Journal of Lie Theory Volume 13 (2003) 189–191 \begin{matrix}c \\ ◯ \end{matrix} 2003 Heldermann Verlag

A Note on the Linear Cycle Space for Groups of Hermitian Type

Joseph A. Wolf^{1} & Roger Zierau^{2}

Communicated by K.-H. Neeb

Abstract. Let G_{0} be a simple Lie group of hermitian type and let B denote the corresponding hermitian symmetric space. The linear cycle space for any nonholomorphic type flag domain of G_{0} is biholomorphic to B × \overline{B}. When G_{0} is a classical group this was proved by the authors in a paper published several years ago. Here we show that the result follows for arbitrary groups of hermitian type. This is done without case by case arguments by combining results from that paper with recent results of A. T. Huckleberry and the first author.

1. Introduction

Let G be a complex semisimple Lie group with noncompact real form G_{0} of hermitian type. Let K_{0} be a maximal compact subgroup of G_{0} and K its complexification. Denote by Q a parabolic subgroup of G and set Z = G/Q, the corresponding flag variety. There is a one to one correspondence between K -orbits and G_{0}-orbits on Z ([2], [8], [9]). Under this correspondence open G_{0}- orbits correspond to closed K -orbits. Following [13, page 529] we let D be an open G_{0}-orbit in Z and Y the corresponding closed K -orbit. The closed K -orbit Y is a maximal compact subvariety of D. The linear cycle space is the connected component of

{g · Y : g ∈ G and g · Y ⊂ D}

containing Y and is denoted by M_{D}. Since G_{0} is noncompact and of hermitian type, it is not transitive on Z [12, Corollary 1.7], and M_{D} is not reduced to a point. Now there are two types of open orbits D, holomorphic type where it is known that M_{D} is biholomorphic to B or to \overline{B}, and nonholomorphic type where we conjectured (and proved in many cases) that M_{D} is biholomorphic to B × \overline{B}. See [13, Definition 1.2] for the definitions and [13, Theorem 1.3] for a number of equivalent conditions.

As M_{D} sits in a holomorphic double fibration with D, it plays a role in the study of representations occurring in cohomology on D. Therefore it is of

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag interest to understand the precise structure of M_{D}. Theorem 1 gives an explicit description of M_{D}.

1Partially supported by NSF Grant DMS 99–88643 and by hospitality from the MSRI.

2Partially supported by NSF Grant DMS 98-01605 and by hospitality from the MSRI.

2. The structure theorem

Let B denote the hermitian symmetric space G_{0}/K_{0}. In the above setting the complete theorem on the structure of the linear cycle space is the following.

Theorem 1. If G_{0} is an arbitrary simple Lie group of hermitian type then

(1) if D is of nonholomorphic type then M_{D} is biholomorphic to B × \overline{B},

(2) if D is of holomorphic type then M_{D} is biholomorphic to B or \overline{B}.

Proof. Statement (2) is known by two different general arguments, [11] and [13].

Statement (1) was proved in [13] for classical groups as follows. The inclusion B × \overline{B} ⊂ M_{D} is proved in [13] for the classical groups by a case by case argument. The other inclusion is proved in [13] (see Section 3 below) for arbitrary hermitian type groups. Now we observe that the other inclusion, B × \overline{B} ⊂ M_{D} for arbitrary hermitian type groups, is essentially contained in the literature.

The containment is fairly direct in [7]: A certain domain Ω_{S}(D), based on Schubert incidence theory, was introduced in [6], and it was recently proved that M_{D} = Ω_{S}(D) [7, Corollary 3.4].

The containment is slightly less direct in [4]. A. T. Huckleberry observed that one can obtain the desired inclusion B × \overline{B} ⊂ M_{D} from a careful look at [4, Proposition 2.4], though it is not actually stated in [4]. □

Corollary 2. For G_{0} a simple Lie group of hermitian type and D any nonholomorphic type orbit A ≃ M_{D}.

In this note, D is measurable in the sense of [10] and [11] because G_{0} is of hermitian type; it has been known for some time [11] that if D is measurable then M_{D} is Stein. The Schubert domain considerations [7], used in our proof of Theorem 1, show that M_{D} is Stein without any measurability condition on D.

