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

Abstract

The orbits of a real form of a complex semisimple Lie group and those of the complexification of its maximal compact subgroup acting on , a homogeneous, algebraic, -manifold, are finite. Consequently, there is an open -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 - and -orbits in the case of an open -orbit and more recently lower-dimensional -orbits. We show that the parameter space associated with the unique closed -orbit in agrees with that of the other orbits characterized as a certain explicitly defined universal domain.

1. Introduction

Let be a noncompact semisimple Lie group which is embedded in its complexification and let be a parabolic subgroup in the sense that it contains a Borel subgroup of , then is a compact, homogeneous, algebraic, rational -flag manifold. Observe that as well as acts naturally on every flag manifold . Since is semisimple, it decomposes as a product of almost simple factors in the sense that the map is surjective with finite kernel. This leads to a corresponding decomposition of the flag manifold as a finite direct product with irreducible factors , where for each , is a parabolic subgroup of the complexification of the simple factors . Hence a -orbit (respectively, -orbit) in is a product of -orbits (respectively, -orbits) in the corresponding factors . As a consequence, when we assume in the sequel that is semisimple; it is with the understanding that the above decomposition is possible.

Let be a Cartan involution of (see Section 2.1), then its fixed point set is a maximal compactly embedded subgroup of . We extend holomorphically to (and still call it ), then its fixed point set is the complexification of the maximal compact subgroup. The maximal subgroup as well as its complexification also acts naturally on . In fact, is a negatively curved Riemannian symmetric space embedded in .

In the sequel, we will follow the notation introduced in [1] or [2], and let (resp., ) denote the set of all -orbits (resp., -orbits) in . It is known that these sets are finite [3]. As a consequence, there is at least one open -orbit since the -action on is algebraic. Indeed, all lower-dimensional -orbits are found in the boundary of an open orbit with the minimal-dimensional orbit being closed. In fact there is only one closed -orbit in .

There exists a duality relation between -orbits and -orbits in the flag manifold : let and , then the pair is said to be a dual pair if is nonempty and compact. Observe that if is an open -orbit, then is a dual pair if and only if ; moreover, such a is unique [3]. This duality relation has been extended to include all - and -orbits [1, 4–7]. For every there exists a unique such that is a dual pair and vice versa.

If is the open -orbit in the flag manifold , let be the dual -orbit and denote by , where , the space of all -dimensional compact cycles in . Observe that is compact and contained in and hence just a point in . By associating to the -translate of , the connected component of the set can be regarded as a family of -dimensional cycles. Since is invariant by the right -action on the right, we often regard it as being in the affine homogenous space .

In the present paper, we generalize this definition to any lower-dimensional -orbit and hence to the case of the unique closed -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 .

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 algebra is semisimple if and only if the killing form defined by is nondegenerate. Note that the real Lie algebra is semisimple if and only if its complexification is semisimple. Here, is the fixed point set of a conjugate linear involution . Indeed, defines a decomposition of into and eigenspaces. The subalgebra is called the real form of and is characterized by the fact that its complexification is .

We note that also has a compact real form which is the fixed point set of a compact real involution . Denote the eigenspace of by , then Also, let be the complexification of . Let be the conjugation on giving the real form . Also, let be the compact real involution giving the compact real form . When restricted to , the involution defines an eigenspace decomposition and stabilizes both real forms and . Hence we have the decompositions Now, set and , then since agrees with on , it follows that and . This leads to the following decompositions: The decomposition is called a Cartan decomposition. It is an orthogonal decomposition with respect to the killing form which is negative definite on and positive definite on ; thus is a maximal compact subalgebra. If is the semisimple Lie group whose Lie algebra is , then is a maximal compact subgroup of .

We can lift the Cartan decomposition from the Lie algebra level to the group level.

Proposition 1. Let be a semisimple Lie group, let be a Cartan involution of its Lie algebra , let be the corresponding Cartan decomposition, and let be the analytic subgroup with Lie algebra . Then, (1)there exist a Lie algebra automorphism of with differential which is an involution.(2) is the subgroup of fixed by .(3)The map given by is a diffeomorphism.(4) is closed and contains the center of .(5)If the center of is finite then is a maximal compact subgroup of .

