Journal of Lie Theory Volume 6 (1996) 191–206 ◯^{C} 1996 Heldermann Verlag

Determinant Functions and the Geometry of the Flag Manifold for SU(p,q)

L. Barchini, S. G. Gindikin^{∗}, H. W. Wong

Communicated by J. Hilgert

Abstract. Let G_{0} = SU(p, q) with q ≤ p, K_{0} = S(U(p)×U(q)) a maximal compact subgroup, and let G, K be their complexifications. Finally, let B be a Borel subgroup of G. We define a number of algebraic functions on G/B × G/K and use them to describe the closures of codimension one K orbits of the flag manifold G/B. We show how the underlying geometry of the flag manifold interacts with these functions. In particular, we shall use these functions to construct a Stein extension of the Riemannian symmetric space G_{0}/K_{0} , whose connected component turns out to be the space of linear cycles in most cases.

1. Introduction

In this paper we define a number of functions called the determinant functions and explain their relationship with the underlying geometry of the flag manifold. We will concentrate on the special case of the Lie group G_{0} = SU(p, q) with q ≤ p. We let K_{0} = S(U(p) × U(q)), a maximal compact subgroup, and G = SL(p + q, ℂ), K be the complexifications. The q + 1 determinant functions D_{j}(x, z) are defined on the product X × Z, where X is the flag manifold associated to G and Z the complex symmetric space G/K. Closely related to these q + 1 determinant functions are the zero sets D_{j}( , z) = 0 for each fixed z ∈ Z. We call these sets 𝔒_{j}(z) the determinant varieties. These determinant varieties contain very interesting geometric information.

We would like to indicate briefly why the determinant functions and varieties are interesting. First, consider the set 𝔒(z) = ∪_{k}𝔒_{k}(z), called a configuration. It turns out to be the complement of the unique open orbit of the stabilizer G_{z} of the point z of the complex symmetric space G/K. Second, the determinant functions are special functions associated to the flag variety. To give an idea why they are important, we wold like to mention that we can write down Szegö kernels for generalized flag manifolds. In principle, the more “degenerate” a generalized flag manifold is, the less is the number of factors involved in the singularities of

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag the kernels. As limiting cases, the Grassmanians G(p, p + q), G(q, p + q) involve only two factors. They are exactly D_{0}, D_{q}. The remaining q − 1 functions are genuienely new. Third, the determinant functions pin down all the codimensional one orbits of G_{z}. Indeed, we expect a suitable G_{z} invariant stratification of 𝔒(z) would yield various higher codimension G_{z} orbits. Fourth, we know that G_{0} admits an Iwasawa decomposition K_{0}A_{0}N_{0}. When we complexify the decomposition, it is no longer true that G = KAN, but the equations D_{k} = 0 encode the obstruction. Finally, as shown in [2] and alluded to just now, the functions D_{k} capture the singularities when we try to extend meromorphically the Szegö kernels.

∗Supported by NSF DMS 9202049.

From a different direction, we can understand the geometric information contained in the determinant functions as follows. From Matsuki’s work ([4]) we know that it is important to study the interaction between K orbits and G_{0} orbits of X. We have now a small variation of the same theme. We have two families of subgroups. On one hand, we have the conjugacy class (in G) of K, and correspondingly the configurations 𝔒(z). On the other hand, we have the conjugacy class of G_{0}, and correspondingly the G translates of X_{0}, the unique closed G_{0} orbit on X. These two classes of objects interact as follows. For a set C ⊆ Z, call its polar Ĉ to be {x ∈ X|(∀z ∈ C)x ∈ 𝔒(z)}. Likewise, for A ⊆ X, the polar Â is {z ∈ Z|(∀x ∈ A)x ∈ 𝔒(z)}. It turns out that X_{0} is its double polar, i.e., ˆ X̂_{0} = X_{0}.

From a third direction, we can appreciate the importance of the determinant functions as follows. For various reasons people are interested in G_{0} invariant Stein extensions of G_{0}/K_{0}. Wolf and Zierau have found, in explicit terms, the Stein extension as the space of linear cycles. We will show that the connected component of X̂_{0} is the space of linear cycles. This suggests another, though equivalent, manner of obtaining the Stein extension.

In [2] we explain the origin of the determinant functions as the singularties of Szegö kernels. An immediate consequence of that paper is that the solutions of Schmid equations on the Riemmanian symmetric space G_{0}/K_{0} extend holormophically to the space of linear cycles.

Here is a brief summary of the paper. In Section 2 we define the q + 1 determinant functions in order to describe the q + 1 codimensional one K orbits of the flag manifold in Sections 3 and 4. In Section 5 we will explain how to find a Stein extension for the Riemannian symmetric space G_{0}/K_{0} using the determinant functions. We relate this extension to the space of linear cycles, first studied by R.O. Wells and J. Wolf, later also by R. Zierau (see the citations in Section 5), in Section 6. In the same section, we also prove that X_{0} = ˆ X̂_{0} . Section 7 is the appendix, in which we explain how to obtain some of our results by Lie-algebraic method, and this should have some bearing on [7, Problem 9, p.740].

We would like to thank Devra Garfinkle, Joe Wolf, and Roger Zierau who explained various aspects of their results to us while we were writing this paper.

2. Determinant Functions

In [2], a number of determinant functions are defined. We would like to repeat the definition here for the reader’s convenience.

The space X denotes the set of all sequences (V_{i}; 1 ≤ i ≤ n) of subspaces of ℂ^{n}, where dim V_{i} = i and V_{i} ⊆ V_{j} if i ≤ j. We can represent a point of X by ω = (ω_{i}, 1 ≤ i ≤ p + q), where ω_{i} ∈ Λ^{i}ℂ^{p+q}. Strictly speaking, we should look at the n−tuple (ω̃_{i}) ∈ Π_{i=1}^{n}ℙ(Λ^{i}ℂ^{n}), where ω̃_{i} is the projective point in ℙ(Λ^{i}ℂ^{n}) represented by ω_{i}. The space Z consists of pairs (L_{p}, L′_{q}) of disjoint subspaces of dimensions p and q. We can likewise think of L_{p}, L′_{q} as two elements in Λ^{•}ℂ^{n}. Sometimes we use an ordered basis (v_{i}, 1 ≤ i ≤ n) to represent a flag, by putting ω_{i} = v_{1} ∧ … ∧ v_{i}.

For the sake of simplicity, we find the following notation ≐ useful. It means that the two sides are equal up to a non-zero scalar multiple.

