Polar orthogonal representations of real reductive algebraic groups

0.32 MB·English

by  Laura Geatti

POLAR ORTHOGONAL REPRESENTATIONS OF
REAL REDUCTIVE ALGEBRAIC GROUPS
LAURA GEATTI AND CLAUDIO GORODSKI
Abstract. We prove that a polar orthogonal representation of a real reductive algebraic
8 group has the same closed orbits as the isotropy representation of a pseudo-Riemannian
0 symmetric space. We also develop a partial structural theory of polar orthogonal repre-
0 sentations of real reductive algebraic groups which slightly generalizes some results of the
2
structural theory of real reductive Lie algebras.

n
a
J

3
1. Introduction

]
T
Arepresentation ofa complex reductive algebraicgroupG ona finite-dimensional complex
R
vector space V is called polar if there exists a subspace c ⊂ V consisting of semisimple
.
h elements such that dimc = dimV//G (the categorical quotient), and for a dense subset of c,
t
a thetangentspacestotheorbitsareparallel[DK85];thenitturnsoutthateveryclosedorbitof
m
Gmeets c(Prop.2.2, loc. cit.). Theclass ofpolarrepresentations wasintroducedandstudied
[
by Dadok and Kac in [DK85], and it is very important in invariant theory because it includes

1 the adjoint actions, the representations associated to symmetric spaces studied by Kostant
v
andRallis[KR71]aswellas,moregenerally, therepresentationsassociatedtoautomorphisms
4
7 of finite order (θ-groups) introduced by Vinberg [Vin76] (see also [Kac80]). At present, there
5 is no complete classification of polar representations although the paper [DK85] contains
0
very important partial results.
.
1
A complex (resp. real) representation admitting a complex-valued (resp. real-valued) in-
0
8 variant non-degenerate symmetric bilinear form is called orthogonal. It is well known that
0 a complex orthogonal representation admits a real form invariant under a maximal com-
:
v pact subgroup. Consider in particular the complex polar orthogonal representations and the
i
X class of compact real forms they originate. Since the complex reductive algebraic groups
r are exactly the complexifications of the compact Lie groups, one can equivalently define
a
directly the concept of a real polar representation of a compact Lie group in the differential-
geometric setting (as in e.g. [PT87]) and obtain the same class. Note that orbits of polar
representations of compact Lie groups are very important in submanifold geometry and
Morse theory [BS58, Con71, Sze84, PT87, DO01, GT03]. Now, such representations were
classified by Dadok in [Dad85], and the following very nice characterization was deduced: A
polar representation of a compact Lie group has the same orbits as the isotropy representation
of a Riemannian symmetric space.
The purpose of this paper is to study non-compact real forms of complex polar orthogonal
representations. Equivalently, we define a representation of a real reductive algebraic group
(in the sense of [BH62, §1]) to be polar if and only if its algebraic complexification is polar.
In section 3 we prove the following theorem.
Date: February 2, 2008.
1
Theorem 1. A polar orthogonal representation of a connected real reductive algebraic group
has the same closed orbits as the isotropy representation of a pseudo-Riemannian symmetric
space.
In section 4, we discuss some aspects of the submanifold geometry of the closed orbits of
the polar orthogonal representations of the real reductive algebraic groups in that we relate
them to a notion of pseudo-Riemannian isoparametric submanifold of a pseudo-Euclidean
space (compare [Hah84, Mag85]).
Finally in section 5, independently of classification results, we develop a partial structural
theory of polar orthogonal representations of real reductive algebraic groups that general-
izes some results of the structural theory of real reductive Lie algebras. In this regard,
we propose to replace adjoint actions by polar orthogonal ones. The results we prove are
slight generalizations of well known results for the adjoint actions, but we believe our proofs
are more geometric. In particular, we show that a polar orthogonal representation of a
real reductive algebraic group admits finitely many pairwise inequivalent so called Cartan
subspaces in standard position with respect to a compact real form such that the union
of those subspaces meets all the closed orbits and always orthogonally (Theorem 15 and
Corollary 18). We also construct the so called Cayley transformations that relate different
equivalence classes of Cartan subspaces (§5.3), and use those to show that the equivalence
classes of Cartan subspaces in the two extremal positions with respect to the compact real
form are unique (Corollary 22).
Unless explicit mention to the Zariski topology is made, we use throughout the classical
topology. We always use lowercase gothic letters to denote Lie algebras. For a given ho-
momorphism of groups, we denote the induced homomorphism on the Lie algebra level by
the same letter whenever the context is clear. Sometimes it is useful to call a representation
orthogonalizable if it admits an invariant non-degenerate symmetric bilinear form but we do
not want to fix such a form.
