Journal of Lie Theory Volume 13 (2003) 565–572 \begin{matrix}c \\ ◯ \end{matrix} 2003 Heldermann Verlag

Stein Extensions of Riemann Symmetric Spaces and some Generalization

Dedicated to Professor Keisaku Kumahara on his 60th birthday

Toshihiko Matsuki

Communicated by S. Gindikin

Abstract. It was proved by Huckleberry that the Akhiezer-Gindikin domain is included in the “Iwasawa domain” using complex analysis. We show here that no complex analysis is necessary for a proof of this result. Indeed we generalize the notions of the Akhiezer-Gindikin domain and the Iwasawa domain for two associated symmetric subgroups in real Lie groups and prove the inclusion. Moreover, by the symmetry of two associated symmetric subgroups, we also give a direct proof of the known fact that the Akhiezer-Gindikin domain is included in all cycle spaces.

1. Introduction

1.1. Akhiezer-Gindikin domain and Iwasawa domain Let G_{ℂ} be a connected complex semisimple Lie group and G_{ℝ} a connected real form of G_{ℂ}. Let K_{ℂ} be the complexification in G_{ℂ} of a maximal compact subgroup K of G_{ℝ}. Let 𝔤_{ℝ} = 𝔨 ⊕ 𝔪 denote the Cartan decomposition of 𝔤_{ℝ} = Lie(G_{ℝ}) with respect to K. Let 𝔱 be a maximal abelian subspace in i𝔪. Put

𝔱^{+} = \{Y ∈ 𝔱 | |α(Y)| < \frac{π}{2} for all α ∈ Σ\}

where Σ is the restricted root system of 𝔤_{ℂ} with respect to 𝔱. Then the “Akhiezer-Gindikin domain” D in G_{ℂ} is defined in [AG] by

D = G_{ℝ}(exp 𝔱^{+})K_{ℂ}.

Let B be a Borel subgroup of G_{ℂ} such that G_{ℝ}B is closed in G_{ℂ}. Then K_{ℂ}B is the unique open dense K_{ℂ}-B double coset in G_{ℂ} ([M2]). Define an open subset

Ω = {x ∈ G_{ℂ} | x^{−1}G_{ℝ}B ⊂ K_{ℂ}B}

in G_{ℂ}. Clearly, Ω is left G_{ℝ}-invariant and right K_{ℂ}-invariant. The connected component Ω_{0} of Ω containing the identity is often called the “Iwasawa domain”.

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Remark 1. Let S_{j} (j ∈ J) be the K_{ℂ}-B double cosets in G_{ℂ} of (complex) codimension one and T_{j} the closure of S_{j}. Then the complement of K_{ℂ}B in G_{ℂ} is ⋃_{j∈J} T_{j}. So we can write

Ω = {x ∈ G_{ℂ} | x^{−1}G_{ℝ} ∩ T_{j} = ϕ for all j ∈ J}
= {x ∈ G_{ℂ} | x ∉ gT_{j}^{−1} for all j ∈ J and g ∈ G_{ℝ}}.

Thus Ω_{0} is Stein because Ω is the complement of an infinite family {gT_{j}^{−1} | j ∈ J, g ∈ G_{ℝ}} of complex hypersurfaces.

The equality

D = Ω_{0}  (1.1)

was proved in [GM] when G_{ℝ} is classical type or exceptional Hermitian type. Independently, Krötz and Stanton proved D ⊂ Ω_{0} for classical cases in [KS]. We should note that the proofs in these two papers are based on elementary linear algebraic computations. (Remark that [FH] did not refer [GM] in their historical reference on (1.1).) On the other hand, Barchini proved the inclusion D ⊃ Ω_{0} by a general but elementary argument in [B]. Recently, Huckleberry ([H], Proposition 2.0.2 in [FH]) gave a proof of the opposite inclusion

D ⊂ Ω_{0}  (1.2)

by using strictly plurisubharmonicness of a function ρ proved in [BHH] (revised version).

But we can see that we need no complex analysis to prove (1.2). In this paper, we will generalize the notions of the Akhiezer-Gindikin domain and the Iwasawa domain for real Lie groups as in the next subsection and prove the inclusion (1.2).

