Invariant Stein domains in Stein symmetric spaces and a non-linear complex convexity theorem

Simon Gindikin^{∗} and Bernhard Krötz^{†}

Abstract

We prove a complex version of Kostant’s non-linear convexity theorem. Applications to the construction of G-invariant Grauert tubes of Riemannian symmetric G/K spaces are given.

Introduction

Let X = G/K be a semisimple non-compact Riemannian symmetric space. We may assume that G is semisimple with finite center and we write G = NAK for an Iwasawa decomposition of G. By our assumption, G sits in its universal complexification G_{ℂ} and so X ⊆ X_{ℂ}: = G_{ℂ}/K_{ℂ}. Note that X_{ℂ} is a Stein symmetric space. Observe that the group G does not act properly (i.e. with compact isotropy subgroups) on G_{ℂ}/K_{ℂ} since K_{ℂ} is not compact. Write U for the compact real form of G_{ℂ}. The maximal connected subdomain of GUK_{ℂ}/K_{ℂ} ⊆ X_{ℂ} on which G acts properly and which contains X was introduced in [AkGi90]. It is given by

Ξ: = G exp(iΩ)K_{ℂ}/K_{ℂ}

where Ω is a polyhedral convex domain in 𝔞: = Lie(A) defined by

Ω: = \{X ∈ 𝔞: (∀α ∈ Σ) |α(X)| < \frac{π}{2} \}.

Here Σ denotes the restricted root system with respect to 𝔞.

One of our principal aims is to construct a broad class of G-invariant Stein subdomains in Ξ. Let us remind the reader that it is a notoriously difficult problem to verify that certain unions of orbits of non-compact groups are Stein. For example the problem posed in [AkGi90] whether Ξ is Stein was unsolved until the last year.

The Iwasawa decomposition on G cannot be holomorphically extended to the whole group G_{ℂ}; we have that N_{ℂ}A_{ℂ}K_{ℂ} ⊊ G_{ℂ} is an open Zariski dense subset. The Iwasawa domain Ξ_{I}

2 Simon Gindikin^{∗} and Bernhard Krötz^{†}

∗ Supported in part by the NSF-grant DMS-0070816 and the MSRI † Supported in part by the NSF-grant DMS-0097314 and the MSRI is defined as the maximal connected G-invariant subdomain in X_{ℂ} which contains X and is contained in N_{ℂ}A_{ℂ}K_{ℂ}/K_{ℂ}:

Ξ_{I}: = ( ⋂_{g∈G} g(N_{ℂ}A_{ℂ}K_{ℂ}/K_{ℂ}))_{0},

where (·)_{0} refers to the connected component containing X. Since Ξ_{I} is the connected component of an open intersection of Stein domains, it is easy to see that Ξ_{I} is Stein (cf. proof of Proposition 3.4).

The domains Ξ and Ξ_{I} again became of recent interest. One has Ξ = Ξ_{I}. Here are the results in chronological order:

· Ξ ⊆ Ξ_{I} for all classical groups G (cf. [KrSt01a]). · Ξ = Ξ_{I} for all classical groups G (cf. [GiMa01]). · Ξ_{I} ⊆ Ξ for all G (cf. [Ba01]).

In [KOS01] it will be shown that Ξ ⊆ Ξ_{I} for the exceptional cases. Also it is announced in [Hu01] that Ξ ⊆ Ξ_{I}. This will then give us Ξ = Ξ_{I} in the general case. In particular, Ξ is Stein. Also, in the preprint [BHH01] a complex-geometric proof of the Steinness of Ξ is given.

Geometrically, the domain Ξ is a rather complicated object. If G is a group of Hermitian type, then Ξ = X × \overline{X} (cf. [BHH01], [GiMa01] or [KrSt01b]). If X = G/K is classical, then there exists a group of Hermitian type S ⊇ G with maximal compact subgroup U ⊇ K such that G/K is a real form of the Hermitian symmetric space S/U. In [BHH01] and [KrSt01b] a G-equivariant subdomain Ξ_{0} ⊆ Ξ was exhibited which is biholomorphic to S/U. Further one has Ξ = Ξ_{0} = S/U if and only if Σ is of type C_{n} or BC_{n}. In particular, if Σ ≠ C_{n}, BC_{n}, then S/U ⊊ Ξ and the explicit geometric structure of Ξ is very intricate. For many exceptional spaces G/K the geometric structure is even more complicated.

