Geometry of invariant domains in complex semi-simple Lie groups
aa r X i v : . [ m a t h . C V ] S e p GEOMETRY OF INVARIANT DOMAINS IN COMPLEX SEMI-SIMPLE LIEGROUPS
CHRISTIAN MIEBACH
Abstract.
We investigate the joint action of two real forms of a semi-simple complex Lie group U C byleft and right multiplication. After analyzing the orbit structure, we study the CR structure of closedorbits. The main results are an explicit formula of the Levi form of closed orbits and the determinationof the Levi cone of generic orbits. Finally, we apply these results to prove q –completeness of certaininvariant domains in U C . Introduction
Let U C be a connected semi-simple complex Lie group with compact real form U which is given bythe Cartan involution θ . Let us assume that there are two anti-holomorphic involutive automorphisms σ and σ of U C which both commute with θ and let G j = Fix( σ j ), j = 1 ,
2, be the correspondingreal forms of U C . The group G × G acts on U C by ( g , g ) · z := g zg − . In this paper we investigatecomplex-analytic properties of certain ( G × G )–invariant domains in U C through the intrinsic Levi formof closed ( G × G )–orbits.If σ = σ = θ , then we discuss the ( U × U )–action on U C by left and right multiplication. Every( U × U )–orbit intersects the set exp( i t ) in the orbit of the Weyl group W := N U ( t ) / Z U ( t ) where t is amaximal torus in u . In [Las78] Lassalle showed that every bi-invariant domain Ω ⊂ U C is of the form U exp( iω ) U for a W –invariant domain ω ⊂ t and that Ω is a domain of holomorphy if and only if ω isconvex. In [AL92] Azad and Loeb proved the stronger statement that a ( U × U )–invariant function Φ onΩ is plurisubharmonic if and only if the W –invariant function ϕ : t → R , ϕ ( η ) := Φ (cid:0) exp( iη ) (cid:1) , is convex.In the case that σ = σ and G is a real semi-simple Lie group of Hermitian type there is a distin-guished ( G × G )–invariant in U C , namely the open complex Ol’shanski˘ı semi-group. According to aresult of Neeb ([Nee98]) the open Ol’shanski˘ı semi-group is a domain of holomorphy.Although the above results are statements about complex-analytic properties of domains in complexStein manifolds, the method of their proofs is representation-theoretic. A different approach to the studyof ( G × G )–invariant domains in U C by analytic methods was made by Fels an Geatti in [FG98]. There,Fels and Geatti gave explicit formulas for the intrinsic Levi form of a closed orbit M z := ( G × G ) · z of maximal dimension in U C (in the following called a generic orbit) and determined the Levi cone of M z , which enabled them to decide whether or not there may exist a bi-invariant domain of holomorphycontaining z in its boundary.The main results in this paper are an explicit formula for the intrinsic Levi form of an arbitrary closed( G × G )–orbit in U C and the determination of the Levi cone of a generic orbit. We use a theorem ofMatsuki ([Mat97]) in order to obtain a parameterization of closed ( G × G )–orbits by certain Cartanalgebras in the Lie algebra u C = Lie( U C ). More precisely, there are finitely many Cartan algebras c j suchthat the closed orbits are precisely those intersecting a set of the form C j = n exp( i c j ), where the element n can be chosen from a fixed torus in U . It turns out that the weight space decomposition of u C withrespect to c j is well-suited to describe the CR structure of closed orbits intersecting C j . In particular,the complex tangent space of such an orbit can be identified with a direct sum of weight spaces and theintrinsic Levi form of a closed orbit is determined by the Lie bracket of certain weight vectors togetherwith a coefficient which depends on the intersection of the orbit with C j (Theorem 3.11). From this fact it can be derived that the CR structures of closed orbits which belong to the same set C j have verysimilar properties.The method used here for the derivation of explicit formulas for the Levi form is different from theone used in [FG98]. While Fels and Geatti found explicit local extensions of complex tangent vectorsto CR vector fields on a generic orbit and computed their Lie brackets, the approach used here avoidsthese technical difficulties by pulling back the CR structure of the orbit into the Lie algebra of G × G where the Levi form can be determined via Lie-theoretic methods. In particular, we obtain a new prooffor their results in the case G = G .A finer analysis of the weight space decomposition of u C with respect to c j reveals that it has propertiesvery close to a root space decomposition. The most important one is the existence of sl (2)–triples whichenables us to determine the Levi cone of generic orbits by essentially the same method as in [FG98](Theorem 3.17).In Section 4 we give several applications of the results obtained so far. First we use the criterionfrom [FG98] together with the knowledge of the Levi cone in order to decide which ( G × G )–invariantdomains containing a generic orbit in their boundary can be Stein. Secondly, we classify and study therank one case in some detail since this case provides a class of examples where the methods and resultsbecome most transparent. This is due to the facts that complex-analytic properties of smooth domainsin Stein manifolds are determined by the classical Levi form of their boundaries and that in the rank onecase the boundaries of almost all invariant domains coincide with orbits of hypersurface type. Finally, wedefine a ( G × G )–invariant domain Ω ⊂ U C which is the right analogon of the open complex Ol’shanski˘ısemi-group in the case G = G . We prove that the classical Levi form of a ( G × G )–invariant smoothfunction at a point z ∈ Ω splits into a contribution coming from the complex tangent space of ( G × G ) · z and a contribution due to a transversal slice. Via this splitting we construct a strictly q –convex functionon Ω which goes to infinity at ∂ Ω, and hence conclude that Ω is q –complete. Notation. If ϕ is an automorphism of a Lie group G , then by abuse of notation we write ϕ also for thederived automorphism of g = Lie( G ). Acknowledgment.
This paper is a modified version of the author’s Ph.D. thesis [Mie07]. The supportby a Promotionsstipendium of the Studienstiftung des deutschen Volkes and by SFB/TR 12 of the DFGis gratefully acknowledged. 2.
The ( G × G ) –Action on U C Compatible real forms.
Let U be a connected semi-simple compact Lie group. Then its universalcomplexification U C is a connected semi-simple complex Lie group, and hence carries a unique structureof a linear algebraic group (compare [Che70]). The map Φ : U × i u → U C , ( u, ξ ) u exp( ξ ), is a real-analytic diffeomorphism, called the Cartan decomposition of U C . Furthermore, the map θ : U C → U C , θ (cid:0) u exp( ξ ) (cid:1) := u exp( − ξ ), is an anti-holomorphic involutive automorphism with U = Fix( θ ), called theCartan involution of U C corresponding to the compact real form U . Proofs of these facts can be founde. g. in [Kna02].Let σ and σ be two anti-holomorphic involutive automorphisms of U C , which both commute withthe Cartan involution θ . The fixed point set G j := Fix( σ j ) is a real form of U C for j = 1 ,
2. Theassumption that σ j commutes with θ implies that the Cartan decomposition of U C restricts to a real-analytic diffeomorphism K j × p j → G j , where K j := G j ∩ U and p j := g j ∩ i u hold. Thus the real form G j is a compatible subgroup of U C in the sense of [HS07]. In particular, K j is a deformation retract of G j . Remark.
Since G j is closed, the group K j is compact and hence a maximal compact subgroup of G j .Thus G j has only finitely many connected components. If the group U C is simply-connected, it followsfrom [Ste68] that G j is connected.The product group G × G acts on U C by left and right multiplication, i. e. we define( g , g ) · z := g zg − where g j ∈ G j and z ∈ U C . EOMETRY OF INVARIANT DOMAINS IN COMPLEX SEMI-SIMPLE LIE GROUPS 3
Definition 2.1.
We say that an element z ∈ U C is regular (with respect to G × G ) if the orbit( G × G ) · z has maximal dimension. The element z is called strongly regular (with respect to G × G ),if it is regular and if ( G × G ) · z is closed. We write U C r and U C sr for the sets of regular and stronglyregular elements, respectively. Finally, we call the orbit ( G × G ) · z generic if z is strongly regular. Remark. (a) If we consider the action of U C on itself given by conjugation, then Definition 2.1 yields theusual notion of (strongly) regular elements in linear algebraic groups (compare [Hum95]).(b) The subsets U C r and U C sr are invariant under G × G . The set U C sr can be proven to be open anddense in U C which justifies the terminology “generic orbit”.(c) In [Mat97] an element z ∈ U C is called regular semi-simple if the automorphism Ad( z − ) σ Ad( z ) σ is semi-simple and if the Lie algebra g ∩ Ad( z − ) g is Abelian. It can be shown that an element isregular semi-simple in Matsuki’s sense if and only if it is strongly regular.2.2. The isotropy representation.
The following proposition is crucial. For convenience of the readerwe give a short proof which makes use of the complex-analytic structure of U C . Proposition 2.2.