1.2. Generalization to real Lie groups

Let G be a connected real semisimple Lie group and σ an involution of G. Take a Cartan involution θ such that σθ = θσ. Put H = G_{0}^{σ} = {g ∈ G | σ(g) = g}_{0} and H′ = G_{0}^{σθ}. Then the symmetric subgroup H′ is called “associated” to the symmetric subgroup H (c.f. [M1]). The structure of the double coset decomposition H′\G/H is precisely studied in [M3] in a general setting for arbitrary two involutions.

Remark 2. (i) For example, if G = G_{ℂ}, then K_{ℂ} and G_{ℝ} are associated.

(ii) We should remark here that all the results on Jordan decompositions and elliptic elements etc. for the decomposition G_{ℝ}\G_{ℂ}/K_{ℂ} in Section 3 of [FH] were already proved in [M3]. In [FH], they are not referred at all.

Let 𝔤 = 𝔨 ⊕ 𝔪 = 𝔥 ⊕ 𝔮 be the +1, −1-eigenspace decompositions of 𝔤 = Lie(G) with respect to θ and σ, respectively. Then the Lie algebra 𝔥′ of H′ is written as 𝔥′ = (𝔨 ∩ 𝔥) ⊕ (𝔪 ∩ 𝔮). Let 𝔱 be a maximal abelian subspace of 𝔨 ∩ 𝔮. Then we can define the root space

𝔤_{ℂ}(𝔱, α) = {X ∈ 𝔤_{ℂ} | [Y, X] = α(Y)X for all Y ∈ 𝔱}

for any linear form α : 𝔱 → iℝ. Here 𝔤_{ℂ} = 𝔤 ⊕ i𝔤 is the complexification of 𝔤. Put

Σ = Σ(𝔤_{ℂ}, 𝔱) = {α ∈ i𝔱^{∗} − {0} | 𝔤_{ℂ}(𝔱, α) ≠ {0}}.

Then Σ satisfies the axiom of the root system ([R]). Since θ(Y) = Y for all Y ∈ 𝔱, we can decompose 𝔤_{ℂ}(𝔱, α) into +1, −1-eigenspaces for θ as

𝔤_{ℂ}(𝔱, α) = 𝔨_{ℂ}(𝔱, α) ⊕ 𝔪_{ℂ}(𝔱, α).  (1.3)

Define a subset Σ(𝔪_{ℂ}, 𝔱) = {α ∈ i𝔱^{∗} − {0} | 𝔪_{ℂ}(𝔱, α) ≠ {0}} of Σ and put

𝔱^{+} = \{Y ∈ 𝔱 | |α(Y)| < \frac{π}{2} for all α ∈ Σ(𝔪_{ℂ}, 𝔱)\}.

Then we define a generalization of the Akhiezer-Gindikin domain D in G by

D = H′T^{+}H.

where T^{+} = exp 𝔱^{+}. (We will show in Proposition 1 that D is open in G.)

Let P be an arbitrary parabolic subgroup of G such that H′P is closed in G. Then HP is open in G by [M2]. Define an open subset

Ω = {x ∈ G | x^{−1}H′P ⊂ HP } = {x ∈ G | H′x ⊂ P H}

in G. Then we may call the connected component Ω_{0} a “generalized Iwasawa domain”. As a complete generalization of (1.2), we can prove the following theorem.

Theorem. D ⊂ Ω_{0}.

Here we should explain the construction of this paper to prove this theorem. In Section 2, we prove properties on the generalized Akhiezer-Gindikin domain D. Lemma 2 is the most important basic technical lemma. It implies (𝔥′ ∩ Ad(a)𝔥) ⊕ (𝔮′ ∩ Ad(a)𝔮) ⊂ 𝔨 for a ∈ T^{+} where 𝔮′ = (𝔨 ∩ 𝔮) ⊕ (𝔪 ∩ 𝔥). Note that H′ ∩ aHa^{−1} is the isotropy subgroup of the action of H′ at the point aH ∈ G/H. So the inclusion 𝔥′ ∩ Ad(a)𝔥 ⊂ 𝔨 is a generalization of Proposition 2 in [AG]. But we do not find statements in [AG] corresponding to the inclusion 𝔮′ ∩ Ad(a)𝔮 ⊂ 𝔨. This inclusion is the key of this paper.