Part of this work was completed while the first author was visiting University of S˜ao Paulo
for which she would like to thank CNPq and CCInt-USP for their generous support, and
part of it was completed while the second author was visiting Universit`a di Roma 2 “Tor
Vergata” for which he would like to thank INdAM for its generous support. The second
author also wishes to thank Ralph Bremigan and Martin Magid for very useful discussions.
2. Preliminaries
Let G be a connected complex reductive algebraic group. Let τ : G → GL(V) be a
complex representation. A vector v ∈ V is called semisimple if the orbit Gv is closed. Not
every orbit of G in V is closed, but the closure of any orbit contains a unique closed orbit.
An element is called regular if it is semisimple and dimGv ≥ dimGx for all semisimple
x ∈ V. The representation τ is called stable or is said to admit generically closed orbits if
there exists an open and dense subset of V consisting of closed orbits. An orthogonalizable
representation is necessarily stable (see [Sch80, Cor. 5.9] or [Lun72, Lun73]).
Let C[V] be the polynomial algebra of V, and let C[V]G be the algebra of G-invariant
polynomials. Itdoes notcontainnilpotents, andis finitely generatedby atheoremofHilbert,
so it is the coordinate ring of an affine algebraic variety denoted by V//G and called the
categoricalquotient ofV byG. TheembeddingC[V]G → C[V]inducesasurjectivemorphism
of affine algebraic varieties π : V → V//G. Every fiber of π contains a unique closed orbit.
2
It follows that V//G can be seen as the parameter set of closed G-orbits in V, and then π(v)
represents the unique closed orbit in the closure of Gv [PV94, §4].
For semisimple v ∈ V, set
c = {x ∈ V | g·x ⊂ g·v}.
v
Then c consists entirely ofsemisimple elements, andtheisotropy subalgebrassatisfy g ⊃ g
v x v
for x ∈ c [DK85, Lem. 2.1]. The representation τ is called polar if a semisimple v can be
v
chosen so that dimc = dimV//G. In this case, c is called a Cartan subspace. The Cartan
v v
subspaces of a polar representation are all G-conjugate [DK85, Thm. 2.3].
The group G can be seen simply as the complexification of a compact connected Lie group
U; compare [Sch80, §5] or [BH62, Rmk. 3.4]. Then U is a maximal compact (necessarily
connected) subgroup of G, and every maximal compact subgroup of G is G-conjugate to
U. It is easy to see that a representation τ is orthogonalizable if and only if it admits
a real form τ : U → GL(W) [Sch80, Prop. 5.7]. The group U must be the fixed point
u
group Gθ of a unique anti-holomorphic involutive automorphism θ of G, which is called a
Cartan involution of G. Also, the subspace W is the fixed point set Vθ˜ of a conjugate-linear
˜ ˜ ˜
involutive automorphism θ of V, the equation θ(g · v) = θ(g) · θ(v) holds for g ∈ G and
v ∈ V, and an invariant form h·,·i can be chosen on V so that it is real-valued on Vθ˜.
More generally, we consider real forms of τ : G → GL(V) given by a pair (σ,σ˜) where σ
is an anti-holomorphic involution of G and σ˜ is a real structure on V satisfying σ˜(g ·v) =
σ(g)·σ˜(v) for g ∈ G, v ∈ V. The fixed point subgroup Gσ is a (not necessarily connected)
real reductive algebraic group, and τ of course restricts to a representation of Gσ → GL(Vσ˜),
where Vσ˜ is the fixed point set of σ˜ in V. We say that two real forms (σ,σ˜) and (σ′,σ˜′)
commute if they commute componentwise. If τ is orthogonal with respect to h·,·i and a real
form (σ,σ˜) is given, then h·,·i is said to be defined over R with respect to σ˜ and (σ,σ˜) is
called an orthogonal real form if h·,·i is real-valued on Vσ˜. Note that the latter condition is
equivalent to having
hσ˜x,σ˜yi = hx,yi
˜
for x, y ∈ V. A Cartan pair of τ is an orthogonal real form (θ,θ) such that θ is a Cartan
involution of G and h·,·i is real-valued and negative-definite on Vθ˜. Note that (θ,θ˜) is a
˜
Cartan pair of τ with respect to h·,·i if and only if (θ,−θ) is a Cartan pair of τ with respect
to −h·,·i. The following result is essentially proved in [Bre93, 7.4], but we find it convenient
to include a proof here because we will need to refer to some of its techniques.
Proposition 2 (Bremigan). Let τ : G → O(V,h·,·i) be an orthogonal representation, and
˜
suppose that (σ,σ˜) is an orthogonal real form. Then there exists a Cartan pair (θ,θ) which
commutes with (σ,σ˜).
Proof. It is well known that there exists a Cartan involution θ of G such that θσ = σθ.