Let z ∈ U C be a point such that the orbit M z := ( G × G ) · z is closed. Then theisotropy group ( G × G ) z is real-reductive and the isotropy representation of ( G × G ) z on T z U C iscompletely reducible.Proof. Since U C is a Stein manifold, there exists a smooth strictly plurisubharmonic exhaustion function ρ : U C → R . By compactness of U we can average ρ using the Haar measure and hence assume that ρ is ( U × U )–invariant. It follows that ω := i∂∂ρ is a ( U × U )–invariant K¨ahler form on U C with respectto which U × U acts in a Hamiltonian fashion. The last statement means that there exists a ( U × U )–equivariant momentum map µ : U C → u ∗ ⊕ u ∗ . Since the group G × G is compatible with the Cartandecomposition of U C × U C , we can restrict µ to the subspace ( i p ) ∗ ⊕ ( i p ) ∗ and obtain the restrictedmomentum map µ i p : U C → ( i p ) ∗ ⊕ ( i p ) ∗ . According to [HS07] this restricted momentum map encodesa lot of information about the ( G × G )–action on U C from which we need the following.(a) A ( G × G )–orbit is closed in U C if and only if it intersects M i p := µ − i p (0) non-trivially (Proposi-tion 11.2 in [HS07]).(b) If z ∈ M i p , then the isotropy group ( G × G ) z is a compatible subgroup of U C × U C and hence real-reductive (Lemma 5.5 in [HS07]). Together with the previous statement this implies that isotropygroups of closed orbits are real-reductive.(c) If z ∈ M i p , then the isotropy representation is completely reducible (Corollary 14.9 in [HS07]).Hence, the proposition is proven. (cid:3) In the rest of this subsection we will have a closer look at the isotropy representation. Every element( ξ , ξ ) ∈ g ⊕ g induces the tangent vector ddt (cid:12)(cid:12)(cid:12)(cid:12) (cid:0) exp( tξ ) z exp( − tξ ) (cid:1) = ddt (cid:12)(cid:12)(cid:12)(cid:12) (cid:0) z exp( t Ad( z − ) ξ ) exp( − tξ ) (cid:1) = ( ℓ z ) ∗ (cid:0) Ad( z − ) ξ − ξ (cid:1) ∈ T z U C , (2.1)where ℓ z denotes left multiplication with z ∈ U C . These tangent vectors span the tangent space of the( G × G )–orbit through z , i. e. we obtain T z M z = ( g ⊕ g ) · z = (cid:8) ( ℓ z ) ∗ ξ ; ξ ∈ g + Ad( z − ) g (cid:9) .Let ρ denote the isotropy representation of ( G × G ) z on T z U C . One checks directly that the isotropygroup at z ∈ U C is given by( G × G ) z = (cid:8) ( zg z − , g ); g ∈ G ∩ z − G z (cid:9) . Consequently, we may identify ( G × G ) z with G ∩ z − G z via the isomorphism Φ : G ∩ z − G z → ( G × G ) z , g ( zgz − , g ). Similarly, we will identify the tangent space T z M z with g + Ad( z − ) g via CHRISTIAN MIEBACH ( ℓ z ) ∗ . We conclude from ρ (cid:0) Φ( g ) (cid:1) ( ℓ z ) ∗ ξ = ddt (cid:12)(cid:12)(cid:12)(cid:12) ( zgz − , g ) · (cid:0) z exp( tξ ) (cid:1) = ddt (cid:12)(cid:12)(cid:12)(cid:12) (cid:0) zg exp( tξ ) g − (cid:1) = ddt (cid:12)(cid:12)(cid:12)(cid:12) z exp (cid:0) t Ad( g ) ξ (cid:1) = ( ℓ z ) ∗ Ad( g ) ξ that the map ( ℓ z ) ∗ intertwines the adjoint representation of G ∩ z − G z on u C with the isotropy repre-sentation of ( G × G ) z on T z U C modulo Φ. We summarize our considerations in the following Proposition 2.3.
Modulo the isomorphism Φ the isotropy representation of ( G × G ) z on T z U C isequivalent to the adjoint representation of G ∩ z − G z on u C . The orbit structure theorem.
We review the main results of [Mat97] in order to describe the orbitstructure of the ( G × G )–action on U C . A proof of Matsuki’s theorem which relies on the momentummap techniques developed in [HS07] can be found in [Mie07].Let a be a maximal Abelian subspace of p ∩ p and let t be a maximal torus in the centralizer of a in k ∩ k . It follows that c := t ⊕ a is a maximally non-compact θ –invariant Cartan subalgebra of g ∩ g . Remark.
By maximality of a the group A c := exp( i a ) is a compact torus in U . Definition 2.4.
A subset of the form C = n exp( i c ) ⊂ U C is called a standard Cartan subset, if n ∈ A c and c = t ⊕ a is a θ –stable Cartan subalgebra of g ∩ Ad( n − ) g such that t ⊃ t , a ⊂ a and dim c = dim c hold. The standard Cartan subset C := exp( i c ) is called the fundamental Cartan subset.We call two standard Cartan subsets equivalent if there is a generic ( G × G )–orbit which intersectsboth non-trivially. Let { C j } j ∈ J be a complete set of representatives for the equivalence classes. For each j ∈ J we define the groups N K × K ( C j ) := (cid:8) ( k , k ) ∈ K × K ; k C j k − = C j (cid:9) , Z K × K ( C j ) := (cid:8) ( k , k ) ∈ K × K ; k zk − = z for all z ∈ C j (cid:9) , and W K × K ( C j ) := N K × K ( C j ) / Z K × K ( C j ). Remark.
The group W K × K ( C j ) is finite for each j ∈ J . Theorem 2.5 (Matsuki) . The set J is finite and we have U C cl = [ j ∈ J G C j G and U C sr = ˙ [ j ∈ J G ( C j ∩ U C sr ) G , where U C cl := { z ∈ U C ; ( G × G ) · z is closed } . Moreover, each generic ( G × G ) –orbit intersects C j ina W K × K ( C j ) –orbit.Remark. If G = G , then let c , . . . , c k be a complete set of representatives for the equivalence classes ofCartan subalgebras of g . We can assume without loss of generality that each c j is θ –stable. Let { n j,l } bea set of representatives for the Weyl group corresponding to c j . It can be shown that the sets n j,l exp( i c j )exhaust the equivalence classes of standard Cartan subsets for the ( G × G )–action on U C . Hence, weobtain Bremigan’s theorem ([Bre96]).2.4. The weight space decomposition.
Let C = n exp( i c ) be a standard Cartan subset. In thissubsection we discuss the weight space decomposition u C = M λ ∈ Λ u C λ of u C with respect to the Cartan subalgebra c ⊂ g ∩ Ad( n − ) g . Here, we have written Λ = Λ( u C , c ) forthe set of weights and u C λ for the weight space corresponding to the weight λ . We say that the weight EOMETRY OF INVARIANT DOMAINS IN COMPLEX SEMI-SIMPLE LIE GROUPS 5 λ is real (respectively imaginary) if λ = 0 and λ ( c ) ⊂ R (respectively λ ( c ) ⊂ i R ) holds. A non-zeroweight which is neither real nor imaginary is called complex. We write Λ r , Λ i and Λ c for the sets of real,imaginary and complex weights, and obtainΛ \ { } = Λ r ˙ ∪ Λ i ˙ ∪ Λ c . Remark.
We extend the weight λ by C –linearity to the complexified Cartan algebra c C . Since λ ( t ) ⊂ i R and λ ( a ) ⊂ R hold for all λ ∈ Λ, we conclude that the weights are real-valued on i t ⊕ a .Since n ∈ A c , the automorphism τ n := Ad( n − ) σ Ad( n ) σ ∈ Aut( u C ) is unitary with respect to theHermitian inner product h ξ , ξ i := − B u C (cid:0) ξ , θ ( ξ ) (cid:1) , where B u C is the Killing form of u C . Consequently, τ n is semi-simple with eigenvalues in the unit circle S . Since τ n leaves c pointwise fixed, each weightspace u C λ is invariant under τ n . Hence, following [Mat97] we obtain the finer decomposition(2.2) u C = M ( λ,a ) ∈ e Λ u C λ,a , where u C λ,a := (cid:8) ξ ∈ u C λ ; τ n ( ξ ) = aξ (cid:9) and e Λ := (cid:8) ( λ, a ) ∈ Λ × S ; u C λ,a = { } (cid:9) . The elements of e Λ arecalled the extended weights, and (2.2) is called the extended weight space decomposition.
Remark.
Since c is a Cartan subalgebra of g ∩ Ad( n − ) g , we conclude u C , = c C .We collect some properties of the extended weight space decomposition in the following Lemma 2.6. (1) The Cartan involution θ maps u C λ,a onto u C − λ,a − . In particular, if ( λ, a ) is an extendedweight, then ( − λ, a − ) is an extended weight, too.(2) We have B u C (cid:0) u C λ,a , u C µ,b (cid:1) = 0 unless ( λ, a ) = ( − µ, b − ) ∈ e Λ .(3) Let ξ λ,a ∈ u C λ,a with k ξ λ,a k = 1 be given and let η λ,a := − (cid:2) ξ λ,a , θ ( ξ λ,a ) (cid:3) . Then we have B u C ( η λ,a , η ) = λ ( η ) for all η ∈ c . In particular, η λ,a does not depend on the element a ∈ S , i. e. η λ,a = η λ,a ′ =: η λ for all ( λ, a ) , ( λ, a ′ ) ∈ e Λ .(4) We have [ ξ λ,a , ξ ] = B u C ( ξ λ,a , ξ ) η λ for all ξ ∈ u C − λ,a − .Proof. In order to prove the first claim let η = η t + η a ∈ t ⊕ a = c and ξ ∈ u C α,λ be given and consider (cid:2) η, θ ( ξ ) (cid:3) = θ (cid:2) θ ( η ) , ξ (cid:3) = θ [ η t − η a , ξ ] = θ (cid:0) λ ( η t ) ξ (cid:1) − θ (cid:0) λ ( η a ) ξ (cid:1) = − λ ( η ) θ ( ξ ) . Here we used the facts that λ ( t ) ⊂ i R while λ ( a ) ⊂ R and that θ is C –anti-linear. Since θ commutes with τ n , we conclude τ n θ ( ξ ) = θτ n ( ξ ) = θ ( aξ ) = aθ ( ξ ) = a − θ ( ξ ) , which proves the first claim.The second claim follows from the fact that the Killing form B u C is invariant under Aut( u C ).In order to prove the third one we compute B u C ( η λ,a , η ) = − B u C (cid:0) [ ξ λ,a , θ ( ξ λ,a )] , η (cid:1) = B u C (cid:0) θ ( ξ λ,a ) , [ ξ λ,a , η ] (cid:1) = − λ ( η ) B u C (cid:0) ξ λ,a , θ ( ξ λ,a ) (cid:1) = λ ( η ) k ξ λ,a k = λ ( η ) . The last claim is proven in the same way as Lemma 2.18(a) in [Kna02]. (cid:3)
Standard arguments from Lie theory (see for example Chapter II.4 in [Kna02]) lead to the followingresult.
