# The Parameter Space of Orbits of a Maximal Compact Subgroup Acting on a Flag Manifold

Authors: B. Ntatin
Journal: Journal of Mathematics
 (2019)

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## Abstract

The orbits of a real formG of a complex semisimple Lie group GC and those of the complexification KC of its maximal compact subgroup K acting on Z=GC/Q, a homogeneous, algebraic, GC-manifold, are finite. Consequently, there is an open G-orbit. Lower-dimensional orbits are on the boundary of the open orbit with the lowest dimensional one being closed. Induced action on the parameter space of certain compact geometric objects (cycles) related to the manifold in question has been characterized using duality relations between G- and KC-orbits in the case of an open G-orbit and more recently lower-dimensional G-orbits. We show that the parameter space associated with the unique closed G-orbit in Z agrees with that of the other orbits characterized as a certain explicitly defined universal domain.


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## Body

## 1. Introduction

LetG be a noncompact semisimple Lie group which is embedded in its complexification GC and let Q be a parabolic subgroup in the sense that it contains a Borel subgroup of GC, then Z=GC/Q is a compact, homogeneous, algebraic, rational GC-flag manifold. Observe that G as well as GC acts naturally on every flag manifold Z=GC/Q. Since G is semisimple, it decomposes as a product of almost simple factors G=G1×G2×⋯×Gk in the sense that the map ΠiGi→G is surjective with finite kernel. This leads to a corresponding decomposition of the flag manifold Z=Z1×Z2×⋯×Zk as a finite direct product with irreducible factors Zi=GiC/Qi, where for each i, Qi=Q∩Gi is a parabolic subgroup of the complexification GiC of the simple factors Gi. Hence a G-orbit (respectively, GC-orbit) in Z is a product of Gi-orbits (respectively, GiC-orbits) in the corresponding factors Zi. As a consequence, when we assume in the sequel that G is semisimple; it is with the understanding that the above decomposition is possible.Letθ be a Cartan involution of G (see Section 2.1), then its fixed point set K≔Gθ is a maximal compactly embedded subgroup of G. We extend θ holomorphically to GC (and still call it θ), then its fixed point set is the complexification KC≔(GC)θ of the maximal compact subgroup. The maximal subgroup K as well as its complexification KC also acts naturally on Z. In fact, G/K is a negatively curved Riemannian symmetric space embedded in GC/KC.In the sequel, we will follow the notation introduced in [1] or [2], and let OrbZ(G) (resp., OrbZ(KC)) denote the set of all G-orbits (resp., KC-orbits) in Z. It is known that these sets are finite [3]. As a consequence, there is at least one open G-orbit since the G-action on Z is algebraic. Indeed, all lower-dimensional G-orbits are found in the boundary of an open orbit with the minimal-dimensional orbit being closed. In fact there is only one closed G-orbit in Z.There exists a duality relation betweenG-orbits and KC-orbits in the flag manifold Z: let κ∈OrbZ(KC)andγ∈OrbZ(G),then the pair(κ,γ)is said to be a dual pair ifκ∩γis nonempty and compact. Observe that if γ is an open G-orbit, then (κ,γ) is a dual pair if and only if κ⊂γ; moreover, such a κ is unique [3]. This duality relation has been extended to include all G- and KC-orbits [1, 4–7].For everyγ∈OrbZ(G)there exists a uniqueκ∈OrbZ(KC)such that(γ,κ)is a dual pair and vice versa.IfD is the open G-orbit in the flag manifold Z, let C0 be the dual KC-orbit and denote by Cq(D), where q≔dimCC0, the space of all q-dimensional compact cycles in D. Observe that C0 is compact and contained in D and hence just a point in Cq(D). By associating g∈GC to the g-translate g(C0) of C0, the connected component ΩW(D) of the set (1)g∈GC:gC0⊂Dcan be regarded as a family of q-dimensional cycles. Since ΩW(D) is invariant by the right KC-action on the right, we often regard it as being in the affine homogenous space GC/KC.In the present paper, we generalize this definition to any lower-dimensionalG-orbit γ∈OrbZ(G) and hence to the case of the unique closed G-orbit. We extend results and methods originally developed in [2], which is the author’s PhD dissertation) for lower-dimensional orbits to the case of the unique closed orbit. For instance, Proposition 7 is implicit in [8] where it was only proven that the intersection of the base cycle with a Schubert slice is finite. Here we prove that this intersection is a single point. We give some interesting examples of Matsuki dual orbits in the Hermitian case in Section 4. This helps in Section 5 to characterize the nature of the unique closed orbit in the full flag manifold in terms of signatures of a Hermitian form (Proposition 19). In Section 6, we use known results from [1, 2, 9] together with a result about cycle ordering (Proposition 21), to prove our main result (Theorem 24) that the cycle space of the closed orbit coincides with the universal domain ΩAG.


## 2. Preliminaries and Some Basic Results

In this section, we discuss some basic results about semisimple Lie groups that we will need in the sequel. A survey of the details of the discussions in this section can be found in [10], [11], [12], or [13].