We introduce two more pieces of notations, for convenience’s sake. On Λ^{•}ℂ^{p+q} we have a “star” operator. Suppose we have an ordered basis (v_{i}, 1 ≤ i ≤ p + q) for ℂ^{p+q}, then ∗v_{k_{1}} ∧ … ∧ v_{k_{i}} := ϵv_{k_{i+1}} ∧ … ∧ v_{k_{p+q}}. Here ϵ is +1 or −1 according to whether (k_{1}, …, k_{p+q}) is an even or odd permutation of (1, …, p + q). Now suppose u ∈ Λ^{a}ℂ^{p+q}, v ∈ Λ^{b}ℂ^{p+q}, and suppose a + b ≥ p + q, define u⊓v := ι(∗u)v. Here ι( ) means the interior product. Geometrically, if u, v represent vector subspaces then u ⊓ v ≠ 0 iff the subspaces intersect transversally and in that case u ⊓ v represents their intersection.

Next, for any forms ω, θ, define ω ⊔ θ := ω ∧ θ. If ω, θ represent subspaces, then ω ⊔ θ ≠ 0 iff they do not intersect, and in that case ω ⊔ θ represents their linear span.

We will take the convention that a⊔b⊔c := (a⊔b)⊔c, likewise for a⊓b⊓c. We will take (e_{1}, …, e_{p+q}) as the standard ordered basis of ℂ^{p+q}. Given any forms u of the top degree, we can identify it as a scalar [u],

provided we have made a choice of a “standard” top form, as we have already done.

We define q + 1 determinant functions on X × Z as follows.

Definition 2.1. For 0 ≤ j ≤ q, define D_{j} = D_{j}(ω_{1}, …, ω_{p+q}; L_{p}, L′_{q}) as follows.

D_{j} = \{ \begin{matrix}[ω_{p} ⊔ L′_{q}] & if j = 0, \\ [(ω_{p+q−j} ⊓ L_{p}) ⊔ ω_{j} ⊔ L′_{q}] & if 1 ≤ j ≤ q − 1, \\ [ω_{q} ⊔ L_{p}] & if j = q. \end{matrix}  (1)

3. Codimension One Orbits

We take as base point l_{p} := e_{1} ⊔ … ⊔ e_{p} and l′_{q} := e_{p+1} ⊔ … ⊔ e_{p+q}. We then define polynomial functions δ_{j} on X by

δ_{j}(ω) := D_{j}(ω; l_{p}, l′_{q}).  (2)

We can now define q + 1 subsets 𝔒_{j}^{0}; 0 ≤ j ≤ q of X. They turn out to be the codimensional one K orbits. For 1 ≤ j ≤ q − 1,

𝔒_{j}^{0} := {(ω_{i}, 1 ≤ i ≤ p + q)|δ_{j}(ω) = 0, δ_{a}(ω) ≠ 0 if a ≠ j}.  (3)

For the case j = 0, we would like to imitate the definition in Equation 3 by requiring δ_{a}(ω) = 0 iff a = 0. However, a moment’s thought reveals that this still consists of several orbits. We have to add more conditions. We define

𝔒_{0}^{0} := {ω|δ_{j}(ω) ≠ 0 if j > 0, ω_{p−1} ⊔ l′_{q} ≠ 0, δ_{0}(ω) = 0}.  (4)

Likewise, define

𝔒_{q}^{0} := {ω|δ_{j}(ω) ≠ 0 if j < q, δ_{q}(ω) = 0, ω_{q−1} ⊔ l_{p} ≠ 0, ω_{q+1} ⊓ l_{p} ≠ 0}. (5)

There are obvious geometric descriptions for the orbits 𝔒_{j}^{0}. When 1 ≤ j ≤ q−1, note that δ_{0}, δ_{q} ≠ 0 implies ω_{p+q−k} ⊓l_{p} ≠ 0 and ω_{k} ⊔l′_{q} ≠ 0 for 1 ≤ k ≤ p−1. So δ_{k} ≠ 0 iff the subspace ω_{p+q−k} ∩ l_{p} (viewing ω_{p+q−k} as a subspace) is disjoint from the subspace ω̃_{k} spanned by ω_{k} and l′_{q}. Hence, ω ∈ 𝔒_{j} iff ω_{p} (ω_{q}) is disjoint from l′_{q} (l_{p}) and that ω_{p+q−k} ∩ l_{p} is disjoint from ω̃_{k} if 1 ≤ k ≤ q − 1 and k ≠ j. (Indeed, when k = j, we can even assume that the intersection of the two subspaces is of dimension one.)

For the case j = 0, q, we can proceed likewise. For all 1 ≤ k ≤ q − 1, ω_{p−1} ⊔ l′_{q} ≠ 0 implies ω_{k} ⊔ l′_{q} ≠ 0, and δ_{q} ≠ 0 means ω_{p+q−k} ⊓ l_{p} ≠ 0. Hence ω ∈ 𝔒_{0}^{0} iff ω_{p−1} is disjoint from l′_{q} and ω_{q} from l_{p}, and that for all 1 ≤ k ≤ q − 1, ω_{p+q−k} ∩ l_{p} is disjoint from ω̃_{k}. Likewise, ω ∈ 𝔒_{q}^{0} iff ω_{p} is disjoint from l′_{q}, ω_{q−1} from l_{p}, and ω_{q+1} is transversal to l_{p}, and for 1 ≤ k ≤ q − 1, ω_{p+q−k} ∩ l_{p} is disjoint from ω̃_{k}.

The following results relate the functions δ_{j} with the geometry of the flag manifold X. Notice that K has a natural action on X. It is well known that there are only finitely many orbits and there is a unique open orbit ([4] and [8]). It is trivial to check that the sets 𝔒_{j}^{0} are K invariant.

Proposition 3.1. The set 𝔒 := {(ω)|δ_{j}(ω) ≠ 0 for all j} is the unique open K orbit.

Proposition 3.2. There are exactly q + 1 codimensional one K orbits. They are 𝔒_{j}^{0}.

Lemma 3.3. The Zariski closure 𝔒_{j} of 𝔒_{j}^{0} is the set {ω|δ_{j}(ω) = 0}. We will prove these results in the next section.

4. Proofs of Results on the Orbits

For the sake of convenience, define K̃ to be the subgroup of GL(p + q) which

consists of matrices of the form ( \begin{matrix}A & 0 \\ 0 & B \end{matrix} ) , as in the case of K ⊆ G. K̃ acts

naturally on X and Z.

First, we prove Proposition 3.1.

Proof. (Proposition 3.1). The set 𝔒 is clearly K invariant and open, we only have to show that K acts transitively on it. Let us pick a base point. Recall we can represent a flag by an ordered basis. The following is a good base point ω_{0} to pick: (e_{1} + e_{p+1}, …, e_{q} + e_{p+q}, \overbrace{e_{q+1}, …, e_{p},} e_{q} − e_{p+q}, …, e_{1} − e_{p+1}) (the part under the overbrace is absent if p = q). It is trivial to verify that such a flag lies inside 𝔒.

