A topic on homogeneous vector bundles over elliptic orbits: A condition for the vector spaces of their cross-sections to be finite dimensional
aa r X i v : . [ m a t h . DG ] J a n A TOPIC ON HOMOGENEOUS VECTOR BUNDLES OVERELLIPTIC ORBITS: A CONDITION FOR THE VECTOR SPACESOF THEIR CROSS-SECTIONS TO BE FINITE DIMENSIONAL
NOBUTAKA BOUMUKI
Abstract.
In this paper we consider the complex vector spaces of holomor-phic cross-sections of homogeneous holomorphic vector bundles over ellipticadjoint orbits, and provide a sufficient condition for the vector spaces to befinite dimensional in view of root systems. Introduction
For a connected real semisimple Lie group G , the adjoint orbit Ad G ( T ) = G/C G ( T ) of G through an elliptic element T ∈ g is called an elliptic ( adjoint ) orbit . Here an element T ∈ g is said to be elliptic , if ad T is a semisimple lineartransformation of g and all the eigenvalues of ad T are purely imaginary. It is knownthat elliptic orbits can be geometrically characterized as follows (cf. Dorfmeister-Guan [4, 5]):Any elliptic orbit G/C G ( T ) is a homogeneous pseudo-K¨ahler man-ifold of G . Conversely, a homogeneous pseudo-K¨ahler manifold M of G is an elliptic orbit whenever G acts on M almost effectively.Accordingly there is no essential difference between elliptic orbits and homogeneouspseudo-K¨ahler manifolds of real semisimple Lie groups. Let us give examples ofelliptic orbits. A complex projective space CP n is one of the Hermitian symmetricspaces of compact type, any Hermitian symmetric space G u /K of compact type isone of the complex flag manifolds (which are also called generalized flag manifoldsor K¨ahler C-spaces), and all complex flag manifolds G C /Q are elliptic orbits. Theseare examples of elliptic orbits which are compact. As a non-compact example, oneknows that all symmetric bounded domains D in C n are elliptic orbits. In thispaper, we deal with such spaces. Elliptic orbits
Hermitian symmetric spacesof compact type • CP n Complex flag manifolds Symmetric bounded domainsin C n Now, let us explain our research background. Let G C be a connected complexsemisimple Lie group, let G be a connected closed subgroup of G C such that g is a Mathematics Subject Classification.
Primary 32M10; Secondary 17B22, 22E45.
Key words and phrases. elliptic (adjoint) orbit, semisimple Lie group, homogeneous vectorbundle, root system, analytic continuation, Bruhat decomposition, continuous representation, K-finite vector.This work was supported by JSPS KAKENHI Grant Number JP 17K05229. real form of g C , and let T be a non-zero elliptic element of g . Setting L := C G ( T ) , g λ := { X ∈ g C | ad T ( X ) = iλX } for λ ∈ R ,Q − := { x ∈ G C | Ad x (cid:0)L µ ≤ g µ (cid:1) ⊂ L µ ≤ g µ } , one has an elliptic orbit G/L , a complex flag manifold G C /Q − and L = G ∩ Q − ;besides, it turns out that ι : G/L → G C /Q − , gL gQ − , is a G -equivariant realanalytic embedding whose image is a simply connected domain in G C /Q − , and that GQ − is a domain in G C . Henceforth, we assume G/L to be a domain in G C /Q − and it to be a homogeneous complex manifold of G via this ι . G/L ι ✲ ι ♯ ( G C × ρ V ) ❄ G C /Q − G C × ρ V ❄ In addition, let V be a finite dimensional complex vector space and let ρ : Q − → GL ( V ), q ρ ( q ), be a holomorphic homomorphism. Denote by G C × ρ V the fiberbundle over the complex flag manifold G C /Q − , with standard fiber V and structuregroup Q − , which is associated to the principal fiber bundle π C : G C → G C /Q − , x xQ − , and denote by ι ♯ ( G C × ρ V ) the restriction of the bundle G C × ρ V to thedomain G/L ⊂ G C /Q − . Then one may assume that V G C /Q − := (cid:26) h : G C → V (1) h is holomorphic , (2) h ( xq ) = ρ ( q ) − ( h ( x )) for all ( x, q ) ∈ G C × Q − (cid:27) and V G/L := (cid:26) ψ : GQ − → V (1) ψ is holomorphic , (2) ψ ( yq ) = ρ ( q ) − ( ψ ( y )) for all ( y, q ) ∈ GQ − × Q − (cid:27) are the complex vector spaces of holomorphic cross-sections of the bundles G C × ρ V and ι ♯ ( G C × ρ V ), respectively. Here, we remark that the vector space V G C /Q − isalways finite dimensional, dim C V G C /Q − < ∞ because G C /Q − is a connected compact complex manifold; but, in contrast, V G/L is not necessarily finite dimensional—for example, dim C V G/L = ∞ in the casewhere G/L is a symmetric bounded domain in C n and V G/L is the vector space O ( T , ( G/L )) of holomorphic vector fields on it. This poses us the following prob-lem: “What is a condition for dim C V G/L < ∞ ?”In this paper we partially solve this problem.The main purpose of this paper is to provide a sufficient condition so that all theholomorphic mappings ψ ∈ V G/L can be continued analytically from GQ − to G C .In view of a root system △ of g C , we assert the following statement (see Subsection3.1, Theorem 3.1): Suppose that (S) there exists a fundamental root system Π △ of △ satisfying (s1) α ( − iT ) ≥ for all α ∈ Π △ , and (s2) g β ⊂ k C for every β ∈ Π △ with β ( T ) = 0 .Then, all the holomorphic mappings ψ ∈ V G/L extend uniquely toholomorphic ones ˆ ψ ∈ V G C /Q − and dim C V G/L = dim C V G C /Q − < ∞ . TOPIC ON HOMOGENEOUS VECTOR BUNDLES OVER ELLIPTIC ORBITS 3
Pay attention to that in the case where the above supposition (S) holds, the vectorspace V G/L is finite dimensional for any complex vector space V of dim C V < ∞ and any holomorphic homomorphism ρ : Q − → GL ( V ). Hence, in particular, onecan deduce that in this case, the group Hol( G/L ) of holomorphic automorphismsof
G/L is a (finite dimensional) Lie group.This paper consists of four sections. In Section 2 we mainly review known factsabout elliptic orbits, generalized Bruhat decompositions and homogeneous holomor-phic vector bundles. In Section 3 we state the main result in this paper (Theorem3.1) and demonstrate it by taking a continuous representation ̺ of G on V G/L , ageneralized Bruhat decomposition of G C and the second Riemann removable sin-gularity theorem into account. Finally in Section 4, we give some examples whichsatisfy the supposition (S) in Theorem 3.1, and give an example which does not so.We will see that the (S) cannot hold for any symmetric bounded domain D in C n ,cf. Example 4.2. 2. Preliminaries
In this section we first fix the notation utilized in this paper, and afterwardsreview known facts about elliptic orbits, generalized Bruhat decompositions andhomogeneous holomorphic vector bundles. We will give two Lemmas 2.6 and 2.14,Corollary 2.20 and Proposition 2.27 especially needed in Section 3.2.1.