### 2.1. Cartan Decomposition and Real Forms

Recall that a Lie algebrag is semisimple if and only if the killing form B:g×g→C defined by B(x,y)=Tr(adxady) is nondegenerate. Note that the real Lie algebra g is semisimple if and only if its complexification g⊗C≅g⊕ig is semisimple. Here, g is the fixed point set of a conjugate linear involution τ:gC→gC. Indeed, τ defines a decomposition g=g+⊕g- of g into +1 and -1 eigenspaces. The subalgebra g=g+ is called the real form of g and is characterized by the fact that its complexification is g⊗C≅gC.We note thatgC also has a compact real form which is the fixed point set of a compact real involution σ. Denote the +1 eigenspace of σ by u, then (2)gC=u⊕iu.Also, let gC≅g⊕ig be the complexification of g. Let τ be the conjugation on gC giving the real form g. Also, let σ:gC→gC be the compact real involution giving the compact real form u. When restricted to g, the involution θ=τσ=στ defines an eigenspace decomposition gC=g+C⊕g-C and stabilizes both real forms g and u. Hence we have the decompositions (3)g=g+⊕g-,u=u+⊕iu-.Now, set k=g+ and p=g-, then since θ agrees with σ on g, it follows that k=u+ and p=iu. This leads to the following decompositions: (4)g=k⊕p,u=k⊕ip.The decomposition g=k⊕p is called a Cartan decomposition. It is an orthogonal decomposition with respect to the killing form B which is negative definite on k and positive definite on p; thus k⊂g is a maximal compact subalgebra. If G is the semisimple Lie group whose Lie algebra is g, then exp⁡(k)=K⊂G is a maximal compact subgroup of G.We can lift the Cartan decomposition from the Lie algebra level to the group level.Proposition 1.
LetG be a semisimple Lie group, let θ be a Cartan involution of its Lie algebra g, let g=k⊕p be the corresponding Cartan decomposition, and let K be the analytic subgroup with Lie algebra k. Then, (1)
there exist a Lie algebra automorphismΘ of G with differential θ which is an involution.(2)
K=Fix(Θ) is the subgroup of G fixed by Θ.(3)
The mapK×p↦G given by k,X→kexp⁡(X) is a diffeomorphism.(4)
K is closed and contains the center of G.(5)
If the center ofG is finite then K is a maximal compact subgroup of G.Example 2.
IfgC=sl(3,C) then the real form sl(3,R) is the fixed point set of the involution defined by complex conjugation. A maximal compact subgroup of the associated Lie group is SO(3,R) or its conjugate. The Cartan involution at the Lie algebra level is given by θ(X)=-X¯t, and at the Lie group level, it is given by Θ(A)=(A¯-1)t. The Cartan decomposition with respect to θ is (5)sl3,R=so3,R⊕p=k⊕p,where p is the set of symmetric matrices with zero trace.


### 2.2. Iwasawa Decomposition and Parabolic Subgroups

Letg be a complex semisimple Lie algebra, θ be a Cartan involution of g, and g=k⊕p be the corresponding Cartan decomposition. Let a be a maximal abelian subspace of p. It follows that, for any Y∈a, adX is a self-adjoint transform of g and so g is an orthogonal direct sum of character spaces (6)gλ=X∈g:adYX=λYX,forallY∈aforλ∈a∗.If gλ≠0 it is called a restricted root space and λ≠0 is referred to as a restricted root. Denote the set of all restricted roots by Σ.Proposition 3.
The Lie algebrag=go⊕⨁λ∈Σgλ is an orthogonal decomposition satisfying:(1)
gλ1,gλ2⊆gλ1+λ2,(2)
θgλ=g-λ, thus if λ∈Σ then -λ∈Σ,(3)
go=a⊕m orthogonally, where m=Zk(a).The dimension ofa is called the real rank of g and if it coincides with the rank of g, then g is a split form.This decomposition is called the restricted root space decomposition. Choose a notion of positivity for the restricted root systemΣ, and let Σ+ be a set of positive roots, then n=⨁λ∈Σ+gλ is a nilpotent Lie subalgebra of g.Proposition 4.
Forg as above, g=k⊕a⊕n is a direct sum of vector spaces, where a is abelian, n is nilpotent, and a⊕n is solvable.The above decomposition is the Iwasawa decomposition.A parabolic subgroupQ of a semisimple Lie group G corresponding to a parabolic subalgebra q is the G-normalizer of q, i.e., (7)Q=g∈G∣Adgq=q.A subalgebra q⊂g is a parabolic subalgebra if and only if qC⊂gC is a parabolic subalgebra. Again, let Σ+⊂Σ be the set of positive roots and consider the corresponding simple roots.With respect to these choices, the minimal parabolic subalgebra is given by(8)b=go⊕⨁λ∈Σ+gλ.The minimal parabolic subalgebra coincides with the Borel subalgebra if g is a split real form. A parabolic subalgebra is one containing a minimal parabolic subalgebra, respectively, a Borel subalgebra if g is split. Up to conjugation, the parabolic subalgebras of g are determined by subsets Π′ of the set of simple roots Π. For instance, let Δ=Σ+∪{α∈Σ:α∈span(Π′)}; we get parabolic subalgebras (9)q=q+=go⊕⨁λ∈Δgλ,q-=go⊕⨁λ∈-Δgλ,which are opposite subalgebras.By exponentiation, we obtain the corresponding parabolic subgroups at the group level. ThusQ=Q+=exp⁡(q+) and Q-=exp⁡(q-) are the corresponding opposite parabolic subgroups.Example 5.
The canonical action ofGC=SL(3,C) on V=C3 induces actions on P(V), G2(V) and the full flag manifold F1,2(V). The parabolic subalgebras are characterized by subsets of the set of simple roots Π={α1,α2}, that is, ∅, {α1}, {α2}, and {α1,α2} corresponding to three nontrivial parabolic subgroups (10)Qα1=∗∗∗0∗∗0∗∗,Qα2=∗∗∗∗∗∗00∗,B=Qα1,α2=∗∗∗0∗∗00∗The corresponding opposite parabolic subgroups are therefore given by (11)Qα1-=∗00∗∗∗∗∗∗,Qα2-=∗∗0∗∗0∗∗∗,B-=Qα1,α2-=∗∗0∗∗0∗∗∗Note that B and B- are the opposite minimal parabolic subgroup; the Borel subgroups.We end this subsection with a statement about the Iwasawa decomposition at the Lie group level.Proposition 6.
LetG be a complex semisimple Lie group and let g=k⊕a⊕n be an Iwasawa decomposition of the Lie algebra. Also let K, A, and N be connected subgroups of G with Lie algebras k, a, and n, respectively. Then the multiplication map .:K×A×N→G given be (k,a,n)↦kan is a diffeomorphism and the groups A and N are simply connected.