In Proposition 1, we show that D is open in G. This is a generalization of Proposition 4 in [AG]. We also give a precise orbit structure H′\D/H ≅ T^{+}/W of D where W = N_{K∩H}(𝔱)/Z_{K∩H}(𝔱) in Proposition 2. This is a generalization of Proposition 8 in [AG]. In Section 3, we construct a left H′-invariant right H - invariant real analytic function ρ on D and prepare the key lemma (Lemma 3) which follows from Lemma 2. In Section 4, we prove Theorem. Basic formulation is the same as Proposition 2.0.2 in [FH]. But we do not need complex analysis.

1.3. Application to cycle spaces Note that we can exchange the roles of H and H′. We can aplly this to the pair of K_{ℂ} and G_{ℝ} in G_{ℂ}. Let P be a parabolic subgroup of G_{ℂ} such that S = K_{ℂ}P is closed in G_{ℂ}. Then S′ = G_{ℝ}P is open in G_{ℂ}. Put

Ω(S) = {x ∈ G_{ℂ} | x^{−1}K_{ℂ}P ⊂ G_{ℝ}P }.

Then by Theorem, we have D^{−1} = K_{ℂ}T^{+}G_{ℝ} ⊂ Ω(S)_{0} by the notation in Section 1.1 and therefore D ⊂ Ω(S)_{0}^{−1}. Since Ω(S)^{−1} = {x ∈ G_{ℂ} | xK_{ℂ}P ⊂ G_{ℝ}P } = {x ∈ G_{ℂ} | xS ⊂ S′}, the domain Ω(S)_{0}^{−1} is usually called the “cycle space” for S′. Hence we have given a direct proof of the known fact:

Corollary. Akhizer-Gindikin domain D is included in all cycle spaces.

(Remark: This fact was known by combining (1.2) and Proposition 8.3 in [GM] because Proposition 8.3 implies that the Iwasawa domain Ω_{0} is included in all the cycle spaces Ω(S)_{0}^{−1}.)

Aknowledgement: The author would like to express his hearty thanks to S. Gindikin for his encouragement and for many useful discussions.

2. A generalization of Akhiezer-Gindikin domain

We will use the notations in Section 1.2 (not in Section 1.1). In this section, we will prepare basic results on a generalization of the Akhiezer-Gindikin domain by extending elementary arguments in [M3] Section 3.

First we should note that we have only to consider the problem on each minimal σ-stable ideals of 𝔤. So we may assume that 𝔤 has no proper σ-stable ideals. We may also assume that 𝔤 is noncompact, otherwise we have P = G and the problem is trivial.

Let [𝔪, 𝔪] denote the linear subspace of 𝔨 spanned by [Y, Z] for Y, Z ∈ 𝔪. Then 𝔤′ = [𝔪, 𝔪] ⊕ 𝔪 becomes a σ-stable ideal of 𝔤 because 𝔤′ is σ-stable and [𝔨, [𝔪, 𝔪]] ⊂ [[𝔨, 𝔪], 𝔪] + [𝔪, [𝔨, 𝔪]] ⊂ [𝔪, 𝔪]. Since 𝔤′ = 𝔤, we have 𝔨 = [𝔪, 𝔪] and therefore

𝔨_{ℂ} = [𝔪_{ℂ}, 𝔪_{ℂ}].  (2.1)

Lemma 1. (i) Every root α ∈ Σ such that 𝔨_{ℂ}(𝔱, α) ≠ {0} is written as a sum of two elements in Σ(𝔪_{ℂ}, 𝔱) ∪ {0}.

(ii) If Y ∈ 𝔱^{+}, then |α(Y)| < π for all α ∈ Σ.

(iii) If Y ∈ 𝔱 satisfies α(Y) = 0 for all α ∈ Σ, then Y = 0.

(iv) If Y, Z ∈ 𝔱^{+} satisfy exp Y ∈ (exp Z)(T ∩H), then Y = Z. Especially, exp is injective on 𝔱^{+}.