Example 2. If then the real form is the fixed point set of the involution defined by complex conjugation. A maximal compact subgroup of the associated Lie group is or its conjugate. The Cartan involution at the Lie algebra level is given by , and at the Lie group level, it is given by . The Cartan decomposition with respect to is where is the set of symmetric matrices with zero trace.

2.2. Iwasawa Decomposition and Parabolic Subgroups

Let be a complex semisimple Lie algebra, be a Cartan involution of , and be the corresponding Cartan decomposition. Let be a maximal abelian subspace of . It follows that, for any , is a self-adjoint transform of and so is an orthogonal direct sum of character spaces If it is called a restricted root space and is referred to as a restricted root. Denote the set of all restricted roots by .

Proposition 3. The Lie algebra is an orthogonal decomposition satisfying:(1),(2), thus if then ,(3) orthogonally, where .

The dimension of is called the real rank of and if it coincides with the rank of , then is a split form.

This decomposition is called the restricted root space decomposition. Choose a notion of positivity for the restricted root system , and let be a set of positive roots, then is a nilpotent Lie subalgebra of .

Proposition 4. For as above, is a direct sum of vector spaces, where is abelian, is nilpotent, and is solvable.

The above decomposition is the Iwasawa decomposition.

A parabolic subgroup of a semisimple Lie group corresponding to a parabolic subalgebra is the -normalizer of , i.e., A subalgebra is a parabolic subalgebra if and only if is a parabolic subalgebra. Again, let be the set of positive roots and consider the corresponding simple roots.

With respect to these choices, the minimal parabolic subalgebra is given by The minimal parabolic subalgebra coincides with the Borel subalgebra if is a split real form. A parabolic subalgebra is one containing a minimal parabolic subalgebra, respectively, a Borel subalgebra if is split. Up to conjugation, the parabolic subalgebras of are determined by subsets of the set of simple roots . For instance, let ; we get parabolic subalgebras which are opposite subalgebras.

By exponentiation, we obtain the corresponding parabolic subgroups at the group level. Thus and are the corresponding opposite parabolic subgroups.

Example 5. The canonical action of on induces actions on , and the full flag manifold . The parabolic subalgebras are characterized by subsets of the set of simple roots , that is, , , , and corresponding to three nontrivial parabolic subgroups The corresponding opposite parabolic subgroups are therefore given by Note that and 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. Let be a complex semisimple Lie group and let be an Iwasawa decomposition of the Lie algebra. Also let , , and be connected subgroups of with Lie algebras , , and , respectively. Then the multiplication map given be is a diffeomorphism and the groups and are simply connected.

2.3. The Akhiezer-Gindinkin Domain

For the real symmetric space , we define below a kind of neighborhood of in 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 be a Cartan decomposition of Lie() with respect to a compact real form of . For this, let be an abelian subalgebra which is maximal with the condition of being contained in and let be the root system on . This gives an Iwasawa decomposition of . Define root hyperplanes as follows: for a root of , let Now define as the connected component containing the neural element of the set which is obtained from by removing all the hyperplanes as runs through the whole set of roots; i.e., The domain is defined to be the open neighborhood of the Riemannian symmetric space in the space given bywhere 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 space [14]. For a survey of some properties of this domain, refer to [14], [15], or [16].

3. Schubert Varieties and Slices

A Borel subgroup of which contains the factor of an Iwasawa decomposition of is called an Iwasawa-Borel subgroup of . For an orbit of such an Iwasawa-Borel subgroup in , we refer to its closure in , that is, , as an Iwasawa-Schubert variety, or just a Schubert variety.

Given a dual pair , an Iwasawa-Borel subgroup , a Schubert variety , and an intersection point , we call the -orbit the associated Schubert slice.

Also let denote the closure of the -orbit dual to the -orbit . It was shown in [8] that the intersection is finite. The following is a refinement of this result.

Proposition 7. For each Schubert slice , the intersection consists of a unique point.