Proposition 2.7. (1) Let λ = 0 . After a suitable normalization the elements η λ , ξ λ,a and θ ( ξ λ,a ) forman sl (2) –triple.(2) If λ = 0 , then we have dim C u C λ,a = 1 and dim C u C mλ,a m = 0 for all m ≥ .(3) The set Λ \ { } of non-zero weights fulfills the axioms of an abstract root system in ( i t ⊕ a ) ∗ .(4) Let λ, µ ∈ Λ \ { } such that λ + µ ∈ Λ \ { } holds. Then we have [ u C λ , u C µ ] = u C λ + µ . CHRISTIAN MIEBACH CR Geometry of Closed Orbits
Preliminaries from CR geometry.
In this subsection we will review the basic definitions andfacts from the theory of CR submanifolds as far as they are needed later on. For more details andcomplete proofs we refer the reader to the textbooks [BER99] and [Bog91].Let Z be a complex manifold with complex structure J . A real submanifold M of Z is called a Cauchy-Riemann or CR submanifold if the dimension of the complex tangent space H p M := T p M ∩ J p T p M doesnot depend on the point p ∈ M . In this case, the set HM := S p ∈ M H p M is a smooth subbundle of thetangent bundle T M invariant under the complex structure J , called the complex tangent bundle of M .A CR submanifold M ⊂ Z is called generic if T p M + J p T p M = T p Z holds for all p ∈ M . For example,every smooth real hypersurface in Z is a generic CR submanifold of Z . Remark.
Since the group G × G acts by holomorphic transformations on U C , each closed ( G × G )–orbit is a CR submanifold of U C . Since the ( G × G )–action extends to a transitive ( U C × U C )–actionon U C , each closed orbit is moreover generic as a CR submanifold.A smooth section in HM is called a CR vector field on M . A smooth map f from M into a CRsubmanifold M ′ ⊂ ( Z ′ , J ′ ) is called a CR map if f ∗ maps HM into HM ′ and if f ∗ J = J ′ f ∗ holds. A CRfunction on M is a CR map M → C , where C is equipped with its usual structure as complex manifold.For each CR submanifold M ⊂ Z one can define the intrinsic Levi form, which generalizes the classicalLevi form of a smooth hypersurface. Definition 3.1.
The Levi form of M at the point p is the map L p : H p M × H p M → T C p M/H C p M definedby L p ( v, w ) := (cid:18) i V, W ] p −
12 [
V, JW ] p (cid:19) mod H C p M, where V and W are CR vector fields on M with V p = v and W p = w . Remark.
One can show that the intrinsic Levi form is well-defined, i. e. that it does not depend on thechoice of CR extensions of v, w ∈ H p M (compare [Bog91]).The Levi cone C p of M at p is by definition the closed convex cone generated by the vectors L z ( v, v )where v runs through H p M . Because of L p ( v, w ) = L p ( w, v ) the Levi cone is contained in T p M/H p M .The Levi cone generalizes the signature of the classical Levi form of a hypersurface. Its significance stemsfrom the fact that it governs the local extension of CR functions on M to holomorphic functions on Z . Theorem 3.2 (Boggess, Polking) . Let M be a generic CR submanifold of a complex manifold Z andlet us assume that the Levi cone at some point p ∈ M satisfies C p ( M ) = T p M/H p M . Then, for eachneighborhood ω of p in M there exists a neighborhood Ω of p in Z satisfying Ω ∩ M ⊂ ω which has theproperty that every CR function on Ω ∩ M extends to a unique holomorphic function on Ω . A proof of this theorem can be found in [Bog91].3.2.
The complex tangent space of a closed orbit.
Let z ∈ U C be given such that the orbit M z = ( G × G ) · z is closed in U C . By Matsuki’s theorem we can assume that there is a standard Cartansubset C = n exp( i c ) which contains z = n exp( iη ). We define e Λ( z ) := (cid:8) ( λ, a ) ∈ e Λ; ae − iλ ( η ) = 1 (cid:9) and set τ z := Ad( z − ) σ Ad( z ) σ ∈ Aut( u C ). Lemma 3.3.
The automorphism τ z is semi-simple and we have Fix( τ z ) = (cid:0) g ∩ Ad( z − ) g (cid:1) C = M ( λ,a ) ∈ e Λ( z ) u C λ,a . EOMETRY OF INVARIANT DOMAINS IN COMPLEX SEMI-SIMPLE LIE GROUPS 7
Proof.
The first equality is a consequence of [Mat97], p. 57. In order to prove the second one let ξ = P ( λ,a ) ξ λ,a be an arbitrary element of u C . Then we have τ z ( ξ ) = Ad( z − ) σ Ad( z ) σ ( ξ ) = Ad (cid:0) exp( − iη ) (cid:1) τ n Ad (cid:0) exp( − iη ) (cid:1) ξ = Ad (cid:0) exp( − iη ) (cid:1) τ n X ( λ,a ) e − iλ ( η ) ξ λ,a = Ad (cid:0) exp( − iη ) (cid:1) X ( λ,a ) ae − iλ ( η ) ξ λ,a = X ( λ,a ) ae − iλ ( η ) ξ λ,a . This proves that τ z is semi-simple. Moreover, τ z ( ξ ) = ξ holds if and only if ξ λ,a = 0 for all ( λ, a ) / ∈ e Λ( z ). (cid:3) Since g ∩ Ad( z − ) g is isomorphic to the Lie algebra of ( G × G ) z , we obtain the following charac-terization of strongly regular elements in terms of the extended weights as a corollary. Theorem 3.4.
We have codim R ( G × G ) · z = dim R c + ( e Λ( z ) − . The element z is strongly regularif and only if e Λ( z ) = (cid:8) (0 , (cid:9) holds. This implies that the codimension of a generic orbit coincides withthe rank of the real-reductive Lie algebra g ∩ g . Finally we describe the tangent space T z M z in terms of the extended weight space decomposition. Theorem 3.5.
Under the map ( ℓ z ) ∗ the tangent space T z M z is isomorphic to g + Ad( z − ) g = (cid:0) g ∩ Ad( z − ) g (cid:1) ⊕ M ( λ,a ) / ∈ e Λ( z ) u C λ,a . In particular, the complex tangent space of ( G × G ) · z is isomorphic to L ( λ,a ) / ∈ e Λ( z ) u C λ,a .Remark. From now on we will identify the quotient T C z M/H C z M with R C z M := ( ℓ z ) ∗ (cid:0) g ∩ Ad( z − ) g (cid:1) C .It follows that these spaces are isomorphic as ( G × G ) z –modules. Proof of Theorem 3.5.
Since τ z is semi-simple, we conclude from Lemma 1(i) in [Mat97] that u C = i (cid:0) g ∩ Ad( z − ) g (cid:1) ⊕ (cid:0) g + Ad( z − ) g (cid:1) holds. Moreover, one checks directly that this decomposition is orthogonal with respect to the real partof the Killing form B u C . Similarly, we have the decomposition u C = Fix( τ z ) ⊕ Fix( τ z ) ⊥ , Fix( τ z ) = (cid:0) g ∩ Ad( z − ) g (cid:1) C , where the orthogonal complement Fix( τ z ) ⊥ with respect to B u C is the sum of the τ z –eigenspaces corre-sponding to eigenvalues = 1. These observations imply g + Ad( z − ) g = (cid:0) g ∩ Ad( z − ) g (cid:1) ⊕ Fix( τ z ) ⊥ . Since the same argument as the one in the proof of Lemma 3.3 implies the equalityFix( τ z ) ⊥ = M ( λ,a ) / ∈ e Λ( z ) u C λ,a , the theorem is proven. (cid:3) Pulling back the Levi form into the Lie algebra.
As abbreviation we put G := G × G inthis subsection. Consequently, we have g := Lie( G ) = g ⊕ g .As we have remarked above, every closed G –orbit M z = G · z is a generic CR submanifold of U C . Let π z : g → g · z = T z M z be the differential of the orbit map. By Equation (2.1) the map π z is given by π z ( ξ , ξ ) = ( ℓ z ) ∗ (cid:0) Ad( z − ) ξ − ξ (cid:1) . In this subsection we will pull back the CR structure of M z into the Lie algebra g and compute the Leviform of M z via this pull back. The following proposition is essential. CHRISTIAN MIEBACH
Proposition 3.6.
We have the G z –invariant decomposition g = g z ⊕ q z , and q z and T z M z are isomorphicas G z –spaces where the isomorphism is given by e π z := π z | q z . Since the complex tangent space H z M z isinvariant under G z , we obtain the G z –invariant decomposition q z = R ( q z ) ⊕ H ( q z ) where H ( q z ) := e π − z ( H z M z ) and R ( q z ) := e π − z ( R z M z ) .Proof. We only have to show that the adjoint representation of G z on g is completely reducible. Thisfollows from Proposition 2.2 since G z is conjugate to a compatible subgroup of U C × U C if the orbit G · z = M z is closed. (cid:3) Proposition 3.7.
The Levi form L z : H z M z × H z M z → R C z M z is given by (3.1) L z ( v, w ) = π z (cid:18) i (cid:2)e π − z ( v ) , e π − z ( w ) (cid:3) − (cid:2)e π − z ( v ) , e π − z ( iw ) (cid:3)(cid:19) mod H C z M z . Proof.
