Coherent state representations of the holomorphic automorphism group of the tube domain over the dual of the Vinberg cone
aa r X i v : . [ m a t h . R T ] J u l COHERENT STATE REPRESENTATIONSOF THE HOLOMORPHIC AUTOMORPHISM GROUPOF THE TUBE DOMAINOVER THE DUAL OF THE VINBERG CONE
KOICHI ARASHI
Abstract.
We classify all irreducible coherent state representa-tions of the holomorphic automorphism group of the tube domainover the dual of the Vinberg cone. The equivalence classes ofthese representations stand in one-one correspondence with thoseof unitarizations of the holomorphic multiplier representations overthe domain except for the one-dimensional representations of thegroup. Introduction
Let G be a connected Lie group, and let ( π, H ) be a unitary rep-resentation of G . Suppose that dim H >
1. We regard the projectivespace P ( H ) as a (possibly infinite-dimensional) K¨ahler manifold. Wecall a G -orbit of P ( H ) a coherent state orbit (CS orbit for short) if itis a complex submanifold of P ( H ), and we call π a coherent state rep-resentation (CS representation for short) if there exists a CS orbit in P ( H ). In this case, we say that π is generic if π is irreducible and ker π is discrete. By Lisiecki [7], the generic CS representations coincide withthe highest weight unitary representations for a semisimple Lie group.Thus CS representations can be considered as generalizations of thehighest weight unitary representations of semisimple Lie groups to awider class of groups. Also the generic CS representations of connectedunimodular Lie groups were studied and classified by Lisiecki [8]. Afterthis remarkable advance, CS representations were also studied in thesetting of Lie groups which have compactly embedded Cartan subalge-bras by Neeb [10].The purpose of the present article is to give classifications of irre-ducible CS representations and generic CS representations for a Liegroup which has not been considered. Let Ω be the dual cone of theVinberg cone, and let D be the tube domain over Ω . Let G be the Key words and phrases.
Coherent state representation; homogeneous boundeddomain; momentum mapping; reproducing kernel; multiplier representation. identity component of the holomorphic automorphism group of D . Wewill show the following theorem Theorem 1.1 (see Theorems 4.1 and 7.1) . Every irreducible CS rep-resentation of G is equivalent with a unitarization of a holomorphicmultiplier representation of G over D . In [1], the author classified all holomorphic multiplier representationsof G over D , and from Theorem 1.1 it follows that the set of equivalenceclasses of irreducible CS representations of G coincides with the one ofunitarizations of the holomorphic multiplier representations of G over D except for the one-dimensional representations of G . Acknowledgements
The author would like to thank Professor H. Ishi for a lot of helpfuladvice on this paper.2.
General theory of CS representations
In this section, we review the theory of CS representations studiedin [7, 8, 9]. Throughout this paper, for a Lie group, we denote its Liealgebra by the corresponding Fraktur small letter.Let G be a connected Lie group. For a G -equivariant holomorphicline bundle L over a complex manifold M , let us denote the naturalrepresentation of G on the space Γ hol ( M , L ) of holomorphic sectionsof L by τ L . We introduce a notion of unitarizability for τ L . Definition 2.1.