Proof. Let be the Iwasawa-Borel subgroup containing the factor of the Iwasawa decomposition . For some base point , let , and where is the corresponding Schubert slice and Schubert variety, respectively. Also let be the Matsuki dual orbit to , then a dimension count shows that . As a consequence therefore of the Iwasawa decomposition , it follows that , hence is open in . Now, let be a decomposition of into its irreducible components. Every such component is -invariant and we already know that every -orbit in intersects , so it follows that every such orbit is open in . This shows that every component , for all is a Schubert slice. This implies that the intersection is a finite set of points, where for each , are the corresponding Schubert slices through .
We now show that the intersection is indeed a single point. Let be the slice through and suppose where is another point different from p. Since is -invariant, as the closure of a -orbit, it follows that there exists such that . Also, there exists such that , since . It therefore follows that , the stabilizer subgroup of the element .
It was shown in [8] that the map is a diffeomorphism. Since is a deformation retract of , it follows that is onto. Thus, , and so . Consequently, .

In order to give a suitable definition of the cycle space associated with a -orbit in any flag manifold , we need the following result (see [8]).

Proposition 8. Let be a dual pair and the associated Schubert variety. Then (1).(2)The map , given by is onto, that is, .

Corollary 9. Given a point in the closure of a -orbit , that is, , there exists some Schubert variety containing .

Proof. Given an Iwasawa decomposition of , let be the Iwasawa-Borel subgroup containing the factor . Let be the base point and let be the Schubert variety through , that is, . Also, let be the associated Schubert slice through . Now suppose , then it follows from the above result, Proposition 8, that there exits an element such that . The -conjugate of and the -conjugate of satisfy the inclusion; . As a consequence, . Set , and , then it follows that , where is some other Schubert variety associated with another Iwasawa decomposition . That is, is the closure of the orbit of another Iwasawa-Borel subgroup which contains the factor .

Lemma 10 ([2], see Lemma 1.7). Let be a dual pair, then .

Proof. Since and , it follows that . Now suppose , then by the first part of Proposition 8, there exists some Schubert variety containing the point , that is, , but this intersection by Proposition 8 is contained in .

Proposition 11. Let be a dual pair, then the identity element is in the interior of the set

Proof. It follows from Lemma 10 that . Let be any distance function on . Since is compact in , it follows that the distance from to , the boundary of , is positive, that is, . Observe that the map defined by is a continuous map. In particular, observe that since , it follows that there exists a neighborhood of the identity such that, for all , . Equivalently, for all , . Now, since and are dual, the intersection is transversal; it therefore follows that the intersection also remains transversal for in a neighborhood of the identity possibly smaller than . Thus, the set is nonempty, and in particular, the identity is and interior point of .

Definition 12. Let be a dual pair. The cycle space associated with a -orbit is defined to be the connected component containing the identity of the interior of the set

Clearly, by Proposition 11 is a nonempty open subset of containing the identity element . This definition agrees with the definition of the cycle domain introduced in [17] for the case when , an open -orbit in Z.

Remark 13. The elements of are clearly transformations since as a set, . Sometimes we will want to think of this set as , the set of -dimensional cycles in . For this purpose, observe that the -action on is algebraic; consequently, for a base cycle , the orbit is identifiable with the -homogeneous space , where is the isotropy subgroup at the base cycle.
It follows therefore that is invariant under right multiplication by . As a result we may think of cycles as being in by replacing by and vice versa.
It turns out that the complex group is always a subgroup of the isotropy subgroup . Indeed, if is not of Hermitian type, then is maximal in in the sense that the only proper subgroups which contain it are finite extensions. Hence in the nonhermitian case, is at most a finite extension of .
On the other hand, in the Hermitian case, is contained in one of the parabolic subgroups , where are the associated compact Hermitian symmetric spaces. It is possible that, in this case, . For instance, if is the base point and is the open -orbit, then the dual -orbit is just the base point, and so .
In the sequel therefore, we will often view as being in , and either by pulling back the fibrations or considering a finite cover for , we will regard it as being in .
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 in certain cases, precisely when is of nonholomorphic type. This conjecture has been shown to be true for the open -orbit by using combinatorial computations involving closures of -orbits in Z.

Fix an open -orbit in , a base cycle in and let denote the corresponding -orbit in the cycle space . The cycle space associated with open -orbits in any flag manifolds has been completely characterized [5]. Furthermore, if is not open (i.e., if is a lower-dimensional orbit), then the following theorem has been proved for not of Hermitian type.

Theorem 14 (see [1]). If is not of Hermitian type, then the cycle space associated with for all .