Let v, w ∈ H z M z be given and let V, W be CR vector fields on M z with V z = v and W z = w .Since the orbit map G → M z is a G z –principal bundle, there exist projectable vector fields e V and f W on G with e V e = e π − z ( v ) and f W e = e π − z ( w ) such that π z e V = V and π z f W = W hold. For a proof of this factand more details about projectable vector fields we refer the reader to [KN63]. Although it is in generalnot possible to choose the vector fields e V and f W to be left-invariant, the same argument which proveswell-definedness of the intrinsic Levi form applies to show that (cid:18) i e V , f W ] e −
12 [ e V , g JW ] e (cid:19) mod H C ( q z )does only depend on the values e V e and f W e (compare the proof of Lemma 1 in Chapter 10.1 of [Bog91]).Therefore we conclude (cid:18) i e V , f W ] e −
12 [ e V , g JW ] e (cid:19) mod H C ( q z ) = (cid:18) i (cid:2)e π − z ( v ) , e π − z ( w ) (cid:3) − (cid:2)e π − z ( v ) , e π − z ( iw ) (cid:3)(cid:19) mod H C ( q z ) , and obtain L z ( v, w ) = (cid:18) i V, W ] z −
12 [
V, JW ] z (cid:19) mod H C z M z = (cid:18) i π z e V , π z f W ] z −
12 [ π z e V , π z g JW ] z (cid:19) mod H C z M z = (cid:18) i π z [ e V , f W ] e − π z [ e V , g JW ] e (cid:19) mod H C z M z = π z (cid:18) i e V , f W ] e −
12 [ e V , g JW ] e mod H C ( q z ) (cid:19) = π z (cid:18) i (cid:2)e π − z ( v ) , e π − z ( w ) (cid:3) − (cid:2)e π − z ( v ) , e π − z ( iw ) (cid:3) mod H C ( q z ) (cid:19) = π z (cid:18) i (cid:2)e π − z ( v ) , e π − z ( w ) (cid:3) − (cid:2)e π − z ( v ) , e π − z ( iw ) (cid:3)(cid:19) mod H C z M z . This finishes the proof. (cid:3)
In the next subsection we will use the weight space decomposition in order to determine the map e π − z explicitely.3.4. The Levi form of a closed orbit.
This rather technical subsection contains the computationswhich are necessary to achieve the final formula of the Levi form.
Lemma 3.8.
We have g z = ker( π z ) = (cid:8) (Ad( z ) ξ, ξ ); ξ ∈ g ∩ Ad( z − ) g (cid:9) . Proof.
One checks directly that (cid:8) (Ad( z ) ξ, ξ ); ξ ∈ g ∩ Ad( z − ) g (cid:9) ⊂ ker( π z ) holds. The other inclusionfollows for dimensional reasons. (cid:3) Lemma 3.9.
The subspace q z = R ( q z ) ⊕ H ( q z ) is determined by the following. EOMETRY OF INVARIANT DOMAINS IN COMPLEX SEMI-SIMPLE LIE GROUPS 9 (i) We have R ( q z ) = e π − z (cid:0) g ∩ Ad( z − ) g (cid:1) = (cid:8) (Ad( z ) ξ, − ξ ); ξ ∈ g ∩ Ad( z − ) g (cid:9) . (ii) We have ( e π z ) − ( u C λ,a ) = (cid:8)(cid:0) Ad( z ) σ ( ξ ) + σ (cid:0) Ad( z ) σ ( ξ ) (cid:1) , ξ + σ ( ξ ) (cid:1) ; ξ ∈ u C λ,a (cid:9) for all ( λ, a ) ∈ e Λ \ e Λ( z ) .Proof. Firstly, we have to show that (cid:8) (Ad( z ) ξ, − ξ ); ξ ∈ g ∩ Ad( z − ) g (cid:9) is contained in q z = g ⊥ z where the orthogonal complement is taken with respect to the Killing form of g ⊕ g . Hence, let ξ, ξ ′ ∈ g ∩ Ad( z − ) g and consider B g ⊕ g (cid:0) (Ad( z ) ξ, ξ ) , (Ad( z ) ξ ′ , − ξ ′ ) (cid:1) = B g (Ad( z ) ξ, Ad( z ) ξ ′ ) − B g ( ξ, ξ ′ )= B u C ( ξ, ξ ′ ) − B u C ( ξ, ξ ′ ) = 0 . A simple computation shows (cid:8) (Ad( z ) ξ, − ξ ); ξ ∈ g ∩ Ad( z − ) g (cid:9) ⊂ R ( q z ). Since the converse inclusionfollows for dimensional reasons, the first claim is proven.A similar argument as above implies that (cid:8)(cid:0) Ad( z ) σ ( ξ ) + σ (cid:0) Ad( z ) σ ( ξ ) (cid:1) , ξ + σ ( ξ ) (cid:1) ; ξ ∈ u C λ,a (cid:9) liesin q z . In order to prove the second assertion let ξ ∈ u C λ,a be given and consider e π z (cid:0) Ad( z ) σ ( ξ ) + σ (cid:0) Ad( z ) σ ( ξ ) (cid:1) , ξ + σ ( ξ ) (cid:1) = σ ( ξ ) + Ad( z − ) σ Ad( z ) σ ( ξ ) − ξ − σ ( ξ )= τ z ( ξ ) − ξ = (cid:0) ae − iλ ( η ) − (cid:1) ξ =: ϕ λ,a ( ξ ) . Since ϕ λ,a ( ξ ) ∈ u C λ,a holds, the lemma is proven. (cid:3) Remark.
Note that the map ϕ λ,a : u C λ,a → u C λ,a is an isomorphism if and only if ( λ, a ) / ∈ e ∆( z ) holds. Inthis case the inverse map is given by ϕ − λ,a ( ξ ) = 1 ae − iλ ( η ) − ξ. Definition 3.10.
A Levi basis of H z M z is a basis ( ξ λ,a ) ( λ,a ) of H z M z such that ξ λ,a ∈ u C λ,a and σ ( ξ λ,a ) = ξ σ ( λ ) ,a hold for all ( λ, a ) ∈ e Λ \ e Λ( z ).From now on we fix a Levi basis ( ξ λ,a ) of H z M z . Theorem 3.11.
We obtain the following formula for the Levi form of M z : L z ( ξ λ,a , ξ µ,b ) = ( iae − iλ ( η ) − (cid:2) ξ λ,a , ξ σ ( µ ) ,b (cid:3) if ( λ + σ ( µ ) , ab ) ∈ e Λ( z )0 else . Proof.
We will start by computing e π − z ( ξ λ,a ) for ξ λ,a ∈ u C λ,a . Lemma 3.9 gives us e π − z ( ξ ) = (cid:16) Ad( z ) σ ( ϕ − λ,a ξ ) + σ (cid:0) Ad( z ) σ ( ϕ − λ,a ξ ) (cid:1) , ϕ − λ,a ξ + σ ( ϕ − λ,a ξ ) (cid:17) = (cid:16) Ad( z ) (cid:0) σ ( ϕ − λ,a ξ ) + τ z ( ϕ − λ,a ξ ) (cid:1) , ϕ − λ,a ξ + σ ( ϕ − λ,a ξ ) (cid:17) = (cid:16) Ad( z ) (cid:0) ϕ − λ,a ξ + σ ( ϕ − λ,a ξ ) (cid:1) + Ad( z ) ξ, ϕ − λ,a ξ + σ ( ϕ − λ,a ξ ) (cid:17) for any ξ ∈ u C λ,a . In the next step we determine the Lie bracket (cid:2)e π − z ( ξ λ,a ) , e π − z ( ξ µ,b ) (cid:3) . Since the Liebracket of g = g ⊕ g is defined component-wise, we consider h Ad( z ) (cid:0) ϕ − λ,a ξ λ,a + σ ( ϕ − λ,a ξ λ,a ) (cid:1) + Ad( z ) ξ α,λ , Ad( z ) (cid:0) ϕ − µ,b ξ µ,b + σ ( ϕ − µ,b ξ µ,b ) (cid:1) + Ad( z ) ξ µ,b i = Ad( z ) h ϕ − λ,a ξ λ,a + σ ( ϕ − λ,a ξ λ,a ) + ξ λ,a , ϕ − µ,b ξ µ,b + σ ( ϕ − µ,b ξ µ,b ) + ξ µ,b i and h ϕ − λ,a ξ λ,a + σ ( ϕ − α,λ ξ λ,a ) , ϕ − µ,b ξ µ,b + σ ( ϕ − µ,b ξ µ,b ) i . The application of π z to the element in g ⊕ g whose components are given by the above gives(3.2) h ϕ − λ,a ξ λ,a + σ ( ϕ − α,λ ξ λ,a ) , ξ µ,b i + h ξ λ,a , ϕ − µ,b ξ µ,b + σ ( ϕ − µ,b ξ µ,b ) i + [ ξ λ,a , ξ µ,b ] . By the same computation we obtain for π z (cid:0) [ e π − z ( ξ ) , e π − z ( iξ )] (cid:1) the following expression:(3.3) h ϕ − λ,a ξ λ,a + σ ( ϕ − α,λ ξ λ,a ) , iξ µ,b i + h ξ λ,a , ϕ − µ,b iξ µ,b + σ ( ϕ − µ,b iξ µ,b ) i + [ ξ λ,a , iξ µ,b ] . To arrive at the Levi form, we have to multiply (3.2) by i and subtract (3.3) multiplied by . Due tothe facts that ϕ λ,a and ϕ µ,b are complex-linear, while σ is anti-linear over C , this leads to i (cid:2) ξ λ,a , σ ( ϕ − µ,b ξ µ,b ) (cid:3) . Inserting the concrete expression for ϕ − µ,b yields π z (cid:18) i (cid:2)e π − z ( ξ λ,a ) , e π − z ( ξ µ,b ) (cid:3) − (cid:2)e π − z ( ξ λ,a ) , e π − z ( iξ µ,b ) (cid:3)(cid:19) = iµ − e iβ ( η ) − (cid:2) ξ λ,a , σ ( ξ µ,b ) (cid:3) . To arrive at the Levi form, we have to project this element onto (cid:0) g ∩ Ad( z − ) g (cid:1) C . Consequently, weonly obtain a nonzero contribution if (cid:2) ξ λ,a , σ ( ξ µ,b ) (cid:3) ∈ Fix( τ z ) holds. By the definition of a Levi basisthis condition translates into the one formulated in the theorem. This finishes the proof. (cid:3) The quadratic Levi form.
In this subsection we will derive explicit formulas for the quadraticLevi form of a generic orbit M z = ( G × G ) · z from Theorem 3.11. For ( λ, a ) ∈ e Λ we define u C [ λ, a ] := u C λ,a + u C σ ( λ ) ,a + u C − λ,a − + u C − σ ( λ ) ,a − . Since the u C [ λ, a ] ⊥ u C [ µ, b ] with respect to the Levi form L z for (cid:0) λ + σ ( µ ) , ab (cid:1) / ∈ e Λ( z ) = (cid:8) (0 , (cid:9) , theLevi form is determined by its restriction to these spaces for which explicit formulas are given in the nextproposition. We make use of the partition Λ \ { } = Λ r ˙ ∪ Λ i ˙ ∪ Λ c . Proposition 3.12.