We say that the representation τ L of G is unitarizable if there exists a nonzero Hilbert space H ⊂ Γ hol ( M , L ) satisfying thefollowing conditions:(i) the inclusion map ι : H ֒ → Γ hol ( M , L ) is continuous withrespect to the open compact topology of Γ hol ( M , L ),(ii) τ L ( g ) H ⊂ H ( g ∈ G ) and k τ L ( g ) s k H = k s k H ( g ∈ G , s ∈H ).In this case, we call the subrepresentation ( τ L , H ) a unitarization ofthe representation ( τ L , Γ hol ( M , L )) of G .A Hilbert space H satisfying the condition (i) is a reproducing kernelHilbert space. We note that a Hilbert space giving a unitarizationof τ L is unique if it exists, and any unitarization is irreducible (see[4, 5, 6]). Thus we write π L instead of ( τ L , H ). Let ( π, H ) be a CSrepresentation of G , and let L be the natural holomorphic line bundleover P ( H ) such that the fiber over [ v ] = C v ∈ P ( H ) is given by the dualspace [ v ] ∗ . Then we can identify the dual space H ∗ with Γ hol ( P ( H ) , L ). OHERENT STATE REPRESENTATIONS OF A LIE GROUP 3
By the following theorem, we can see that if π is irreducible, then π is equivalent with π L for a G -equivariant holomorphic line bundle L over a CS orbit. Theorem 2.2 ([5], [8, Proposition 2]) . Suppose that π is irreducible,and let M ⊂ P ( H ) be a CS orbit. Then the map H ∗ → Γ hol ( M, L ) givenby the composition of the map H ∗ → Γ hol ( P ( H ) , L ) and the restrictionmap Γ hol ( P ( H ) , L ) → Γ hol ( M, L ) is injective. Let M be a CS orbit, let α : G × M → M be the action of G on M , and let Z g be the center of g . When π is generic, it holds that(2.1) Lie(ker α ) = Z g , where ker α = { g ∈ G ; α ( g, x ) = x for all x ∈ M } .Next let us see the relationship between CS orbits and coadjointorbits. Let µ π : P ( H ∞ ) → g ∗ be a moment map defined by h µ π ([ v ]) , x i = − i ( dπ ( x ) v, v ) H ( v, v ) H ( v ∈ H ∞ \{ } , x ∈ g ) . Then the image of M under µ π coincides with a coadjoint orbit. Wenote that M has the natural structure of a K¨ahler manifold which isinduced by the Fubini-Stdy metric on P ( H ). As a consequence of thisproperty, we have the following theorem. Theorem 2.3 ([12, Theorem 2.17]) . The isotropy subgroup of G atany point of µ π ( M ) is connected. In particular, the coadjoint orbit µ π ( M ) is simply connected, and µ π defines a diffeomorphism of M onto a coadjoint orbit. The holomorphic automorphism group of the tubedomain over the dual of the Vinberg cone
In this section, we review the explicit description of the holomorphicautomorphism group of the tube domain over the dual of the Vinbergcone studied in [3].Let V = x x x x x x x ∈ M ( R ); x , · · · , x ∈ R , and let Ω = V ∩ P (3 , R ), where P (3 , R ) denotes the homogeneous con-vex cone consists of all 3-by-3 real positive-definite symmetric matrices. KOICHI ARASHI
We consider the following Siegel domain D in the complexification V C of V : D = z = z z z z z z z ∈ V C ; Im z ∈ Ω . Let Aut hol ( D ) be the holomorphic automorphism group of D . Wenote that D is holomorphically equivalent to a complex bounded do-main, and Aut hol ( D ) has the unique structure of a Lie group compati-ble with the compact open topology. Let G be the identity componentof Aut hol ( D ). We shall see a description of G which was determinedby Geatti [3]. For y , y , y , y , y ∈ R with y , y , y > x , x , x , x , x ∈ R , let A y ,y ,y ,y ,y , B x ,x ,x ,x ,x ∈ M ( R )be the matrices given by A y ,y ,y ,y ,y = y y y y y ,B x ,x ,x ,x ,x = x x x x x x x . For A = A y ,y ,y ,y ,y , B = B x ,x ,x ,x ,x ∈ M ( R ), let gl A : D ∋ z Az t A ∈ D , t B : D ∋ z z + B ∈ D , and for ϑ, τ ∈ R , and z ∈ D , let k ϑ,τ ( z )= sin ϑ + z cos ϑ cos ϑ − z sin ϑ z cos ϑ − z sin ϑ sin τ + z cos τ cos τ − z sin τ z cos τ − z sin τz cos ϑ − z sin ϑ z cos τ − z sin τ z + sin ϑ ( z ) cos ϑ − z sin ϑ + sin τ ( z ) cos τ − z sin τ . Let G iI be the isotropy subgroup of G at iI ∈ D . Then we have thefollowing theorem. Theorem 3.1 (Geatti, [3]) . The group G is generated by gl A y ,y ,y ,y ,y , t B x ,x ,x ,x ,x , and k ϑ,τ ( y , y , y > , y , y , x , x , x , x , x , ϑ, τ ∈ R ) , and we have the equality G iI = { k ϑ,τ ; ϑ, τ ∈ R } . OHERENT STATE REPRESENTATIONS OF A LIE GROUP 5