Suppose we have any ω ∈ 𝔒, it suffices to find an element k ∈ K̃ such that kω_{0} = ω. Now such an element k must be of the form

k = ( \begin{matrix}u_{1} & ⋯ & u_{p} & 0_{p} & ⋯ & 0_{p} \\ 0_{q} & ⋯ & 0_{q} & v_{1} & ⋯ & v_{q} \end{matrix} ) ,  (6)

where u_{i} ∈ ℂ^{p}, v_{j} ∈ ℂ^{q} are column vectors, and 0_{p}, 0_{q} are the zero vectors. We will show how to choose the vectors u_{i}, v_{j}.

First of all, pick any ordered basis (z_{i}, 1 ≤ i ≤ p + q) that represents ω.

Suppose z_{i} = ( \begin{matrix}u_{i} \\ v_{i} \end{matrix} ) for 1 ≤ i ≤ q. The requirement that δ_{q}(ω) ≠ 0 implies

that v_{1}, …, v_{q} form a basis for ℂ^{q}. Hence we can assume that z_{i} = ( \begin{matrix}u_{i} \\ 0_{q} \end{matrix} ) for

q + 1 ≤ i ≤ p (if p = q, there is nothing to do). The fact that δ_{0}(ω) ≠ 0 implies that u_{1}, …, u_{p} form a basis of ℂ^{p}.

We would like to argue that we can assume that z_{i} = ( \begin{matrix}u_{p+q−i+1} \\ −v_{p+q−i+1} \end{matrix} ) for

p + 1 ≤ i ≤ p + q. We can do this by an inductive argument. We will argue carefully for i = p + 1, the general inductive step involves a similar reasoning and will be omitted.

We can assume that z_{p+1} = ( \begin{matrix}u_{p+1} \\ 0_{q} \end{matrix} ). The fact that δ_{i} ≠ 0 for i = q−1 im-

plies that u_{1} ∧… u_{q−1} ∧(u_{q+1} … u_{p}∧)u_{p+1} ≠ 0. So u_{1}, …, u_{q−1}, (u_{q+1}, … u_{p},)u_{p+1} form a basis of ℂ^{p}, as do u_{1}, … u_{p} (in both cases, if p = q, the portions inside the parentheses are absent). We can assume that u_{p+1} ≐ u_{q} + ∑_{1≤i≤p,i≠q} a_{i}u_{i}. Now replace u_{q} by u_{q} + ∑_{1≤i≤q−1} a_{i}u_{i} and v_{q} by v_{q} + ∑_{1≤i≤q−1} a_{i}v_{i}. (This replacement does not change the flag represented!) Hence we can rewrite ω_{p+1} as u_{p+1} ≐ u_{q} + ∑_{i=q+1}^{p} a_{i}u_{i} (if p = q, the summation from q + 1 to p is treated as

zero). Thus z_{p+1} ≐ \frac{1}{2}(( \begin{matrix}u_{q} \\ v_{q} \end{matrix} ) + ( \begin{matrix}u_{q} \\ −v_{q} \end{matrix} )) + ∑_{i=q+1}^{p} a_{i}( \begin{matrix}u_{i} \\ 0_{q} \end{matrix} ). Thus, without loss

of generality, we can redefine z_{p+1} = ( \begin{matrix}u_{q} \\ −v_{q} \end{matrix} ) and we still represent the same flag.

By a similar argument, we know that we can choose z_{i} = ( \begin{matrix}u_{p+q−i+1} \\ −v_{p+q−i+1} \end{matrix} )

when p + 1 ≤ i ≤ p + q. Thus the element k ∈ K̃ moves ω_{0} to ω. Hence, K̃ acts transitively on 𝔒.

The following elementary observation can be proved by direct verification, and we will omit the proof.

Lemma 4.1. The following ordered basis represents a base point ω_{0}(j) of 𝔒_{j}^{0}. For 1 ≤ j ≤ q − 1, it is (e_{1} + e_{p+1}, …, e_{q} + e_{p+q}, \overbrace{e_{q+1}, …, e_{p},} e_{q} − e_{p+q}, …, e_{j+2} − e_{p+j+2}, e_{j} − e_{p+j}, e_{j+1} − e_{p+j+1}, e_{j−1} − e_{p+j−1}, …, e_{1} − e_{p+1}). For j = 0 it is (e_{1} + e_{p+1}, …, \overbrace{e_{q} + e_{p+q}, e_{q+1}, …, e_{p−1}}, e_{p+q}, e_{p}, e_{q−1} − e_{p+q−1}, …, e_{1} − e_{p+1}). For j = q, it is (e_{1}+e_{p+1}, …, e_{q−1}+e_{p+q−1}, e_{q}, \overbrace{e_{q+1} + e_{p+q}, e_{q+2}, …, e_{p},} e_{q}−e_{p+q}, …, e_{1}− e_{p+1}). (When p = q, the portions under the overbraces are skipped.)

Lemma 4.2. The group K̃ acts transitively on 𝔒_{j}^{0}.

Proof. We will constantly refer to the proof of Proposition 3.1. We will write ω_{0} instead of ω_{0}(j). We would like to find an element k ∈ K̃ that moves ω_{0} to any given element ω ∈ 𝔒^{0}, where k is as in Equation 6.

First, we prove the case j = q. The case j = 0 is similarly proved and

thus omitted. Pick any ordered basis (z_{i}) that represents ω. Suppose z_{i} = ( \begin{matrix}u_{i} \\ v_{i} \end{matrix} )

for 1 ≤ i ≤ q − 1. The fact that ω_{q−1} ⊔ l_{p} ≠ 0 implies that v_{1}, … v_{q−1} is

linearly independent. As δ_{q}(ω) = 0, we can assume that z_{q} = ( \begin{matrix}u_{q} \\ 0_{q} \end{matrix} ). We also let

z_{q+1} = ( \begin{matrix}u_{q+1} \\ v_{q} \end{matrix} ) (if p = q, let u_{q+1} = 0_{p} and we may as well let z_{q+1} = ( \begin{matrix}0_{p} \\ −v_{q} \end{matrix} )).

The fact that ω_{q+1} ⊓ l_{p} ≠ 0 implies that v_{1}, …, v_{q} form a basis of ℂ^{q}. We can

therefore assume that z_{i} = ( \begin{matrix}u_{i} \\ 0_{q} \end{matrix} ) for q + 2 ≤ i ≤ p (there is nothing to do if

p = q or p = q + 1). The fact that δ_{0} ≠ 0 implies that u_{1}, … u_{p} form a basis of ℂ^{p}.