As a note added in proof in [1], the case for when is of Hermitian type has also been considered.

Proposition 15 (see [1]). If is of Hermitian type, then either or the base cycle is -invariant and is either or its complex conjugate 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 where is an open -orbit.

Example 16. Let , , and be the subgroup of the isometry group of the Hermitian form defined for by . Since multiplication by a constant has no effect in the projective space, it follows that the orbits of and coincide on . Consequently, there are three orbits by Witt’s theorem, namely, Now fix a Cartan decomposition of determined by the decomposition where and . Let , whose complexification is given by . Observe that the group stabilizes the decomposition and so has 4 orbits in , namely, , , , and . These orbits are all -stable hence there are three -orbits in , namely, Observe that duality is given by the fact that , , and ; consequently, , , and are dual pairs.
The action of on also induces an action on the full flag manifold . Let represent the standard basis for ; define , , and , then is still a basis for . The Iwasawa-Borel subgroup is the Borel subgroup that stabilizes the full flag . The -orbits are as follows: The orbits , , and are open in . Indeed, the fibrations and are holomorphic with fibers equal to . Since the ball can be retracted to a point, these fibration are trivial, hence . As for the orbit, , consider the fibration , and observe that if any holomorphic function defined on is restricted to , it must be constant by the maximum principle since every point in can be joined by a projective line.
The orbit is the unique closed orbit. This orbit is totally real since .
To determine the orbits of , observe that the orbits of the group coincide with the -orbits in the full flag manifold . These orbits are specified by considering dimensions , , , and , where is a full flag. Thus forThe three orbits , , and are the -orbits which correspond via Matsuki duality to the three open -orbits , , and , respectively, listed above. Thus, , , and are Matsuki dual pairs.
Observe that is the unique open -orbit in

5. Nature of the Unique Closed -Orbit

In this section, we will mostly focus our attention in particular on the unique closed -orbit in . First observe that when is closed, the dual -orbit is open. Thus, duality in this case becomes the statement that ; i.e., The following facts about the closed -orbit are mostly a reformulation of Corollary 3.2 in [3] emphasizing the fact that the closed -orbit, , is unique.

As usual, let be a real form of a complex semisimple Lie group and a complex flag manifold. Let be the base point, then the following statements about the closed orbit are equivalent: is unique, contained in the closure of every -orbit in , contained in every -stable closed subset of , and is the lowest-dimensional -orbit in . Furthermore, some maximal compact subgroup and hence every maximal compact subgroup of , acts transitively on [21].

Proposition 17. Let , the closed -orbit in , then there is an Iwasawa-Borel subgroup such that .

Proof. Let be an Iwasawa decomposition of the real group , then the factor is a simply connected algebraic group. Since its maximal compact subgroup is trivial, it follows that an -orbit that is compact most is a point. Consequently, the minimal -orbit in is a single point with the -isotropy at the Borel group containing the factor . Since the maximal compact subgroup acts transitively on it follows that to every point we can associate an Iwasawa-Borel subgroup; the -isotropy group at .

The unique closed -orbit in the full flag manifold parameterizes the set of Iwasawa-Borel subgroups of . Indeed, every Iwasawa-Borel subgroup stabilizes a point in the closed -orbit.

Example 18. Let us end this section with a characterization of the closed orbit of the group acting as a real form of = on the full flag manifold . Actually, is the subgroup of the isometry group of the Hermitian form defined for by that is, . Consequently the orbits of are determined by the signatures of , with positive eigenvalues and negative eigenvalues. We give a characterization of the closed -orbit in the flag manifold below.

Proposition 19. The unique closed -orbit in 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 a -invariant. So let denote the signature of the Hermitian form restricted to the subspace , where denotes the dimension of the degeneracy. It follows that the maximal isotropic flags are precisely the flags with signature: for all , ; , for all , ; and for all , . Given two such flags, there exists an element of taking one to the other. This follows by Witt’s theorem, and consequently, acts transitively on the set of all these flags.

6. Cycle Space of the Closed -Orbit

As usual, let denote an Iwasawa-Borel subgroup. Let and be the base point, then is open in . The complement of the open -orbit in is therefore a finite union of complex -invariant hypersurfaces. Let be such a hypersurface, then the family of -translates of is the same as since is -invariant and contains the factor of an Iwasawa decomposition of . Now define the -invariant domain as the connected component containing the base point in .