Proof. (i) Let X be a nonzero element in ∈ 𝔨_{ℂ}(𝔱, α). Then by (2.1), we can write X = ∑_{j}[Y_{j}, Z_{j}] with some nonzero elements Y_{j} ∈ 𝔪_{ℂ}(𝔱, β_{j}) and Z_{j} ∈ 𝔪_{ℂ}(𝔱, γ_{j}) such that β_{j} + γ_{j} = α.

(ii) is clear by (i) and the definition of 𝔱^{+}.

(iii) If α(Y) = 0 for all α ∈ Σ, then Y commutes with 𝔤_{ℂ}(𝔱, α) for all α ∈ Σ ∪ {0}. Hence Y commutes with 𝔤 and therefore Y = 0 because 𝔤 is semisimple.

(iv) Suppose exp Y = (exp Z)h with some Y, Z ∈ 𝔱^{+} and h ∈ T ∩ H. Applying σ, we have exp(−Y) = (exp(−Z))h. So we have exp 2Y = exp 2Z. Put a = exp 2Y = exp 2Z and apply Ad(a) to 𝔤_{ℂ}(𝔱, α) for α ∈ Σ(𝔪_{ℂ}, 𝔱). Then we have e^{2α(Y)} = e^{2α(Z)} and therefore α(Y) − α(Z) ∈ πiℤ. Since |α(Y)| < π/2 and |α(Z)| < π/2 by the definition of 𝔱^{+}, we have α(Y − Z) = α(Y) − α(Z) = 0 for all α ∈ Σ(𝔪_{ℂ}, 𝔱). This implies α(Y − Z) = 0 for all α ∈ Σ by (i) and therefore Y − Z = 0 by (iii).

Let Y be an element of 𝔱 and put a = exp Y . Consider the conjugate σ_{a} = Ad(a)σAd(a)^{−1} = σAd(a)^{−2} of σ by Ad(a). Then σ_{a} is an involution of 𝔤 such that 𝔤^{σ_{a}} = Ad(a)𝔥 and G_{0}^{σ_{a}} = aHa^{−1}. Put τ = σθ = θσ. The key idea of [M3] was to consider the automorphism τσ_{a} (which is not involutive in general) of 𝔤. Since τσ_{a} = τσAd(a)^{−2} = θAd(a)^{−2} and since θ and Ad(a) commutes, the automorphism τσ_{a} is semisimple. Hence by Lemma 1 in [M3], we have a direct sum decomposition

𝔤 = (𝔥′ + Ad(a)𝔥) ⊕ (𝔮′ ∩ Ad(a)𝔮)  (2.2)

where 𝔮′ = (𝔨 ∩ 𝔮) ⊕ (𝔪 ∩ 𝔥). On the other hand, we have the +1, −1-eigenspace decomposition of 𝔤^{τσ_{a}} for τ by 𝔤^{τσ_{a}} = (𝔤^{τσ_{a}} ∩ 𝔥′) ⊕ (𝔤^{τσ_{a}} ∩ 𝔮′) = (𝔥′ ∩ Ad(a)𝔥) ⊕ (𝔮′ ∩ Ad(a)𝔮) as in [M3] Section 3.

Lemma 2. If Y ∈ 𝔱^{+}, then 𝔤^{τσ_{a}} = 𝔷_{𝔨}(Y) = {X ∈ 𝔨 | [Y, X] = 0}.

Proof. By (1.3), we have a direct sum decomposition

𝔤_{ℂ} = ⨁_{α∈Σ∪{0}}(𝔨_{ℂ}(𝔱, α) ⊕ 𝔪_{ℂ}(𝔱, α)).  (2.3)

If X ∈ 𝔨_{ℂ}(𝔱, α), then we have τσ_{a}X = θAd(a)^{−2}X = e^{−2α(Y)}X. On the other hand, if X ∈ 𝔪_{ℂ}(𝔱, α), then we have τσ_{a}X = θAd(a)^{−2}X = −e^{−2α(Y)}X. Hence the decomposition (2.3) is the eigenspace decomposition of 𝔤_{ℂ} for τσ_{a}. We have only to verify whether every direct summand in (2.3) is contained in 𝔤_{ℂ}^{τσ_{a}} or not.