At this point we can carry over the argument for the proof of Proposition 3.1

verbatim and conclude that we can choose z_{p+1} = ( \begin{matrix}u_{q+1} \\ −v_{q} \end{matrix} ) (if p = q, this step

is redundant) and then let z_{i} = ( \begin{matrix}u_{p+q−i+1} \\ −v_{p+q−i+1} \end{matrix} ) for p + 2 ≤ i ≤ p + q (if q ≥ 2,

otherwise there is nothing to do). The proof is complete at this point. When 1 ≤ j ≤ q − 1, we pick any ordered basis representing ω. Let

z_{i} = ( \begin{matrix}u_{i} \\ v_{i} \end{matrix} ) for 1 ≤ i ≤ q. The fact that δ_{q} ≠ 0 implies that v_{1}, … v_{q} form

a basis of ℂ^{q}. Therefore we can choose z_{i} = ( \begin{matrix}u_{i} \\ 0_{q} \end{matrix} ) for q + 1 ≤ i ≤ p.

Since δ_{0} ≠ 0, we know that u_{1}, …, u_{p} form a basis of ℂ^{p}. By the same sort

of argument as in Proposition 3.1, we can assume that z_{i} = ( \begin{matrix}u_{p+q−i+1} \\ −v_{p+q−i+1} \end{matrix} ) for

p + 1 ≤ i ≤ p + q − (j + 2) + 1. Next, let z_{p+q−j} = ( \begin{matrix}u_{p+q−j} \\ 0_{q} \end{matrix} ). The fact

that δ_{j} = 0 implies that u_{p+q−j} is a linear combination of u_{1}, …, u_{j}, u_{j+2}, …, u_{p}. However, δ_{j−1} ≠ 0 implies that the coefficient of u_{j} is non-zero. Therefore, without loss of generality, we can assume that u_{p+q−j} ≐ u_{j} + ∑_{m=1}^{j−1} a_{m}u_{m} + ∑_{k=j+2}^{p} b_{k}u_{k}. Redefine u_{j} to be u_{j} + ∑_{m=1}^{j−1} a_{m}u_{m} and v_{j} to be v_{j} + ∑_{m=1}^{j−1} a_{m}v_{m}. Hence we have

2( \begin{matrix}u_{p+q−j} \\ 0_{q} \end{matrix} ) = ( \begin{matrix}u_{j} \\ v_{j} \end{matrix} ) + ( \begin{matrix}u_{j} \\ −v_{j} \end{matrix} ) + ∑_{k=j+2}^{p} b_{k}(( \begin{matrix}u_{k} \\ v_{k} \end{matrix} ) + ( \begin{matrix}u_{k} \\ −v_{k} \end{matrix} )). Therefore,

without loss of generality, we can choose z_{p+q−j} = ( \begin{matrix}u_{j} \\ −v_{j} \end{matrix} ).

We can argue in a similar fashion that we can assume z_{p+q−j+1} = ( \begin{matrix}u_{j+1} \\ −v_{j+1} \end{matrix} ),

and z_{i} = ( \begin{matrix}u_{p+q−i+1} \\ −v_{p+q−i+1} \end{matrix} ) for p + q − j + 2 ≤ i ≤ p + q.

We can now prove Proposition 3.2.

Proof. of Proposition 3.2. To show that 𝔒_{j}^{0} are orbits, it is enough to see that K̃ acts transitively on them, which we have shown. The q+1 orbits 𝔒_{j}^{0} are clearly distinct and have codimension 1. So we know there are at least q + 1 orbits.

We need to see that the complement of ∪_{j}𝔒_{j}^{0} is a union of orbits with codimension other than 1. The complement is contained inside the union of 𝔒 with ∪_{j}𝔒_{j}\𝔒_{j}^{0}. The orbit 𝔒 is open (Proposition 3.1). It remains to check that 𝔒_{j}\𝔒_{j}^{0} has higher codimension. Essentially, the only way for a point ω to be contained in 𝔒_{j}\𝔒_{j}^{0} is to satisfy, apart from δ_{j}(ω) = 0, at least some additional K invariant system of polynomial equations. For example, for j = q, it may be, amongst others, δ_{q−1} = 0 or the system ω_{q+1} ⊓ l_{p} = 0. The proof is complete at this point.

Proof. Proof of Lemma 3.3. We can view the flag manifold as a projective subvariety of ℙ_{n−1} × 𝔾(1, n − 1) × ⋯ × 𝔾(n − 2, n − 1) in the standard way.

For any j, we can find homogeneous polynomials P_{0}, P_{1}, …, P_{m} (m depends on j), such that the set 𝔒_{j}^{0} = {ω|P_{0}(ω) = 0, P_{j}(ω) ≠ 0 for i > 0}. For example, when j = q, put P_{0} = δ_{q}, P_{i} = δ_{i} for 1 ≤ i ≤ q − 1, and P_{q} = δ_{0}. Note that ω_{q−1} ⊔ l_{p} and ω_{q+1} ⊓ l_{p} have various components, and we let P_{i}, for i > q, be these components.

Since K is connected, the K -orbits 𝔒_{j}^{0} are irreducible. Now the lemma follows as a basic result of algebraic geometry.

5. Stein Extension and Determinant Functions

Let G_{0} be a general connected semisimple linear Lie group, K_{0} its maximal compact subgroup, and let G and K be the complexifications. For various reasons, we are interested in G_{0} invariant Stein extensions of G_{0}/K_{0} ⊆ G/K. Work done in this direction includes [1], [11], [9], [10]. Recently, J. Wolf and R. Zierau, ([12]) have shown that if G_{0}/K_{0} admits a Hermitian symmetric structure, then G_{0}/K_{0} × \overline{G_{0}/K_{0}} can be viewed as sitting inside G/K and this embedding is a G_{0} invariant Stein extension of G_{0}/K_{0} ⊆ G/K ([12]). The motivation behind their results is to find concrete realization of the space of linear cycles of a measurable open G_{0} orbit of a generalized flag manifold associated to G. We will now show, in the special case of G_{0} := SU(p, q) and K_{0} := S(U(p) × U(q)), another way of obtaining the same Stein extension.

First of all, we identify G_{0}/K_{0} as

Z_{0} := {L_{p} ∈ G(p, p + q)|⟨ , ⟩ is positive definite on L_{p}}.

Here we define ⟨z, z⟩ := ∑_{i=1}^{p} |z_{i}|^{2} − ∑_{j=p+1}^{p+q} |z_{j}|^{2}. We shall identify the conjugate manifold \overline{G_{0}/K_{0}} as

\overline{Z_{0}} := \{L′_{q} ∈ G(q, p + q)|⟨ , ⟩ is negative definite on L′_{q}\}.