Let U = Gθ be the associated maximal compact subgroup of G. Consider the realification
Vr of V, and denote the invariant complex structure on Vr by J so that V = (Vr,J). Note
that
σ˜τ(g)σ˜−1 = τ(σ(g)), Jτ(g)J−1 = τ(g) and Jσ˜J−1 = −σ˜
for g ∈ G. Let G∗ be the subgroup of GL(Vr) generated by τ(G), σ˜ andJ. Then G∗ contains
τ(G) as a normal subgroup of finite index. Due to θσ = σθ, we have also that σ˜ normalizes
τ(U). Let U∗ be the subgroup of G∗ generated by τ(U), σ˜ and J. Then U∗ is a compact
3
subgroup of G∗, so we can find an U∗-invariant positive-definite real inner product on Vr
which we denote by “·”. Set
(x,y) = x·y +i(x·Jy)
forx, y ∈ Vr. Thenit iseasily checked that(·,·)isanU-invariant positive-definite Hermitian
form on (Vr,J) = V which is real-valued on Vσ˜. In particular, iu acts on V by Hermitian
˜
endomorphisms. Next, define a conjugate-linear automorphism θ of V by setting
˜
(1) (x,θy) = −hx,yi
for x, y ∈ V. Then
(2) (x,θ˜(gy)) = −hx,gyi = −hg−1x,yi = (g−1x,θ˜y) = (x,θ(g)θ˜(y)),
˜ ˜
so θτ(g) = τ(θ(g))θ for g ∈ G. Moreover
˜ ˜ ˜
(3) (x,θσ˜y) = −hx,σ˜yi = −hσ˜x,yi = (σ˜x,θy) = (x,σ˜θy)
˜ ˜
implying that θσ˜ = σ˜θ. We also have that
˜2 ˜2 ˜ ˜
(θ x,y) = (y,θ x) = −hy,θxi = −hθx,yi
˜ ˜ ˜ ˜
= (θx,θy) = (θy,θx) = ···
˜2
= (x,θ y)
for x, y ∈ V. It follows that θ˜2 : V → V is a G∗-equivariant C-linear Hermitian automor-
phism. Hence there exists a h·,·i- and (·,·)-orthogonalG∗-invariant decomposition V = ⊕ V
j j
such that θ˜2| = λ id where λ ∈ R\{0} and the λ ’s are pairwise distinct.
Vj j Vj j j
˜ ˜
Note that λ (x,x) = (θx,θx) > 0 if x ∈ V \ {0}, so we also have λ > 0. If we change
j j j
1/2 ˜ −1/2
(·,·) by a factor of λ on V × V , as we do, θ is changed by a factor of λ on V , and
j j j j j
˜ ˜2
then the resulting θ satisfies θ = id . Note that equations (2) and (3) are unchanged. Now
V
˜
(θ,θ) is a real form of (G,V) commuting with (σ,σ˜). Further,
˜ ˜ ˜ ˜2 ˜ ˜
hθx,θyi = −(θx,θ y) = −(θx,y) = −(y,θx) = hy,xi = hx,yi
for x, y ∈ V and
˜
hx,xi = −(x,θx) = −(x,x) < 0
for x ∈ Vθ˜\{0}. This completes the proof. (cid:3)
Proposition 3. Let τ : G → O(V,h·,·i) be an orthogonal representation, and suppose that θ
˜
is a Cartan involution of G. Then there can be at most one real structure θ on V such that
˜
(θ,θ) is a Cartan pair of τ.
˜ ˜′
Proof. Suppose that (θ,θ) and (θ,θ ) are two Cartan pairs of τ. Define
h(x,y) = −hx,θ˜yi and h′(x,y) = −hx,θ˜′yi
for x, y ∈ V. It is easy to see that h and h′ are two U-invariant positive-definite Hermitian
forms. Diagonalizing h′ with respect to h, we get a U-invariant, h-orthogonal splitting
˜′ ˜
V = ⊕ V such that θ = λ θ on V , where λ > 0 and the λ ’s are pairwise distinct. Using
j j j j j j
(θ˜′)2 = θ˜2 = 1, we finally see that θ˜′ = θ˜. (cid:3)
4
Corollary 4. Let τ : G → O(V,h·,·i) be an orthogonal representation. Then any two Cartan
pairs of τ are G-conjugate; moreover, if the underlying Cartan involutions commute with a
real form σ of G, then the Cartan pairs are (Gσ)◦-conjugate.
˜ ˜
Proof. Let (θ ,θ ) and (θ ,θ ) be two Cartan pairs of τ. It is known that there exists
1 1 2 2
g ∈ G such that θ = Inn θ Inn−1. Of course, (Inn θ Inn−1,gθ˜ g−1) is also a Cartan pair.