We recall that the Iwasawa-Borel subgroup acts on the symmetric spaces .

Since are simply connected, irreducible compact Hermitian symmetric spaces, they have second Betti number equal to 1. As a consequence, there is a unique -invariant hypersurface in the complement of .

It is crucial in the characterization of the cycle space (see [1] or [2]) if the -invariant hypersurface in the complement of the cycle space is a lift of this hypersurface.

Theorem 20 (see [1, 2]). If the maximal -invariant hypersurface in the complement of is not a lift, then

We will now consider the cycle space associated with the unique closed -orbit for which its Matsuki dual -orbit is open in . Here the condition for duality reduces to the requirement that as mentioned earlier. Recall that the lower-dimensional -orbits belong in the closure of the open orbit and so we may define a partial ordering of these orbits thus: for -orbits , , , we will write if , . We will need the following result about orbit orderings.

Proposition 21. Suppose the orbit pairs are dual pairs, then if and only if .

Proof. Let be the gradient flow of the norm of the moment map introduced in [4, 22], and suppose that . It follows by definition of duality that for each . This flow is invariant with respect to and is tangent to both the -orbits and the -orbits.
So, for , suppose belongs to in the sense that given any neighborhood of there exists an increasing unbounded sequence such that belongs to this neighborhood for every . Since is invariant under the flow, it follows that . By hypothesis, for . Consequently, , that is .
Since the flow is also -invariant, we may start with -orbits and repeat the argument in a similar fashion to prove the converse.

Again, let denote the open -orbit whose Matsuki dual is the unique closed -orbit . Let be the boundary of decomposed as a union of irreducible components. Observe that, for , contains a Zariski open -orbit, , which is dense in , i.e., for each . Let be the corresponding Matsuki dual -orbit.

Corollary 22. For as above for each , it follows that , a disjoint union.

Proof. If is a -orbit contained in then its dual -orbit satisfies , i.e., . The only -orbit with this property is the one dual to the -orbit which is , the open -orbit. Hence is the unique closed -orbit.

Now, observe that if for some , then . Consequently, . This implies in particular that , and we have proved the following.

Corollary 23. For as above,

Theorem 24. Let be the unique closed orbit in , then it follows that

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 runs through the whole set and runs over the set of all parabolic subgroups of , the intersection of all the cycle spaces is the same as the intersection of all the cycle spaces for all open -orbits in , where is a Borel subgroup of . That is, one has the following.

Lemma 25 (see [9]).

Corollary 26. If is of Hermitian type and is open then

Proof. We know that, in the Hermitian case, the domain coincides with naturally embedded in [23]. Also, recall that, in this case, is properly contained in one of the opposite parabolic subgroups or , where and are the associated compact Hermitian symmetric spaces (for convenience, we will just write ). Let be the neutral point in , then are bounded symmetric spaces realizable as -orbits. Let us denote by and by its dual . The pull-back of these orbits by the fibrations gives which is embedded in as the domain .

Now the following result can be deduced from results in [5, 8] and Lemma 25.

Corollary 27. For any ,

Proof of Theorem 24. First of all, assume that, for all , or . This implies that, for all , is -invariant. In particular, this would imply that is -invariant.
Observe that the open -orbit, , cannot be -invariant. Indeed, if this were the case, then would be -invariant as well. Since , it would imply that .
Since the action of is algebraic, it follows that the -orbit of a generic point in is open in . That is, if were -invariant, it would follow that would contain an open subset of . This is contrary to the fact that the complement of is a union of invariant hypersurfaces, thus nonempty and -invariant. This proves the result in this case since for all , it follows from Corollary 27 that It also follows that if for some , then we are done.
Furthermore, if we are in the Hermitian case and for , say, we have that and , then clearly, , and we are done as well.

Data Availability

No data were used to support this study.

Disclosure

The author discloses that some parts of this work are a direct follow-up to the author’s PhD thesis at the Ruhr Universität, Bochum, Germany.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The author would like to thank Prof. Dr. Dr. h.c.mult. Alan T. Huckleberry for initially introducing him to the general area of Lie group theory and Global Analysis and this topic in particular.