### 2.3. The Akhiezer-Gindinkin DomainΩAG

For the real symmetric spaceM=G/K, we define below a kind of neighborhood of M in Z which is related to the parameter space of interest. We will show that this universally defined domain agrees with the cycle space we seek. Let g=k⊕p be a Cartan decomposition of Lie(G) with respect to a compact real form of g. For this, let a⊂p be an abelian subalgebra which is maximal with the condition of being contained in p and let Φ be the root system on a. This gives an Iwasawa decomposition g=k+a+n of g. Define root hyperplanes Hα as follows: for α a root of a, let (12)Hα≔ξ∈a:αξ=π2.Now define ωAG as the connected component containing the neural element 0∈a of the set which is obtained from a by removing all the hyperplanes Hα as α runs through the whole set of roots; i.e., (13)ωAG=a∖⋃α∈ΠHα0=⋂α∈Πξ∈a:αξ<π2.The domain ΩAG is defined to be the open neighborhood of the Riemannian symmetric space G/K in the space Ω≔GC/KC given by(14)ΩAG=G.exp⁡iωAG.z0where z0=e.KC∈GC/KC is the base point.This domain has been called the universal domain by some authors and others have referred to it as the complex crown of the real symmetric spaceG/K [14]. For a survey of some properties of this domain, refer to [14], [15], or [16].


## 2.1. Cartan Decomposition and Real Forms

Recall that a Lie algebrag is semisimple if and only if the killing form B:g×g→C defined by B(x,y)=Tr(adxady) is nondegenerate. Note that the real Lie algebra g is semisimple if and only if its complexification g⊗C≅g⊕ig is semisimple. Here, g is the fixed point set of a conjugate linear involution τ:gC→gC. Indeed, τ defines a decomposition g=g+⊕g- of g into +1 and -1 eigenspaces. The subalgebra g=g+ is called the real form of g and is characterized by the fact that its complexification is g⊗C≅gC.We note thatgC also has a compact real form which is the fixed point set of a compact real involution σ. Denote the +1 eigenspace of σ by u, then (2)gC=u⊕iu.Also, let gC≅g⊕ig be the complexification of g. Let τ be the conjugation on gC giving the real form g. Also, let σ:gC→gC be the compact real involution giving the compact real form u. When restricted to g, the involution θ=τσ=στ defines an eigenspace decomposition gC=g+C⊕g-C and stabilizes both real forms g and u. Hence we have the decompositions (3)g=g+⊕g-,u=u+⊕iu-.Now, set k=g+ and p=g-, then since θ agrees with σ on g, it follows that k=u+ and p=iu. This leads to the following decompositions: (4)g=k⊕p,u=k⊕ip.The decomposition g=k⊕p is called a Cartan decomposition. It is an orthogonal decomposition with respect to the killing form B which is negative definite on k and positive definite on p; thus k⊂g is a maximal compact subalgebra. If G is the semisimple Lie group whose Lie algebra is g, then exp⁡(k)=K⊂G is a maximal compact subgroup of G.We can lift the Cartan decomposition from the Lie algebra level to the group level.Proposition 1.
LetG be a semisimple Lie group, let θ be a Cartan involution of its Lie algebra g, let g=k⊕p be the corresponding Cartan decomposition, and let K be the analytic subgroup with Lie algebra k. Then, (1)
there exist a Lie algebra automorphismΘ of G with differential θ which is an involution.(2)
K=Fix(Θ) is the subgroup of G fixed by Θ.(3)
The mapK×p↦G given by k,X→kexp⁡(X) is a diffeomorphism.(4)
K is closed and contains the center of G.(5)
If the center ofG is finite then K is a maximal compact subgroup of G.Example 2.
IfgC=sl(3,C) then the real form sl(3,R) is the fixed point set of the involution defined by complex conjugation. A maximal compact subgroup of the associated Lie group is SO(3,R) or its conjugate. The Cartan involution at the Lie algebra level is given by θ(X)=-X¯t, and at the Lie group level, it is given by Θ(A)=(A¯-1)t. The Cartan decomposition with respect to θ is (5)sl3,R=so3,R⊕p=k⊕p,where p is the set of symmetric matrices with zero trace.