We take a basis { E , E , E , E , , E , , A , A , A , A , , A , , W , W } of g given by E = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 B t, , , , ,E , = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 B , , ,t, ,A = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 A ,e t , , , ,A , = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 A , , , ,t , E = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 B ,t, , , ,E , = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 B , , , ,t A = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 A , ,e t , , ,W = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 k − t, , E = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 B , ,t, , ,A = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 A e t , , , , ,A , = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 A , , ,t, ,W = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 k , − t . Then g has the bracket relation[ E , A ] = − E , [ E , A , ] = − E , , [ E , W ] = A , [ E , A ] = − E , [ E , A , ] = − E , , [ E , W ] = A , [ E , A ] = − E , [ E , , A ] = − E , , [ E , , A ] = − E , , [ E , , A , ] = − E , [ E , , W ] = A , , [ E , , A ] = − E , , [ E , , A ] = − E , , [ E , , A , ] = − E , [ E , , W ] = A , , [ A , A , ] = − A , , [ A , W ] = − W + 2 E ) , [ A , A , ] = − A , , [ A , W ] = − W + 2 E ) , [ A , A , ] = A , , [ A , A , ] = A , , [ A , , W ] = − E , , [ A , , W ] = − E , . CS orbits of generic CS representations
In this section, we see that every generic CS representation of G isrealized as a unitarization of a holomorphic multiplier representationover D .Let M be a CS orbit of a generic CS representation π of G , andlet K be the isotropy subgroup of G at some point m of M . By(2.1), there exists no nonzero ideals of g contained in k since g hastrivial center. Considering the adjoint operators ad( x ) ( x ∈ g ) and theinvariant subspace h E i , it follows that k ⊂ [ g , g ] = h E , E , E , E , , E , , A , A , A , , A , , W , W i . Indeed, for any x ∈ k , the operator ad( x ) : g → g is semisimple andhas only purely imaginary eigenvalues. Let Int [ g , g ] = exp(ad [ g , g ]) ⊂ KOICHI ARASHI GL ([ g , g ]), and let G J = a b µ ′ a b µ ′ λ λ µ µ κc d − λ ′ c d − λ ′ ∈ M ( R ); a i , b i , c i , d i , λ i , λ ′ i , µ i , µ ′ i , κ ∈ R ,a i d i − b i c i = 1 , ( λ i , µ i ) = ( λ ′ i , µ ′ i ) (cid:20) a i b i c i d i (cid:21) ( i = 1 , . The group G J is a semidirect product of the Heisenberg group H ( R )and SL (2 , R ) × SL (2 , R ). Then Int [ g , g ] ⊂ GL ([ g , g ]) is an algebraicsubgroup and is isomorphic to G J /Z G J . Here for a group G , we denoteby Z G the center of G . Hence exp(ad k ) ⊂ Int [ g , g ] is a compact sub-group. By a generalization of the Iwasawa decomposition [11, Chapter4, Theorems 4.7 and 4.9], we see that every maximal compact subgroupof H ( R ) ⋊ SL (2 , R ) × SL (2 , R ) /Z H ( R ) is conjugate to a maximal com-pact subgroup of Z H ( R ) × SL (2 , R ) × SL (2 , R ) /Z H ( R ) , and hence k iscontained in Ad( g )( h E , W , W i ) for some g ∈ G . Taking a conju-gation if necessarily, we may and do assume that k ⊂ h E , W , W i .Considering the adjoint operators ad( x ) ( x ∈ h E , W , W i ) and the in-variant subspace h E , A i , we obtain k ⊂ h W , W i . We then have k = 0or h W , W i because M is an even-dimensional differentiable manifold.Now we shall show that k must equal h W , W i . Arguing contradic-tion, assume that k = 0. Then M is diffeomorphic to G . The followinglinear group gives an explicit realization of G . a b µ ′ a b µ ′ λ λ a µ µ κc d − λ ′ c d − λ ′ a − ∈ M ( R ); a i , b i , c i , d i , λ i , λ ′ i , µ i , µ ′ i , κ ∈ R ,a ∈ R > , a i d i − b i c i = 1 , ( λ i , µ i ) = a ( λ ′ i , µ ′ i ) (cid:20) a i b i c i d i (cid:21) ( i = 1 , . The above group is the product of three subgroups which are iso-morphic to H ( R ), R > , and SL (2 , R ) × SL (2 , R ). Then we have π ( G, e ) = Z , which contradicts that M is simply connected. Hencewe conclude that k = h W , W i .Now we have a G -equivariant diffeomorphism ϕ : D → M . Letus consider the K¨ahler structure (˜ j, ˜ g ) on D which is the pullback,by the diffeomorphism ϕ , of the K¨ahler structure on M . Also we canregard D as a K¨ahler manifold by means of the Bergman metric on D .Then it follows from [2, Theorem 6.1] that there exists a G -equivariantbiholomorphism D → M since G acts on ( D , ˜ j, ˜ g ) by holomorphicisometries. Thus by Theorem 2.2, the CS representation π is unitary OHERENT STATE REPRESENTATIONS OF A LIE GROUP 7 equivalent with π L for a G -equivariant holomorphic line bundle L over D . We note that by the Oka-Grauert principle, every holomorphicline bundle over D is trivial. Hence the representation τ L can berealized as the space O ( D ) of holomorphic functions on D , and thisrepresentation of G is called a holomorphic multiplier representation of G over D . Therefore we get the following theorem. Theorem 4.1.