2 g 1 g g 1 g 1
Proposition 3 implies that θ˜ = gθ˜ g−1. Further, if both θ and θ commute with σ, it is
2 1 1 2
known that g can be taken in the identity component of Gσ. (cid:3)
3. The classification
LetGˆR/GR beasemisimple pseudo-Riemanniansymmetric space. HereGˆR isaconnected
real semisimple Lie group, τˆ is a non-trivial involutive automorphism of GˆR and GR is an
open subgroup of the fixed point group of τˆ. The automorphism τˆ induces an automorphism
of the Lie algebra ˆgR of GˆR which we denote by the same letter. Let gˆR = gR + VR be
the decomposition into ±1-eigenspaces of τˆ. Of course, gR is the Lie algebra of GR. The
restriction of the Killing form of gˆR to VR×VR is AdGR-invariant and non-degenerate, so it
induces a GˆR-invariant pseudo-Riemannian metric on GˆR/GR. The adjoint action of GR on
VR is equivalent to the isotropy representation of GˆR/GR at the base-point.
Next, extend τˆ complex-linearly to an automorphism of the complexification gˆ = (gˆR)c
denoted by the same letter and consider the corresponding decomposition ˆg = g + V into
±1-eigenspaces. Let Gˆ bethe simply-connected complex LiegroupwithLiealgebragˆ, view τˆ
ˆ ˆ
as an involution ofG, and let G be the fixed point group of τˆin G. Note that G is connected.
The adjoint action of G on V is a complex polar action whose Cartan subspaces coincide
with the maximal Abelian subspaces of V consisting of semisimple elements (indeed, this is
a θ-group (see [DK85, Introd.] or [PV94, 8.5, 8.6]; no relation here to the aforementioned
Cartan involution θ). Further, it is an orthogonal action with respect to the restriction of
the Killing form of gˆ to V. By passing from GˆR to a finite covering if necessary, we may
assume that GˆR embeds into Gˆ and GR embeds into G, so we can view the adjoint action of
GR on VR as an orthogonal real form of the adjoint action of G on V. We deduce that the
isotropy representation of a pseudo-Riemannian symmetric space is a polar representation.
In this section, we prove Theorem 1 which is essentially a converse to this result.
Before giving the proof of Theorem 1, we prove four lemmas. In the remaining of this
section, let G be a complex reductive algebraic group defined over R and denote by GR the
identity component of its real points.
Lemma 5. Let τ : G → GL(V) be a polar representation, where V = V ⊕V is a G-invariant
1 2
decomposition. Assume that the induced representations τ : G → GL(V ) are stable. Then:
i i
(a) τ is polar, i = 1, 2.
i
(b) Every Cartan subspace of τ is of the form c = c ⊕c , where c is a Cartan subspace
1 2 i
of τ , i = 1, 2.
i
(c) The closed orbits of G on V coincide with those of G , where v is any semisimple
2 v1 1
vector of V . In particular, V and V are inequivalent representations.
1 1 2
(d) Fix a Cartan subspace c = c ⊕ c , let h (resp. h ) denote the centralizer of c
1 2 1 2 2
(resp. c ) in g, and denote by H the connected subgroup of G corresponding to h .
1 i i
Then the closed orbits of τ coincide with those of τˆ : H ×H → GL(V ⊕V ), where
1 2 1 2
τˆ(g ,g )(v +v ) = τ (g )v +τ (g )v .
1 2 1 2 1 1 1 2 2 2
5
Proof. Parts (a) and (b) are Prop. 2.14 in [DK85], and (c) is Cor. 2.15 of that paper. Let
us prove (d). Select a regular element v +v ∈ c ⊕c for τ. Then h = g , h = g and
1 2 1 2 1 v2 2 v1
(c) implies that h ·v = g·v and h ·v = g·v (note that v is semisimple for τ since c is a
1 1 1 2 2 2 i i i
Cartan subspace). This implies that g = h +h , so G = H ·H = H ·H by connectedness
1 2 1 2 2 1
of G. For any u +u ∈ c ⊕c , we now have that G(u +u ) ⊂ (H ×H )(u +u ). Since
1 2 1 2 1 2 1 2 1 2
g = h +h , and G(u +u ), H u ×H u are closed and connected, it follows that the two
1 2 1 2 1 1 2 2
(cid:3)
orbits coincide.
Lemma 6. Let ρ : GR → GL(VR) be a polar representation, where VR = (VR)1⊕(VR)2 is a
GR-invariant decomposition. Assume that the induced representations ρi : GR → GL((VR)i)
are orthogonalizable. Then there exist closed connected subgroups H′ of GR, i = 1, 2, such
i
that the restricted representations ρi|Hi′ : Hi′ → GL((VR)i) are polar and the closed orbits
of ρ coincide with those of ρˆ : H1′ × H2′ → GL((VR)1 ⊕ (VR)2), where ρˆ(g1,g2)(v1 + v2) =
ρ (g )v +ρ (g )v .