## 2.2. Iwasawa Decomposition and Parabolic Subgroups

Letg be a complex semisimple Lie algebra, θ be a Cartan involution of g, and g=k⊕p be the corresponding Cartan decomposition. Let a be a maximal abelian subspace of p. It follows that, for any Y∈a, adX is a self-adjoint transform of g and so g is an orthogonal direct sum of character spaces (6)gλ=X∈g:adYX=λYX,forallY∈aforλ∈a∗.If gλ≠0 it is called a restricted root space and λ≠0 is referred to as a restricted root. Denote the set of all restricted roots by Σ.Proposition 3.
The Lie algebrag=go⊕⨁λ∈Σgλ is an orthogonal decomposition satisfying:(1)
gλ1,gλ2⊆gλ1+λ2,(2)
θgλ=g-λ, thus if λ∈Σ then -λ∈Σ,(3)
go=a⊕m orthogonally, where m=Zk(a).The dimension ofa is called the real rank of g and if it coincides with the rank of g, then g is a split form.This decomposition is called the restricted root space decomposition. Choose a notion of positivity for the restricted root systemΣ, and let Σ+ be a set of positive roots, then n=⨁λ∈Σ+gλ is a nilpotent Lie subalgebra of g.Proposition 4.
Forg as above, g=k⊕a⊕n is a direct sum of vector spaces, where a is abelian, n is nilpotent, and a⊕n is solvable.The above decomposition is the Iwasawa decomposition.A parabolic subgroupQ of a semisimple Lie group G corresponding to a parabolic subalgebra q is the G-normalizer of q, i.e., (7)Q=g∈G∣Adgq=q.A subalgebra q⊂g is a parabolic subalgebra if and only if qC⊂gC is a parabolic subalgebra. Again, let Σ+⊂Σ be the set of positive roots and consider the corresponding simple roots.With respect to these choices, the minimal parabolic subalgebra is given by(8)b=go⊕⨁λ∈Σ+gλ.The minimal parabolic subalgebra coincides with the Borel subalgebra if g is a split real form. A parabolic subalgebra is one containing a minimal parabolic subalgebra, respectively, a Borel subalgebra if g is split. Up to conjugation, the parabolic subalgebras of g are determined by subsets Π′ of the set of simple roots Π. For instance, let Δ=Σ+∪{α∈Σ:α∈span(Π′)}; we get parabolic subalgebras (9)q=q+=go⊕⨁λ∈Δgλ,q-=go⊕⨁λ∈-Δgλ,which are opposite subalgebras.By exponentiation, we obtain the corresponding parabolic subgroups at the group level. ThusQ=Q+=exp⁡(q+) and Q-=exp⁡(q-) are the corresponding opposite parabolic subgroups.Example 5.
The canonical action ofGC=SL(3,C) on V=C3 induces actions on P(V), G2(V) and the full flag manifold F1,2(V). The parabolic subalgebras are characterized by subsets of the set of simple roots Π={α1,α2}, that is, ∅, {α1}, {α2}, and {α1,α2} corresponding to three nontrivial parabolic subgroups (10)Qα1=∗∗∗0∗∗0∗∗,Qα2=∗∗∗∗∗∗00∗,B=Qα1,α2=∗∗∗0∗∗00∗The corresponding opposite parabolic subgroups are therefore given by (11)Qα1-=∗00∗∗∗∗∗∗,Qα2-=∗∗0∗∗0∗∗∗,B-=Qα1,α2-=∗∗0∗∗0∗∗∗Note that B and B- are the opposite minimal parabolic subgroup; the Borel subgroups.We end this subsection with a statement about the Iwasawa decomposition at the Lie group level.Proposition 6.
LetG be a complex semisimple Lie group and let g=k⊕a⊕n be an Iwasawa decomposition of the Lie algebra. Also let K, A, and N be connected subgroups of G with Lie algebras k, a, and n, respectively. Then the multiplication map .:K×A×N→G given be (k,a,n)↦kan is a diffeomorphism and the groups A and N are simply connected.


## 2.3. The Akhiezer-Gindinkin DomainΩAG

For the real symmetric spaceM=G/K, we define below a kind of neighborhood of M in Z which is related to the parameter space of interest. We will show that this universally defined domain agrees with the cycle space we seek. Let g=k⊕p be a Cartan decomposition of Lie(G) with respect to a compact real form of g. For this, let a⊂p be an abelian subalgebra which is maximal with the condition of being contained in p and let Φ be the root system on a. This gives an Iwasawa decomposition g=k+a+n of g. Define root hyperplanes Hα as follows: for α a root of a, let (12)Hα≔ξ∈a:αξ=π2.Now define ωAG as the connected component containing the neural element 0∈a of the set which is obtained from a by removing all the hyperplanes Hα as α runs through the whole set of roots; i.e., (13)ωAG=a∖⋃α∈ΠHα0=⋂α∈Πξ∈a:αξ<π2.The domain ΩAG is defined to be the open neighborhood of the Riemannian symmetric space G/K in the space Ω≔GC/KC given by(14)ΩAG=G.exp⁡iωAG.z0where z0=e.KC∈GC/KC is the base point.This domain has been called the universal domain by some authors and others have referred to it as the complex crown of the real symmetric spaceG/K [14]. For a survey of some properties of this domain, refer to [14], [15], or [16].