Let ( π, H ) be a generic CS representation of G . Then π is realized as a unitarization of a holomorphic multiplier representationover D . Holomorphic multiplier representations over D In this section, we review the classification of the unitarizations ofholomorphic multiplier representations of G over D studied in [1].Let g − be the complex subalgebra of the complexification g C of g given by g − = (cid:26) x + iy ∈ g C ; ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 e tx iI + i ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 e ty iI ∈ T , iI D (cid:27) , where T , iI D denotes the antiholomorphic tangent vector space at iI .By Tirao and Wolf [13], the isomorphism classes of G -equivariant holo-morphic line bundles over D stand in one-one correspondence withthe one-dimensional complex representations of g − whose restrictionsto g iI lift to representations of G iI . For a basis { x λ } of g , we shalldenote the dual basis by { x ∗ λ } . Let M be the set consists of all linearforms ξ on g given by ξ = ξ ( ξ , η , n, n ′ ) = ξ E ∗ + η A ∗ + n W ∗ − E ∗ ) + n ′ W ∗ − E ∗ ) , with ξ , η ∈ R and n, n ′ ∈ Z ≥ . Then any one-dimensional complexrepresentation of g − whose restriction to g iI lifts to a representation of G iI is given by ξ | g − ( ξ ∈ M ), where ξ is extended to a complex linearform on g C . For ξ ∈ M , let L be a G -equivariant line bundle over D whose isomorphism class corresponds to ξ , and put τ ξ = τ L . Also weput π ξ = π L when τ L is unitarizable. LetΘ G ( n, n ′ ) = { ξ ( ξ , η , n, n ′ ); ξ < , η ∈ R } ( n, n ′ ∈ Z > ) , Θ G ( η , n, n ′ ) = { ξ (0 , η , n, n ′ ) } ( η ∈ R , n, n ′ ∈ Z ≥ ) . (5.1)Then we have the following theorem. Theorem 5.1 ([1]) . (i) For ξ ∈ M , the representation τ ξ is uni-tarizable if and only if ξ belongs to any of the sets in (5.1) . KOICHI ARASHI (ii)
For ξ, ξ ′ ∈ M with τ ξ , τ ξ ′ unitarizable, the representations π ξ and π ξ ′ are unitary equivalent if and only if ξ and ξ ′ belongs tothe same set in (5.1) . From now on, for ξ ∈ M such that τ ξ is unitarizable, we think of π ξ as any of the holomorphic multiplier representations over D . We shallmention the converse of Theorem 4.1. Let H ξ be the representationspace of π ξ , let K ξ : D × D → C be the reproducing kernel of H ξ , andlet K ξiI ∈ H ξ be the function given by K ξiI ( z ) = K ξ ( z, iI ) ( z ∈ D ).If the representation dπ ξ of g is extended to a complex representation,then we have dπ ξ ( x ) K ξiI = iξ ( x ) K ξiI ( x ∈ g − ) , which implies that π ξ is an irreducible CS representation of G if dim H ξ >
1. 6. generic CS representations
In this section, we classify all generic CS representations of G .Let us consider the set of equivalence classes of irreducible unitaryrepresentations of G . For a unitary representation π of G , we denotethe equivalence class of π by [ π ]. For n, n ′ ∈ Z > , let ξ n,n ′ be any ofthe elements of Θ G ( n, n ′ ). Theorem 6.1.