Hence there is the obvious identification of Z_{0} × \overline{Z_{0}} ⊆ Z = G/K. If p = q we have also the following identification:

\overline{Z_{0}} × Z_{0} = \{(L_{p}, L′_{q}) ∈ G(p, p + q) × G(q, p + q)|

⟨ , ⟩ is negative definite on L_{p} and positive definite on L′_{q}}.

We sometimes identify

Z_{0} = \{(L_{p}, (L_{p})^{⊥})|⟨ , ⟩ is positive definite on L_{p}\} ⊆ Z_{0} × \overline{Z_{0}}.

Here ⊥ denotes the orthogonal complement with respect to ⟨ , ⟩. This identification sets Z_{0} as the real form of Z.

It is well known that X has a unique closed G_{0} orbit X_{0} which is K_{0} homogeneous ([8]). It can be realized geometrically as follows. On ℙ_{n−1} are three G_{0} orbits, two of them are open (those points on which ⟨ , ⟩ is positive, resp. negative), the remaining one Σ is closed. A flag ω = (ω_{j}, 1 ≤ j ≤ n) belongs to X_{0} iff the following is true: ω_{q} ⊆ Σ (here, we view ω_{q} as a projective plane of dimension q−1 and Σ a subset in ℙ_{n−1}) and ω_{n−1}, when viewed as a hyperplane of ℙ_{n−1}, is tangent to Σ at any point of ω_{q}. Indeed, the family of flags just described is G_{0}- invariant. On the other hand, it is easy to check that its diemnsion coincides with the diemnsion of the closed orbit, X_{0}.

Now consider the set

Definition 5.1. Define X̂_{0} ⊆ Z as follows.

X̂_{0} := {(L_{p}, L′_{q})|D_{j}(ω; L_{p}, L′_{q}) ≠ 0 for all j and for all ω ∈ X_{0}}.
If D := Π_{j}D_{j}, then
X̂_{0} = {(L_{p}, L′_{q})|D(ω; L_{p}, L′_{q}) ≠ 0 for all ω ∈ X_{0}}.  (7)

Since X_{0} is G_{n}ot-invariant, X̂_{0} is a G_{0} invariant subset of Z. We want to show that X̂_{0} is an invariant Stein space containing Z_{0}. In [2], we give a proof whose method we believe is capable of generalization. Here, instead, we will compute X̂_{0} explicitly, and it will be manifest that the space is Stein.

Lemma 5.2. The set Z_{0} × \overline{Z_{0}} is contained in X̂_{0}. If p = q, then we also have \overline{Z_{0}} × Z_{0} ⊆ X̂_{0}. In particular, X̂_{0} ⊆ Z is non-empty.

Proof. While it is not strictly necessary, it is simpler to recall that X_{0} is indeed a K_{0} orbit. In fact, if we let U(p, q) act on X and Z in the obvious way, U(p) × U(q) preserves X_{0}. We use the ordered basis (e_{1} + e_{p+1}, …, e_{q} + e_{p+q}, \overbrace{e_{q+1}, …, e_{p},} e_{q} − e_{p+q}, …, e_{1} − e_{p+1}) to represent a base point on X_{0} (for p = q, omit the overbrace). Since X_{0} is a K_{0}-orbit, a point on X_{0} is of the form k_{0}˙(e_{1} + e_{p+1}, …, e_{q} + e_{p+q}, \overbrace{e_{q+1}, …, e_{p},} e_{q} − e_{p+q}, …, e_{1} − e_{p+1}) with k_{0} ∈ K_{0} as in (6). Thus, any point on X_{0} can be represented by

\begin{matrix}( \begin{matrix}u_{1} \\ v_{1} \end{matrix} ), & …, & ( \begin{matrix}u_{q} \\ v_{q} \end{matrix} ); \\ ( \begin{matrix}u_{q+1} \\ 0_{q} \end{matrix} ), & …, & ( \begin{matrix}u_{p} \\ 0_{q} \end{matrix} ); \\ ( \begin{matrix}u_{q} \\ −v_{q} \end{matrix} ), & …, & ( \begin{matrix}u_{1} \\ −v_{1} \end{matrix} ), \end{matrix}  (8)

where u_{1}, …, u_{p} form an orthonormal basis of ℂ^{p} (with respect to the standard Hermitian inner product) and v_{1}, …, v_{q} form an orthonormal basis for ℂ^{q} (for p = q, omit the middle row).

Suppose (ω_{j}) represents a point in X_{0} and (L_{p}, L′_{q}) ∈ Z_{0} × \overline{Z_{0}}. Since ω_{q} represents a totally null subspace (i.e., the restriction of ⟨ , ⟩ to ω_{q} is identically zero) and L_{p} a positive definite subspace, ω_{q} ∩ L_{p} = 0, hence D_{q} = [ω_{q} ⊔ L_{p}] ≠ 0. Likewise, ω_{p} is a positive semi-definite subspace whereas L′_{q} is negative definite. By similar reasoning D_{0} ≠ 0.

For 1 ≤ j ≤ q − 1, assume, for the moment, that ω_{p+q−j} ⊓ L_{p} ≠ 0. Under this assumption, ω_{p+q−j} ⊓ L_{p} represents a positive definite subspace. However, ω_{j} is a totally null subspace, and L′_{q} a negative definite subspace. Hence, these three subspaces intersect trivially pairwisely, hence D_{j} = [(ω_{p+q−j} ⊓ L_{p}) ⊔ ω_{j} ⊔ L′_{q}] ≠ 0.

It remains to show that ω_{p+q−j} ⊓ L_{p} ≠ 0. In other words, we want to show that the dimension of ω_{p+q−j} ∩ L_{p} is p − j. By general consideration, it is at least p − j. On the other hand, the maximum possible dimension of a positive definite subspace of ω_{p+q−j} is p−j, as ω_{p+q−j} represents a subspace containing the totally null subspace ω_{q} of dimension q. As L_{p} is positive definite, the dimension of ω_{p+q−j} ∩ L_{p} cannot be more than p − j.

The additional possibility when p = q is similarly proved and we omit it.

Proposition 5.3. If p ≠ q, X̂_{0} = Z_{0}×\overline{Z_{0}}. If p = q, X̂_{0} = (Z_{0}×\overline{Z_{0}})∪(\overline{Z_{0}}×Z_{0}).

Proof. In view of Lemma 5.2, it suffices to show that X̂_{0} ⊂ Z_{0} × \overline{Z_{0}} if p ≠ q and X̂_{0} ⊂ (Z_{0} × \overline{Z_{0}}) ∪ (\overline{Z_{0}} × Z_{0}) if p = q.

