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