3. Remark

It is announced in [4] that [13] contains a gap in the proof of M_{D} ⊂ B × \overline{B} (Theorem 3.8) for general hermitian type groups. At the very end of our proof of Case 1 in the proof of that Theorem 3.8, we made a few inaccurate statements which tend to obscure the argument. Referring to [13], the following adjustments should clarify that argument.

1. Omit ‘c_{Γ} =’ in line 2 of page 537.

2. Omit the sentence ‘In particular c_{Γ}(z) ∉ D.’ in line 3, page 537. This is true by Lemma 3.6, but is irrelevant in view of (1) above.

3. In view of (1) above, we must change ‘c_{Γ}’ to ‘c_{Γ∩Δ(𝔯_{+})}’ in line 4, page 537. The corresponding change of g_{0} in the same line to g′_{0} = ∏_{ψ∈Δ(𝔯_{+})\Γ} g_{0,ψ} is also necessary.

Now make the following simple observation. Let ψ ∈ Ψ^{𝔤} ⊂ Δ(𝔰_{+}), and suppose that g ∈ G[ψ] has decomposition g = exp(ξ_{+})k exp(ξ_{−}), ξ_{±} ∈ 𝔤_{±ψ} and k ∈ K ∩ G[ψ]. Then k ∈ H ⊂ Q and

(1) if ψ ∈ Δ(𝔮) = Δ(𝔩 + 𝔯_{−}) then ξ_{+} ∈ 𝔤_{ψ} ⊂ 𝔮 so exp(ξ_{+})(z) = z, and

(2) if ψ ∈ Δ(𝔯_{+}) then ξ_{−} ∈ 𝔤_{−ψ} ⊂ 𝔯_{−} so exp(ξ_{+})(z) = exp(ξ_{+})k exp(ξ_{−})(z) = g(z).

References

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

[2] Bremigan, R. J. & J. D. Lorch, Matsuki duality for flag manifolds, to appear.

[3] Burns, D, S. Halverscheid & R. Hind, The geometry of Grauert tubes and complexification of symmetric spaces, to appear.

[4] Gindikin, S. & T. Matsuki, Stein extensions of riemannian symmetric spaces and dualities of orbits on flag manifolds, MSRI Preprint 2001–028.

[5] Halverscheid, S., Maximal domains of definition of adapted complex structures for symmetric spaces of non-compact type, Thesis, Ruhr–Universität Bochum, 2001.

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

[7] Huckleberry, A. T., & J. A. Wolf, Schubert varieties and cycle spaces, to appear.

[8] Matsuki, T., Orbits of affine symmetric spaces under the action of parabolic subgroups, Hiroshima Math. J. 12 (1982), 307–320.

[9] Mirkovič, I., K. Uzawa, & K. Vilonen, Matsuki correspondence for sheaves, Invent. math. 109 (1992), 231–245.

[10] Wolf, J. A., The action of a real semisimple group on a complex flag manifold, I: Orbit structure and holomorphic arc components, Bull. Amer. Math. Soc. 75 (1969), 1121–1237.

[11] Wolf, J. A., The Stein condition for cycle spaces of open orbits on complex flag manifolds, Annals of Math. 136 (1992), 541–555.

[12] Wolf, J. A., Real groups transitive on complex flag manifolds, Proc. Amer. Math. Soc., 129 (2001), 2483–2487.

[13] Wolf, J. A., & R. Zierau, Linear cycle spaces in flag domains, Math. Annalen 316 (2000), 529–545.

Joseph A. Wolf Department of Mathematics University of California Berkeley, CA 94720–3840, USA jawolf@math.berkeley.edu Roger Zierau Mathematics Department Oklahoma State University Stillwater, OK 7407, USA zierau@math.okstate.edu

Received January 4, 2002 and in final form June 4, 2002

Editorial Note. Journal of Lie Theory publishes “Short Notes” on communications containing short or elegant proofs of known facts, relevant comments on ongoing research within the scope of Journal of Lie Theory, remarks on the history of the subject, or other items deemed constructive for the clarification of pending issues in the area of Lie Theory. Short Notes are not instruments for setting priorities but for furthering a scholarly discourse. The length of a Short Note should not exceed four pages.