## 3. Schubert Varieties and Slices

A Borel subgroupB of GC which contains the factor AN of an Iwasawa decomposition G=KAN of G is called an Iwasawa-Borel subgroup of GC. For an orbit O of such an Iwasawa-Borel subgroup in Z, we refer to its closure in Z, that is, S=cl(O)⊂Z, as an Iwasawa-Schubert variety, or just a Schubert variety.Given a dual pair(γ,κ)∈OrbZ(G)×OrbZ(KC), an Iwasawa-Borel subgroup B⊂GC, a B Schubert variety S, and an intersection point z0∈κ∩S, we call the AN-orbit Σ=AN.zo the associated Schubert slice.Also letcl(κ) denote the closure of the KC-orbit dual to the G-orbit γ. It was shown in [8] that the intersection cl(κ)∩Σ is finite. The following is a refinement of this result.Proposition 7.
For each Schubert sliceΣ, the intersection cl(κ)∩Σ consists of a unique point.Proof.
LetB be the Iwasawa-Borel subgroup containing the factor AN of the Iwasawa decomposition G=KAN. For some base point p∈Z, let Σp=AN.p, and S=cl(O) where O=B.p is the corresponding Schubert slice and Schubert variety, respectively. Also let γ be the Matsuki dual orbit to κ, then a dimension count shows that dimCγ=dimCΣ+dimCcl(κ). As a consequence therefore of the Iwasawa decomposition G=KAN, it follows that dimRS=dimRAN.p, hence Σ=AN.p is open in S. Now, let S∩γ=A1∪A2∪⋯∪Ak be a decomposition of S∩γ into its irreducible components. Every such component is AN-invariant and we already know that every AN-orbit in γ intersects cl(κ), so it follows that every such orbit Σ=AN.p is open in S∩γ. This shows that every component Aj, for all j is a Schubert slice. This implies that the intersection S∩γ={p1,…,pk} is a finite set of points, where for each j, Σj=Σpj=AN.pj are the corresponding Schubert slices through pj.
We now show that the intersectionΣ∩γ is indeed a single point. Let Σ be the slice through p and suppose Σ∩cl(κ)={p,p′} where p′≠p is another point different from p. Since cl(κ) is K-invariant, as the closure of a K-orbit, it follows that there exists k∈K such that k.p′=p. Also, there exists (an)∈AN such that (an).p=p, since p∈Σ. It therefore follows that g=kan∈StabG(p), the stabilizer subgroup of the element p.
It was shown in [8] that the map (15)α:StabKp×StabANp→StabGpgivenbyk,an↦g=kanis a diffeomorphism. Since γ∩κ is a deformation retract of γ, it follows that α is onto. Thus, k∈StabK(p), and so k.p′=p′=p. Consequently, (kan).p′=p.In order to give a suitable definition of the cycle space associated with aG-orbit in any flag manifold Z=GC/Q, we need the following result (see [8]).Proposition 8.
Letγ,κ)∈OrbZ(G)×OrbZ(GC) be a dual pair and S the associated Schubert variety. Then (1)
S∩cl(κ)⊂κ∩γ.(2)
The mapK×cl(Σ)→cl(γ), given by (κ,z)↦k(z) is onto, that is, K.cl(Σ)=cl(γ).Corollary 9.
Given a pointp in the closure of a G-orbit γ, that is, p∈cl(γ), there exists some Schubert variety S containing p.Proof.
Given an Iwasawa decompositionG=KAN of G, let B be the Iwasawa-Borel subgroup containing the factor AN. Let z0∈κ∩γ be the base point and let S be the Schubert variety through z0, that is, S=cl(B.z0). Also, let Σ=AN.z0 be the associated Schubert slice through z0. Now suppose p∈cl(γ), then it follows from the above result, Proposition 8, that there exits an element k∈K such that p∈K.cl(Σ). The K-conjugate K(AN)K-1 of AN and the K-conjugate KBK-1 of B satisfy the inclusion; K(AN)K-1⊂KBK-1. As a consequence, p∈cl(K(AN)K-1.z0)=cl((KAK-1)(KNK-1).z0). Set A~=K(A)K-1, and N~=K(N)K-1, then it follows that p∈cl(A~N~.z0)⊂cl(B~.z0)=S~, where S~ is some other Schubert variety associated with another Iwasawa decomposition G=KA~N~. That is, S is the closure cl(O~=cl(B~) of the orbit O~ of another Iwasawa-Borel subgroup which contains the factor A~N~.Lemma 10 ([2], see Lemma 1.7).
Let(γ,κ)∈OrbZ(G)×OrbZ(GC) be a dual pair, then cl(κ)∩cl(γ)=κ∩γ.Proof.
Sinceκ∈cl(κ) and γ∈cl(γ), it follows that cl(κ)∩cl(γ)⊂κ∩γ. Now suppose p∈cl(κ)∩cl(γ), then by the first part of Proposition 8, there exists some Schubert variety S containing the point p, that is, p∈cl(κ)∩S, but this intersection by Proposition 8 is contained in κ∩γ.Proposition 11.
Let(γ,κ)∈OrbZ(G)×OrbZ(GC) be a dual pair, then the identity element is in the interior of the set(16)C≔g∈GC:gκ∩γisnon-emptyandcompact.Proof.
It follows from Lemma10 that cl(κ)∩cl(γ)=κ∩γ⊂γ. Let d:Z×Z→R≥0 be any distance function on Z. Since Y≔cl(κ) is compact in Z, it follows that the distance from Y to bd(γ), the boundary of γ, is positive, that is, d(Y,bd(γ))>0. Observe that the map α:GC→R≥0 defined by g↦d(g(Y),bd(γ)) is a continuous map. In particular, observe that since α(Id)>0, it follows that there exists a neighborhood U of the identity Id∈GC such that, for all g∈U, d(Y,bd(γ))>0. Equivalently, for all g∈U, g(Y)∩cl(γ)⊂γ. Now, since γ and κ are dual, the intersection γ∩κ is transversal; it therefore follows that the intersection g(κ)∩γ) also remains transversal for g in a neighborhood of the identity possibly smaller than U. Thus, the set C is nonempty, and in particular, the identity Id∈GC is and interior point of C.Definition 12.
Let(γ,κ)∈OrbZ(G)×OrbZ(GC) be a dual pair. The cycle space C(γ) associated with a G-orbit γ is defined to be the connected component containing the identity of the interior of the set (17)Cγ=g∈GC:gκ∩γisnon-emptyandcompact.Clearly, by Proposition11C(γ) is a nonempty open subset of GC containing the identity element Id∈GC. This definition agrees with the definition of the cycle domain ΩW(D) introduced in [17] for the case when γ=D, an open G-orbit in Z.Remark 13.
The elements ofC(γ) are clearly transformations since as a set, C(γ)⊂GC. Sometimes we will want to think of this set as Cq(Z), the set of q-dimensional cycles in Z. For this purpose, observe that the GC-action on Cq(Z) is algebraic; consequently, for a base cycle C0, the orbit GC.C0 is identifiable with the GC-homogeneous space GC/GC0C, where GC0C is the isotropy subgroup at the base cycle.
It follows therefore thatC(γ) is invariant under right multiplication by GC0C. As a result we may think of cycles as being in Cq(Z) by replacing C(γ) by C(γ)/GC0C and vice versa.
It turns out that the complex groupKC is always a subgroup of the isotropy subgroup GC0C. Indeed, if G is not of Hermitian type, then KC is maximal in GC in the sense that the only proper subgroups which contain it are finite extensions. Hence in the nonhermitian case, GC0C is at most a finite extension of KC.
On the other hand, in the Hermitian case,KC is contained in one of the parabolic subgroups P±, where GC/P± are the associated compact Hermitian symmetric spaces. It is possible that, in this case, GC0C=P-orP+. For instance, if x-∈GC/P- is the base point and γ=G.x- is the open G-orbit, then the dual KC-orbit κ=C0 is just the base point, and so GC0C=P-.
In the sequel therefore, we will often viewC(γ) as being in Cq(Z)=GC/GC0C, and either by pulling back the fibrations GC/KC→GC/P± or considering a finite cover for KC, we will regard it as being in GC/KC.
We will make use of some results and methods developed in [5, 6, 8, 9, 12, 13, 15, 17–19] to prove our results.It was conjectured in [20] that the cycle space coincided with the domain ΩAG in certain cases, precisely when Z=GC/Q is of nonholomorphic type. This conjecture has been shown to be true for the open G-orbit by using combinatorial computations involving closures of KC-orbits in Z.Fix an openG-orbit D in Z=GC/Q, a base cycle C0 in D and let Ω≔GC.C0 denote the corresponding GC-orbit in the cycle space Cq(D). The cycle space ΩW(D) associated with open G-orbits in any flag manifolds Z has been completely characterized [5]. Furthermore, if γ∈OrbZ(G) is not open (i.e., if γ is a lower-dimensional orbit), then the following theorem has been proved for G not of Hermitian type.Theorem 14 (see [1]).
IfG is not of Hermitian type, then the cycle space associated with γ(18)Cγ=ΩAGfor all γ∈OrbZ(G).As anote added in proof in [1], the case for when G is of Hermitian type has also been considered.Proposition 15 (see [1]).
IfG is of Hermitian type, then either (19)Cγ=ΩAG=B×B¯or the base cycle cl(κ) is P-invariant and C(γ) is either B or its complex conjugate B¯ depending on the choice of sign.