The domain Ξ (in [Gi98] it is called complex crown of X) is universal in the sense that many analytical and geometrical constructions on the Riemannian symmetric space X extend to Ξ. In [KrSt01a] it was shown that the inclusion Ξ ⊆ Ξ_{I} implies that all eigenfunctions on X for the algebra of G-invariant differential operators 𝔻(X) extend holomorphically to Ξ. In [GiMa01] the problem Ξ ⊆ Ξ_{I} is included in a broad class of geometrical problems connected with Matsuki duality. These geometrical problems include in particular the problem of the parametrization of compact complex cycles in flag domains (cf. [Wo92]).

Let us now come to the contents of this paper. Our first main result is a complex version of Kostant’s non-linear convexity theorem (cf. [Kos73]). Write G → A, g ↦ a(g) for the middle projection in G = NAK.

Theorem. (Kostant) Let X ∈ 𝔞. Then

a(K exp(X)) = exp(conv(WX)),

where W = N_{K}(𝔞)/Z_{K}(𝔞) denotes the Weyl group and conv(·) refers to the convex hull of (·).

One shows that the middle projection a: G → A holomorphically extends to

(1)  a: G exp(iΩ)K_{ℂ} → A_{ℂ}.

Then our main result is:

Theorem A. (Complex Convexity Theorem) For all X ∈ Ω we have that

a(G exp(iX)) ⊆ A exp(i conv(WX)).

Our convexity theorem features interesting applications to the geometry of the domain Ξ and its generic subdomains which are defined as follows: Let ω ⊆ Ω be a non-empty convex W -invariant open subset. Then we can form the domains

Ξ(ω): = G exp(iω)K_{ℂ}/K_{ℂ}.

Note that Ξ = Ξ(Ω) and that the Iwasawa projection (1) naturally factors to a holomorphic mapping a: Ξ → A_{ℂ}. Finally we define for every g ∈ G the horospherical tube T (g, ω) ⊆ X_{ℂ} by

T (g, ω): = g(N_{ℂ}A exp(iω)K_{ℂ}/K_{ℂ}).

As an application of Theorem A we now obtain:

Theorem B. Let ω be an open convex Weyl group invariant subset of Ω. Then the following assertions hold:

(i) a(Ξ(ω)) = A exp(iω). (ii) Ξ(ω) = ( ⋂_{g∈G} T (g, ω))_{0} . (iii) The domain Ξ(ω) is Stein.

Let us emphasize that in the the case of Ξ(ω) = Ξ, the inclusion Ξ ⊆ Ξ_{I} means only the existence of an Iwasawa projection a: Ξ → A_{ℂ}; (i) in Theorem B gives a more precise information on the image of this projection. Further (ii) in the above theorem is a much stronger statement than (iii); in particular, (ii) implies (iii) since all horospherical tubes T (g, ω) are Stein.

Theorem B can be considered as an analogue of Lassalle’s results for compact symmetric spaces (cf. [La78]). We can interpret Theorem A and Theorem B as statements for G-orbits in X_{ℂ} intersecting A exp(iΩ)K_{ℂ}/K_{ℂ} which in the case of compact symmetric spaces G/K are true for arbitrary G-orbits (cf. [La78]).

It is our pleasure to thank the MSRI, Berkeley, for its hospitality during the Integral geometry program where this work was accomplished. We thank Dmitri Akhiezer for his careful screening of the manuscript and his worthy suggestions. Further we would like to thank Karl-Hermann Neeb for his kindness to proofread the paper and Laura Geatti for sharing her knowledge with us on the boundary of Ξ.

1. Notation

Let G be a connected semisimple Lie group sitting inside a complexification G_{ℂ}. We denote by 𝔤 and 𝔤_{ℂ} the Lie algebras of G and G_{ℂ}, respectively. Let K < G be a maximal compact subgroup and 𝔨 its Lie algebra. Denote by θ: G → G a Cartan involution which has K as a fixed point set.