Let ( λ, a ) ∈ e Λ be given.(1) For λ ∈ Λ r we obtain b L z ( r λ ξ λ,a + r − λ ξ − λ,a − ) = − (cid:18) r λ r − λ ae − iλ ( η ) − (cid:19) [ ξ λ,a , ξ − λ,a − ] . (2) For λ ∈ Λ i and a = 1 we obtain b L z ( r λ ξ λ, + r − λ ξ − λ, ) = (cid:18) | r λ | e − iλ ( η ) − − | r − λ | e iλ ( η ) − (cid:19) i [ ξ λ, , ξ − λ, ] . (3) For λ ∈ Λ i and a = − we obtain b L z ( r λ ξ λ, − + r − λ ξ − λ, − ) = − (cid:18) | r λ | e − iλ ( η ) + 1 − | r − λ | e iλ ( η ) + 1 (cid:19) i [ ξ λ, − , ξ − λ, − ] . (4) For λ ∈ Λ i and a = ± we obtain b L z ( r λ,a ξ λ,a + r λ,a − ξ λ,a − + r − λ,a ξ − λ,a + r − λ,a − ξ − λ,a − )= 2 Re (cid:18) ir λ,a r λ,a − ae − iλ ( η ) − ξ λ,a , ξ − λ,a − ] (cid:19) + 2 Re (cid:18) ir − λ,a r − λ,a − ae iλ ( η ) − ξ − λ,a , ξ λ,a − ] (cid:19) . (5) For λ ∈ Λ c and a = 1 we obtain b L z (cid:0) r λ ξ λ, + s λ σ ( ξ λ, ) + r − λ ξ − λ, + s − λ σ ( ξ − λ, ) (cid:1) = 2 Re (cid:18)(cid:18) ir λ s − λ e − iλ ( η ) − − ir − λ s λ e iλ ( η ) − (cid:19) [ ξ λ, , ξ − λ, ] (cid:19) . EOMETRY OF INVARIANT DOMAINS IN COMPLEX SEMI-SIMPLE LIE GROUPS 11 (6) For λ ∈ Λ c and a = − we obtain b L z (cid:0) r λ ξ λ, − + s λ σ ( ξ λ, − ) + r − λ ξ − λ, − + s − λ σ ( ξ − λ, − ) (cid:1) = 2 Re (cid:18)(cid:18) ir − λ s λ e iλ ( η ) + 1 − ir λ s − λ e − iλ ( η ) + 1 (cid:19) [ ξ λ, − , ξ − λ, − ] (cid:19) . (7) For λ ∈ Λ c and a = ± we obtain b L z (cid:0) r λ,a ξ λ,a + s λ,a σ ( ξ λ,a ) + r − λ,a − ξ − λ,a − + s − λ,a − σ ( ξ − λ,a − ) (cid:1) = 2 Re (cid:18) ir λ,a s − λ,a − ae − iλ ( η ) − ξ λ,a , ξ − λ,a − ] (cid:19) + 2 Re (cid:18) ir − λ,a − s λ,a a − e iλ ( η ) − ξ − λ,a − , ξ λ,a ] (cid:19) . Proof.
The proof is a straightforward application of Theorem 3.11. As illustration we will prove the firstassertion. If λ is a real weight, we have σ ( λ ) = λ and therefore u C [ λ, a ] = u C λ,a ⊕ u C − λ,a − . For arbitrarynumbers r λ , r − λ ∈ C we obtain b L z ( r λ ξ λ,a + r − λ ξ − λ,a − ) = | r λ | b L z ( ξ λ,a ) + r λ r − λ L z ( ξ λ,a , ξ − λ,a − )+ r λ r − λ L z ( ξ − λ,a − , ξ λ,a ) + | r − λ | b L z ( ξ − λ,a − )= 2 Re (cid:0) r λ r − λ L z ( ξ λ,a , ξ − λ,a − ) (cid:1) = 2 Re (cid:18) ir λ r − λ ae − iλ ( η ) − ξ λ,a , ξ − λ,a − ] (cid:19) = 2 Re (cid:18) ir λ r − λ ae − iλ ( η ) − (cid:19) [ ξ λ,a , ξ − λ,a − ] , since σ [ ξ λ,a , ξ − λ,a − ] = [ ξ λ,a , ξ − λ,a − ] for real weights λ . (cid:3) Reduction to the ( σ , σ ) –irreducible case. In this subsection we will introduce the appropriatereduction method in order to facilitate the determination of the Levi cone.
Definition 3.13.
We say that u C is ( σ , σ )–irreducible if there is no non-trivial ideal in u C which isinvariant under σ and σ . Remark.
Let n ∈ A c ⊂ U and σ ′ := Ad( n − ) σ Ad( n ). Then σ ′ is again a C –anti-linear involutiveautomorphism of u C commuting with θ and u C is ( σ ′ , σ )–irreducible if and only if it is ( σ , σ )–irreducible.The next lemma characterizes ( σ , σ )–irreducibility in terms of the set of weights Λ = Λ( u C , c ) where c = t ⊕ a is the fundamental Cartan subalgebra of g ∩ g . Lemma 3.14.
The Lie algebra u C is ( σ , σ ) –irreducible if and only if the root system ∆ := Λ \ { } ⊂ ( i t ⊕ a ) ∗ is irreducible.Proof. Let us assume that u C is ( σ , σ )–irreducible. If the root system ∆ is not irreducible, there isa decomposition ∆ = ∆ ˙ ∪ ∆ into non-empty subsystems ∆ , ∆ such that for all λ j ∈ ∆ j neither of λ ± λ is a root. It follows that u C j := u C ,j ⊕ M λ ∈ ∆ j u C λ , where u C ,j := Span (cid:8) [ u C λ , u C − λ ]; λ ∈ ∆ j (cid:9) , is a non-trivial ideal invariant under σ and σ , which contradictsthe fact that u C is ( σ , σ )–irreducible.In order to prove the converse, let us assume that u C is a non-trivial ideal in u C invariant under σ and σ . Consequently, its orthogonal complement u C with respect to the Killing form B u C is also a non-trivial σ – and σ –stable ideal and u C = u C ⊕ u C . It is not hard to see that this decomposition induces similardecompositions of g ∩ g , c , and hence also of the root system ∆ which contradicts the fact that ∆ isirreducible. (cid:3) Since the computation of the Levi form is local and the Levi form is invariant under local biholomor-phisms, it does no harm to go over to coverings. Hence, we assume that U C is simply connected. Theorem 3.15.
There exists an up to re-ordering unique decomposition u C = u C ⊕ · · ·⊕ u C N into ( σ , σ ) –irreducible ideals. If U C is simply-connected, we have the corresponding decomposition of U C , of the realforms G and G , and of the orbits and their (complex) tangent spaces. This decomposition of the complextangent space of a closed orbit is orthogonal with respect to its Levi form. Consequently, the Levi cone isthe direct product of the Levi cones of each factor.Proof. Since u C is semi-simple, it is the direct sum of its simple ideals, and each of these is θ –invariant.Since σ and σ are automorphisms of u C , they map simple ideals onto simple ideals. This observationproves that u C has a unique decomposition into ( σ , σ )–irreducible ideals. Moreover, the simple idealswhich appear in one ( σ , σ )–irreducible ideal must be all isomorphic.Let u C = u C ⊕ · · · ⊕ u C N denote this decomposition and let U C k be the subgroup of U C with Lie algebra u C k . Since U C is simply-connected, we obtain U C ∼ = U C × · · · × U C N , and since each semi-simple normal subgroup U C k is invariant under σ and σ , we have similar decompo-sitions G j ∼ = ( G j ) × · · · × ( G j ) N for j = 1 ,
2. Here, ( G j ) k is the fixed point set of σ j | U C k . It follows that the ( G × G )–orbits are alsodirect products of their intersections with the normal subgroups U C k . Since the u C k are ideals, we havea corresponding decomposition of the set of weights into strongly orthogonal subsystems. Finally, thecomputation of the Levi form in Theorem 3.11 tells us that the respective parts of the complex tangentspaces are Levi-orthogonal. (cid:3) The Levi cone.
In this subsection we will determine the full Levi cone of a generic ( G × G )–orbit.We assume that U C is simply-connected and that u C is ( σ , σ )–irreducible.Let z = n exp( iη ) be a regular element contained in the standard Cartan slice C := n exp( i c ). Since z is regular, we have e iλ ( η ) = 1 for all ( λ, ∈ e Λ. Hence, we conclude λ ( η ) = 0 for all ( λ, ∈ e Λ i . Thefollowing lemma is then a direct consequence of Proposition 3.12. Lemma 3.16.