## 4. Example of Dual Orbits

In this section, we present an example in which we determine all the orbits of a real form of a Lie group and their corresponding Matsuki duals, specifying which orbits are open and which ones are closed. This same example was considered in ([20], Example 1.5) with the purpose of understanding the Akhierzer-Gindikin domain D/KC where D is an open G-orbit.Example 16.
LetV=C3, GC=SL3(C), and G=SU(1,2)={g∈GC:b(v,w)=b(gv,gw),v,w∈C3}⊂GC be the subgroup of the isometry group of the Hermitian form b:C3→C3 defined for z=(z0,z1,z2)∈C3 by b(z)=-|z0|2+|z1|2+|z2|2. Since multiplication by a constant has no effect in the projective space, it follows that the orbits of SU(1,2) and U(1,2) coincide on P2(C). Consequently, there are three orbits by Witt’s theorem, namely, (20)γ-=l∈PC3:bl<0,γ+=l∈PC3:bl>0,γ0=l∈PC3:bl=0,Now fix a Cartan decomposition of su(1,2) determined by the decomposition V=V-⊕V+ where V-={λe1:λ∈C} and V+={λe1+μe2:λ,μ∈C}. Let K={g∈SU(1,2):g(V±)=V±}, whose complexification is given by KC=(GL(V-)×G(L(V+))∩G. Observe that the group GL(V-)×GL(V+) stabilizes the decomposition V=V-⊕V+ and so has 4 orbits in V, namely, 0, V-∖{0}, V+∖{0}, and {(v-,v+):v-≠0,v+≠o}. These orbits are all C∗-stable hence there are three KC-orbits in P(v-,v+), namely, (21)κ-=PV-=KC.l1≅ptforp=1,q=2,κ+=PV+=KC.l2≅P1,κ-+=Pv-⊕v+∖Pv-∪Pv+≅C2∖0.Observe that duality is given by the fact that γ-∩P(V-)=P(V-)=K.l1, γ+∩P(V+)=P(V+)=K.l2, and γ0∩K.l3=γ0=K.l3; consequently, (γ-,κ-), (γ+,κ+), and (γ-,κ-) are dual pairs.
The action ofGC on C3 also induces an action on the full flag manifold F1,2,3(C3). Let {e1,e2,e3} represent the standard basis for C3; define v1=e2-e1, v2=e2+e1, and v3=e3, then {v1,v2,v3} is still a basis for C3. The Iwasawa-Borel subgroup is the Borel subgroup that stabilizes the full flag Cv2⊂(v2⊕v3)⊂C3. The G-orbits are as follows: (22)γ+,++=l,E∈FC3:bl<0,signb,E=++γ+,+-=l,E∈FC3:bl>0,signb,E=+-γ-,+-=l,E∈FC3:bl<0,signb,E=+-γ0,+-=l,E∈FC3:bl=0,signb,E=+-γ+,+0=l,E∈FC3:bl>0,signb,E=+0γ0,+0=l,E∈FC3:bl=0,signb,E=+0The orbits γ+,++, γ+,+-, and γ-,+- are open in F1,2,3(C3). Indeed, the fibrations π1:γ-,+-→γ-≅B2 and π2:γ+,++→γ++≅B2 are holomorphic with fibers equal to P1. Since the ball can be retracted to a point, these fibration are trivial, hence γ+,++≅π-1(B)=B2×P1≅γ++,+. As for the orbit, γ+,+-, consider the fibration π:γ+,+-→γ+≅P2∖B¯2, and observe that if any holomorphic function defined on γ+,+- is restricted to π-1(P1∩γ+,+-), it must be constant by the maximum principle since every point in γ+ can be joined by a projective line.
The orbitγ0,+0 is the unique closed orbit. This orbit is totally real since γ0,+0≅K/S1≅S3.
To determine the orbits ofKC, observe that the orbits of the group GL(V-)×GL(V+) coincide with the KC-orbits in the full flag manifold F(C3). These orbits are specified by considering dimensions dim(V-∩l), dim(V-∩l), dim(V-∩E), and dim(V+∩E), where l⊂E⊂C3 is a full flag. Thus for(23)dim⁡V+∩E=2,letz1=Ce2⊂V+bethebasepoint,thenκ+,++=KC.z1=l⊂V+≅PV+.dim⁡V+∩l=1=dim⁡V-∩E,letz2=Ce2⊂V-⊕Ce2bethebasepoint,thenκ+,-+=KC.z2=l⊂V-⊕l:l∈V+≅PV+.dim⁡V+∩l=1,letz3=V-⊂V-⊕Ce2bethebasepoint,thenκ-,-+=KC.z3≅PV+.dim⁡V+∩l=0=dim⁡V+∩l,anddim⁡V-∩E=1,letz4=Cv1⊂Ce1⊕Ce2bethebasepoint,thenκ0,-+=KC.z4=ϕl⊂V+⊕ϕl:ϕ:V-→V+islineardim⁡V+∩l=1,dim⁡V-∩E=0,letz5=Cv3⊂Cv1⊕C2bethebasepoint,thenκ+,0+=KC.z5=ker⁡ϕl⊂ϕl:ϕ:V-→V+islinear≅C2∖0.dim⁡V+∩l=0=dim⁡V+∩l=anddim⁡V-∩E=0,letz6=Cv1⊂Cv1⊕Cv2bethebasepoint,thenκ0,0+=KC.z6≅Q3.The three orbits κ+,++, κ+,-+, and κ-,-+ are the KC-orbits which correspond via Matsuki duality to the three open G-orbits γ+,++, γ+,-+, and γ-,-+, respectively, listed above. Thus, (κ+,++,γ+,++), (κ+,-+,γ+,-+), and (κ-,-+,γ-,-+) are Matsuki dual pairs.
Observe thatκ0,0+ is the unique open KC-orbit in F1,2(C3)


