε ≤ , the integral (6.9) does not converge,although the Bargmann type transformation is defined for such polynomials. Hence by defining aninner product on the finite dimensional space P k +44+2 ε ≤ P k [ X O ] in a suitable way, the Bargmanntype transformation behave in the same way as the case of (2) ( see Remark 7 ) . Remark 10.
The result in the above theorem differs from the original Bargmann transformationand other cases of the spheres, complex projective spaces and quaternion projective spaces for whichthe Bargmann type transformations are always isomorphisms ([Ba], [Ra2], [Fu1], [FY]) without amodification factor in the weight for defining an inner product in the Fock-like space.
Some additional results
Reproducing kernel of the Fock-like space F ε . As an application of the explicit determi-nation of the constant b k we show our Fock-like space F ε has the reproducing kernel.Since the operator A k is an isomorphism from H k to P k [ X O ] and the operator B k ◦ A k ≡ b k , thecomposition A k ◦ B k ≡ b k too. The kernel function (we put it as R k ( A, B ), (
A, B ) ∈ X O × X O ) of thecomposition A k ◦ B k b k , ARGMANN TYPE TRANSFORMATION 57 which is the identity operator on P k [ X O ], is expressed as R k ( A, B )= R P O (tr X ◦ A ) k (tr X ◦ B ) k dv P O · e − √ π ( || A || / + || B || / ) ( || A || · || B || ) ε b k = R P O (tr ( X ◦ A/ || A || )) k (tr ( X ◦ B/ || B || )) k · dv P O · e − √ π ( || A || / + || B || / ) ( || A || · || B || ) k +14+ ε b k . Hence the sum R ( A, B )) := ∞ X k =0 R k ( A, B ) ∞ X k =0 R P O (tr X ◦ A/ || A || ) k (tr X ◦ B/ || B || ) k dv P O · e − √ π ( || A || / + || B || / ) ( || A || · || B || ) k +14+ ε b k . is estimated as (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X k =0 R P O (tr X ◦ A/ || A || ) k (tr X ◦ B/ || B || ) k · dv P O · e − √ π ( || A || / + || B || / ) ( || A || · || B || ) k +14+ ε b k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Vol ( P O )2 επ ε Vol ( S ( P O )) · e − √ π ( || A || / + || B || / ) ( || A || · || B || ) ε · X k π k · ( || A |||| B || ) k Γ(4 k + 44 + 2 ε ) ×× (cid:18) Γ(24 + k )Γ( k + 1) + 2 Γ(23 + k )Γ( k ) + 2 Γ(22 + k )Γ( k −
1) + Γ(21 + k )Γ( k − (cid:19) This inequality implies that the series converges locally uniformly on the space X O × X O and thefunction R ( A, B ) is holomorphic there. So R ( A, B ) is the reproducing kernel of the Hilbert space F ε ( ε > − Geodesic flow and eigenspaces of Laplacian on P O . Let φ t ( t ∈ R ) be an action on X O defined by X O ∋ A φ t ( A ) = e √− t · A. Then this is an interpretation of the geodesic flow action onto the space X O through the map τ O .Let p ∈ P k [ X O ]. Then(10.1) φ t ∗ ( p · t ⊗ Ω O )( A ) = e √− t (11+ k ) · p ( A ) · t ( A ) ⊗ Ω O ( A ) . Let p ∈ P k [ X O ] and q ∈ P ℓ [ X O ] with k = ℓ , then Lemma 10.1. ( p, , q ) ε = Z X O p · q · g · || A || ε · Ω O ∧ Ω O = 0 . Proof.
The transformation φ t ∗ on Γ G ( L ⊗ K G , X O ) is unitary, hence ( φ t ∗ ( p ) , φ t ∗ ( q ) ) ε ≡ ( p, q ) ε for any t ∈ R . On the other hand φ t ∗ (cid:0) p · q · < t , t > L Ω O ∧ Ω O (cid:1) = e √− k − ℓ ) t · (cid:0) p · q · < t , t > L Ω O ∧ Ω O (cid:1) . Hence ( p, q ) ε = 0. (cid:3) Let ∆ P O be the Laplacian on P O . Then Proposition 10.2.
The geodesic flow action on X O and the action given by the one parameter group { e √− t √ ∆ P O +11 } of unitary transformations consisting of the Fourier integral operators commutethrough the Bargmann type transformation. Proof.
This is shown based on the data that the eigenvalues of the Laplacian ∆ P O is given by k + 11 k and the Bargmann type transformation on each subspace P k [ X O ] maps to H k which coincides withthe k -th eigenspace of the Laplacian (Propositions 8.23, 8.24). (cid:3) Appendix A. Generating functions of Poincar´e series
In this Appendix we consider the generating functions of the Poincar´e series of(1) the polynomial algebra : P P ( t ) = X dim P k t k , (2) the algebra of invariant polynomials : P H ( t ) = X dim H k t k and(3) the space of the Cayley harmonic polynomials : P I ( t ) = X dim I k t k , and prove the inequality : dim H k +1 > dim H k . In fact, these formal power series converge for | t | <
1, which will be seen by explicitly determiningtheir generating functions.The generating function
P I ( t ) of the Poincar´e series of the dimensions of invariant polynomials I = P I k is determined as P I ( t ) = ∞ X k =0 dim I k t k = ∞ X k =0 [ k/ X ℓ =0 (cid:18) (cid:20) k − ℓ (cid:21) + 1 (cid:19) t k = ∞ X k =0 X i +2 i +3 i = k, i ,i ,i ∈ N t k = X ( i ,i ,i ) ∈ N × N × N t i +2 i +3 i = 11 − t · − t · − t . The generating function
P P ( t ) of the polynomial algebra C [ s , · · · , s N ] = P P k is given by P P ( t ) = X dim P k t k = ∞ X k =0 N + k − C k t k = X ( r ,r ,...,r N ) ∈ N N t r + r + ··· + r N = (cid:18) − t (cid:19) N , in which N = 27 for our case . Let
P H ( t ) be the generating function of the Poincar´e series of the dimensions of Cayley harmonicpolynomials, then by Lemma 8.16 and Proposition 8.17(A.1) P P ( t ) = P H ( t ) · P I ( t )and we have P H ( t ) = (cid:18) − t (cid:19) · (1 + t )(1 + t + t )= ∞ X k =0 24+ k − C k t k · (1 + 2 t + 2 t + t ) = ∞ X k =0 dim H k t k . (A.2)Then Proposition A.1. dim H k < dim H k +1 . This can be proved by the following elementary fact:
Lemma A.2.
Let f ( t ) = P a k t k and g ( t ) = P b k t k be formal power series with positive coefficientsand satisfies the condition that for all n, b n ≤ b n +1 . Then the coefficients of the product formal power series f · g is increasing. ARGMANN TYPE TRANSFORMATION 59
Proof.
Since the n -th coefficient c n of the product f g is c n = n X i =0 a n − i b i c n +1 − c n = a n +1 b + a n ( b − b ) + · · · + a ( b n +1 − b n ) . In the above expression, each term is non-negative by assumption so that c n +1 − c n ≥
0. In addition if { b n } is strictly increasing, then { c n } is also strictly increasing at least one of the coefficient a k > (cid:3) Proof of Proposition A.1.
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K. Baba: Department of Mathematics, Faculty of Science and Technology, Tokyo University ofScience, 2641 Yamazaki, Noda, Chiba, 278-8510, Japan.
Email address : [email protected] K. Furutani: Osaka City University Advanced Mathematical Institute (=OCAMI) 3-3-138 Sugimoto,Sumiyoshi-ku, Osaka, 558-8585, Japan.
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