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Journal of Lie Theory Volume 12 (2002) 617–618 c 2002 Heldermann Verlag Remark on the Complexified Iwasawa Decomposition Andrew R. Sinton & Joseph A. Wolf Communicated by K.-H. Neeb Abstract. Let GR be a real form of a complex semisimple Lie group GC . We identify the complexification KC AC NC ⊂ GC of an Iwasawa decomposition GR = KR AR NR as {g ∈ GC | gB ∈ Ω} where B ⊂ GC is a Borel subgroup of GC that contains AR NR and Ω is the open KC –orbit on GC /B . This is done in the context of subsets KC RC ⊂ GC , where RC is a parabolic subgroup of GC defined over R , and the open KC –orbits on complex flag manifolds GC /Q. 1. Details Let GR be a real form of a complex semisimple Lie group GC . For simplicity we assume that GC is connected and that the adjoint representation of GR maps GR into the group of inner automorphisms of the Lie algebra of GC . Choose an Iwasawa decomposition GR = KR AR NR and a corresponding minimal parabolic subgroup PR = MR AR NR of GR . Every parabolic subgroup of GR is conjugate to one that contains PR . Let RC ⊂ GC be the complexification of a (real) parabolic subgroup RR ⊂ GR . Passing to a GR –conjugate we may, and do, assume PR ⊂ RR . Let KC ⊂ GC be the complexification of the maximal compactly embedded subgroup KR ⊂ GR . Then GR = KR RR . In particular the Lie algebras satisfy gR = kR +rR , so their complexifications satisfy gC = kC +rC , and thus KC RC is open in GC . Since the projection GC → GC /RC is an open map, ΩX := KC (1RC ) is an open KC –orbit in X = GC /RC . If g ∈ GC then g ∈ KC RC if and only if gRC ∈ KC (1RC ) ⊂ GC /RC . Now Lemma 1.1. An element g ∈ GC is contained in KC RC if and only if gRC ∈ ΩX . Let Q be a parabolic subgroup of GC and write ΩZ for the open orbit KC (1Q) on Z = GC /Q. The argument of Lemma 1.1 shows: Lemma 1.2. Suppose KC Q = KC RC . Then g ∈ KC RC if and only if gQ ∈ ΩZ . There is a unique closed GR –orbit on X , and the other GR –orbits are finite in number and higher in dimension. See [4]. By (GR , KC )–duality ([3]; see [2] for a geometric proof), there is a unique open KC –orbit on X (which thus must be ΩX ), and the other KC –orbits are finite in number and lower in dimension. Thus ΩX is a dense open subset of X . Now Lemma 1.1 gives us c Heldermann Verlag ISSN 0949–5932 / $2.50 618 Sinton & Wolf Lemma 1.3. KC RC is a dense open subset of GC . Let MC , AC and NC denote the respective complexifications of MR , AR and NR . Here MC is the centralizer of AC in KC . Let B denote a Borel subgroup of GC that contains AR NR ; in other words B = BM AC NC where BM is a Borel subgroup of MC . The phenomenon KC Q = KC RC of Lemma 1.2 occurs, for example, when RR is the minimal parabolic subgroup PR of GR and Q = B . In that case MR ⊂ KR , so BM ⊂ MC ⊂ KC , and thus KC = KC BM . As B = BM AC NC now KC AC NC = KC B . We have proved Proposition 1.4. Let B be a Borel subgroup of GC that contains AR NR . Let g ∈ GC . Then g is contained in KC AC NC if and only if gB belongs to the open KC –orbit on GC /B . In particular KC AC NC is a dense open subset of GC . Proposition 1.4 describes the complexified Iwasawa decomposition set KC AC NC in terms of the open KC –orbit on the full flag manifold GC /B . Usually it is easy to decide whether a given group element belongs to KC AC NC , and that then describes the open KC –orbit on GC /B : Example 1.5. Let GR = SLn (R) and GC = SLn (C). For g ∈ GC , let sk be the k × k submatrix in the upper left corner of t gg . Then g ∈ KC AC NC if and only if det(sk ) 6= 0 for 1 5 k 5 n. Added in proof. After this note was accepted for publication, we learned that there is a nontrivial overlap with material in [1, Appendix B], specifically [1, Theorem B.1.2] and some of the immediately preceding discussion. References [1] [2] [3] [4] van den Ban, E. P., A convexity theorem for semisimple symmetric spaces, Pacific J. Math. 124 (1986), 21–55. Bremigan, R. and J. Lorch, Matsuki duality for flag manifolds, to appear. Matsuki, T., Orbits on affine symmetric spaces under the action of parabolic subgroups, Hiroshima Math J. 12 (1982), 307–326. 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. Andrew R. Sinton Department of Mathematics University of California Berkeley, CA 94720-3840, USA sinton@math.berkeley.edu Received December 6, 2001 and in final form February 2, 2002 Joseph A. Wolf Department of Mathematics University of California Berkeley, CA 94720-3840, USA jawolf@math.berkeley.edu