The Levi cone C z of the generic orbit ( G × G ) · z is generated by(1) ± [ ξ λ,a , ξ − λ,a − ] for ( λ, a ) ∈ e Λ r ,(2) − i [ ξ λ, , ξ − λ, ] for ( λ, ∈ e Λ i with λ ( η ) > ,(3) i [ ξ λ, , ξ − λ, ] for ( λ, ∈ e Λ i with λ ( η ) < ,(4) ± i [ ξ λ, − , ξ − λ, − ] for ( λ, − ∈ e Λ i ,(5) ± Re (cid:0) [ ξ λ,a , ξ − λ,a − ] (cid:1) and ± Im (cid:0) [ ξ λ,a , ξ − λ,a − ] (cid:1) for ( λ, a ) ∈ e Λ i with λ = ± , and(6) ± Re (cid:0) [ ξ λ,a , ξ − λ,a − ] (cid:1) and ± Im (cid:0) [ ξ λ,a , ξ − λ,a − ] (cid:1) for ( λ, a ) ∈ e Λ c .Remark. Since we have defined the real structure on c C via σ , we obtainRe (cid:0) [ ξ λ,a , ξ − λ,a − ] (cid:1) = [ ξ λ,λ , ξ − λ,a − ] + σ (cid:0) [ ξ λ,a , ξ − λ,a − ] (cid:1) = [ ξ λ,a , ξ − λ,a − ] + [ ξ σ ( λ ) ,λ , ξ − σ ( λ ) ,λ − ] . The imaginary part Im (cid:0) [ ξ λ,a , ξ − λ,a − ] (cid:1) can be expressed by an analogous formula.In order to state the main theorem we have to review some properties of real simple Lie algebras ofHermitian type. For a more detailed exposition of these topics we refer the reader to [HN93] and [Nee00].Recall that a simple real Lie algebra g = k ⊕ p is said to be of Hermitian type if the center of k isnon-trivial. This condition implies that a maximal torus t ⊂ k is a Cartan subalgebra of g . Then everyroot α in ∆ = ∆( g C , t ) is imaginary, and either g C α ⊂ k C or g C α ⊂ p C holds. In the first case we call α acompact root, while in the second case α is said to be non-compact. We write ∆ k and ∆ p for the sets ofcompact and non-compact roots, respectively. Since g is Hermitian, the root system ∆ possesses a goodordering, i. e. there is a choice of the set ∆ + of positive roots such that each positive non-compact rootis larger than every compact root. This is equivalent to the fact that the set ∆ + p is invariant under the EOMETRY OF INVARIANT DOMAINS IN COMPLEX SEMI-SIMPLE LIE GROUPS 13
Weyl group W (∆ k ). Therefore there are two natural W (∆ k )–invariant cones C min ⊂ C max , where C min is the closed convex cone generated by (cid:8) − i (cid:2) ξ α , σ ( ξ α ) (cid:3) ; ξ α ∈ g C α , α ∈ ∆ + p (cid:9) ⊂ t and C max := (cid:8) η ∈ t ; iα ( η ) ≥ α ∈ ∆ + p (cid:9) . Let C be the interior of C max . Then the open subset G exp( iC ) G ⊂ U C is closed under multipli-cation and hence a semi-group, called the open complex Ol’shanski˘ı semi-group. Theorem 3.17.
Let u C be ( σ , σ ) –irreducible and let ( G × G ) · z be a generic orbit where z = n exp( iη ) lies in the standard Cartan slice C := n exp( i c ) .(1) If the standard Cartan subset c is non-compact, then C z = c holds.(2) If c is compact and if a = 1 for some ( λ, a ) ∈ e Λ , then we have C z = c .(3) If c is compact and if a = 1 for all generalized weights, then σ = σ holds and there are the followingcases.(i) If g = g =: g is of Hermitian type and if η lies in C max , then the Levi cone C z is isomorphicto the dual of the positive Weyl chamber defined by Λ + . In particular, the Levi cone is pointed.(ii) If g is of Hermitian type and η / ∈ C max , then C z = c .(iii) If g is not of Hermitian type, then C z = c .Remark. The reader will note that the statement of Theorem 3.17 differs also for the case σ = σ fromthe corresponding Theorem 5.3 in [FG98]. Indeed, as L. Geatti has kindly pointed out, the formulationof the third part of Theorem 5.3 in [FG98] is not correct. The correct statement in Theorem 3.17 and itsproof in the case σ = σ are due to an unpublished erratum written by L. Geatti.It will turn out to be convenient to express the generators of the Levi cone in terms of the coroots η λ ∈ i t ⊕ a . Therefore we will identify c = t ⊕ a with i t ⊕ a via the map ( η , η ) ( iη , η ). By abuseof notation, we denote the image of the Levi cone under this map again by C z ⊂ i t ⊕ a . According toLemma 2.6 we have [ ξ λ,a , ξ − λ,a − ] = B u C ( ξ λ,a , ξ − λ,a − ) η λ ∈ C η λ . Hence, we can normalize the ξ λ,a such that [ ξ λ,a , ξ − λ,a − ] = η λ holds for all λ ∈ Λ + \ Λ i and [ ξ λ,a , ξ − λ,a − ] = ± η λ holds for λ ∈ Λ + i depending on the sign of B u C ( ξ λ,a , ξ − λ,a − ). Remark.
In the case where g = g =: g and t is a compact Cartan subalgebra of g , we obtain after theabove normalization [ ξ α , ξ − α ] = ( η α for α ∈ ∆ + p − η α for α ∈ ∆ + k , since the real part of B u C is positive definite on p and negative definite on k . Proof of Theorem 3.17. (1)
Let c be non-compact. Since Λ \ { } satisfies the axioms for an abstract rootsystem, we may choose a set Π ⊂ Λ + of simple weights. By Lemma 3.16 we know that ± η λ lies in C z for λ ∈ Λ \ Λ i , and we have to show that ± η λ ∈ C z holds for all λ ∈ Λ. It is enough to prove this fact for all η λ with λ ∈ Π i := Π ∩ Λ i .If λ, µ ∈ Π i with λ + µ ∈ Λ are given, then λ + µ ∈ Λ + i holds. Since c is non-compact, this observationimplies that there exists an element µ ∈ Π \ Π i . Let λ ∈ Π i be arbitrary (if Π i = ∅ , the proof is finished).Since Λ \ { } is irreducible by Lemma 3.14, its Dynkin diagram is connected and hence we find a sequence λ = λ , . . . , λ N = µ of simple roots which are adjacent in the Dynkin diagram. Consequently, we obtain λ j + · · · + λ N ∈ Λ \ Λ i for all 0 ≤ j ≤ N −
1. This implies ± η λ j + ··· + λ N = ± ( η λ j + ··· + λ N − + η λ N ) ∈ C z for all 0 ≤ j ≤ N −
1. Since ± η λ N lies in C z , we conclude ± η λ j + ··· + λ N − ∈ C z for all j . Iterating thisargument we finally arrive at ± η λ ∈ C z which was to be shown. (2) Let us assume that c is compact and that there exists ( λ, a ) ∈ e Λ with a = 1. In this case we haveΛ = Λ i and ± η λ ∈ C z for all λ such that there exists a = 1 with ( λ, a ) ∈ e Λ. If there are two weights λ , λ ∈ Λ such that ( λ j , a ) ∈ e Λ implies a = 1 for j = 1 , λ + λ is again a weight, thenwe conclude from Proposition 2.7 that ( λ + λ , a ) ∈ e Λ = ⇒ a = 1holds. Consequently, each set Π ⊂ Λ + of simple roots must contain a root µ with ± η µ ∈ C z . Now theclaim follows from the same argument as above. (3) Let c = t be a compact Cartan subalgebra of g ∩ Ad( n − ) g such that a = 1 holds for eachextended weight ( λ, a ) ∈ e Λ. It is enough to prove that this assumption implies g = g since then theclaim follows from [FG98] and Geatti’s erratum.The proof of g = g relies on the comparison of the weight space decompositions u C = t C ⊕ M λ ∈ Λ \{ } u C λ, and (cid:0) g ∩ Ad( n − ) g ) C = t C ⊕ M λ ∈ Λ ′ u C λ, , where Λ ′ denotes the set of non-zero weights λ for which u C λ, is contained in (cid:0) g ∩ Ad( n − ) g (cid:1) C . Notethat this is well-defined since dim u C λ, = 1 by Proposition 2.7. Since the weight space decomposition is inboth cases defined with respect to t , a basis of Λ ′ has to be a basis of Λ \ { } , too. Since the root systemΛ \ { } is reduced by Proposition 2.7, we conclude u C = (cid:0) g ∩ Ad( n − ) g (cid:1) C , and hence, that g ∩ Ad( n − ) g is a real form of u C . For dimensional reasons this implies g = g ∩ Ad( n − ) g = Ad( n − ) g , i. e. σ = Ad( n − ) σ Ad( n ).By the definition of a standard Cartan subset, the fundamental Cartan subalgebra c ⊂ g ∩ g hasthe same dimension as t . Therefore, we obtainrk( g ∩ g ) = dim c = dim t = rk (cid:0) g ∩ Ad( n − ) g (cid:1) C = rk u C , which implies in the same way as above that g ∩ g is a real form of u C and hence that g = g holds. (cid:3) Applications
The criterion of Fels and Geatti.
We restate Corollary 5.6 from [FG98] whose proof relies onTheorem 3.2.
Theorem 4.1 (Fels, Geatti) . Let Z be a complex manifold on which the Lie group G acts by holomorphictransformations. Let the orbit M z = G · z be a generic CR submanifold such that C z = T z M z /H z M z holds.Then there exists no G –invariant Stein domain in Z which contains M z in its boundary. Furthermore,there is no non-constant G –invariant plurisubharmonic function which is defined in a neighborhood of M z . Theorem 4.1 gives a necessary condition for an invariant domain with a generic orbit in its boundaryto be Stein. In our situation we obtain the following result.
Theorem 4.2.
Let C = n exp( i c ) be a standard Cartan subset and let Ω be a connected component ofthe open set G ( C ∩ U C sr ) G . Then Ω does not contain any proper ( G × G ) –invariant Stein subdomainunless c is compact and τ n has a = 1 as only eigenvalue. In this case G = G must be of Hermitian typeand Ω is a translate of the open Ol’shanski˘ı semi-group in U C . Consequently, we see that in the case G = G there are no invariant Stein subdomains in U C in whoseboundary a generic orbit lies. The reader should note that there are only finitely many ( G × G )–invariantdomains whose boundaries consist entirely of non-generic orbits. EOMETRY OF INVARIANT DOMAINS IN COMPLEX SEMI-SIMPLE LIE GROUPS 15 q –pseudo-convex functions and q –completeness. In this subsection we review quickly the no-tions of q –pseudo-convex functions and q –complete complex manifolds. Let Ω be a domain in a complexmanifold Z . We call a smooth function on Ω strictly q –pseudo-convex if its Levi form has at least n − q positive eigenvalues, n := dim C Ω, at each point of Ω. Hence, a strictly 0–pseudo-convex function is thesame as a strictly plurisubharmonic function. If Ω admits a strictly q –pseudo-convex exhaustion function,we say that Ω is q –complete. The solution of the Levi problem implies that a domain is Stein if and onlyif it is 0–complete. For more properties of q –complete complex spaces we refer the reader to [AG62]. Remark.