Let 𝔤 = 𝔨 ⊕ 𝔭 the Cartan decomposition attached to 𝔨. Take 𝔞 ⊆ 𝔭 a maximal Abelian subspace and let Σ = Σ(𝔤, 𝔞) ⊆ 𝔞^{∗} be the corresponding root system. Related to this root system is the root space decomposition according to the simultaneous eigenvalues of ad(H), H ∈ 𝔞 :

𝔤 = 𝔞 ⊕ 𝔪 ⊕ ⨁_{α∈Σ} 𝔤^{α},

here 𝔪 = 𝔷_{𝔨}(𝔞) and 𝔤^{α} = {X ∈ 𝔤: (∀H ∈ 𝔞) [H, X] = α(H)X}. For the choice of a positive system Σ^{+} ⊆ Σ one obtains the nilpotent Lie algebra 𝔫 = ⨁_{α∈Σ^{+}} 𝔤^{α}. Then one has the Iwasawa decomposition on the Lie algebra level

𝔤 = 𝔫 ⊕ 𝔞 ⊕ 𝔨.

We write A, N for the analytic subgroups of G corresponding to 𝔞 and 𝔫. For these choices one has for G the Iwasawa decomposition, namely, the multiplication map

N × A × K → G, (n, a, k) ↦ nak

In particular, every element g ∈ G can be written uniquely as g = n(g)a(g)κ(g) with each of the maps κ(g) ∈ K, a(g) ∈ A, n(g) ∈ N depending analytically on g ∈ G. The last piece of structure theory we shall recall is the little Weyl group. We denote by W = N_{K}(𝔞)/Z_{K}(𝔞) the Weyl group of Σ(𝔞, 𝔤).

Finally we define the domain

Ω = \{X ∈ 𝔞: (∀α ∈ Σ) |α(X)| < \frac{π}{2} \}.

Clearly Ω is convex and W -invariant.

2. The complex convexity theorem

Let us first give the relevant notation. Denote by K_{ℂ}, A_{ℂ}, N_{ℂ} the complexifications of K, A and N. Then N_{ℂ}A_{ℂ}K_{ℂ} is a proper Zariski-open, hence dense subset of G_{ℂ}. Throughout this paper we will assume that Ξ ⊆ Ξ_{I} holds, i.e.,

G exp(iΩ) ⊆ N_{ℂ}A_{ℂ}K_{ℂ}

(see the introduction for proofs of this statement). We now set T_{Ω}: = A exp(iΩ) ⊆ A_{ℂ}. Then GT_{Ω} ⊆ N_{ℂ}A_{ℂ}K_{ℂ}.

One can show that one has a well defined holomorphic middle projection N_{ℂ}A_{ℂ}K_{ℂ} → A_{ℂ}/(A_{ℂ} ∩ K_{ℂ}). Due to the simple connectedness of T_{Ω} , this projection restricted to GT_{Ω} lifts to A_{ℂ} (cf. [KrSt01a, proof of Th. 1.8(iii)]) and we obtain an analytic mapping

G × T_{Ω} → A_{ℂ}, (g, a) ↦ a(ga)

holomorphic in the second variable such that

ga ∈ N_{ℂ}a(ga)K_{ℂ}

holds.

If V is a vector space and E ⊆ V is a subset, then we denote by conv(E) the convex hull of E in V .

Theorem 2.1. (Complex Convexity Theorem) Assume that G is classical. Then we have for all X ∈ Ω that

a(G exp(iX)) ⊆ A exp(i conv(WX)).

The proof of Theorem 2.1 will be given in several steps.

Fix a ∈ T_{Ω} and consider the function

f_{a}: K → 𝔞_{ℂ}, k ↦ log a(ka).

Further we write p_{𝔞_{ℂ}}: 𝔤_{ℂ} → 𝔞_{ℂ} for the projection along 𝔨_{ℂ} + 𝔫_{ℂ}. For x ∈ GT_{Ω}K_{ℂ} ⊆ N_{ℂ}A_{ℂ}K_{ℂ} we write b(x) = n(x)a(x) for the triangular part of x.

Lemma 2.2. For any a ∈ T_{Ω} , k ∈ K and X ∈ 𝔨 we have

\frac{d}{dt}|_{t=0}f_{a}(exp(tX)k) = p_{𝔞_{ℂ}}(Ad(b(ka))^{−1}X).

Proof. This result can be easily deduced from the known case for a ∈ A by analytic continuation. However, for conveniene for the reader, we briefly recall the proof. We have

\frac{d}{dt}|_{t=0}f_{a}(exp(tX)k) = \frac{d}{dt}|_{t=0} log a(exp(tX)ka)

= \frac{d}{dt}|_{t=0} log a(exp(tX)b(ka))

= \frac{d}{dt}|_{t=0}[log a(b(ka)^{−1} exp(tX)b(ka)) + log a(ka)]

= \frac{d}{dt}|_{t=0} log a(exp(t Ad(b(ka))^{−1}X)) = p_{𝔞_{ℂ}}(Ad(b(ka))^{−1}X),

proving the lemma.