Note that α(Y) is pure imaginary. Let X be a nonzero element in 𝔪_{ℂ}(𝔱, α). Since |α(Y)| < π/2 by the assumption, we have −e^{−2α(Y)} ≠ 1. Hence X ∉ 𝔤_{ℂ}^{τσ_{a}}. On the other hand, let X be a nonzero element in 𝔨_{ℂ}(𝔱, α). Since |α(Y)| < π by Lemma 1 (ii), we have e^{−2α(Y)} = 1 ⟺ α(Y) = 0. Hence X ∈ 𝔤_{ℂ}^{τσ_{a}} ⟺ α(Y) = 0. Thus we have proved

𝔤_{ℂ}^{τσ_{a}} = ⨁_{α∈Σ∪{0}, α(Y)=0} 𝔨_{ℂ}(𝔱, α) = 𝔷_{𝔨_{ℂ}}(Y)

and therefore 𝔤^{τσ_{a}} = 𝔷_{𝔨}(Y).

Since 𝔤^{τσ_{a}} = 𝔷_{𝔨}(Y) is a compact Lie algebra and since 𝔱 is maximal abelian in 𝔮′ ∩ Ad(a)𝔮 = 𝔷_{𝔨∩𝔮}(Y), we have 𝔮′ ∩ Ad(a)𝔮 = Ad(H′ ∩ aHa^{−1})_{0}𝔱. Moreover if U is a neighborhood of the origin 0 in 𝔱, then

V = Ad(H′ ∩ aHa^{−1})_{0}U is a neighborhood of 0 in 𝔮′ ∩ Ad(a)𝔮.  (2.4)

Proposition 1. D is open in G.

Proof. By the left H′-action and the right H -action on D, we have only to show that a neighborhood of a = exp Y for Y ∈ 𝔱^{+} is contained in D. Take a neighborhood U = {Z − Y | Z ∈ 𝔱^{+}} of 0 in 𝔱. Then we have T^{+}a^{−1} = exp U. Hence

e ∈ H′T^{+}Ha^{−1} = H′T^{+}a^{−1}aHa^{−1} = H′(exp U)aHa^{−1}
= H′ exp(Ad(H′ ∩ aHa^{−1})_{0}U)aHa^{−1} = H′(exp V)aHa^{−1}.  (2.5)

Since V is a neighborhood of 0 in 𝔮′ ∩ Ad(a)𝔮 by (2.4), it follows from (2.2) that H′T^{+}Ha^{−1} contains a neighborhood of e. Hence H′T^{+}H contains a neighborhood of a.

Proposition 2. Let a and b be elements of T^{+}. Then b = ℓah^{−1} for some ℓ ∈ H′ and h ∈ H ⟺ b = waw^{−1} for some w ∈ N_{K∩H}(𝔱). Here N_{K∩H}(𝔱) is the normalizer of 𝔱 in K ∩ H. (Write K = G^{θ} as usual ).

Proof. Since the implication ⟸ is clear, we have only to prove ⟹. Suppose b = ℓah^{−1} for some ℓ ∈ H′ and h ∈ H. Put 𝔱′ = Ad(ℓ)𝔱 = Ad(ℓa)𝔱 = Ad(bh)𝔱. Then 𝔱′ is a maximal abelian subspace of 𝔮′ ∩ Ad(b)𝔮. Since (𝔥′ ∩ Ad(b)𝔥, 𝔮′ ∩ Ad(b)𝔮) is a compact symmetric pair by Lemma 2, there is an x ∈ H′ ∩ bHb^{−1} such that Ad(x)𝔱′ = 𝔱. Put ℓ′ = xℓ ∈ H′ and h′ = b^{−1}xbh ∈ H. Then we have ℓ′ah′^{−1} = xℓah^{−1}b^{−1}x^{−1}b = xbhh^{−1}b^{−1}x^{−1}b = b and

ℓ′T h′^{−1} = xℓT h^{−1}b^{−1}x^{−1}b = xℓaT h^{−1}b^{−1}x^{−1}b
= xbhT h^{−1}b^{−1}x^{−1}b = xT′x^{−1}b = T b = T  (2.6)

where T = exp 𝔱 and T′ = exp 𝔱′.