The set of unitary equivalence classes of generic CSrepresentations of G is given by { [ π ξ n,n ′ ]; n, n ′ ∈ Z > } .Proof. We shall show that(i) For any n, n ′ ∈ Z > , and ξ ∈ Θ G ( n, n ′ ), the CS representation π ξ is generic,(ii) For any η ∈ R , n, n ′ ∈ Z ≥ , and ξ ∈ Θ G ( η , n, n ′ ), the CSrepresentation π ξ is not generic.For ξ ∈ M with τ ξ unitarizable, we have µ π ξ ([ K ξiI ]) = ξ , and hencewe can identify the coadjoint orbit through ξ ∈ g ∗ with the CS orbitthrough [ K ξiI ] ∈ P ( H ξ ). We denote by α the action of G on the coad-joint orbit through ξ . Let G ξ be the isotropy subgroup of G at ξ . Wenote that g ξ = { x ∈ g ; ξ ([ x, y ]) = 0 for all y ∈ g } .(i) A direct calculation shows that g ξ = h W , W i . Now we haveker π ξ ⊂ ker α = { e } , and hence π ξ is generic.(ii) We have E ∈ g ξ . Thus dim ker α ≥
1, which implies π ξ is notgeneric. (cid:3) OHERENT STATE REPRESENTATIONS OF A LIE GROUP 9 Irreducible non-generic CS representations
In this section, we see that every irreducible CS representation of G is realized as a unitarization of a holomorphic multiplier representationover D .By the definition of CS representation, if all generic CS represen-tations of the quotient groups of G are given, then we can obtainall irreducible CS representations of G by composing the quotientmaps. Let h = h E , E , , E , , A , A , , A , i , h = h E , E , , A , i , h ′ = h E , E , , A , i , a = h A i , s = h E , A , W i , s ′ = h E , A , W i .Figure 1 gives the Hasse diagram for nontrivial ideals of g . Thus it isenough to consider the Lie groups with the following Lie algebras:(i) R , (ii) sl (2 , R ) , (iii) R ⊕ sl (2 , R ) , (iv) sl (2 , R ) ⊕ sl (2 , R ) , (v) R ⊕ sl (2 , R ) ⊕ sl (2 , R ) , (vi) h ⊕ a ⊕ s / h E i , (vii) h ⊕ a ⊕ s ⊕ s ′ / h E i , (viii) g / h E i . h ⊕ s ⊕ s ′ h ⊕ a ⊕ s h ⊕ a ⊕ s ′ h ⊕ s h ⊕ a h ⊕ s ′ h ⊕ s h h ′ ⊕ s ′ h h ′ h E i Figure 1.
Hasse diagram for nontrivial ideals of g However the cases (vi)-(viii) are impossible. We shall prove this forthe case (viii). For the other cases, this can be proved in the sameway. Suppose that M is a CS orbit of a generic CS representationof a connected Lie group ˜ G with Lie algebra ˜ g = g / h E i . Let K bethe isotropy subgroup of ˜ G at some point m of M . Considering theadjoint operators ad( x ) ( x ∈ ˜ g ) and the invariant subspace h / h E i , itfollows that k ⊂ h E , E , E , E , , E , , A , A , A , , A , , W , W i / h E i . By an argument similar to that of Section 4, we may assume that k ⊂ h E , W , W i / h E i , and we then have dim k = 1. The group Int ˜ g =exp(ad ˜ g ) ⊂ GL (˜ g ) is isomorphic to G/Z G . Now π (Int ˜ g , e ) = Z .Since M is diffeomorphic to a coadjoint orbit, the group Int ˜ g actstransitively on M , and the isotropy subgroup (Int ˜ g ) m at m is a one-dimensional torus. Thus π ((Int ˜ g ) m , e ) = Z . This contradicts that M is simply connected.Consequently, every irreducible CS representation of a quotient groupof G comes from the external tensor product of a one-dimensional uni-tary representation of R > and two highest weight representations of SL (2 , R ). We shall see an explicit description of the holomorphic mul-tiplier representation in which the external tensor product of the repre-sentations is realized. We fix a triple ( η , n, n ′ ) with η ∈ R and n, n ′ ∈ Z ≥ . Let D be the unit disc in C , let ˜ G = SL (2 , R ) × SL (2 , R ) × R > ,and let ˜ m : ˜ G × D × D → C × be the holomorphic multiplier given by˜ m (( g , g , γ ) , ( w , w )) = ( γ w + δ ) n ( γ w + δ ) n ′ γ − iη (( g , g , γ ) ∈ ˜ G, ( w , w ) ∈ D × D ) , where g i = (cid:20) α i β i γ i δ i (cid:21) ∈ SL (2 , R ) for i = 1 ,
2. We denote by D ∋ w i g i w i ∈ D the action of SL (2 , R ) by linear fractional transfor-mations for i = 1 ,
2. Then we can define the following holomorphicmultiplier representation τ ˜ m of ˜ G on the space O ( D × D ) of holomor-phic functions on D × D : τ ˜ m ( g ) f (( w , w )) = m ( g − , ( w , w )) − f ( g − w , g − w )( g = ( g , g , γ ) ∈ ˜ G, ( w , w ) ∈ D × D , f ∈ O ( D × D )) . Using the realization of G as a linear group in Section 4, we shalldefine a holomorphic multiplier representation of G . Let m : G × D → C × be the holomorphic multiplier given by m ( g, z ) = ( c z + d ) n ( c z + d ) n ′ γ − iη ( g ∈ G, z ∈ D ) , and let τ m be the holomorphic multiplier representation given by τ m ( g ) f ( g, z ) = m ( g − , z ) − f ( g − z ) ( g ∈ G, z ∈ D , f ∈ O ( D )) . If we regard τ ˜ m as a representation of G which H ( R ) acts by thetrivial representation, then the map O ( D ×D ) ∋ f F f ∈ O ( D ) de-fined by F f ( z ) = f ( z , z ) ( z ∈ D ) intertwines τ ˜ m with τ m . Thereforewe get the following theorem. OHERENT STATE REPRESENTATIONS OF A LIE GROUP 11
Theorem 7.1.