Write 𝔤_{ℂ}^{ℝ} for 𝔤_{ℂ} considered as a real Lie algebra and κ_{ℝ} for the Cartan-Killing form on 𝔤_{ℂ}^{ℝ}. Let κ be the Cartan-Killing form on 𝔤 and recall the following relation between κ and κ_{ℝ}:

(∀X, X′, Y, Y′ ∈ 𝔤) κ_{ℝ}(X + iY, X′ + iY′) = 2(κ(X, X′) − κ(Y, Y′)).

For every λ ∈ (𝔤_{ℂ}^{ℝ})^{∗} we define H_{λ} ∈ 𝔤_{ℂ}^{ℝ} by λ(X) = κ_{ℝ}(X, H_{λ}) for all X ∈ 𝔤_{ℂ}^{ℝ}.

For every λ ∈ (𝔞_{ℂ}^{ℝ})^{∗} and a ∈ T_{Ω} we now define the function

f_{a,λ}: K → ℝ, k ↦ λ(f_{a}(k)).

Lemma 2.3. Let a ∈ T_{Ω} and k ∈ K. Then the following assertions hold:

(i) For all X ∈ 𝔨 one has \frac{d}{dt} |_{t=0}f_{a,λ}(exp(tX)k) = κ_{ℝ}(X, Ad(n(ka))H_{λ}).

(ii) We have df_{a,λ}(k) = 0 if and only if

𝔨⊥_{κ_{ℝ}} Ad(n(ka))H_{λ}.

Proof. (i) is immediate from Lemma 2.2 and the notations introduced from above. (ii) follows from (i).

Write i𝔞_{ℝ}^{∗} for the subspace of (𝔞_{ℂ}^{ℝ})^{∗} which vanishes on 𝔞. Our next goal is to determine the critical set of f_{a,λ} for λ ∈ i𝔞_{ℝ}^{∗}.

We also denote by θ the holomorphic extension of the Cartan involution to G_{ℂ}. Further we write G_{ℂ} → G_{ℂ}, x ↦ \overline{x} for the complex conjugation with respect to the real form G.

Lemma 2.4. Suppose that X ∈ Ω is regular. Set a = exp(iX) and let k ∈ K. Then the following implication holds

ka ∈ NA_{ℂ}K_{ℂ} ⇒ k ∈ N_{K}(𝔞).

Proof. We write ka = nbk′ for n ∈ N, b ∈ A_{ℂ} and k′ ∈ K_{ℂ}. Then

ka^{2}k^{−1} = ka(θ(ka)^{−1}) = nbk′(θ(nbk′))^{−1} = nb^{2}θ(n)^{−1}.

Set x: = ka^{2}k^{−1} . Then

\overline{x} = ka^{−2}k^{−1} = x^{−1}.

Thus we obtain that

\overline{x} = n\overline{b}^{2}θ(n)^{−1} = θ(n)b^{−2}n^{−1}.

Therefore x = θ(n)\overline{b}^{−2}n^{−1} and so

x^{2} = θ(n)\overline{b}^{−2}n^{−1}nb^{2}θ(n)^{−1} ∈ A_{ℂ}θ(N_{ℂ}).

Now let 0 < t ≤ 1 and write a_{t}: = exp(itX) and accordingly we define x_{t}. Then we obtain that

log x_{t}^{2} = log(k exp(i4tX)k^{−1}) = Ad(k)(i4tX) ∈ 𝔞_{ℂ} + θ(𝔫_{ℂ})

for small t and hence for all t by analyticity. But Ad(k)(i4tX) ∈ i𝔭. Hence by the regularity of X we obtain that k ∈ N_{K}(𝔞), as was to be shown.

Proposition 2.5. Suppose that X ∈ Ω and set a = exp(iX). Let λ ∈ i𝔞_{ℝ}^{∗} be such that H_{λ} ∈ i𝔞 is regular. Then we have

df_{a,λ}(k) = 0 ⟺ k ∈ N_{K}(𝔞).

