Journal of Lie Theory Volume 12 (2002) 617–618 \begin{matrix}c \\ ◯ \end{matrix} 2002 Heldermann Verlag

Remark on the Complexified Iwasawa Decomposition

Andrew R. Sinton & Joseph A. Wolf

Communicated by K.-H. Neeb

Abstract. Let G_{ℝ} be a real form of a complex semisimple Lie group G_{ℂ} . We identify the complexification K_{ℂ}A_{ℂ}N_{ℂ} ⊂ G_{ℂ} of an Iwasawa decomposition G_{ℝ} = K_{ℝ}A_{ℝ}N_{ℝ} as {g ∈ G_{ℂ} | gB ∈ Ω} where B ⊂ G_{ℂ} is a Borel subgroup of G_{ℂ} that contains A_{ℝ}N_{ℝ} and Ω is the open K_{ℂ} –orbit on G_{ℂ}/B. This is done in the context of subsets K_{ℂ}R_{ℂ} ⊂ G_{ℂ} , where R_{ℂ} is a parabolic subgroup of G_{ℂ} defined over ℝ, and the open K_{ℂ} –orbits on complex flag manifolds G_{ℂ}/Q.

1. Details

Let G_{ℝ} be a real form of a complex semisimple Lie group G_{ℂ} . For simplicity we assume that G_{ℂ} is connected and that the adjoint representation of G_{ℝ} maps G_{ℝ} into the group of inner automorphisms of the Lie algebra of G_{ℂ} . Choose an Iwasawa decomposition G_{ℝ} = K_{ℝ}A_{ℝ}N_{ℝ} and a corresponding minimal parabolic subgroup P_{ℝ} = M_{ℝ}A_{ℝ}N_{ℝ} of G_{ℝ} . Every parabolic subgroup of G_{ℝ} is conjugate to one that contains P_{ℝ} . Let R_{ℂ} ⊂ G_{ℂ} be the complexification of a (real) parabolic subgroup R_{ℝ} ⊂ G_{ℝ} . Passing to a G_{ℝ}–conjugate we may, and do, assume P_{ℝ} ⊂ R_{ℝ} . Let K_{ℂ} ⊂ G_{ℂ} be the complexification of the maximal compactly embedded subgroup K_{ℝ} ⊂ G_{ℝ} . Then G_{ℝ} = K_{ℝ}R_{ℝ} . In particular the Lie algebras satisfy 𝔤_{ℝ} = 𝔨_{ℝ}+𝔯_{ℝ} , so their complexifications satisfy 𝔤_{ℂ} = 𝔨_{ℂ}+𝔯_{ℂ} , and thus K_{ℂ}R_{ℂ} is open in G_{ℂ}. Since the projection G_{ℂ} → G_{ℂ}/R_{ℂ} is an open map, Ω_{X} := K_{ℂ}(1R_{ℂ}) is an open K_{ℂ}–orbit in X = G_{ℂ}/R_{ℂ} . If g ∈ G_{ℂ} then g ∈ K_{ℂ}R_{ℂ} if and only if gR_{ℂ} ∈ K_{ℂ}(1R_{ℂ}) ⊂ G_{ℂ}/R_{ℂ} . Now

Lemma 1.1. An element g ∈ G_{ℂ} is contained in K_{ℂ}R_{ℂ} if and only if gR_{ℂ} ∈ Ω_{X}.

Let Q be a parabolic subgroup of G_{ℂ} and write Ω_{Z} for the open orbit K_{ℂ}(1Q) on Z = G_{ℂ}/Q. The argument of Lemma 1.1 shows:

Lemma 1.2. Suppose K_{ℂ}Q = K_{ℂ}R_{ℂ} . Then g ∈ K_{ℂ}R_{ℂ} if and only if gQ ∈ Ω_{Z}.

There is a unique closed G_{ℝ}–orbit on X, and the other G_{ℝ}–orbits are finite in number and higher in dimension. See [4]. By (G_{ℝ}, K_{ℂ})–duality ([3]; see [2] for a geometric proof), there is a unique open K_{ℂ}–orbit on X (which thus must be Ω_{X}), and the other K_{ℂ}–orbits are finite in number and lower in dimension. Thus Ω_{X} is a dense open subset of X. Now Lemma 1.1 gives us

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Lemma 1.3. K_{ℂ}R_{ℂ} is a dense open subset of G_{ℂ} .

Let M_{ℂ} , A_{ℂ} and N_{ℂ} denote the respective complexifications of M_{ℝ} , A_{ℝ} and N_{ℝ} . Here M_{ℂ} is the centralizer of A_{ℂ} in K_{ℂ} . Let B denote a Borel subgroup of G_{ℂ} that contains A_{ℝ}N_{ℝ} ; in other words B = B_{M}A_{ℂ}N_{ℂ} where B_{M} is a Borel subgroup of M_{ℂ} . The phenomenon K_{ℂ}Q = K_{ℂ}R_{ℂ} of Lemma 1.2 occurs, for example, when R_{ℝ} is the minimal parabolic subgroup P_{ℝ} of G_{ℝ} and Q = B. In that case M_{ℝ} ⊂ K_{ℝ} , so B_{M} ⊂ M_{ℂ} ⊂ K_{ℂ} , and thus K_{ℂ} = K_{ℂ}B_{M} . As B = B_{M}A_{ℂ}N_{ℂ} now K_{ℂ}A_{ℂ}N_{ℂ} = K_{ℂ}B. We have proved

Proposition 1.4. Let B be a Borel subgroup of G_{ℂ} that contains A_{ℝ}N_{ℝ} . Let g ∈ G_{ℂ} . Then g is contained in K_{ℂ}A_{ℂ}N_{ℂ} if and only if gB belongs to the open K_{ℂ}–orbit on G_{ℂ}/B. In particular K_{ℂ}A_{ℂ}N_{ℂ} is a dense open subset of G_{ℂ} .

Proposition 1.4 describes the complexified Iwasawa decomposition set K_{ℂ}A_{ℂ}N_{ℂ} in terms of the open K_{ℂ}–orbit on the full flag manifold G_{ℂ}/B. Usually it is easy to decide whether a given group element belongs to K_{ℂ}A_{ℂ}N_{ℂ} , and that then describes the open K_{ℂ}–orbit on G_{ℂ}/B:

Example 1.5. Let G_{ℝ} = SL_{n}(ℝ) and G_{ℂ} = SL_{n}(ℂ). For g ∈ G_{ℂ} , let s_{k} be the k × k submatrix in the upper left corner of ^{t}gg. Then g ∈ K_{ℂ}A_{ℂ}N_{ℂ} if and only if det(s_{k}) ≠ 0 for 1 ≦ k ≦ 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] van den Ban, E. P., A convexity theorem for semisimple symmetric spaces, Pacific J. Math. 124 (1986), 21–55.

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

[3] Matsuki, T., Orbits on affine symmetric spaces under the action of parabolic subgroups, Hiroshima Math J. 12 (1982), 307–326.

[4] 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 Joseph A. Wolf Department of Mathematics University of California Berkeley, CA 94720-3840, USA jawolf@math.berkeley.edu

Received December 6, 2001 and in final form February 2, 2002