Since ℓ′h′^{−1} = ℓ′eh′^{−1} ∈ T by (2.6) and since ℓ′T ℓ′^{−1}ℓ′h′^{−1} = T also by (2.6), we have ℓ′T ℓ′^{−1} = T. Write ℓ′ = k exp X with k ∈ K ∩ H and X ∈ 𝔪 ∩ 𝔮. Then it is well-known that [X, 𝔱] = {0} by a standard argument. (Suppose Ad(k exp X)Y = Y′ for some Y, Y′ ∈ 𝔨. Then applying θ, we have Ad(k exp(−X))Y = Y′ and therefore Ad(exp 2X)Y = Y . Since adX : 𝔤 → 𝔤 is expressed by a real symmetric matrix, we have [X, Y] = 0.)

Hence we have kT k^{−1} = T and b = ℓ′aℓ′^{−1}ℓ′h′^{−1} = kak^{−1}ℓ′h′^{−1}. We have only to prove c = ℓ′h′^{−1} = e. Write h′ = k′ exp X′ with k′ ∈ K∩H and X′ ∈ 𝔪∩𝔥. Then it follows from ℓ′ = ch′ that k exp X = ck′ exp X′. Since c ∈ T ⊂ K, we have k = ck′ and X = X′ = 0. Hence c ∈ T ∩ H. Since kak^{−1}, b ∈ exp t^{+}, we have c = e by Lemma 1 (iv).

3. Construction of a function ρ

Write W = N_{K∩H}(𝔱)/Z_{K∩H}(𝔱). Let ρ_{0} be a W -invariant real analytic function on 𝔱^{+} which has no critical points except the origin. For the sake of later use, we should also assume ρ_{0}(Z) tends to +∞ when Z goes to the boundary of 𝔱^{+}. For example, we may put

ρ_{0}(Z) = ∑_{α∈Σ(𝔪_{ℂ},𝔱)} \frac{1}{π − 2iα(Z).}  (3.1)

This function (3.1) is clearly convex and therefore it has no critical points except the origin. By Proposition 2 and Lemma 1 (iv), we can define a function ρ on D by

ρ(ℓ(exp Z)h) = ρ_{0}(Z) for ℓ ∈ H′, h ∈ H and Z ∈ 𝔱^{+}.

Proposition 3. ρ is real analytic on D.

Proof. By the left H′-action and the right H -action on D, we have only to show ρ is real analytic at every a = exp Y ∈ T^{+}. Consider the right a-translate ρ_{a}(x) = ρ(xa) for x ∈ Da^{−1} of ρ. Since ρ_{a} is left H′-invariant and right aHa^{−1}- invariant, we have only to show that the function ρ′_{a}(X) = ρ_{a}(exp X) for X ∈ V is real analytic at 0 by (2.2) and (2.5) where U ⊂ 𝔱 and V = Ad(H′∩aHa^{−1})_{0}U ⊂ 𝔮′ ∩ Ad(a)𝔮 are as in the proof of Proposition 1. Since ρ_{a}(gxg^{−1}) = ρ_{a}(x) for g ∈ H′ ∩ aHa^{−1} and x ∈ exp V , ρ′_{a} is Ad(H′ ∩ aHa^{−1})-invariant. Note that (H′ ∩ aHa^{−1})_{0} = (Z_{K∩H}(Y))_{0} by Lemma 2 and that ρ′_{a}(Z) = ρ((exp Z)a) = ρ_{0}(Z +Y) for Z ∈ U is invariant under the action of w ∈ W such that w(Y) = Y . Since we can easily extend the well-known Chevalley’s restriction theorem to real analytic functions at 0, the function ρ′_{a} on V = Ad(H′∩aHa^{−1})_{0}U is real analytic at 0.

Remark 3. (i) The function ρ on D has no critical points outside H′H = H′eH by the assumption on ρ_{0}.

(ii) If ρ_{0} is a W -invariant smooth function on 𝔱^{+}. Then we can show that ρ is a smooth function on D by using [S].

The tangent space T_{a}(G) of G at a is identified with 𝔤 = T_{e}(G) by the right a-action. In other words, we identify T_{a}(G) with the left infinitesimal action of 𝔤 at a. Now we have the following key lemma.