A standard argument of complex analysis (compare Corollary XIII.5.4 in [Nee00] for the case q = 0 and [Dem] for the generalization to q >
0) shows that a domain Ω in a Stein manifold Z is q –complete if and only if there exists a strictly q –pseudo-convex function Φ on Ω with the propertyΦ( z n ) → ∞ whenever z n → z ∈ ∂ Ω.A domain Ω ⊂ Z with smooth boundary is called Levi– q –convex, if the boundary ∂ Ω can locally bedefined by a function whose Levi form has at most q negative eigenvalues when restricted to the complextangent space at any point of ∂ Ω.By a theorem of Docquier and Grauert ([DG60]) a domain Ω with smooth boundary in a Stein manifold Z is Stein if and only if it is Levi–0–convex. In [ES80] this result is generalized to arbitrary q . Theorem 4.3 (Oka, Docquier-Grauert, Eastwood-Suria) . Let Z be a Stein manifold and let Ω ⊂ Z be adomain with smooth boundary. Then Ω is strictly q –complete if and only if Ω is Levi– q –convex. The rank one case.
If the closed orbit M z = ( G × G ) · z is a hypersurface in U C , its intrinsicLevi form coincides with the classical Levi form of that hypersurface, and hence the signature of b L z isdefined. According to Theorem 4.3 this signature encodes information about complex-analytic propertiesof the domains bounded by M z . In this subsection we will use Matsuki’s classification of pairs of involutiveautomorphisms of simply-connected compact Lie groups in order to classify the triples ( U C , G , G ) where U C is simply-connected and the generic ( G × G )–orbit is a hypersurface. Moreover, we will determinethe signature of the Levi form of each generic hypersurface orbit.In [Mat02] pairs of involutive automorphisms of simply-connected semi-simple compact Lie groups areclassified under the following notion of equivalence. Definition 4.4.
Let U be a simply-connected semi-simple compact Lie group. Two pairs of involutiveautomorphisms ( σ , σ ) and ( σ ′ , σ ′ ) are called equivalent if there exist an automorphism ϕ ∈ Aut( U )and an element u ∈ U such that σ ′ = ϕσ ϕ − and σ ′ = Int( u ) ϕσ ϕ − Int( u ) − hold.Since in our case the involutive automorphisms σ , σ : U C → U C commute with θ and are anti-holomorphic, they are completely determined by their restrictions to U . Therefore, we may apply theclassification result from [Mat02]. Theorem 4.5 (Matsuki) . Let U C be simply-connected. If the generic ( G × G ) –orbit is a hypersurfacein U C , then U C is of the form U C = S × · · · × S | {z } k times , where S is a θ –stable normal subgroup of U C either isomorphic to SL(2 , C ) or SL(3 , C ) . Let σ and τ beanti-holomorphic involutive automorphisms of S commuting with θ | S . If k is odd, then we have σ ( g , . . . , g k ) = (cid:0) σ ( g ) , θ ( g ) , θ ( g ) , . . . , θ ( g k ) , θ ( g k − ) (cid:1) σ ( g , . . . , g k ) = (cid:0) θ ( g ) , θ ( g ) , . . . , θ ( g k − ) , θ ( g k − ) , τ ( g k ) (cid:1) , and if k is even, then σ ( g , . . . , g k ) = (cid:0) σ ( g ) , θ ( g ) , θ ( g ) , . . . , θ ( g k − ) , θ ( g k − ) , τ ( g k ) (cid:1) σ ( g , . . . , g k ) = (cid:0) θ ( g ) , θ ( g ) , . . . , θ ( g k ) , θ ( g k − ) (cid:1) holds. If S = SL(2 , C ) , then the pair ( σ, τ ) is equivalent to one of (cid:8) ( σ , , σ , ) , ( σ , , θ ) , ( θ, θ ) (cid:9) , where σ , is the involution defining the non-compact real form SU(1 , of SL(2 , C ) . If S = SL(3 , C ) , then theonly possibility for ( σ, τ ) up to equivalence is (cid:0) σ ( g ) , τ ( g ) (cid:1) = (cid:0) g, I , θ ( g ) I , (cid:1) with I , := (cid:16) − (cid:17) .Proof. Since the semi-simple complex Lie group U C is assumed to be simply-connected, we can identifythe automorphism group Aut( U C ) with Aut( u C ). By Proposition 2.2 in [Mat02] there exists a θ –invariantdecomposition u C = u C ⊕ · · · ⊕ u C N into σ – and σ –invariant semi-simple ideals u C j . Moreover, each u C j is of the form u C j = s j ⊕ · · · ⊕ s j | {z } k j times , where s j is a θ –stable simple ideal in u C j , such that the restriction of the pair ( σ , σ ) (or ( σ , σ )) to u C j is equivalent to one of the following three types:(1) The number k j is even and σ ( ξ , . . . , ξ k ) = (cid:0) ϕ ( ξ k ) , θ ( ξ ) , θ ( ξ ) , . . . , θ ( ξ k − ) , θ ( ξ k − ) , ϕ − ( ξ ) (cid:1) σ ( ξ , . . . , ξ k ) = (cid:0) θ ( ξ ) , θ ( ξ ) , . . . , θ ( ξ k ) , θ ( ξ k − ) (cid:1) , for some C –anti-linear automorphism ϕ of s j commuting with θ | s j .(2) The number k is even and σ ( ξ , . . . , ξ k ) = (cid:0) σ ( ξ ) , θ ( ξ ) , θ ( ξ ) , . . . , θ ( ξ k − ) , θ ( ξ k − ) , τ ( ξ k ) (cid:1) σ ( ξ , . . . , ξ k ) = (cid:0) θ ( ξ ) , θ ( ξ ) , . . . , θ ( ξ k ) , θ ( ξ k − ) (cid:1) , where σ and τ are C –anti-linear involutive automorphisms of s j commuting with θ | s j .(3) The number k is odd and σ ( ξ , . . . , ξ k ) = (cid:0) σ ( ξ ) , θ ( ξ ) , θ ( ξ ) , . . . , θ ( ξ k ) , θ ( ξ k − ) (cid:1) σ ( ξ , . . . , ξ k ) = (cid:0) θ ( ξ ) , θ ( ξ ) , . . . , θ ( ξ k − ) , θ ( ξ k − ) , τ ( ξ k ) (cid:1) , where σ and τ are C –anti-linear involutive automorphisms of s j commuting with θ | s j .The condition that the generic ( G × G )–orbit is a hypersurface is equivalent to rk( g ∩ g ) = 1. Inparticular, this condition implies that u C is ( σ , σ )–irreducible.If u C is of the first type, one checks directly that g ∩ g ∼ = ( s θϕj ) R holds. Consequently rk( g ∩ g ) is even and in particular larger than 1. This excludes the first type.Let u C be of the second or the third type. Again it is not hard to see that g ∩ g ∼ = s σj ∩ s τj holds. It follows that the simple complex Lie algebra s j contains the complex subalgebra ( g ∩ g ) C whichis given as the set of fixed points of the C –linear semi-simple automorphism στ . Using the classificationof semi-simple automorphisms of simple complex Lie algebras we see that the only possibilities for s are sl (2 , C ) and sl (3 , C ). Then the claim follows from Proposition 2.1 in [Mat02] where the pairs of involutionson the classical Lie algebras are classified up to equivalence. (cid:3) Remark.
Let U C = S × · · · × S ( k times) with S = SL(2 , C ), and let us consider the involutions σ and σ on U C corresponding to ( σ , , θ ) in the way described in Theorem 4.5. In this case we seethat g ∩ g =: t ∼ = so (2 , R ) is one-dimensional and compact. Hence, the fundamental Cartan subset C = exp( i t ) is an exact slice for the ( G × G )–action on U C , i. e. every ( G × G )–orbit intersects C in exactly one point. In particular, we conclude that each element z ∈ U C is strongly regular and that G × G acts properly on U C . EOMETRY OF INVARIANT DOMAINS IN COMPLEX SEMI-SIMPLE LIE GROUPS 17
In the following let us consider a point z ∈ U C such that the orbit M z = ( G × G ) · z is a closedhypersurface in U C . Without loss of generality we take z to be of the form z = exp( iη ) ∈ C for somestandard Cartan subset C = exp( i c ). Because of rk( g ∩ g ) = 1 the Cartan subalgebra c ⊂ g ∩ g is one-dimensional and hence either c = t is a maximal torus in k ∩ k or c = a is a maximal Abeliansubspace of p ∩ p . Let u C = M ( λ,a ) ∈ e Λ u C λ,a be the extended weight space decomposition of u C with respect to c .Let us first assume that c = a is non-compact. In this case every weight is real and we conclude fromTheorem 3.11 that the only non-zero contributions to the Levi form of M z stem from the terms L z ( ξ λ,a , ξ − λ,a − ) = iae − iλ ( η ) − ξ λ,a , ξ − λ,a − ] , where λ = 0 and ξ λ,a is a non-zero element in u C λ,a with σ ( ξ λ,a ) = ξ λ,a . Consequently, the restriction ofthe Levi form to u C [ λ, a ] = u C λ,a ⊕ u C − λ,a − , λ ∈ Λ + , has with respect to the bases ( ξ λ,a , ξ − λ,a − ) of u C [ λ, a ] and [ ξ λ,a , ξ − λ,a − ] of a the matrix (cid:18) iae − iλ ( η ) − − ia − e iλ ( η ) − (cid:19) , which has the eigenvalues ± | ae − iλ ( η ) − | . Hence, we obtain a pair of one positive and one negativeeigenvalue of the Levi form on u C [ λ, a ] for each λ ∈ Λ + .If c = t is compact, each weight is imaginary and we have to handle the cases a = ± a = ± a = ±
1, then a = a − and consequently u C [ λ, a ] = u C λ,a ⊕ u C − λ,a , λ ∈ Λ + , holds. As basis of u C [ λ, a ] we choose ( ξ λ,a , ξ − λ,a ) with σ ( ξ λ,a ) = ξ − λ,a , and as basis of t we take i [ ξ λ,a , ξ − λ,a ]. Then the Levi form has with respect to these bases the matrix (cid:18) ae − iλ ( η ) − − ae iλ ( η ) − (cid:19) . If a = 1, then both eigenvalues have the same sign, and if a = −
1, then the eigenvalues have differentsign.For a = ± u C [ λ, a ] = u C λ,a ⊕ u C − λ,a − ⊕ u C λ,a − ⊕ u C − λ,a and take ( ξ λ,a , ξ − λ,a − , ξ λ,a − , ξ − λ,a ) as a basis of u C [ λ, a ]. Under the assumption B u C ( ξ λ,a , ξ − λ,a − ) = B u C ( ξ λ,a − , ξ − λ,a ) we obtain i [ ξ λ,a , ξ − λ,a − ] = i [ ξ λ,a − , ξ − λ,a ] which we take as a basis of t . With respectto these bases the restriction of the Levi form has the matrix ae − iλ ( η ) − a − e − iλ ( η ) − a − e − iλ ( η ) − ae − iλ ( η ) − , whose eigenvalues are given by ± | ae − iλ ( η ) − | and ± | a − e − iλ ( η ) − | .We summarize these results in the following Theorem 4.6. If c = a is non-compact, then each generic orbit M z with z ∈ exp( i a ) is Levi– q –convexwith q = + . If c = t is compact, let us choose an ordering on the set of weights such that λ ( η ) < for all λ ∈ Λ + and z = exp( iη ) . Then each generic orbit M z with z ∈ exp( i t ) is Levi– q –convex with q = (cid:8) ( λ, − ∈ e Λ; λ ∈ Λ + (cid:9) + (cid:8) ( λ, a ) ∈ e Λ; a = ± (cid:9) Moreover, this numbers for q are sharp, i. e. M z is not Levi– q ′ –convex for any q ′ < q .Proof. The only claim which is left to show is the fact that the multiplicity of the eigenvalue 0 of L z isgiven by rk( u C ) − M (0 ,a ) ∈ e Λ: a =1 u C ,a According to Lemma 5.1 in [Mat02] the subalgebra u C is a Cartan subalgebra of u C . Since we assume thatthe generic ( G × G )–orbit is a hypersurface, we conclude dim C u C , = 1 which finishes the proof. (cid:3) Theorem 4.3 and Theorem 4.6 yield the following result.