1 1 1 2 2 2
Proof. The complexification τ = ρc : G → GL(V) is polar and each τ = ρc is or-
i i
thogonalizable, hence stable. By Lemma 5, the closed orbits of τ coincide with those of
τˆ : H1 × H2 → GL(V1 ⊕ V2), where Vi = (VR)ci, the group H1 (resp. H2) is the connected
centralizer of c (resp. c ) in G, and c = c ⊕ c ⊂ V ⊕ V is a Cartan subspace of τ.
2 1 1 2 1 2
As usual, suppose that ρ is defined by (σ,σ˜). Now c can be taken to be σ˜-stable due to
Lemma 13 below. In this case, Hi is σ-stable; set Hi′ to be subgroup of GR given by the
identity component of (H )σ. It is clear that the groups H′ have the desired properties. (cid:3)
i i
Given a representation ρ : GR → GL(VR), denote by ρ∗ : GR → GL(VR∗) the dual
representation. Note that ρ⊕ρ∗ : GR → GL(VR⊕VR∗) is always orthogonal with respect to
(4) h(v ,v∗),(v ,v∗)i = v∗(v )+v∗(v ).
1 1 2 2 1 2 2 1
The proof of the following lemma is simple and we omit it.
Lemma 7. Let ρ : GR → GL(VR) be orthogonalizable. Then there exists an irreducible
decomposition
VR = (VR)1 ⊕···⊕(VR)r ⊕(VR)r+1 ⊕(VR)∗r+1 ⊕···⊕(VR)s ⊕(VR)∗s,
where (VR)1,...,(VR)r are orthogonalizable and (VR)r+1,...,(VR)s are not orthogonalizable.
The following lemma will be used to show that certain polar representations have the same
closed orbits as a the isotropy representation of a symmetric space.
Lemma 8. Suppose τ : G → GL(V) is a polar orthogonalizable representation, U is a
maximal compact subgroup of G and τ : U → GL(W) is a real form. Suppose also that U′
u
is a connected closed subgroup of U and G′ ⊂ G is the complexification of U′. If τu|U′ has
the same orbits in W as τu, then τ|G′ has the same closed orbits in V as τ. If, in addition,
ρ : GR → GL(VR) is a real form of τ and G′R ⊂ GR is a connected real form of G′, then
ρ|G′R has the same closed orbits in VR as ρ.
Proof. The assertion about ρ immediately follows from that about τ and the facts that
GRv is closed if and only if Gv is closed [Bir71] and dimRGRv = dimGv for v ∈ VR. Let us
prove the assertion about τ. We first claim that if v ∈ V and Gv is closed, then G′v = Gv.
Of course, we already have that G′v ⊂ Gv. In the case in which v ∈ W, we have that both
Gv and G′v are connected, closed and have dimension equal to dimRUv = dimRU′v, so the
result follows. In the general case, fix a U-invariant positive-definite Hermitian form (·,·)
6
and choose v ∈ Gv of minimal length [DK85, p.508]. Of course, Gv = Gv and v is also of
1 1 1
minimal length in G′v . It follows that G′v is also closed [DK85, Thm. 1.1] and G , G′
1 1 v1 v1
are θ-stable, where θ is the Cartan involution of G associated to U [DK85, Prop. 1.3]. Let
L = (G )θ and L′ = (G′ )θ. Now we can choose w ∈ W such that U = L by the same
v1 v1 w
argument as in [Sch80, Prop. 5.8], and it easily follows that U′ = L′. We have established
w
that U (resp. U′ ) is a real form of G (resp. G′ ). Therefore
w w v1 v1
dimGv = dimG−dimG
1 v1
= dimRU −dimRUw
= dimRUw
= dimRU′w
= dimRU′ −dimRU′
w
= dimG′ −dimG′
v1
= dimG′v ,
1
which implies that G′v = Gv . Since Gv = Gv, we also have G′v = Gv, proving the claim.
1 1 1
Let c ⊂ V be a Cartan subspace of τ. In view of the claim proved above, c consists of
semisimple elements of τ|G′. Also, dimc = dimV//G = dimV//G′, where the last equality
follows from the fact that τu and τu|U′ have the same co-homogeneity in W. By [DK85,
Prop. 2.2], every closed G′-orbit meets c, from which it follows that τ|G′ has the same closed
orbits in V as τ. (cid:3)
In order to prove Theorem 1, we will use the explicit lists of polar representations of
compact Lie groups that have been obtained in [EH99] (irreducible case) and [Ber99, Ber01]
(reducible case); see also [GT00] (both cases). For brevity, an isotropy representation of a
semisimple symmetric space will be called an s-representation. Let ρ : GR → GL(VR) be
a polar orthogonal representation. Let τ = ρc : G → GL(V), and suppose that ρ is given
by (σ,σ˜) so that GR is the identity component of Gσ. Let (θ,θ˜) be a Cartan pair as in
Proposition 2, U = Gθ, W = Vθ˜, and τ : U → GL(W) the associated real form. Then
u
τ is a polar representation of a compact Lie group. By Dadok’s theorem quoted in the
u
introduction and the results in [EH99, Ber01], τ is either a Riemannian s-representation or
u
one of the exceptions listed in those papers. We need the following fundamental lemma.