Notation.
Throughout this paper, for a Lie group G , we denote its Lie algebraby the corresponding Fraktur small letter g , and utilize the following notation:(n1) i := √− G , g ,(n3) C G ( T ) := { g ∈ G | Ad g ( T ) = T } for an element T ∈ g ,(n4) N G ( m ) := { g ∈ G | Ad g ( m ) ⊂ m } for a vector subspace m ⊂ g ,(n5) m ⊕ n : the direct sum of vector spaces m and n ,(n6) GL ( V ) : the general linear group on a complex vector space V .Besides, we sometimes denote by f | A the restriction of a mapping f to a set A .2.2. Elliptic orbits.
Kobayashi [7] has introduced the notion of elliptic orbit,which is as follows:
Definition 2.1 (cf. Kobayashi [7, p.5]) . Let g be a real semisimple Lie algebra and G a connected Lie group with Lie algebra g . An element T ∈ g is said to be elliptic ,if ad T is a semisimple linear transformation of g and all the eigenvalues of ad T arepurely imaginary. The adjoint orbit Ad G ( T ) = G/C G ( T ) of G through an ellipticelement T ∈ g is called an elliptic ( adjoint ) orbit .Now, let G C be a connected complex semisimple Lie group, let G be a connectedclosed subgroup of G C such that g is a real form of g C , and let T be a non-zeroelliptic element of g . Then we set(2.2) L := C G ( T ) , L C := C G C ( T ) , g λ := { X ∈ g C | ad T ( X ) = iλX } for λ ∈ R , u ± := L λ> g ± λ , U ± := exp u ± , Q ± := N G C ( l C ⊕ u ± ) , where g λ = { } in the case where λ is different from the eigenvalues of ad T , wedenote by exp : g C → G C the exponential mapping, and u ± T will stand for the N. BOUMUKI above u ± for once in Lemma 2.6. Since T ∈ g is elliptic, there exists a Cartandecomposition g = k ⊕ p of g such that(2.3) T ∈ k , where k is a maximal compact subalgebra of g . Noting that the center Z ( G ) of G isfinite due to Z ( G ) ⊂ Z ( G C ) and that g u := k ⊕ i p is a compact real form of g C , wedenote by K and G u the maximal compact subgroups of G and G C correspondingto the subalgebras k ⊂ g and g u ⊂ g C , respectively. In addition, we denote by the(anti-holomorphic) Cartan involution ¯ θ of G C such that(2.4) G u = { g u ∈ G C | ¯ θ ( g u ) = g u } . Let us give easy lemmas and review a known fact.
Lemma 2.5.
In the setting (2.2);(1) L C is a connected closed complex subgroup of G C with l C = g . (2) g C = L λ ∈ R g λ = u + ⊕ l C ⊕ u − . (3) l C ⊕ u + = L µ ≥ g µ and l C ⊕ u − = L µ ≤ g µ . (4) Ad L C ( g λ ) ⊂ g λ for all λ ∈ R . (5) [ g λ , g µ ] ⊂ g λ + µ for all λ, µ ∈ R . (6) u s is a complex nilpotent subalgebra of g C such that Ad L C ( u s ) ⊂ u s , foreach s = ± .In the setting (2.2) , (2.3) and (2.4);(7) ¯ θ ∗ ( g λ ) = g − λ for all λ ∈ R . (8) ¯ θ ( L C ) = L C , ¯ θ ∗ ( u + ) = u − , ¯ θ ∗ ( u − ) = u + . Proof.
Since G C is connected semisimple and T ∈ g is an elliptic element of g C also, one shows that L C = C G C ( T ) is connected. The rest of proof is trivial. (cid:3) Lemma 2.6 (cf. [2]) . Let G C be a connected complex semisimple Lie group, let G be a connected closed subgroup of G C such that g is a real form of g C , and let T bea non-zero elliptic element of g . Fix a Cartan decomposition g = k ⊕ p with T ∈ k ,and take a maximal torus i h R of g u = k ⊕ i p containing T . Then, there exists anelliptic element T ′ ∈ g such that (i) all the eigenvalues of ad iT ′ are integer, (ii) C G ( T ) = C G ( T ′ ) , (iii) u + T = u + T ′ , u − T = u − T ′ and (iv) T ′ ∈ i h R .Here, we refer to (2.2) for u ± T , u ± T ′ . Proof.
One can conclude this lemma by the proof of Theorem 2.3 in [2, p.66]. (cid:3)
From Lemma 2.5 we deduce
Proposition 2.7.
In the setting (2.2);(1) U s is a simply connected, closed complex nilpotent subgroup of G C whoseLie algebra coincides with u s , and exp : u s → U s is biholomorphic, for each s = ± . (2) Q s is a connected, closed complex parabolic subgroup of G C such that Q s = L C ⋉ U s ( semidirect ) and q s = ( l C ⊕ u s ) = L µ ≥ g sµ , for each s = ± . TOPIC ON HOMOGENEOUS VECTOR BUNDLES OVER ELLIPTIC ORBITS 5 (3) U + × Q − ∋ ( u, q ) uq ∈ G C is a holomorphic embedding whose image isa dense, domain in G C . (4) L is a connected closed subgroup of G , and the homogeneous space G/L issimply connected. (5) L = G ∩ Q − . (6) ι : G/L → G C /Q − , gL gQ − , is a G -equivariant real analytic embeddingwhose image is a simply connected domain in G C /Q − . (7) GQ − is a domain in G C .In the setting (2.2) , (2.3) and (2.4);(8) ¯ θ ( U + ) = U − , ¯ θ ( U − ) = U + , ¯ θ ( Q + ) = Q − , ¯ θ ( Q − ) = Q + and ¯ θ ( L ) = L . Proof. e.g. Warner [14] or [2, Paragraph 2.4.2]. (cid:3)
Remark 2.8.
In general, there are several kinds of invariant complex structureson the elliptic orbit
G/L . In this paper we deal with the complex structure on
G/L induced by ι : G/L → G C /Q − , gL gQ − . Here, the imaginary unit i ∈ C givesrise to a G C -invariant complex structure J on the complex flag manifold G C /Q − ina natural way.Proposition 2.7-(3), (7) leads to Corollary 2.9.
In the setting (2.2); the following two items hold for given finiteelements x , x , . . . , x j ∈ G C :(1) The intersection GQ − ∩ x U + Q − ∩ · · · ∩ x j U + Q − is a non-empty opensubset of G C . (2) The union GQ − ∪ x U + Q − ∪ · · · ∪ x j U + Q − is a dense, domain in G C . Root systems and generalized Bruhat decompositions.
We review fun-damental results about root systems and modify a generalized Bruhat decomposi-tion of G C for our situation (see Proposition 2.18-(3)). The setting (2.2), (2.3) and(2.4) remains valid in this subsection.2.3.1. Root systems and Weyl groups.