Theorem 4.7.
Let M z be a closed hypersurface orbit where z ∈ C = exp( i c ) and let Ω be an invariantdomain with ∂ Ω = M z .(1) If c = a is non-compact, then Ω or U C \ Ω is strictly q –complete with q = + , and this q is optimal.(2) If c = t is compact, then Ω or U C \ Ω is strictly q –complete with q = (cid:8) ( λ, − ∈ e Λ; λ ∈ Λ + (cid:9) + (cid:8) ( λ, a ) ∈ e Λ; a = ± (cid:9) and this q is optimal. The Levi form of invariant functions and q –complete domains. Let us assume in this sub-section that the intersection g ∩ g contains a compact Cartan subalgebra t . By Matsuki’s result ageneric orbit M z = ( G × G ) · z intersects the corresponding standard Cartan subset C = exp( i t ) in anorbit of the group W K × K ( C ). Remark.
It can be shown that there exists a group isomorphism from W := W K ∩ K ( t ) := N K ∩ K ( t ) / Z K ∩ K ( t )onto W K × K ( C ) such that the diffeomorphism t → C , η exp( iη ), intertwines the W –action on t withthe W K × K ( C )–action on C modulo this isomorphism ([Mie07]).We say that a non-zero weight λ ∈ Λ = Λ( u C , t ) is compact, if the reflection with respect to thehypersurface ( iη λ ) ⊥ ⊂ t belongs to W . Otherwise, λ is called non-compact. Let us assume that thereexists a good ordering Λ + on the set of non-zero weights, i. e. that that each positive non-compact weightis larger than every compact weight. This implies that the convex cone C max := (cid:8) η ∈ t ; iλ ( η ) ≥ λ ∈ Λ + (cid:9) ⊂ t is W –invariant.Let Ω := G exp( iC ) G , where C is the interior of C max . Then G × G acts properly on Ω.Moreover, the mapping R : C ∞ (Ω) G × G → C ∞ ( C ) W ,Φ ϕ : η Φ (cid:0) exp( iη ) (cid:1) , is an isomorphism (compare [FJ78]). The inverse E := R − is called the extension operator.One would expect that the Levi form of an invariant smooth function on Ω is determined by thedirection tangent to the ( G × G )–orbits and by a direction transversal to the orbit. The followingpropositions explains how the Levi form L (Φ)( z ) is influenced by the complex tangent space of ( G × G ) · z = M z . Lemma 4.8.
Let Φ ∈ C ∞ (Ω) G × G be given. If v, w ∈ H z M z ⊂ T z Ω , then we have L (Φ)( z )( v, w ) = − d c Φ( z ) L z ( v, w ) , where L z is the Levi form of M z . EOMETRY OF INVARIANT DOMAINS IN COMPLEX SEMI-SIMPLE LIE GROUPS 19
Proof.
By definition, the Levi form of Φ ∈ C ∞ (Ω) at the point z ∈ Ω is the Hermitian form L (Φ)( z ) on T z Ω associated to the (1 , ω := − dd c Φ. We use the formula dω ( V, W ) = V (cid:0) ω ( W ) (cid:1) − W (cid:0) ω ( V ) (cid:1) − ω (cid:0) [ V, W ] (cid:1) and extend v to a CR vector field V on M z to compute as follows: − dd c Φ( z )( v, J z v ) = − v (cid:0) d c Φ( JV ) (cid:1) + J z v (cid:0) d c Φ( V ) (cid:1) + d c Φ( z )[ V, JV ]= v (cid:0) d Φ( V ) (cid:1) + J z v (cid:0) d Φ( JV ) (cid:1) + d c Φ( z )[ V, JV ]= v (cid:0) V (Φ) (cid:1) + J z v (cid:0) JV (Φ) (cid:1) + d c Φ( z )[ V, JV ] . Since the vector fields V and JV are tangent to the orbit and since Φ is constant along the orbit, weobtain L (Φ)( z )( v, v ) = − dd c Φ( z )( v, Jv ) = d Φ( z ) J [ V, JV ] z = − d Φ( z ) J b L z ( v ) . Thus the claim follows from the polarization identities. (cid:3)
Proposition 4.9.
Let z ∈ Ω ∩ U C sr and let T z Ω = T z U C be identified with (4.1) u C = M ( λ,a ) ∈ e Λ u C λ,a via ( ℓ z ) ∗ . Let ϕ ∈ C ∞ ( C ) W be given and let Φ := E ( ϕ ) be its extension to a smooth ( G × G ) –invariantfunction on Ω . Then the decomposition (4.1) is orthogonal with respect to the Levi form L (Φ)( z ) .Proof. In view of Lemma 4.8 it is enough to show that t C and H z M z ∼ = L ( λ,a ) =(0 , u C λ,a are orthogonalwith respect to L (Φ)( z ). Thus let v ∈ t and w ∈ u C λ,a be given. Since Jv and w are tangent to M z = ( G × G ) · z , there are elements η, ξ ∈ q z ⊂ g ⊕ g such that Jv = η Ω ( z ) and w = ξ Ω ( z ) hold,where η Ω and ξ Ω are the corresponding vector fields on Ω. Using the same arguments as in the proof ofLemma 4.8 together with the invariance of Φ we obtain L (Φ)( z )( v, w ) = d c Φ( z )[ η Ω , ξ Ω ]( z ) − id c Φ( z )[ η Ω , ξ Ω ]( z ) . Since [ η Ω , ξ Ω ]( z ) = [ η, ξ ] Ω ( z ) ∈ H z M z , the invariance of Φ implies d c Φ( z )[ η Ω , ξ Ω ]( z ) = 0, which finishesthe proof. (cid:3) We will apply Proposition 4.9 in order to establish existence of a strictly q –pseudo-convex exhaustionfunction on Ω. The following theorem extends Neeb’s result on open complex Ol’shanski˘ı semi-groups tothe case G = G . Theorem 4.10.
The domain Ω is q –complete for q = rk( g ∩ g ) + (cid:8) ( λ, ∈ e Λ; λ ∈ Λ + (cid:9) + (cid:8) ( λ, − ∈ e Λ; λ ∈ Λ + (cid:9) + (cid:8) ( λ, a ) ∈ e Λ; a = ± (cid:9) . Proof.
Let ϕ : C → R ≥ be smooth, W –invariant, and strictly convex with the property that ϕ ( x n ) →∞ whenever x n → x ∈ ∂C , and let Φ := E ( ϕ ) be the corresponding smooth ( G × G )–invariantfunction in Ω. Let z ∈ Ω ∩ U C sr . Due to Proposition 4.9 we may compute the Levi form L (Φ)( z ) on each u C [ λ, a ] separately.We start with the case a = 1. Then our considerations from the rank one case imply together withLemma 4.8 that we obtain for each u C [ λ, −
1] a pair of one positive and one negative eigenvalue and foreach u C [ λ, a ], a = −
1, two pairs of positive and negative eigenvalues in the Levi form L (Φ)( z ).Thus let a = 1. In this case all computations take place in ( g ∩ g ) C and hence the whole question isreduced to the case that Ω is an open Ol’shanski˘ı semi-group in ( G ∩ G ) C . In [Nee00] it is proven thatin this case the extension of a strictly convex function is strictly plurisubharmonic. Hence, we see thatin our case the extension Φ is strictly q –pseudo-convex for q = dim t C + (cid:8) ( λ, a ) ∈ e Λ; a = 1 (cid:9) + (cid:8) ( λ, − ∈ e Λ; λ ∈ Λ + (cid:9) + (cid:8) ( λ, a ) ∈ e Λ; a = ± (cid:9) If z n → z ∈ ∂ Ω, then Φ( z n ) → ∞ holds by construction and hence we conclude that Ω is q –complete forthe above q . (cid:3) References [AG62] Aldo Andreotti and Hans Grauert,
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