Lemma 9. Ifτ isan irreducibleRiemannians-representation, thenρ isa pseudo-Riemannian
u
s-representation.
Proof. By assumption, uˆ = u+W admits a real Lie algebra structure extending that of u
such that [HZ96, p.182]
[X,w] = τ (X)w and hX,[w,w′]i = hτ (X)w,w′i
u u u
for X ∈ u and w, w′ ∈ W, where
hX,Yi = trace (ad ad )
u ˆu X Y
for X, Y ∈ u and ad (Z) = [X,Z] for X ∈ u and Z ∈ uˆ. Denote the Killing form of uˆ
X
by β; note that it is nondegenerate as uˆ is semisimple. Also, it turns out that h·,·i is the
u
restriction of β to u.
Now, since β| and h·,·i| are both positive-definite real-valued symmetric bilinear
W×W W×W
forms which are u-invariant, and τ is irreducible, there exists λ > 0 such that β(x,y) =
u
7
λhx,yi for x, y ∈ W. By C-bilinearity, βc(x,y) = λhx,yi for x, y ∈ V, where βc is the
Killing form of uˆc = g+V. It suffices to prove that gˆR := gR+VR is a real subalgebra of uˆc.
It is clear that [gR,gR] ⊂ gR and [gR,VR] ⊂ VR. We claim that also [VR,VR] ⊂ gR. In fact,
(5) βc(σ˜x,σ˜y) = λhσ˜x,σ˜yi = λhx,yi = λhx,yi = βc(x,y)
for x, y ∈ V. If Z , Z ∈ g, then also
1 2
βc(σZ ,σZ ) = trace (ad ad )
1 2 ˆuc σZ1 σZ2
= trace (ad ad )+trace (ad ad )
g σZ1 σZ2 V σZ1 σZ2
(6) = trace (σad ad σ)+trace (σ˜ad ad σ˜)
g Z1 Z2 V Z1 Z2
= trace (ad ad )+trace (ad ad )
g Z1 Z2 V Z1 Z2
= trace (ad ad )
ˆuc Z1 Z2
= βc(Z ,Z ),
1 2
where we used in the third equality that
ad x = σZ ·x = σ˜(Z ·σ˜x) = σ˜ad σ˜x
σZ1 1 1 Z1
for x ∈ V. Therefore
βc(Z,σ[x,y]) = βc(σZ,[x,y]) by (6)
= βc(σZ ·x,y)
= βc(σ˜(σZ ·x),σ˜y) by (5)
= βc(Z ·σ˜x,σ˜y)
= βc(Z,[σ˜x,σ˜y])
for all Z ∈ g and x, y ∈ V. Hence σ[x,y] = [σ˜x,σ˜y], proving the claim. Of course, ρ is
now the isotropy representation of the pseudo-Riemannian symmetric space GˆR/GR, where
GˆR := Int(gˆR) and GR is the connected subgroup associated to gR. (cid:3)
Proof of Theorem 1. In view of Lemmas 6 and 7, it is enough to consider the following
two cases:
(a) ρ is irreducible.
(b) ρdecomposesasρ0⊕ρ∗0,whereρ0 : GR → GL(V0)isirreducibleandnon-orthogonalizable,
VR = V0 ⊕V0∗, and the inner product on VR is given by (4).
(a.1) Suppose first that ρ is absolutely irreducible. Then τ is an absolutely irreducible
u
polar representation of a compact Lie group, so it is either a Riemannian s-representation
and then the result follows from Lemma 9, or it is listed in [EH99]. In the latter case, it
must be (SO(3) × Spin(7),R3 ⊗ R8), where R8 denotes the spin representation; accord-
ing to [Oni04, Table 5, p.79], GR equals SO0(1,2) × Spin(7) (resp. SO(3) × Spin0(3,4),
SO0(1,2)×Spin0(3,4); here the subscript denotes the identity component), and ρ : GR →
GL(R3 ⊗R8) is the tensor product of the standard representation and the spin representa-
tion. Since Spin(7) ⊂ SO(8) and Spin (3,4) ⊂ SO (4,4) [Har90, Thm. 14.2], and ρ extends
0 0
to a pseudo-Riemannian s-representation ρ′ of SO (1,2)×SO(8) (resp. SO(3)×SO (4,4),
0 0
SO (1,2) × SO (4,4)) on R3 ⊗ R8, it follows from Lemma 8 that ρ has the same closed
0 0
orbits as ρ′, so this case is checked.