Let i h R be a maximal torus of g u = k ⊕ i p containing the element T , let △ = △ ( g C , h C ) be the (non-zero) root system of g C relative to h C , where h C is the complex vector subspace of g C generated by i h R , andlet g α be the root subspace of g C for α ∈ △ . For each root α ∈ △ , there exists aunique H α ∈ h C such that α ( H ) = B g C ( H α , H ) for all H ∈ h C , where B g C is theKilling form of g C . Then h R = span R { H α | α ∈ △} , and for every α ∈ △ thereexists a vector E α ∈ g α satisfying(2.10) ( E α − E − α ) , i ( E α + E − α ) ∈ g u and [ E α , E − α ] = (2 /α ( H α )) H α (cf. Helgason [6, Lemma 3.1, p.257–258]). Here, it is immediate from (2.4) that g u = i h R ⊕ L α ∈△ span R { E α − E − α } ⊕ span R { i ( E α + E − α ) } , and(2.11) ¯ θ ∗ ( E α ) = − E − α for all α ∈ △ . Define a Weyl group W of G C and an action ζ of W on the dual space ( h C ) ∗ by(2.12) (cid:26) W := N G u ( i h R ) /C G u ( i h R ) ,ζ ([ w ]) η := t Ad w − ( η ) for [ w ] ∈ W and η ∈ ( h C ) ∗ , where [ w ] stands for the left coset wC G u ( i h R ). By use of E α in (2.10) we set(2.13) w α := exp( π/ E α − E − α ) for α ∈ △ . N. BOUMUKI
Needless to say, w α belongs to N G u ( i h R ) and so [ w α ] ∈ W for every root α ∈ △ ;besides, ζ ([ w α ]) is the reflection along α which leaves △ invariant. We need Lemma 2.14.
Let k C be the complex subalgebra of g C generated by k . For a root β ∈ △ = △ ( g C , h C ) with β ( T ) = 0 , the following (a) , (b) and (c) are equivalent :(a) g β ⊂ k C , (b) E β ∈ k C , (c) ( E β − E − β ) ∈ k . Therefore, w β = exp( π/ E β − E − β ) belongs to K ∩ N G u ( i h R ) whenever one ofthe (a) , (b) and (c) holds. Proof.
Since (a) ⇔ (b) is obvious, we only confirm (b) ⇔ (c).(b) ⇒ (c): This follows by (2.11), ¯ θ ∗ ( k C ) ⊂ k C and k = { Y ∈ k C | ¯ θ ∗ ( Y ) = Y } .(c) ⇒ (b): Suppose that ( E β − E − β ) ∈ k . Then, from (2.3) one obtains β ( T )( E β + E − β ) = [ T, E β − E − β ] ∈ [ k , k ] ⊂ k ;and so 0 = β ( T ) ∈ i R yields ( E β + E − β ) ∈ i k . Hence E β = (1 / E β − E − β + E β + E − β ) ∈ k + i k ⊂ k C . (cid:3) Generalized Bruhat decompositions.
We continue to obey the setting of Para-graph 2.3.1.Our first aim in this paragraph is to state Proposition 2.17 which is a result ofKostant [9, 10] and the second one is to modify a generalized Bruhat decompositionof G C for our situation. For the aim, we are going to fix two Iwasawa decompositionsof G C first.Let Π △ be a fundamental root system of △ = △ ( g C , h C ) satisfying(s1) α ( − iT ) ≥ α ∈ Π △ . Relative to this Π △ we fix the set △ + of positive roots, and put △ − := −△ + . Then(s1) yields β ( − iT ) ≥ β ∈ △ + . Setting n s := L β ∈△ s g β and b s := h C ⊕ n s ( s = ± ) one has Iwasawa decompositions g C = g u ⊕ h R ⊕ n ± of g C ; moreover, itfollows from (2.2) and g C = n + ⊕ h C ⊕ n − that(2.15) ( u + = L λ> g λ ⊂ L β ∈△ + g β = n + ⊂ b + ⊂ L µ ≥ g µ = ( l C ⊕ u + ) = q + , u − ⊂ n − ⊂ b − ⊂ q − . Denote by G C = G u H R N ± the Iwasawa decompositions of G C corresponding tothe g C = g u ⊕ h R ⊕ n ± , respectively.Following Kostant [9, 10], we set(2.16) △ ( u ± ) := { β ∈ △ ± | β ( T ) = 0 } (cid:0) = { α ∈ △ | ± α ( − iT ) > } (cid:1) , △ ( l C ) := { γ ∈ △ | γ ( T ) = 0 } , △ ± ( l C ) := △ ( l C ) ∩ △ ± , Φ [ w ] := { β ∈ △ + | ζ ([ w ]) − β ∈ △ − } for [ w ] ∈ W , W := { [ σ ] ∈ W | Φ [ σ ] ⊂ △ ( u + ) } , W := N L u ( i h R ) /C L u ( i h R ) , where L u := C G u ( T ). Note that u ± = L α ∈△ ( u ± ) g α due to (2.15), that Φ [ w ] isa closed subsystem of △ for any [ w ] ∈ W , and that W is a Weyl group of L C .Hereafter, let us assume that W is a subgroup of W via N L u ( i h R ) /C L u ( i h R ) ∋ τ C L u ( i h R ) τ C G u ( i h R ) ∈ N G u ( i h R ) /C G u ( i h R ). Now, we are in a position to statethe proposition: Proposition 2.17 (cf. Kostant [9, p.359–361], [10, p.121]) . In the setting above ; There is such a system with (s1)—for example, consider the lexicographic linear ordering onthe dual space ( h R ) ∗ associated with a real base − iT =: A , A , . . . , A ℓ of h R . TOPIC ON HOMOGENEOUS VECTOR BUNDLES OVER ELLIPTIC ORBITS 7 (1)
For any [ w ] ∈ W , it follows that △ + = Φ [ w ] ∪ Φ [ wκ ] ( disjoint union ) , where [ κ ] is the unique element of W such that ζ ([ κ ]) △ − = △ + . (2) If [ σ ] ∈ W , then ζ ([ σ ]) − (cid:0) △ + ( l C ) (cid:1) ⊂ △ + and ζ ([ σ ]) − (cid:0) △ − ( l C ) (cid:1) ⊂ △ − . (3) For each [ w ] ∈ W there exists a unique ([ τ ] , [ σ ]) ∈ W × W such that [ w ] = [ τ σ ] . (4) For a [ σ ] ∈ W , the following items (4.i) and (4.ii) hold :(4.i) n [ σ ] = 0 if and only if [ e ] = [ σ ] . (4.ii) n [ σ ] = 1 if and only if there exists a β ∈ Π △ satisfying β ( T ) = 0 and [ w β ] = [ σ ] .Here n [ σ ] is the cardinal number of the set Φ [ σ ] , and e is the unit element of G C . Proposition 2.17 enables us to establish
Proposition 2.18.