Proof. The implication “⇐” follows immediately from Lemma 2.3 (ii). “⇒”: Suppose that df_{a,λ}(k) = 0. Then Lemma 2.3 (ii) implies that 𝔨⊥_{κ_{ℝ}} Ad(n(ka))H_{λ}, or equivalently Ad(n(ka))H_{λ} ∈ i𝔤 + 𝔭. Assume that k ∉ N_{K}(𝔞). By Lemma 2.4 we have n(ka) = exp(Y) for some Y ∈ 𝔫_{ℂ}\𝔫. Hence

Ad(n(ka))H_{λ} = H_{λ} + \underbrace{[Y, H_{λ}]}_{∈𝔫_{ℂ}} +\frac{1}{2} \underbrace{[Y, [Y, H_{λ}]]}_{∈[𝔫_{ℂ},𝔫_{ℂ}]} + ….

Thus the fact that H_{λ} ∈ i𝔞 is regular implies that Ad(n(ka))H_{λ} ∈ i𝔞 + (𝔫_{ℂ}\i𝔫). But this contradicts Ad(n(ka))H_{λ} ∈ i𝔤 + 𝔭, proving our proposition.

Proof of Theorem 2.1. First we observe that it is sufficient to prove

a(K exp(iX)) ⊆ A exp(i conv(WX))

for all X ∈ Ω. By a simple density/continuity argument we may further assume that X is regular. Suppose that there exists a k ∈ K such that a(k exp(iX)) ∉ A exp(i conv(WX)), or equivalently

Im log a(k exp(iX)) ∉ conv(WX).

Then we find a regular element λ ∈ i𝔞_{ℝ}^{∗} such that

f_{exp(iX),λ}(k) > max_{Y ∈i conv(WX)} λ(Y).

But Proposition 2.5 implies that f_{exp(iX),λ} takes its maximum at an element k ∈ N_{K}(𝔞). Hence f_{exp(iX),λ}(k) = λ(Ad(k)iX); a contradiction to our inequality from above.

3. Applications

Let now ω ⊆ Ω be an open Weyl group invariant convex set. Then we define the domain

Ξ(ω): = G exp(iω)K_{ℂ}/K_{ℂ}.

If ω = Ω, then we set Ξ: = Ξ(Ω). Write ∂Ξ(ω) for the toplogical boundary of Ξ(ω) in G_{ℂ}/K_{ℂ}. Note the following properties of Ξ(ω):

· Ξ(ω) is open in G_{ℂ}/K_{ℂ} (cf. [AkGi90]). · Ξ(ω) is connected and G-invariant. · G acts properly on Ξ (cf. [AkGi90]). · One has G exp(i∂ω)K_{ℂ}/K_{ℂ} ⊆ ∂Ξ(ω). Moreover if \overline{ω} ⊆ Ω, then we have ∂Ξ(ω) =

G exp(i∂ω)K_{ℂ}/K_{ℂ} (cf. [KrSt01b, Prop. 4.1]).

From our discussions in the previous section it is clear that we have a holomorphic projection

a: Ξ_{ω} → A_{ℂ}, x ↦ a(x)

with x ∈ N_{ℂ}a(x)K_{ℂ}/K_{ℂ} for all x ∈ Ξ. Finally we define the Abelian tube domain

T_{ω}: = A exp(iω) ⊆ A_{ℂ}.

An immediate consequence of our complex convexity theorem then is:

Lemma 3.1. We have

a(Ξ(ω)) = T_{ω}.

Remark 3.2. From Lemma 3.1 we obtain in particular that a(Ξ) ⊆ T_{Ω} . It is interesting to observe that this inclusion for G = Sp(n, ℝ) extends a result of Siegel.

Consider the vector space V: = Symm(n, ℝ) of real symmetric matrices with its subcone of positive definite matrices V_{+} . Then we have the symmetric Siegel domain

S^{+}: = V + iV_{+} ⊆ V_{ℂ}.

Recall that S^{+} ≅ Sp(n, ℝ)/U(n).

If we write Δ_{j} for the j -th principal minor on V_{ℂ}, then Siegel’s Lemma says

(3.1)  Δ_{j}(z) ≠ 0  for z ∈ S^{+}

and all 1 ≤ j ≤ n. Now consider the rational functions

χ_{j}(z) = \frac{Δ_{j}(z)}{Δ_{j−1}(z)}

on V_{ℂ}. Then the inclusion a(Ξ) ⊆ T_{Ω} for G = Sp(n, ℝ) implies

(3.2)  Im χ_{j}(z) > 0  for z ∈ S^{+}

and for all j. Note that (3.2) implies (3.1). To see this one identifies S^{+} with the symmetric space Sp(n, ℝ)/U(n). Then with S^{−}: = \overline{S^{+}} one has a biholomorphism Ξ ≅ S^{+} × S^{−} (cf. [GiMa01] or [KrSt01b]). Realizing A as diagonal matrices in G, one then easily shows that a(Ξ) ⊆ T_{Ω} implies (3.2).