8
(a.2) Suppose now that ρ is irreducible but not absolutely irreducible. Then VR admits
an invariant complex structure.
(a.2.1) If τ is irreducible, then W admits an U-invariant complex structure, and by
u
Lemma 9 we have only to consider the cases in which it is not an s-representation. According
to [EH99], those are
(SO(2)×G ,R2 ⊗R7)
2
(SO(2)×Spin(7),R2 ⊗R8)
(7) (SU(p)×SU(q),(Cp ⊗Cq)r) (p 6= q)
(SU(n),(Λ2Cn)r) (n odd)
(Spin(10),C16)
We do only the first and third cases, the others being similar in spirit. In the first case,
GR must be SO(2)× G∗, where G∗ is the automorphism group of the split octonions and
2 2
ρ is the real tensor product of the standard representation of SO(2) and the 7-dimensional
representation of G∗2 since VR admits an invariant complex structure. Now G∗2 ⊂ SO0(3,4)
and there exists an obvious pseudo-Riemannian s-representation ρ′ of SO(2)×SO (3,4) on
0
R2 ⊗ R7. It follows from Lemma 8 that ρ and ρ′ have the same closed orbits and we are
done with this case. In the third case, viewing ρ as a complex representation, its conjugate
representation ρ¯ with respect to GR must be equivalent to ρ∗ because ρ ⊕ ρ¯ = (τu)c. The
only possibility is that ρ equals (SU(r,p−r)×SU(s,q−s),(Cp⊗Cq)r), which has the same
closed orbits as the s-representation of the pseudo-Riemannian symmetric space
SU(r +s,p+q −r−s)/S(U(r,p−r)×U(s,q −s)).
(a.2.2) If τ is not irreducible, then there exists an U-irreducible decomposition W =
u
W ⊕ W , where (τ ) : U → GL(W ) is absolutely irreducible. Now V = Wc ⊕ Wc is a
1 2 u i i 1 2
G-irreducible decomposition, where Wc and Wc are inequivalent by polarity (Lemma 5(c))
1 2
and W2c must be the conjugate representation to W1c with respect to GR. Denote τi = (τu)ci :
G → GL(Wc). It follows that
i
(8) σ˜Wc = Wc and τ (g) = τ (σ(g)) = σ˜τ (g)σ˜
1 2 2 1 1
for g ∈ G. Since σ commutes with θ, we can view σ as an automorphism of U. Suppose first
that τ is splitting, that is U = U ×U and τ is the outer direct product of (τ ) | and
u 1 2 u u 1 U1
(τ ) | . On the level of Lie algebras, (8) implies that u = ker(τ ) = σ(ker(τ ) ) = σ(u ).
u 2 U2 1 u 2 u 1 2
Now we can assume that u = u , (τ ) = (τ ) , g = g ⊕g¯ , where g = uc and ¯g is the
1 2 u 1 u 2 1 1 1 1 1
conjugate Lie algebra of g , and σ : g → g is given by σ(Z′,Z¯′′) = (Z′′,Z¯′). Moreover,
1
V = W1c ⊕ Wc1 and σ˜: V → V is given by σ˜(w′,w¯′′) = (w′′,w¯′). Hence gR = {(Z′,Z¯′′) ∈
g1 ⊕¯g1 : Z′ = Z′′}, VR = {(w′,w¯′′) ∈ W1c ⊕W1c : w′ = w′′}, and ρ is just the realification of
the complexification of the real polar absolutely irreducible representation (τ ) | : U →
u 1 U1 1
GL(W ). If (τ ) | is an s-representation, this means that ρ is the s-representation of a
1 u 1 U1
complex symmetric space viewed as a real representation. The only other possibility is that
(τ ) | equals (SO(3)× Spin(7),R3 ⊗ R8). In this case, ρ : SO(3,C)r × Spin(7,C)r →
u 1 U1
GL((C3⊗C8)r) has the same closed orbits as SO(3,C)r×SO(8,C)r → GL((C3⊗C8)r), so
wearedone. Supposenowthatτ isnotsplitting. ThenU = U ×U ×U ,whereU (resp.U )
u 1 0 2 2 1
coincideswithker(τ ) (resp.ker(τ ) )uptosomediscretepart. Sinceσ(ker(τ ) ) = ker(τ ) ,
u 1 u 2 u 1 u 2
the automorphism σ : U → U must restrict to isomorphisms U → U and U → U . It
1 2 0 0
follows that U is essential for (τ ) if and only if it is essential for (τ ) . Therefore τ is
0 u 1 u 2 u
not almost splitting in the sense of [GT00, p.58]; we use the classification given there: due
9
to the facts that the (τ ) admits no invariant complex structure and dimW = dimW ,
u i 1 2
we need only to consider the case in which U = Spin(8), U = U = {1}, and W , W
0 1 2 1 2
are two 8-dimensional inequivalent representations of Spin(8). Referring to [Oni04, Table 5,
p.80], GR must be either Spin0(3,5) or Spin0(1,7), and ρ must be the realification of an
8-dimensional complex representation of GR which is not of real type (indeed, in each case
there exist two such representations and they are conjugate to one another). Since τ is
(SO(8,C),C8 ⊕ C8) (where C8 denote the half-spin representations) with compact real
+ − ±
form (Spin(8),R8 ⊕ R8) having the same orbits as (SO(8)×SO(8),R8 ⊕ R8), it follows
+ −
that ρ has the same closed orbits as the standard action of (SO(8,C))r on (C8)r. Now the
latter is a pseudo-Riemannian s-representation, so this case is also dealt with.