## 5. Nature of the Unique ClosedG-Orbit

In this section, we will mostly focus our attention in particular on the unique closedG-orbit in Z. First observe that when γ=γcl∈OrbZ(G) is closed, the dual KC-orbit κop is open. Thus, duality in this case becomes the statement that κop⊃γcl; i.e., (24)Cγcl=g∈GC:gκop⊃γcl.The following facts about the closed G-orbit are mostly a reformulation of Corollary 3.2 in [3] emphasizing the fact that the closed G-orbit, γcl, is unique.As usual, letG be a real form of a complex semisimple Lie group GC and Z=GC/Q a complex flag manifold. Let z0∈Z be the base point, then the following statements about the closed orbit γcl=G.z0 are equivalent: γcl is unique, contained in the closure of every G-orbit in Z, contained in every G-stable closed subset of Z, and is the lowest-dimensional G-orbit in Z. Furthermore, some maximal compact subgroup and hence every maximal compact subgroup of G, acts transitively on γcl [21].Proposition 17.
Letz∈γcl, the closed G-orbit in Z=GC/B, then there is an Iwasawa-Borel subgroup B∈GC such that B=Bz.Proof.
LetG=KAN be an Iwasawa decomposition of the real group G, then the factor AN is a simply connected algebraic group. Since its maximal compact subgroup is trivial, it follows that an AN-orbit that is compact most is a point. Consequently, the minimal AN-orbit in γcl is a single point z with the B-isotropy at z the Borel group containing the factor AN. Since the maximal compact subgroup K acts transitively on γcl it follows that to every point z∈γcl we can associate an Iwasawa-Borel subgroup; the B-isotropy group at z.The unique closedG-orbit in the full flag manifold GC/B parameterizes the set of Iwasawa-Borel subgroups of GC. Indeed, every Iwasawa-Borel subgroup stabilizes a point in the closed G-orbit.Example 18.
Let us end this section with a characterization of the closed orbit of the groupG=SU(p,q) acting as a real form of GC= SLn(C) on the full flag manifold GC/B. Actually, G is the subgroup of the isometry group of the Hermitian form hp,q defined for z=(z1,…,zp,zp+1,…,zp+q)∈Cp+q by (25)hp,qz=z12+⋯+zp2-zp+12-⋯-zp+q,that is, G=Isom(hp,q)∩GC. Consequently the orbits of G are determined by the signatures of hp,q, with p positive eigenvalues and q negative eigenvalues. We give a characterization of the closed G-orbit in the flag manifold GC/B below.Proposition 19.
The unique closedG-orbit in GC/B is the set of all maximally isotropic full flags.Proof.
The dimensions of the null spaces, i.e., the maximally isotropic subspaces, called the Witt index is aG-invariant. So let (ni-,ni+,ni0) denote the signature of the Hermitian form h restricted to the subspace Λi, where ni0 denotes the dimension of the degeneracy. It follows that the maximal isotropic flags are precisely the flags with signature: (0,0,ni0) for all Λi, 1≤ni0≤q; (ni-,0,q), for all Λq+i, 1≤ni-≤p-q; and (p-q+ni+,ni+,q-ni+) for all Λp+i, 1≤ni+≤q. Given two such flags, there exists an element of G taking one to the other. This follows by Witt’s theorem, and consequently, G acts transitively on the set of all these flags.