Assume (L_{p}, L′_{q}) ∈ X̂_{0}. We first show that both are definite subspaces. If not, the one which is not must contain a null vector v. We can always find a flag in X_{0} such that, if (ω_{j}) represents the flag, v is a representative of ω_{1}. This means either ω_{q} ⊔ L_{p} = 0 or ω_{p} ⊔ L′_{q} = 0 and we have a contradiction.

Next, we show that L_{p} and L′_{q} cannot have the same sign. We first consider the case when q = 1. The case U(1, 1) is well understood. Thus, we can assume that p > 1. If L_{p} and L′_{1} have the same sign, since we know that L_{p} is positive, then L′_{1} is also positive. Pick a point a ∈ L′_{1}. We can choose a base point in X_{0} such that ω_{p} contains a. (See Equation 8). At this base point D_{0} = 0 and we have a contradiction.

If q > 1 and both L_{p} and L′q have the same sign, when p > q we know L_{p} must be positive, and hence L′_{q} is positive also; when p = q, since the signs are symmetric, we can without loss of generality assume that both L_{p}, L′_{q} are positive.

Under this assumption, we have a ∈ L_{p}, b ∈ L′_{q} such that the restriction of ⟨ , ⟩ on the span of a, b has mixed signatures (or else, we can find a subspace of dimension p + 1 on which ⟨ , ⟩ is positive definite). We can find an orthonormal basis for this span, i.e., {u, v} such that ⟨u, u⟩ = 1, ⟨v, v⟩ = −1, ⟨u, v⟩ = 0. We can always find a flag in X_{0} which can be represented by an ordered basis (z_{i}, 1 ≤ i ≤ p + q) in which z_{q−1} = u + v, z_{p+1} = u − v (see Equation 8).

In that case ω_{p+1} contains both a, b, hence ω_{p+1} ∩ L_{p} contains a. We may as well assume that ω_{p+1} ⊓ L_{p} ≠ 0, or else, D_{q−1} = 0 and we already have a contradiction. So the span of ω_{p+1} ⊓ L_{p} and L′_{q} contains a, b and hence u + v, which is also contained in ω_{q−1}, hence (ω_{p+1} ⊓ L_{p}) ⊔ L′_{q} ⊔ ω_{q−1} = 0, i.e., D_{q−1} = 0, and we have a contradiction.

In other words L_{p}, L′_{q} must be definite of opposite signs, and we are done.

6. Linear Cycles

Next, we would like to relate our Stein extension X̂_{0} with the one obtained by Wolf and Zierau. Their results can be briefly summarized as follows. Fix an open G_{0} orbit D of the flag manifold. In D is a special maximally compact subvariety Γ. In details, if we express D = G_{0}/T_{0}, where T_{0} is a compact Cartan subgroup, then Γ = K_{0}/T_{0}. Now consider the set of all those G translates of Γ which stay inside D, they are called linear cycles. It is known that the space of linear cycles admits the structure of a Stein manifold ([9]). It is now known that if G_{0}/K_{0} admits a Hermitian symmetric structure, and if D is not of Hermitian type (i.e., it does not fibrate holomorphically over G_{0}/K_{0} if the latter is given either of the two Hermitian symmetric structures), then the space of linear cycles is exactly G_{0}/K_{0} × \overline{G_{0}/K_{0}}.

In the case of SU(p, q), we can describe the space of linear cycles explicitly as follows. First of all, we need a way of parametrizing the flag domain D. It turns out that we can parametrize them by a non-decreasing sequence of integers 0 ≤ a_{1} ≤ … ≤ a_{q} ≤ p. We have a related sequence 1 ≤ b_{1} < … < b_{q} ≤ (p+q), by putting b_{i} := a_{i} + i. There are various ways to specify the orbit once a sequence (a_{i}) or (b_{i}) is given. We can pin down the open orbit by specifying the signatures of the restrictions of ⟨ , ⟩ to various subspaces in a flag belonging to an orbit. Given (b_{i}), let the orbit determined to be 𝔒(b), then ω ∈ 𝔒(b) iff for all j, ω_{j} has

signature (\overbrace{+, …, +}^{j−l}, \overbrace{−, …, −}^{l}). Here we define l := max_{i}\{i|b_{i} ≤ j\}.

A G_{n}ot orbit ,S, and a K -orbit, Q, are dual to each other (in the sense of Matsuki) if the intersection S ∪Q consists of exactly one K_{n}ot-orbit. The K -orbit dual to an open G_{n}ot-orbit is closed. We can pin down the Matsuki dual (closed) K orbit as follows. Recall the subspaces l_{p}, l′_{q} in Section 3. A flag ω lies in the K orbit dual to 𝔒(b) iff dim(ω_{j} ∩ l_{p}) = j − l and dim(ω_{j} ∩ l′_{q}) = l, where l retains the same meaning as just now. Finally, we can specify the orbit 𝔒(b) by picking a “nice” base point. We can use the following ordered basis (v_{j}) to pin down the “nice” base point. Let σ be the permutation of {1, …, p+q} which sends p+i to b_{i} for 1 ≤ i ≤ q and sends i to i + n(i) for 1 ≤ i ≤ p. Here n(i) := ♯{j|a_{j} < i}. With this notation in hand, define v_{i} = e_{σ^{−1}(i)}.

Secondly, we view Z_{0} × \overline{Z_{0}} = G_{0}/K_{0} × \overline{G_{0}/K_{0}} as the space:

{(L_{p}, L′_{q}) ∈ G(p, p + q) × G(q, p + q)|  ⟨ , ⟩ is positive
(negative) definite on L_{p}(L′_{q})}.

The space Z_{0} × \overline{Z_{0}} parametrizes the space of linear cycles associated to D. Suppose D is pinned down by the sequence (a_{i}). Suppose we choose (L_{p}, L′_{q}) ∈ Z_{0} × \overline{Z_{0}}, then the linear cycle thus parametrized consists of flags (ω_{j}) where dim(ω_{j} ∩ L_{p}) = j − l, dim(ω_{j} ∩ L′_{q}) = l. (Keep the same notations as just now.)

For the sake of completeness, we consider the two flag domains of Hermitian type. The flag domains are either (a_{j} = p; 1 ≤ j ≤ q) or (a_{j} = 0; 1 ≤ j ≤ q). In the former case, the space of linear cycle is Z_{0}, and in the latter case \overline{Z_{0}}. In the former case, suppose we have L_{p} ∈ Z_{0} ⊆ G(p, p + q), the linear cycle thus parametrized is given by ω_{j} ⊂ L_{p}; 1 ≤ j ≤ p and L_{p} ⊆ ω_{j}; p + 1 ≤ j ≤ p + q. Interchange p and q and replace Z_{0} by \overline{Z_{0}} in the previous sentence and we have the latter case.