With the same notation and setting as in Proposition let r := dim C u + . (1) For each [ σ ] ∈ W we set △ σ := { γ ∈ Φ [ σ − κ ] | ζ ([ σ ]) γ ∈ △ ( u + ) } , U + σ := exp (cid:0)L γ ∈△ σ g ζ ([ σ ]) γ (cid:1) . Then, U + σ is a simply connected closed complex nilpotent subgroup of U + and it is biholomorphic to the ( r − n [ σ ] ) -dimensional complex Euclideanspace ; furthermore, N + σ − Q − = σ − U + σ Q − . (2) For a [ σ ] ∈ W , the following items (2.i) and (2.ii) hold :(2.i) dim C U + σ = r = dim C U + if and only if [ e ] = [ σ ] . (2.ii) dim C U + σ = r − if and only if there exists a β ∈ Π △ satisfying β ( T ) = 0 and [ w β ] = [ σ ] . (3) G C = S [ σ ] ∈ W N + σ − Q − = S [ σ ] ∈ W σ − U + σ Q − ( disjoint unions ) . Proof. (1) We only prove that dim C U + σ = r − n [ σ ] and N + σ − Q − = σ − U + σ Q − for any [ σ ] ∈ W . In view of (2.16), ζ ([ κ ]) △ − = △ + and △ ( u + ) ⊂ △ + we see ζ ([ σ ]) (cid:0) △ σ (cid:1) = { ζ ([ σ ]) γ ∈ △ ( u + ) | γ ∈ Φ [ σ − κ ] } = { ζ ([ σ ]) γ ∈ △ ( u + ) | γ ∈ △ + , ζ ([ σ − κ ]) − γ ∈ △ − } = { ζ ([ σ ]) γ ∈ △ ( u + ) | γ ∈ △ + } = { ζ ([ σ ]) γ ∈ △ ( u + ) | ζ ([ σ ]) − (cid:0) ζ ([ σ ]) γ (cid:1) ∈ △ + } = △ ( u + ) − Φ [ σ ] . This implies that △ σ consists of ( r − n [ σ ] )-elements, so that dim C U + σ = r − n [ σ ] .The Proposition 2.17-(1) above and Lemma 6.2 in Kostant [10, p.124] yield N + σ − Q − = exp (cid:0)L γ ∈ Φ [ σ − κ ] g γ ⊕ L β ∈ Φ [ σ − g β (cid:1) σ − Q − = exp (cid:0)L γ ∈ Φ [ σ − κ ] g γ (cid:1) exp (cid:0)L β ∈ Φ [ σ − g β (cid:1) σ − Q − = σ − exp (cid:0)L γ ∈ Φ [ σ − κ ] g ζ ([ σ ]) γ (cid:1) exp (cid:0)L β ∈ Φ [ σ − g ζ ([ σ ]) β (cid:1) Q − = σ − exp (cid:0)L γ ∈ Φ [ σ − κ ] g ζ ([ σ ]) γ (cid:1) Q − = σ − exp (cid:0)L γ ∈△ σ g ζ ([ σ ]) γ (cid:1) Q − = σ − U + σ Q − , where we remark that L β ∈ Φ [ σ − g ζ ([ σ ]) β ⊂ n − ⊂ q − , and that either ζ ([ σ ]) γ ∈△ + ( l C ) or ζ ([ σ ]) γ ∈ △ ( u + ) holds for every γ ∈ Φ [ σ − κ ] ; besides, the above compu-tation is independent of the choice of representative σ ∈ [ σ ].(2) comes from (1) and Proposition 2.17-(4).(3) The arguments below will be similar to those in the proof of Lemma 5.6 inTakeuchi [12, p.21] or Proposition 6.1 in Kostant [10, p.123]. N. BOUMUKI
By virtue of (1), it suffices to confirm that G C = S [ σ ] ∈ W N + σ − Q − (dis-joint union). Setting B + := N G C ( b + ) we fix the Bruhat decomposition G C = S [ w ] ∈ W N + w − B + (disjoint union). Then, it follows from ζ ([ κ ]) △ − = △ + that G C = κ − G C = S [ w ] ∈ W N − ( wκ ) − B + = S [ w ] ∈ W N − w − B + , namely(2.19) G C = S [ w ] ∈ W N − w − B + (disjoint union) . In a similar way we have L C = S [ τ ] ∈ W N − τ − B +1 , where n ± := L α ∈△ ± ( l C ) g α , N − := exp n − and B +1 := N L C ( h C ⊕ n +1 ). This, togetherwith Q + = L C U + and B + = B +1 U + , assures that for any [ σ ] ∈ W , N − σ − Q + = N − σ − L C U + = S [ τ ] ∈ W N − σ − N − τ − B +1 U + = S [ τ ] ∈ W N − σ − N − τ − B + = S [ τ ] ∈ W N − ( τ σ ) − B + , where σ − N − ⊂ N − σ − follows from [ σ ] ∈ W and Proposition 2.17-(2). Conse-quently, (2.19) and Proposition 2.17-(3) allow us to assert that G C = S [ σ ] ∈ W N − σ − Q + (disjoint union) . Thus G C = S [ σ ] ∈ W N + σ − Q − (disjoint union) because of ¯ θ ( G C ) = G C , ¯ θ ( N − ) = N + , ¯ θ ( σ ) = σ and ¯ θ ( Q + ) = Q − . (cid:3) The following corollary will play a role later (recall (2.13) for w β ): Corollary 2.20.
Let G C be a connected complex semisimple Lie group, let G be aconnected closed subgroup of G C such that g is a real form of g C , and let T be anon-zero elliptic element of g . Set U + , Q − as (2.2) , fix a Cartan decomposition g = k ⊕ p with T ∈ k , and take a maximal torus i h R of g u = k ⊕ i p containing T .Consider the root system △ = △ ( g C , h C ) and a fundamental root system Π △ of △ such that (s1) α ( − iT ) ≥ for all α ∈ Π △ . Now, let O := U + Q − ∪ (cid:0)S β ∈ Π △ with β ( T ) = 0 w − β U + Q − (cid:1) . Then, O is a dense domain in G C . Furthermore, any holomorphic function f on O can be continued analytically to the whole G C . Proof.
Proposition 2.7-(3) implies that O is a dense domain in G C .Proposition 2.18 tells us that e − U + e Q − ∪ (cid:0)S β ∈ Π △ with β ( T ) = 0 w − β U + w β Q − (cid:1) isa subset of O , and moreover, G C − O must be of complex codimension 2 or more.Therefore any holomorphic function f on O can be continued analytically to thewhole G C , by the second Riemann removable singularity theorem (which is some-times called Hartogs’s continuation theorem). Here dim C G C ≥
3, since G C iscomplex semisimple. (cid:3) Homogeneous holomorphic vector bundles.