## 6. Cycle Space of the ClosedG-Orbit

As usual, letB⊂GC denote an Iwasawa-Borel subgroup. Let Ω≔GC/KC and x0=eKC be the base point, then B.x0 is open in Ω. The complement of the open B-orbit in Ω is therefore a finite union of complex B-invariant hypersurfaces. Let H be such a hypersurface, then the family {gH}g∈G of G-translates of H is the same as {kH}k∈K since H is B-invariant and B contains the factor AN of an Iwasawa decomposition of G. Now define the G-invariant domain (26)ΩH≔Ω∖⋃g∈GgH0=Ω\⋃k∈KkH0as the connected component containing the base point x0 in Ω.We recall that the Iwasawa-Borel subgroupB acts on the symmetric spaces X±=GC/P±.SinceX± are simply connected, irreducible compact Hermitian symmetric spaces, they have second Betti number equal to 1. As a consequence, there is a unique B-invariant hypersurface in the complement of B.It is crucial in the characterization of the cycle space (see [1] or [2]) if the B-invariant hypersurface H in the complement of the cycle space C(γ) is a lift of this hypersurface.Theorem 20 (see [1, 2]).
If the maximalB-invariant hypersurface H in the complement of C(γ) is not a lift, then (27)ΩAG=Cγ=ΩH.We will now consider the cycle space associated with the unique closedG-orbit γcl for which its Matsuki dual KC-orbit κop is open in Z. Here the condition for duality reduces to the requirement that γcl⊂κop as mentioned earlier. Recall that the lower-dimensional G-orbits belong in the closure of the open orbit and so we may define a partial ordering of these orbits thus: for G-orbits O1, O2, O3,…, we will write O1<O2<O3<⋯ if O2⊂cl(O1), O3⊂cl(O2),…. We will need the following result about orbit orderings.Proposition 21.
Suppose the orbit pairs(γ1,κ1),(γ2,κ2),…∈OrbZ(G)×OrbZ(KC) are dual pairs, then γ1<γ2<⋯ if and only if κ1<κ2<⋯.Proof.
Letϕt be the gradient flow of the norm of the moment map introduced in [4, 22], and suppose that γ1<γ2<⋯. It follows by definition of duality that γi∩κi≠0 for each i=1,2,…. This flow is invariant with respect to K and is tangent to both the G-orbits and the KC-orbits.
So, forp∈γj+1∩κj, suppose q=limt→∞ϕt(p) belongs to γj∩κj in the sense that given any neighborhood of q there exists an increasing unbounded sequence {tn}n∞⊂R such that ϕtn(p) belongs to this neighborhood for every n. Since cl(κj) is invariant under the flow, it follows that κj=KC.q⊂cl(κj+1). By hypothesis, γi≠γj for i≠j. Consequently, κj⊂cl(κj+1)∖κj+1, that is κj<κj+1.
Since the flow is alsoG-invariant, we may start with G-orbits and repeat the argument in a similar fashion to prove the converse.Again, letκop denote the open KC-orbit whose Matsuki dual is the unique closed G-orbit γcl. Let (28)bdκop=Z∖κop=C1∪C2∪⋯∪Cnbe the boundary of κop decomposed as a union of irreducible components. Observe that, for j=1,2,…,n, Cj contains a Zariski open KC-orbit, κj, which is dense in Cj, i.e., cl(κj)=Cj for each j. Let γj be the corresponding Matsuki dual G-orbit.Corollary 22.
Forγj as above for each j, it follows that cl(γj)=γj∪γcl, a disjoint union.Proof.
Ifγ is a G-orbit contained in cl(γj)∖γj then its dual KC-orbit satisfies κj⊂cl(κj)∖κ, i.e., κj<κ. The only KC-orbit with this property is the one dual to the G-orbit γ which is κ=κop, the open KC-orbit. Hence γ=γcl is the unique closed G-orbit.Now, observe that ifg∈bd(Cγj) for some j, then g(cl(κj))∩bd(γj)≠∅. Consequently, g(cl(κj))∩γcl≠∅. This implies in particular that g∉C(γcl), and we have proved the following.Corollary 23.
Forγj as above, (29)Cγcl⊂Cγj,foreachj.Theorem 24.
Letγcl be the unique closed G orbit in Z=GC/Q, then it follows that (30)Cγcl=ΩAG.A version of this theorem was originally proved in [2], and other versions have also appeared in [1, 15].For the proof of this result, we will need the following [9]: as γ runs through the whole set OrbZ(G) and Q runs over the set of all parabolic subgroups of GC, the intersection of all the cycle spaces C(γ) is the same as the intersection of all the cycle spaces C(γop) for all open G-orbits in GC/B, where B is a Borel subgroup of GC. That is, one has the following.Lemma 25 (see [9]).
(31) ⋂ γ ∈ O r b Z G Z = G C / Q C γ = ⋂ γ = γ o p Z = G C / B C γ .Corollary 26.
IfG is of Hermitian type and γ∈OrbZ(G) is open then (32)⋂γ=γopZ=GC/BCγ=ΩAG.Proof.
We know that, in the Hermitian case, the domainΩAG coincides with B×B¯ naturally embedded in GC/KC [23]. Also, recall that, in this case, KC is properly contained in one of the opposite parabolic subgroups P+ or P-, where X+=GC and X-=GC are the associated compact Hermitian symmetric spaces (for convenience, we will just write X±). Let x±=e.P± be the neutral point in X±, then X± are bounded symmetric spaces realizable as G-orbits. Let us denote G.x+ by B and G.x- by its dual B¯. The pull-back of these orbits by the fibrations π±:GC/KC→GC/P± gives (33)π--1B∩π+-1B¯=B×B¯which is embedded in GC/KC as the domain ΩAG.Now the following result can be deduced from results in [5, 8] and Lemma 25.Corollary 27.
For anyγ∈OrbZ(G), (34)⋂γ=γopZ=GC/BCγ=ΩAG⊂Cγ.Proof of Theorem24.
First of all, assume that, for allj, C(γj)=π--1(B) or π+-1(B¯). This implies that, for all j, cl(κj) is P±-invariant. In particular, this would imply that κop is P±-invariant.
Observe that the openKC-orbit, κop, cannot be P±-invariant. Indeed, if this were the case, then C(γcl)⊂κop would be P±-invariant as well. Since ΩAG⊂C(γcl), it would imply that P±.ΩAG⊂ΩAG.
Since the action ofGC⊃P± is algebraic, it follows that the P±-orbit of a generic point in ΩAG is open in Ω. That is, if κop were P±-invariant, it would follow that C(γcl) would contain an open subset of Ω. This is contrary to the fact that the complement of C(γcl) is a union of B invariant hypersurfaces, thus nonempty and G-invariant. This proves the result in this case since for all j, it follows from Corollary 27 that (35)ΩAG⊂Cγcl⊂Cγj.It also follows that if for some j, C(γj)=ΩAG then we are done.
Furthermore, if we are in the Hermitian case and forj≠i, say, we have that C(γj)=π--1(B) and C(γi)=π+-1(B¯), then clearly, C(γj)∩C(γi)=B×B¯=ΩAG, and we are done as well.



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