The example discussed above admits a natural generalization to all tube domains V + iV_{+} associated to an Euclidean Jordan algebra V and cone V_{+} .

For every g ∈ G we define the horospherical tube associated to ω by

T (g, ω): = g(N_{ℂ}T_{ω}K_{ℂ}/K_{ℂ}) ⊆ G_{ℂ}/K_{ℂ}.

Note that T (1, ω) is biholomorphic to N_{ℂ} × T_{ω}. In particular, all horospherical tubes T (g, ω) are Stein.

Theorem 3.3. For any non-empty open convex W -invariant set ω ⊆ Ω the domain Ξ(ω) is the connected component of the intersection of horospherical tubes:

Ξ(ω) = ( ⋂_{g∈G} T (g, ω))_{0}.

Proof. From Lemma 3.1 we obtain that

Ξ(ω) ⊆ N_{ℂ}T_{ω}K_{ℂ}/K_{ℂ} = a^{−1}(T (1, ω)).

Thus the fact that Ξ(ω) is G-invariant and connected implies that Ξ(ω) ⊆ ( ⋂_{g∈G} T (g, ω))_{0}. On the other hand we obtain from Ξ = Ξ_{I} (cf. the discussion in the introduction) and Lemma 3.1 that Ξ = ( ⋂_{g∈G} T (g, Ω))_{0} . From this and Lemma 3.1 we hence get

( ⋂_{g∈G} T (g, ω))_{0} ⊆ Ξ ∩ T (1, ω) = Ξ(ω),

completing the proof of the theorem.

An interesting application of Theorem 3.3 is the following:

Proposition 3.4. Let ω ⊆ Ω be an open convex W -invariant set. Then the intersection

I(ω): = ⋂_{g∈G} T (g, ω)

is open. In particular, I(ω) and every connected component of I(ω) is Stein. In particular, Ξ(ω) is Stein. Proof. (following a suggestion of Dmitri Akhiezer) Since G = KAN and since AN leaves T (1, ω) invariant, we obtain

I(ω) = ⋂_{k∈K} T (k, ω).

Hence I(ω) is an intersection of open sets over the “compact parameter space” K. In particular I(ω) is open. Since all horospherical tubes T (k, ω) are Stein, we obtain that I(ω) is Stein. With I(ω) all its connected components are Stein, concluding the proof of the theorem.

As a final application of the complex convexity theorem we prove a result on the characterization of the boundary of Ξ.

Proposition 3.5. Let ω ⊆ Ω be an open convex W -invariant set. Let (z_{n})_{n∈ℕ} be a sequence with z_{n} → z_{0} ∈ ∂Ξ(ω). Then

(Im log a(z_{n}))_{n∈ℕ}

is a sequence in ω and every accumulation point of this sequence lies in ∂ω. Proof. Set X_{n}: = Im log a(z_{n}). By Lemma 3.1 we have X_{n} ∈ ω. Since \overline{ω} is compact in 𝔞, we may assume that X_{n} → X_{0} with X_{0} ∈ \overline{ω}.

If X_{0} ∉ ∂ω, then we find a convex Weyl group invariant open set ω_{1} such that \overline{ω_{1}} ⊆ ω and X_{n} ∈ ω_{1} . Then Theorem 3.3 implies that z_{n} ∈ Ξ(ω_{1}) and so z_{0} ∈ \overline{Ξ(ω_{1})}. Now \overline{ω}_{1} ⊆ Ω and so ∂Ξ(ω_{1}) = G exp(i∂ω_{1})K_{ℂ}/K_{ℂ}. Thus z_{0} ∈ Ξ(ω), contradicting the assumption z_{0} ∈ ∂Ξ(ω). This concludes the proof of the proposition.

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Simon Gindikin Department of Mathematics Rutgers University New Brunswick, NJ 08903 USA

gindikin@math.rutgers.edu

Bernhard Krötz

The Ohio State University Department of Mathematics 231 West 18th Avenue Columbus, OH 43210–1174 USA

                              kroetz@math.ohio-state.edu