In this subsection we recallelementary facts about homogeneous holomorphic vector bundles.Let G C be a connected complex semisimple Lie group, let G be a connected closedsubgroup of G C such that g is a real form of g C , and let T be a non-zero ellipticelement of g . Define the closed subgroups L ⊂ G and Q − ⊂ G C by (2.2). Then,we assume that the elliptic orbit G/L is a domain in the complex flag manifold G C /Q − and is a homogeneous complex manifold of G via ι : G/L → G C /Q − , gL gQ − . Now, for a complex vector space V of dim C V < ∞ and a holomorphichomomorphism ρ : Q − → GL ( V ), q ρ ( q ), we denote by G C × ρ V the fiber bundle TOPIC ON HOMOGENEOUS VECTOR BUNDLES OVER ELLIPTIC ORBITS 9 over G C /Q − , with standard fiber V and structure group Q − , which is associated tothe principal fiber bundle π C : G C → G C /Q − , x xQ − , and denote by ι ♯ ( G C × ρ V )the restriction of the bundle G C × ρ V to the domain G/L ⊂ G C /Q − . In this setting,one may assume that V G C /Q − := (cid:26) h : G C → V (1) h is holomorphic , (2) h ( xq ) = ρ ( q ) − ( h ( x )) for all ( x, q ) ∈ G C × Q − (cid:27) , (2.21) V G/L := (cid:26) ψ : GQ − → V (1) ψ is holomorphic , (2) ψ ( yq ) = ρ ( q ) − ( ψ ( y )) for all ( y, q ) ∈ GQ − × Q − (cid:27) (2.22)are the complex vector spaces of holomorphic cross-sections of the bundles G C × ρ V and ι ♯ ( G C × ρ V ), respectively. Remark 2.23.
One knows that dim C V G C /Q − < ∞ because G C /Q − is a connectedcompact complex manifold. e.g. Kodaira [8, p.161].From now on, we are going to set a topology for the V G/L . Since G C is connected,it satisfies the second countability axiom. Hence GQ − satisfies the same axiom alsoand is a locally compact Hausdorff space, since GQ − is open in G C . Consequentlythere exist non-empty open subsets O n ⊂ GQ − such that (i) GQ − = S ∞ n =1 O n (countable union) and (ii) the closure O n in GQ − is compact for each n ∈ N . Takinga norm k · k on the vector space V , we define d n by d n ( ψ , ψ ) := sup {k ψ ( a ) − ψ ( a ) k : a ∈ O n } for n ∈ N , ψ , ψ ∈ V G/L ; and furthermore we define(2.24) d ( ψ , ψ ) := ∞ X n =1 n d n ( ψ , ψ )1 + d n ( ψ , ψ ) for ψ , ψ ∈ V G/L . This d is called the Fr´echet metric on V G/L . Then, one can show the lemma below(e.g. refer to [2, Paragraph 2.4.4]), where ̺ : G → GL ( V G/L ), g ̺ ( g ), is ahomomorphism defined by(2.25) (cid:0) ̺ ( g ) ψ (cid:1) ( y ) := ψ ( g − y ) for ψ ∈ V G/L , y ∈ GQ − . Lemma 2.26.
In the setting (2.22) , (2.24) and (2.25); the following four itemshold for the Fr´echet metric d on V G/L :(1) ( V G/L , d ) is a complete metric space. (2) The metric topology for ( V G/L , d ) coincides with the topology of uniformconvergence on compact sets ; and besides it also coincides with the locallyconvex topology determined by a countable number of seminorms { p n } n ∈ N ,where p n ( ψ ) := d n ( ψ, for n ∈ N , ψ ∈ V G/L . (3) Both V G/L × V
G/L ∋ ( ψ , ψ ) ψ + ψ ∈ V G/L and C × V G/L ∋ ( α, ψ ) αψ ∈ V G/L are continuous mappings. (4) G × V G/L ∋ ( g, ψ ) ̺ ( g ) ψ ∈ V G/L is a continuous mapping.
Lemma 2.26 implies that V G/L = ( V G/L , d ) is a Fr´echet space and that ̺ is acontinuous representation of the Lie group G on V G/L . Therefore
Proposition 2.27 (e.g. van den Ban [13, p.24]) . In the setting (2.22) , (2.24) and (2.25); for a compact subgroup K ′ of G we set ( V G/L ) K ′ := (cid:8) ϕ ∈ V G/L (cid:12)(cid:12) dim C span C { ̺ ( k ) ϕ : k ∈ K ′ } < ∞ (cid:9) . Then, ( V G/L ) K ′ is dense in V G/L with respect to the metric topology for ( V G/L , d ) . We end this section with the following remark: the set ( V G/L ) K ′ in Proposition2.27 accords with the set of K ′ - finite vectors in V G/L for the continuous represen-tation ̺ of G on V G/L .3.
The main result in this paper (Theorem 3.1)
This section consists of two subsections. In Subsection 3.1 we state Theorem 3.1which is the main result in this paper; and in Subsection 3.2 we demonstrate thetheorem.3.1.
The statement of Theorem . . The setting of Theorem 3.1 is as follows: • G C is a connected complex semisimple Lie group, • G is a connected closed subgroup of G C such that g is a real form of g C , • T is a non-zero elliptic element of g , • g = k ⊕ p is a Cartan decomposition of g with T ∈ k , • i h R is a maximal torus of g u = k ⊕ i p containing T , • △ = △ ( g C , h C ) is the root system of g C relative to h C , where h C is thecomplex vector subspace of g C generated by i h R , • g α is the root subspace of g C for α ∈ △ , • L = C G ( T ), • Q − is the closed complex subgroup of G C defined by (2.2), • k C is the complex subalgebra of g C generated by k , • V is a finite dimensional complex vector space, • ρ : Q − → GL ( V ), q ρ ( q ), is a holomorphic homomorphism, • V G C /Q − and V G/L are the complex vector spaces defined by (2.21) and(2.22), respectively.Now, we are in a position to state
Theorem 3.1.
In the setting of Subsection . suppose that (S) there exists afundamental root system Π △ of △ satisfying (s1) α ( − iT ) ≥ for all α ∈ Π △ , and (s2) g β ⊂ k C for every β ∈ Π △ with β ( T ) = 0 .Then, the complex vector space V G C /Q − is linear isomorphic to V G/L via F : V G C /Q − → V G/L , h h | GQ − , and therefore dim C V G/L = dim C V G C /Q − < ∞ .Here h | GQ − stands for the restriction of h to GQ − ⊂ G C . Proof of Theorem . . The setting of Theorem 3.1 remains valid in thissubsection. In addition, we take the closed complex subgroup U + defined by (2.2)and the maximal compact subgroup K ⊂ G corresponding to the subalgebra k ⊂ g into consideration.Our goal in Subsection 3.2 is to complete the proof of Theorem 3.1. We aregoing to show two Lemmas 3.5 and 3.6, Proposition 3.7 and Corollary 3.10, andobtain the goal from them. Remark 3.2.