Finally, we would like to discuss briefly how the various minimal K orbits sit in the boundaries of 𝔒_{j}. These minimal orbits are dual (via Matsuki duality) to the flag domains, indeed, we have described them already. We have the following.

Proposition 6.1. Any minimal K orbit dual to a flag domain lies in 𝔒_{j} for the following ranges of j. If it is not of Hermitian type, we have 0 ≤ j ≤ q. If it is dual to the domain given by a_{k} = p, 1 ≤ k ≤ q, it is 1 ≤ j ≤ q. If it is given by a_{k} = 0, 1 ≤ k ≤ q, then it is 0 ≤ j ≤ q − 1.

Proof. As all 𝔒_{j} are K invariant, it suffices to pick a nice base point of the orbit and check which 𝔒_{j} it belongs to. To do that, we only have to consider which equations D_{j}( ; l_{p}, l′_{q}) = 0 it satisfies. The base point chosen earlier in the section suffices. We will only argue carefully that all minimal orbits lie in 𝔒_{j} for 1 ≤ j ≤ q − 1. The rest can be similarly argued.

Observe that for any 1 ≤ j ≤ q − 1, ω_{p+q−j}, viewed as a linear subspace, must contain e_{1}, so e_{1} ∈ ω_{p+q−1} ∩l_{p}. On the other hand, either ω_{j} ⊔l′_{q} = 0, in that case D_{j}(ω; l_{p}, l′_{q}) = 0, or ω_{j} ⊆ l_{p} (when viewed as a linear subspace) and e_{1} ∈ ω_{j}, in that case, we are forced to conclude D_{j} = (ω_{p+q−j} ⊓ l_{p}) ⊔ (ω_{j} ⊔ l′_{q}) = 0 also.

We know that higher codimensional orbits occur as certain strata of a K invariant stratification of the boundary of the configuration ∪_{j=0}^{q}𝔒_{j}. Proposition 6.1 implies that this stratification is more than mere intersections of the boundaries of a subcollection of 𝔒_{j}.

Finally, we would like to prove a result which makes use of knowledge of the orbit structure. Recall the notions of polar and double polar introduced in Section 1.

Proposition 6.2. For any subset A ⊂ X, we have A ⊆ ˆ Â. When A = X_{0}, we have ˆ X̂_{0} = X_{0}.

Proof. The fact that A ⊆ ˆ Â in general is basically a tautology. When A = X_{0}, since X_{0} is G_{0} invariant, so is ˆ X̂_{0} . So the latter is a G_{0} invariant subset, hence a finite union of G_{0} orbits. According to Matsuki’s duality, a K orbit Γ and a G_{0} orbit S are dual iff Γ ∩ S consists of exactly one K_{0} orbit. Suppose ˆ X̂_{0} contains the G_{0} orbit S. Let Γ be the K orbit dual to S. Now Γ ∩ S ⊂ S ⊂ ˆ X̂_{0} . As (l_{p}, l′_{q}) ∈ X̂_{0}, we conclude that Γ ∩ S is in the open K orbit. In othe words Γ intersects non-trivially with the open K orbit. Hence Γ is the open K orbit, and therefore S = X_{0}.

7. Appendix

Propositions 3.1 and 3.2 can be proved by using the well known correspondence between K orbits of X and the set of equivalence classes (under conjugation by G_{0}) of real data \overline{D}^{ℝ} as defined in [7, p. 107]. In this appendix we explain briefly this approach.

We recall that a set of real data for G_{0} consists of a pair (H_{0}, Δ^{+}(𝔤, 𝔥)) where H_{0} ⊂ G_{0} is a Cartan subgroup, and Δ^{+} a choice of positive system for 𝔥 in 𝔤. To each equivalence class corresponds a unique K orbit 𝔒 in X. Fix 𝔒 and denote by (H_{0,r}, Δ_{r}^{+}(𝔤, 𝔥)) a choice of representative for the corresponding equivalence class in \overline{D}^{ℝ}. We may assume that H_{0,r} is θ stable.

There is a formula which gives the dimension of 𝔒 in terms of the number of real, imaginary, and complex roots in Δ_{r}. Real, imaginary, and complex roots are defined in the obvious way. We denote by Δ_{r,ℝ}, Δ_{r,I}, Δ_{r,ℂ}, and Δ_{r,CI} the set of real, imaginary, complex, and compact imaginary roots. If θ_{r} is the Cartan involution on Δ_{r} (chosen as in [6, pp. 147-148]), then set

D_{+}(𝔒) = {α ∈ Δ_{r}^{+}|θ_{r}α ∈ Δ_{r}^{+}, θ_{r}α ≠ α}

and set d(𝔒) = |D_{+}(𝔒)|. We have (see [6, Lemma 5.6]):

Lemma 7.1. Let 𝔒 be a K orbit in X, then

dim 𝔒 = \frac{1}{2}(|Δ_{r,CI}| + |Δ_{r,ℝ}| + |Δ_{r,ℂ}| − d(𝔒)).

The dimension of the K orbits associated to H_{0,r} lies between ([6])

\frac{1}{2}(|Δ_{r,CI}| + |Δ_{r,ℝ}| + \frac{1}{2}|Δ_{r,ℂ}|)

and

\frac{1}{2}(|Δ_{r,CI}| + |Δ_{r,ℝ}| + |Δ_{r,ℂ}|).

The K orbit of maximal dimension associated to H_{0,r} is called the Langlands orbit attached to H_{0,r}. The K orbit of minimal dimension associated to H_{0,r} is called the Zuckerman orbit attached to H_{0,r}. It is well known that if X is connected, then there exists a unique open K orbit and it is attached to the maximally split Cartan. ([4])

When G_{0} = SU(p, q), there is a convenient choice of representatives from the equivalence classes of \overline{D}^{ℝ}. Here we follow [3, §1.3]. Let 𝔱_{0} denote the maximally compact Cartan subalgebra of 𝔤_{0}. Set α_{i}^{0} = e_{i}^{0} − e_{i+1}^{0}, 1 ≤ i ≤ p + q − 1. The fixed choice of positive system for Δ(𝔤, 𝔱) for which α_{i}^{0} are simple roots will be denoted by Δ_{0}^{+}. For 1 ≤ r ≤ q, set

β_{r}^{0} = α_{p−r+1}^{0} + α_{p−r+2}^{0} + … + α_{p+r−1}^{0} = e_{p−r+1}^{0} − e_{p+r}^{0}.