Lemma 3. Let a = exp Y with Y ∈ 𝔱^{+} − {0}. Then the hyperplane in T_{a}(G) defined by dρ = 0 is orthogonal, with respect to the Killing form on 𝔤, to a nonzero vector Z in 𝔨.

Proof. Taking the right a-translate ρ_{a} of ρ as in the proof of Proposition 3, we have only to consider the hyperplane in the tangent space T_{e}(G) ≅ 𝔤 defined by dρ_{a} = 0.

Since ρ_{a} is left H′-invariant and right aHa^{−1}-invariant, the differential dρ_{a} vanishes on 𝔥′+Ad(a)𝔥. Hence the normal vector Z is contained in the orthogonal complement 𝔮′ ∩ Ad(a)𝔮 (⊂ 𝔨 by Lemma 2) of 𝔥′ + Ad(a)𝔥.

Proof of Theorem. The basic formulation is the same as Proposition 2.0.2 in [FH]. Suppose that D ⊄ P H. We will deduce a contradiction.

Let P xH be a P -H double coset with the least dimension among the P -H double cosets intersecting D. So the intersection P xH∩D is relatively closed in D. Since H′P = (K∩H)P by [M2], we have H′ = (K∩H)(P ∩H′) = (P ∩H′)(K∩H). Hence P xH intersects (K ∩H)T^{+} and the image of ρ|_{P xH∩D} is equal to the image of ρ|_{P xH∩(K∩H)T^{+}} . The set {x ∈ (K∩H)T^{+} | ρ(x) ≤ m} is compact for any m ∈ ℝ because we carefully assumed that ρ_{0} is +∞ on the boundary of 𝔱^{+}. Hence the function ρ|_{P xH∩D} attains its minimum on some point ka with k ∈ K ∩ H and a ∈ T^{+}. Replacing P by the k-conjugate k^{−1}P k, we may assume k = e. Since a ∈ P xH and P xH ∩ P H = ϕ, we have a ≠ e.

By Lemma 3, there is a nonzero element Z in 𝔨 such that Z is orthogonal to 𝔭 = Lie(P). But this leads a contradiction because Z ∈ 𝔨 is also orthogonal to θ𝔭 and therefore Z is orthogonal to 𝔭 + θ𝔭 = 𝔤 which cannot happen since the Killing form is nondegenerate on 𝔤.

References

[AG] Akhiezer, D. N., and S. G. Gindikin, On Stein extensions of real symmetric spaces, Math. Ann. 286 (1990), 1–12.

[B] Barchini, L., Stein extensions of real symmetric spaces and the geometry of the flag manifold, preprint.

[BHH] Burns, D., S. Halverscheid, and R. Hind, The geometry of Grauert tubes and complexification of symmetric spaces, preprint (CV/0109186).

[FH] Fels, G., and A. Huckleberry, Characterization of cycle domains via Kobayashi hyperbolicity, preprint (AG/0204341).

[GM] Gindikin, S., and T. Matsuki, Stein extensions of Riemann symmetric spaces and dualities of orbits on flag manifolds, preprint (MSRI-Preprint 2001-028).

[H] Huckleberry, A., On certain domains in cycle spaces of flag manifolds, Math. Annalen, to appear.

[KS] Krötz, B., and R. Stanton, Holomorphic extension of a representation I: automorphic functions, preprint.

[M1] Matsuki, T., The orbits of affine symmetric spaces under the action of minimal parabolic subgroups, J. Math. Soc. Japan 31 (1979), 331–357.

[M2] —, Closure relations for orbits on affine symmetric spaces under the action of parabolic subgroups. Intersections of associated orbits, Hiroshima Math. J. 18 (1988), 59–67.

[M3] —, Double coset decompositions of reductive Lie groups arising from two involutions, J. of Algebra 197 (1997), 49–91.

[R] Rossmann, W., The structure of semisimple symmetric spaces, Canad. J. Math. 31 (1979), 157–180.

[S] Schwarz, G., Smooth functions invariant under the action of a compact Lie group, Topology 14 (1975), 63–68.

Toshihiko Matsuki Faculty of Integrated Human Studies Kyoto University Kyoto 606-8501, Japan matsuki@math.h.kyoto-u.ac.jp

Received August 28, 2002 and in final form September 6, 2002