For the element T concerning Theorem 3.1, one may assume that(3.3) all the eigenvalues of ad iT are integer . Let us explain the reason why. Let T ′ be the element in Lemma 2.6. Then forany root α ∈ △ , one can assert that α ( iT ) > α ( iT ) = 0 and α ( iT ) < α ( iT ′ ) > α ( iT ′ ) = 0 and α ( iT ′ ) <
0, respectively. In particular, for any
TOPIC ON HOMOGENEOUS VECTOR BUNDLES OVER ELLIPTIC ORBITS 11 β ∈ Π △ , β ( T ) = 0 if and only if β ( T ′ ) = 0. Accordingly there are no changes inthe topological group structures on L and Q − , and no change in the supposition(S) even if one substitutes T ′ for T . For this reason, we assume (3.3) hereafter.Suppose that △ ( u + ) consists of r -elements γ , γ , . . . , γ r ∈ △ , where r = dim C u + .Then, { E γ j } rj =1 is a complex base of u + = L α ∈△ ( u + ) g α = L rj =1 g γ j , and by virtueof (3.3) there exist n , n , . . . , n r ∈ N satisfying γ j ( T ) = in j for each 1 ≤ j ≤ r , sothat(3.4) Ad(exp λT ) E γ j = e in j λ E γ j (1 ≤ j ≤ r )for all λ ∈ R . cf. (2.10) for E γ j , (2.16) for △ ( u + ). Lemma 3.5.
The mapping F : V G C /Q − → V G/L , h h | GQ − , is injective linear. Proof.
It is enough to confirm that F is injective. That comes from the theoremof identity, since h : G C → V is holomorphic, G C is connected and GQ − is open in G C . (cid:3) Lemma 3.6. (1)
Let ϕ be a K -finite vector in V G/L for the representation ̺ defined by (2.25) ,and let V ϕ be the complex vector subspace of V G/L generated by { ̺ ( k ) ϕ : k ∈ K } . Then, there exist a complex base { ϕ a } m ϕ a =1 of V ϕ and µ , µ , . . . , µ m ϕ ∈ R such that ̺ (exp λT ) ϕ a = e iµ a λ ϕ a for all ≤ a ≤ m ϕ = dim C V ϕ and λ ∈ R . (2) There exist a complex base { v b } kb =1 of V and θ , θ , . . . , θ k ∈ R such that ρ (exp λT ) v b = e iθ b λ v b for all ≤ b ≤ k = dim C V and λ ∈ R . Proof. (1) Let S := { exp λT | λ ∈ R } . Then, it follows from T ∈ k that S is a real one-dimensional torus and S ⊂ K . Therefore, since V ϕ is ̺ ( K )-invariantand m ϕ = dim C V ϕ < ∞ , there exist ̺ ( S )-invariant complex vector subspaces V , V , . . . , V m ϕ ⊂ V ϕ and µ , µ , . . . , µ m ϕ ∈ R such that V ϕ = V ⊕ V ⊕ · · · ⊕ V m ϕ ,dim C V a = 1 and ̺ (exp λT ) = e iµ a λ id on V a for all 1 ≤ a ≤ m ϕ and λ ∈ R . Hence we can get the conclusion by taking anon-zero element of V a for each 1 ≤ a ≤ m ϕ .(2) One can conclude (2) by arguments similar to those above. Indeed; thereexist ρ ( S )-invariant complex vector subspaces V , . . . , V k ⊂ V and θ , . . . , θ k ∈ R such that V = V ⊕ · · · ⊕ V k , dim C V b = 1 and ρ (exp λT ) = e iθ b λ id on V b for all1 ≤ b ≤ k and λ ∈ R , because of S ⊂ Q − and k = dim C V < ∞ . (cid:3) Proposition 3.7.
Let { ϕ a } m ϕ a =1 and { v b } kb =1 be the complex bases of V ϕ and V inLemma , respectively. For y ∈ GQ − we express ϕ a ( y ) ∈ V as ϕ a ( y ) = ϕ a ( y ) v + ϕ a ( y ) v + · · · + ϕ ka ( y ) v k . Then, for each ≤ b ≤ k there exists a unique polynomial ( holomorphic ) function ϕ ba ′ on C r ∼ = U + of finite degree such that ϕ ba = ϕ ba ′ | U + ∩ GQ − . Therefore, for a given φ ∈ V ϕ there exists a unique holomorphic mapping φ ′ : U + → V such that φ = φ ′ | U + ∩ GQ − . Proof.
Denote by z , z , . . . , z r the canonical coordinates of the first kind as-sociated with the complex base { E γ j } rj =1 of u + (see Remark 3.2 for E γ j ). Here, itturns out that U + ∼ = C r via U + ∋ exp( z E γ + z E γ + · · · + z r E γ r ) ( z , z , . . . , z r ) ∈ C r . Noting that U + ∩ GQ − is an open subset of U + containing the unit e ∈ G C and that the restriction ϕ ba | U + ∩ GQ − is a holomorphic function on U + ∩ GQ − ,we obtain an R > O := { u ∈ U + : | z j ( u ) | < R , 1 ≤ j ≤ r } :(i) O is an open subset of U + ∩ GQ − containing e , and(ii) on O we can express ϕ ba | U + ∩ GQ − as ϕ ba ( z , z , . . . , z r ) = X m ,m ,...,m r ≥ α bm m ··· m r ( z ) m ( z ) m · · · ( z r ) m r (the Taylor expansion of ϕ ba | U + ∩ GQ − at e = (0 , , . . . , O is stable under every inner automorphism of S = { exp λT | λ ∈ R } ,cf. (3.4). For any λ ∈ R and u ∈ O we have k X b =1 e iθ b λ ϕ ba ( u ) v b = ρ (exp λT )( k X b =1 ϕ ba ( u ) v b ) ( ∵ Lemma 3.6-(2))= ρ (exp λT )( ϕ a ( u )) = ϕ a ( u exp( − λT )) ( ∵ exp λT ∈ Q − , (2.22)-(2))= (cid:0) ̺ (exp λT ) ϕ a (cid:1)(cid:0) (exp λT ) u exp( − λT ) (cid:1) ( ∵ exp λT ∈ G , (2.25))= ( e iµ a λ ϕ a ) (cid:0) (exp λT ) u exp( − λT ) (cid:1) ( ∵ Lemma 3.6-(1))= k X b =1 e iµ a λ ϕ ba (cid:0) (exp λT ) u exp( − λT ) (cid:1) v b , and hence e iθ b λ ϕ ba ( u ) = e iµ a λ ϕ ba (cid:0) (exp λT ) u exp( − λT ) (cid:1) . This, together with (ii) and(3.4), yields X m ,m ,...,m r ≥ e i ( θ b − µ a ) λ α bm m ··· m r ( z ) m ( z ) m · · · ( z r ) m r = e i ( θ b − µ a ) λ ϕ ba ( z , z , . . . , z r ) = e i ( θ b − µ a ) λ ϕ ba ( u )= ϕ ba (cid:0) (exp λT ) u exp( − λT ) (cid:1) = ϕ ba ( e in λ z , e in λ z , . . . , e in r λ z r )= X m ,m ,...,m r ≥ e i ( n m + n m + ··· + n r m r ) λ α bm m ··· m r ( z ) m ( z ) m · · · ( z r ) m r in case of u = exp( z E γ + z E γ + · · · + z r E γ r ). Therefore one shows that e i ( θ b − µ a ) λ α bm m ··· m r = e i ( n m + n m + ··· + n r m r ) λ α bm m ··· m r . Differentiating this equation at λ = 0, we deduce that(3.8) ( θ b − µ a ) α bm m ··· m r = ( n m + n m + · · · + n r m r ) α bm m ··· m r for all 1 ≤ b ≤ k and m , m , . . . , m r ≥
0. It follows from n , n , . . . , n r ∈ N and(3.8) that for each b , all the coefficients α bm m ··· m r vanish whenever θ b − µ a N ∪ { } ; and that with respect to m , m , . . . , m r with θ b − µ a = n m + n m + TOPIC ON HOMOGENEOUS VECTOR BUNDLES OVER ELLIPTIC ORBITS 13 · · · + n r m r , the coefficient α bm m ··· m r vanishes even if θ b − µ a ∈ N ∪ { } . Theseimply that ϕ ba ( z , z , . . . , z r ) = X m ,m ,...,m r ≥ α bm m ··· m r ( z ) m ( z ) m · · · ( z r ) m r must be a polynomial function on O of finite degree. Consequently, for each 1 ≤ b ≤ k , ϕ ba ( z , . . . , z r ) can extend uniquely to a polynomial function ϕ ba ′ ( z , . . . , z r )on C r ∼ = U + of finite degree. (cid:3) Remark 3.9.