Let c_{r} be the Cayley transform through β_{r}^{0}. Set 𝔥_{0} = 𝔱 and for r > 0, set inductively 𝔥_{r} = c_{r}(𝔥_{r−1}), and let H_{0,r} be the Cartan subgroup corresponding to 𝔥_{r} ∩ 𝔤_{0}. Write W_{r}(𝔤, 𝔥_{r}) for the Weyl group of Δ(𝔤, 𝔥_{r}). Define Δ_{r}^{+}, α_{i}^{r}, β_{i}^{r} inductively.

Let s_{i}^{r} ∈ W_{r} denote the simple reflection through α_{i}^{r}. Any Cartan subalgebra 𝔥̃_{0} ⊂ 𝔤_{0} is G_{0} conjugated to one of the 𝔥_{0,r}, so we assume that the representatives of the equivalence classes in \overline{D}^{ℝ} are of the form (H_{0,r}, Δ_{r}^{+}).

Lemma 7.2. Let G_{0} = SU(p, q), with p ≥ q.

1. If p > q, then the codimension one K orbits in X (if any) are associated to the maximally split Cartan H_{0,q}.

2. If p = q, then the codimension one K orbits in X (if any ) are associated either to the maximally split Cartan or to H_{0,q−1}( The Cartan subgroup with split rank q − 1).

Proof. We will only sketch the proof. Consider H_{0,q−1}, the Cartan subgroup with split rank q − 1. It is not difficult to show that

\begin{aligned}|Δ_{q−1,ℝ}| & = |Δ_{q,ℝ}| − 2 \\ |Δ_{q−1,CI}| & = |Δ_{q,CI}| + 2(p − q) \\ |Δ_{q−1,ℂ}| & = |Δ_{q,ℂ}| − 4(p − q). \end{aligned}

By formula 7.1, the dimension of the Langlands orbit associated to H_{0,q−1} is equal to dim 𝔒open − (p − q) − 1. Thus, if p > q, then the dimensions of the K orbits associated to H_{0,q−1} are always smaller than dim 𝔒open − 1. We can treat the other Cartan subgroups H_{0,i} in a similar way.

Next, we use Vogan’s parameter set S(p + q, p, r), as explained in [3, §1.6], to parametrize K -orbits. A parameter set consists of pairs (x, y) ∈ {1, …, p + q}×{1, …, p+q} or (x, y) ∈ {1, …, p+q}×{+, −} satisfying certain conditions. There exist bijections

\frac{|W (𝔤, 𝔥_{r})|}{|W (G_{0}, H_{0,r})|} →^{g_{1}} S(p + q, p, r) →^{g_{2}} D_{r} = \{(H_{0,r}, Δ^{+})\}.

In particular, S(p + q, p, r) classifies the K orbits attached to H_{0,r}.

Lemma 7.3. The parameter set associated to the open K orbit in X is

P_{open} = (p, +)(p − 1, +) … (q + 1, +)(1, p + q)(2, p + q − 1) … (q, p + 1). (9)

Proof. We will only sketch the proof. The bijection g_{1} associates to P_{open} (as in Formula 9) an equivalence class in \frac{W (𝔤,𝔥_{q})}{W (G_{0},H_{0,q})} . Let w ∈ W (𝔤, 𝔥_{q}) be a representative of such an equivalence class. Consider wΔ^{+}(𝔤, 𝔥_{q}) and use the dimension formula 7.1 to show that the corresponding K orbit is a Langlands orbit.

There is a well defined action of the Weyl group elements s_{i} in D_{r} ([3, §1.8]). We have

s_{i} × (H_{r}, wΔ^{+}_{r}) = (H_{r}, ws_{i}^{r}Δ_{r}^{+}).

On the other hand, there is a compatible definition for the action of s_{i} on the parameter set ([3, §1.9]). Indeed, if P ∈ S(p + q, p, r) corresponds to (H_{r}, wΔ^{+}_{r}), then s_{i} × P corresponds to (H_{r}, ws_{i}^{r}Δ_{r}^{+}).

Lemma 7.4. 1. If p > q, then the K orbit corresponding to

s_{i} × P_{open}

are codimension one orbits.

2. If p > q, then there are exactly q + 1 co-dimensional one K orbits of X.

3. If p = q, then the K orbits associated to

s_{i} × P_{open}

are codimensional one orbits. There are exactly q − 1 orbits of this type.

4. If p = q, then the Langlands orbits associated to H_{0,q−1} are codimensional one orbits. There are two orbits of this type. If p = q, then the total number of codimension one orbits is q + 1.

Proof. We will only sketch the proof. We use [3, §1.9] and the bijection between K orbits in X and the parameter set to show that the K orbits attached to the maximally split Cartan correspond to wP_{open} ∈ S(p + q, p, q) with w ∈ ⟨s_{p}^{q}; s_{i}^{q}, 1 ≤ i ≤ q⟩ ⊂ W (𝔤, 𝔥_{q}) if p > q, and w ∈ ⟨s_{i}^{p}, 1 ≤ i ≤ q − 1⟩ if p = q. Here, s_{i}^{r} is the reflection with respect to the root α_{i}^{r}.

If (H_{0,q}, Δ_{open}^{+}) denotes the pair that corresponds to P_{open} in the parameter set, then (H_{q}, s_{i}^{r}Δ_{open}^{+}) corresponds to s_{i} × P_{open}. We observe that the new positive system differs from Δ_{open}^{+} by one complex root. Thus, by the dimension formula 7.1, the K orbits associated to (H_{0,r}, s_{i}^{r}Δ_{open}^{+}) are co-dimensional one orbits. Any other orbit is attached to a pair (H_{0,q}, wΔ_{open}^{+}) with the length of w bigger than 1 and thus its dimension is lower.

If p = q, then

P_{1} = (p, +)(p + 1, −)(1, 2p) … (p − 1, p + 2)

and

P_{2} = (p + 1, +)(p, −)(1, 2p) … (p − 1, p + 2)

parametrize two Langlands orbits associated to H_{0,q−1}. All other orbits attached to H_{0,q−1} are obtained by letting elements in W (𝔤, 𝔥_{q−1}) act on P_{1} and P_{2}. It can be easily verified that, in this case, the same procedure will produce orbits of lower dimensions.

Remark 7.5. The method described in this section selects the elements in the parameter set that are associated to codimensional one orbits. Associated to that element is an equivalence class in W (𝔤, 𝔥_{r})/W (G_{0}, H_{0,r}). Let w be a representative of the equivalence class. We can build the pair (H_{r}, wΔ_{open}^{+}) that is attached to the K orbit. The corresponding Borel subalgebra is then a base point of the orbit. This is an idea of Vogan that is explained in [3].

References

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Department of Mathematics Temple University Broad and Montgomery Philadelphia PA 19122 Department of Mathematics Rutgers University New Brunswick NJ 08903

Received March 22, 1996 and in final form July 18, 1996