Let ( π, H ) be an irreducible CS representation of G .Then π is realized as a unitarization of a holomorphic multiplier rep-resentation over D . The representation τ m is given by a G -equivalent holomorphic linebundle over D whose isomorphism class corresponds to ξ (0 , η , n, n ′ ) ∈M . Finally we have the following theorem. Theorem 7.2.
The set of unitary equivalence classes of irreduciblenon-generic CS representations of G is given by { [ π ξ η ,n,n ′ ]; η ∈ R , n, n ′ ∈ Z ≥ }\{ [ π ξ η , , ]; η ∈ R } . References [1] K. Arashi, Holomorphic multiplier representations for bounded homogeneousdomain, to appear in Jounal of Lie Theory, Heldermann Verlag.[2] I. G. Dotti, Rigidity of invariant complex structures. Trans. Amer. Math. Soc.338 (1993), no. 1, 159–172.[3] L. Geatti, Holomorphic automorphisms of some tube domains over nonselfad-joint cones. Rend. Circ. Mat. Palermo (2) 36 (1987), no. 2, 281–331.[4] H. Ishi, Unitary holomorphic multiplier representations over a homogeneousbounded domain. Adv. Pure Appl. Math. 2 (2011), no. 3–4, 405–419.[5] S. Kobayashi, Irreducibility of certain unitary representations. J. Math. Soc.Japan 20 (1968), 638–642.[6] R. A. Kunze, On the irreducibility of certain multiplier representations. Bull.Amer. Math. Soc. 68 (1962), 93–94.[7] W. Lisiecki, Kaehler coherent state orbits for representations of semisimple Liegroups. Ann. Inst. H. Poincar´e Phys. Th´eor. 53 (1990), no. 2, 245–258.[8] W. Lisiecki, A classification of coherent state representations of unimodularLie groups. Bull. Amer. Math. Soc. (N.S.) 25 (1991), no. 1, 37–43.[9] W. Lisiecki, Coherent state representations. A survey. Mathematics as lan-guage and art (Bia lowie˙za, 1993). Rep. Math. Phys. 35 (1995), no. 2–3, 327–358.[10] K.-H. Neeb, Holomorphy and convexity in Lie theory. De Gruyter Expositionsin Mathematics, 28. Walter de Gruyter & Co., Berlin, 2000. xxii+778 pp.[11] A. L. Onishchik, `E. B. Vinberg, Lie groups and Lie algebras, III. Structureof Lie groups and Lie algebras. Encyclopaedia of Mathematical Sciences, 41.Springer-Verlag, Berlin, 1994.[12] J. Rosenberg, M. Vergne, Harmonically induced representations of solvable Liegroups. J. Funct. Anal. 62 (1985), no. 1, 8–37.[13] J. A. Tirao, J. A. Wolf, Homogeneous holomorphic vector bundles. IndianaUniv. Math. J. 20 (1970/71), 15–31.[14] `E. B.Vinberg, S. G. Gindikin, I. I. Pjatecki˘ı-ˇSapiro, Classification and canon-ical realization of complex homogeneous bounded domains. (Russian) TrudyMoskov. Mat. Ob. 12 1963 359–388.
K. Arashi: Graduate School of Mathematics, Nagoya University,Chikusa-ku, Nagoya, 464-8602 Japan
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