In Proposition 3.7 we have concluded that for any φ ∈ V ϕ , therestriction φ | U + ∩ GQ − can be continued analytically to U + , without the supposition(s2) in Theorem 3.1.Proposition 3.7 leads to Corollary 3.10.
Let ϕ be any K -finite vector in V G/L for the representation ̺ defined by (2.25) , and let V ϕ be the complex vector subspace of V G/L generated by { ̺ ( k ) ϕ : k ∈ K } . Suppose that (S) there exists a fundamental root system Π △ of △ satisfying (s1) α ( − iT ) ≥ for all α ∈ Π △ , and (s2) g β ⊂ k C for every β ∈ Π △ with β ( T ) = 0 .Then, it follows that ϕ ∈ V ϕ ⊂ F ( V G C /Q − ) . Proof.
Take any φ ∈ V ϕ . By Proposition 3.7 there exists a unique holomorphicmapping φ ′ : U + → V such that φ = φ ′ | U + ∩ GQ − . Proposition 2.7-(3) enables us toconstruct the holomorphic extension φ ′′ : U + Q − → V of φ ′ from φ ′′ ( uq ) := ρ ( q ) − (cid:0) φ ′ ( u ) (cid:1) for ( u, q ) ∈ U + × Q − . Here, it follows from ( U + Q − ∩ GQ − ) = ( U + ∩ GQ − ) Q − , φ = φ ′ | U + ∩ GQ − , φ ∈ V G/L and (2.22)-(2) that φ = φ ′′ on U + Q − ∩ GQ − . Now, Lemma 2.14 and (s2) assure that w β ∈ K for every β ∈ Π △ with β ( T ) = 0.This enables us to obtain ̺ ( w β ) φ ∈ V ϕ , since V ϕ is ̺ ( K )-invariant. Accordingly for each β ∈ Π △ with β ( T ) = 0, thereexists a unique holomorphic mapping ( ̺ ( w β ) φ ) ′′ : U + Q − → V such that ̺ ( w β ) φ = ( ̺ ( w β ) φ ) ′′ on U + Q − ∩ GQ − . Then, we define a holomorphic mapping ˆ φ from O = U + Q − ∪ (cid:0)S β ∈ Π △ with β ( T ) = 0 w − β U + Q − (cid:1) into V as follows:(3.11) ˆ φ ( x ) := ( φ ′′ ( x ) if x ∈ U + Q − , ( ̺ ( w β ) φ ) ′′ ( w β x ) if x ∈ w − β U + Q − . Here O is a dense, domain in G C (cf. Corollary 2.20). Let us confirm that thedefinition (3.11) is well-defined. Corollary 2.9-(1) implies that the intersection GQ − ∩ U + Q − ∩ (cid:0)T β ∈ Π △ with β ( T ) = 0 w − β U + Q − (cid:1) is a non-empty open subset of G C . For any element y of the intersection above andany β ∈ Π △ with β ( T ) = 0 we have w β y ∈ U + Q − and w β y ∈ KGQ − ⊂ GQ − ; andthus ( ̺ ( w β ) φ ) ′′ ( w β y ) = ( ̺ ( w β ) φ )( w β y ) (2.25) = φ ( y ) = φ ′′ ( y )in terms of w β y, y ∈ U + Q − ∩ GQ − . For this reason (3.11) is well-defined bythe theorem of identity and it follows that φ = ˆ φ on O ∩ GQ − . From Corollary2.20, there exists the analytic continuation ˆ φ ′ : G C → V of ˆ φ : O → V . This ˆ φ ′ satisfies ˆ φ ′ ( xq ) = ρ ( q ) − ( ˆ φ ′ ( x )) for all ( x, q ) ∈ G C × Q − , by the theorem of identity, φ = ˆ φ ′ | GQ − , (2.22)-(2) and φ ∈ V G/L . Consequently it is immediate from (2.21)that ˆ φ ′ ∈ V G C /Q − , so that φ = ˆ φ ′ | GQ − = F ( ˆ φ ′ ) ∈ F ( V G C /Q − ). This provides uswith V ϕ ⊂ F ( V G C /Q − ). (cid:3) Now, let us demonstrate Theorem 3.1.
Proof of Theorem . By Lemma 3.5 and Remark 2.23 it suffices to conclude(3.12) V G/L ⊂ F ( V G C /Q − ) . Let ( V G/L ) K be the set of K -finite vectors in V G/L for the representation ̺ definedby (2.25). From Corollary 3.10 we obtain(3.13) ( V G/L ) K ⊂ F ( V G C /Q − ) . Now, let ψ be an arbitrary element of V G/L . On the one hand; Proposition 2.27assures that there exists a sequence { ϕ n } ∞ n =1 ⊂ ( V G/L ) K satisfyinglim n →∞ d ( ψ, ϕ n ) = 0 . On the other hand; since V G/L = ( V G/L , d ) is a Hausdorff topological vector spaceand dim C F ( V G C /Q − ) = dim C V G C /Q − < ∞ , it turns out that F ( V G C /Q − ) is closedin V G/L . Thus, it follows from (3.13) that ψ = lim n →∞ ϕ n ∈ F ( V G C /Q − ), so that(3.12) holds. (cid:3) Examples
Let us give some examples which satisfy the supposition (S) in Theorem 3.1 andan example which does not so. Recall that the supposition is as follows:(S) there exists a fundamental root system Π △ of △ satisfying(s1) α ( − iT ) ≥ α ∈ Π △ , and(s2) g β ⊂ k C for every β ∈ Π △ with β ( T ) = 0. Example 4.1 ( G/L = SU ( p, q ) /S ( U ( h ) × U ( p − h, q )), p + q ≥
2, 0 < h < p ) . Let G C := SL ( p + q, C ), G := SU ( p, q ), g u := su ( p + q ) and h R := x O . . . O x p + q x l ∈ R , p + q X l =1 x l = 0 , where p + q ≥
2. Denote by △ = △ ( g C , h C ) the root system of g C relative h C , definesimple roots α k ∈ △ (1 ≤ k ≤ p + q −
1) as α k (cid:16) z O . . . O z p + q (cid:17) := z k − z k +1 , TOPIC ON HOMOGENEOUS VECTOR BUNDLES OVER ELLIPTIC ORBITS 15 and set Π △ := { α k } p + q − k =1 . Here, the dual base { Z k } p + q − k =1 of Π △ = { α k } p + q − k =1 is Z k = 1 p + q (cid:18) ( p + q − k ) I k OO − kI p + q − k (cid:19) for 1 ≤ k ≤ p + q − , where I n is the unit matrix of degree n . Let T h := iZ h , 0 < h < p , and k := (cid:26)(cid:18) A p OO D q (cid:19) ∈ g A p : p × p matrix, D q : q × q matrix (cid:27) , p := (cid:26)(cid:18) O B p × q C q × p O (cid:19) ∈ g B p × q : p × q matrix, C q × p : q × p matrix (cid:27) . In the setting above, it follows that T h is an elliptic element of g , i h R is a maximaltorus of g u containing T h , g = k ⊕ p , g u = k ⊕ i p and (s1) α ( − iT h ) ≥ α ∈ Π △ . Moreover,(1) for β ∈ Π △ = { α k } p + q − k =1 , β ( T h ) = 0 if and only if β = α h ,(2) g α h = span C { E h,h +1 } ,where g α h is the root subspace of g C for α h and E h,h +1 is the matrix whose ( h, h +1)-element is 1 and whose other elements are all 0. Since 0 < h < p , we have (s2) g α h ⊂ k C . For this reason, the supposition (S) in Theorem 3.1 holds for this example. In-cidentally, L = C G ( T h ) = S ( U ( h ) × U ( p − h, q )), and Theorem 3.1 implies thatthe complex Lie algebra O ( T , ( G/L )) of holomorphic vector fields on
G/L = SU ( p, q ) /S ( U ( h ) × U ( p − h, q )) is isomorphic to sl ( p + q, C ), where p + q ≥ < h < p .Unfortunately, there are examples of elliptic orbits to which we cannot applyTheorem 3.1. Example 4.2.
The supposition (S) in Theorem 3.1 cannot hold for any symmetricbounded domain D in C n at all.Let us explain the reason why. In order to do so, we consider an elliptic orbit G/L = G/C G ( T ) in the setting of Subsection 3.1, and put u := [ T, g ]. Sincead T ∈ End( g ) is semisimple and l = c g ( T ) one can decompose g as g = l ⊕ u , andfurthermore decompose it as follows: g = ( k ∩ l ) ⊕ ( p ∩ l ) ⊕ ( k ∩ u ) ⊕ ( p ∩ u )because of T ∈ k . Then, Lemma 2.14 tells us that k ∩ u = { } is a necessary condition for the (s2) to hold. However, if G/L is a symmetricbounded domain in C n (where G is the identity component of Hol( G/L )), then k ∩ l = k , p ∩ l = { } , k ∩ u = { } and p ∩ u = p . For this reason, the supposition(S) cannot hold for the D at all.The following example is interesting, we think: Example 4.3 ( G/L = G / ( SL (2 , R ) · T )) . Let g C be the exceptional complexsimple Lie algebra ( g ) C of the type G . Assume that the Dynkin diagram of △ = △ ( g C , h C ) is as follows (cf. Bourbaki [3, p.289] ): There is a minor misprint in [3]: p.289, ↓
9, Add α to (II) Positive roots. α ❡ (cid:0)(cid:0)❅❅ α ❡ g C :First of all, let us set a non-compact real form g of g C . Define a compact real form g u of g C by h R := span R { H α | α ∈ △} , g u := i h R ⊕ L α ∈△ span R { E α − E − α } ⊕ span R { i ( E α + E − α ) } , and denote by { Z , Z } ⊂ h R the dual base of Π △ = { α , α } (cf. Paragraph 2.3.1 for H α , E α ). By use of this Z we set(4.4) θ := exp π ad( iZ ) . Then θ is an involutive automorphism of the complex Lie algebra g C such that θ ( g u ) ⊂ g u , and we define a non-compact real form g ⊂ g C in the following way: k := { X ∈ g u | θ ( X ) = X } , i p := { Y ∈ g u | θ ( Y ) = − Y } , g := k ⊕ p . Remark here that g u = k ⊕ i p , k = sp (1) ⊕ sp (1) and g = g ; besides, k C = { Z ∈ g C | θ ( Z ) = Z } , where k C is the complex subalgebra of g C generated by k . α ❡ − α − α ❡ k C :In this setting, a given T ∈ i h R is an elliptic element of g and we know that for l := c g ( T ),(a) l = sl (2 , R ) ⊕ t in case of T = i ( Z − Z ),(b) l = sl (2 , R ) ⊕ t in case of T = i ( Z − Z ).cf. Proposition 5.5 [1, p.1157]. We investigate the cases (a) and (b), individually.Case (a): Let T := i ( Z − Z ) and Π a := { α + α , − α − α } . Then Π a is afundamental root system of △ such that (s1) α ( − iT ) ≥ α ∈ Π a . Indeed, itfollows from α k ( Z j ) = δ kj that (2 α + α )( − iT ) = 0 and ( − α − α )( − iT ) = 1.2 α + α ❡ (cid:0)(cid:0)❅❅ − α − α ❡ Π a :Since (4.4) yields θ ( E − α − α ) = E − α − α , we have (s2) g − α − α ⊂ k C . There-fore the supposition (S) in Theorem 3.1 holds in this case.Case (b): Let T := i ( Z − Z ) and Π b := { α , − α − α } . Then, Π b is afundamental root system of △ such that (s1) α ( − iT ) = 1 and ( − α − α )( − iT ) =0. α ❡ (cid:0)(cid:0)❅❅ − α − α ❡ Π b :From (4.4) one obtains (s2) θ ( E α ) = E α . Hence the supposition (S) in Theorem3.1 holds in this case, also.We end this paper with a comment on Example 4.3, G/L = G / ( SL (2 , R ) · T ).In both the cases (a) and (b), the supposition (S) in Theorem 3.1 holds. So,in each case Theorem 3.1 implies that the complex Lie algebra O ( T , ( G/L )) ofholomorphic vector fields on
G/L is isomorphic to O ( T , ( G C /Q − )). Then,a. O ( T , ( G/L )) is isomorphic to ( g ) C in case (a); but, in contrast,b. O ( T , ( G/L )) is isomorphic to so (7 , C ) in case (b).cf. the proof of Theorem 7.1 in Oniˇsˇcik [11, p.238–239]. TOPIC ON HOMOGENEOUS VECTOR BUNDLES OVER ELLIPTIC ORBITS 17
Acknowledgements
The author would like to express his sincere gratitude to Professor Soji Kaneyukifor the valuable suggestions in Kyoto, 24 October 2005.
References
Division of Mathematical Sciences, Faculty of Science and TechnologyOita University, 700 Dannoharu, Oita-shi, Oita 870-1192, JAPAN
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