(b.1) Consider now the case in which ρ = ρ ⊕ρ∗, where ρ is absolutely irreducible and
0 0 0
non-orthogonalizable. Then ρc = ρc ⊕ (ρ∗)c is polar and ρc is irreducible. By polarity,
0 0 0
ρc and (ρ∗)c = (ρc)∗ are inequivalent, so ρc is not self-dual implying that it is not of real
0 0 0 0
type with respect to U. Recall that (ρc)∗ is the conjugate representation ρˆc with respect
0 0
to U. It follows that θ˜(v′,vˆ′′) = (v′′,vˆ′) for (v′,vˆ′′) ∈ Vc ⊕ Vˆc, the space W = {(v′,vˆ′′) ∈
0 0
Vc ⊕ Vˆc : v′ = v′′}, and τ : U → GL(W) is irreducible, not absolutely irreducible, and
0 0 u
equivalent to (ρc)r| : U → GL((Vc)r). In other words, W admits a U-invariant complex
0 U 0
structure J and ρ is just a real form of the holomorphic extension of τ : U → GL(W,J)
0 u
to a representation of G on (W,J). By Lemma 9, it suffices to consider the case in which
τ is not an s-representation, namely, given in (7). We do only the case (SU(n),(Λ2Cn)r)
u
for n odd, the others being similar in spirit. Since n is odd, [Oni04, Table 5] gives that
GR = SL(n,R) and ρ0 is the representation on Λ2Rn. Now ρ = ρ0⊕ρ∗0 has the same closed
orbits as (GL+(n,R),Λ2Rn ⊕ (Λ2Rn)∗), which turns out to be the s-representation of the
pseudo-Riemannian symmetric space SO(n,n)/GL+(n,R) [Ber57, Tableau II].
(b.2) Finally, suppose that ρ = ρ ⊕ρ∗, where ρ is irreducible, not absolutely irreducible
0 0 0
and non-orthogonalizable. Then ρ , ρ∗ can be viewed as complex representations, and ρc =
0 0
ρ ⊕ρ¯ ⊕ρˆ ⊕ρˆ¯ is an irreducible decomposition with pairwise inequivalent summands, where
0 0 0 0
ρ¯0 (resp. ρˆ0 = ρ∗0) is the conjugate representation to ρ0 with respect to GR (resp. U). We
must have τ = (τ ) ⊕ (τ ) : U → GL(W ⊕ W ), where (τ ) is polar irreducible, not
u u 1 u 2 1 2 u i
absolutely irreducible. Moreover, τ = ρ ⊕ρˆ and τ = ρ¯ ⊕ρˆ¯ , where we have set τ = (τ )c.
1 0 0 2 0 0 i u i
(b.2.1) Suppose τ is splitting. Then U = U ×U and τ is the outer direct product of
u 1 2 u
(τ ) | and (τ ) | , where each (τ ) | is irreducible and not absolutely irreducible. The
u 1 U1 u 2 U2 u i Ui
automorphism σ : U → U must take U to U , so we can assume U = U and (τ ) =
1 2 1 2 u 1
(τ ) . Write G = G × G where g = uc. Then ρ is equivalent to the realification of
u 2 1 2 i i
τ | : G → GL(Vc ⊕ Vc∗), and τ | is the complexification of a polar irreducible, not
1 G1 1 0 0 1 G1
absolutely irreducible representation (τ ) | : U → GL(W ). We have only to consider
u 1 U1 1 1
the case in which it is not an s-representation, namely, given in (7). We do only the case
(Spin(10),(C16)r), the others being similar in spirit. Here τ is(Spin(10,C),C16⊕C16∗) and
1
ρ is (Spin(10,C)r,(C16)r ⊕(C16∗)r), which turns out to have the same closed orbits as the
pseudo-Riemannian s-representation given by the realification of (C× ×Spin(10,C),C16 ⊕
C16∗).
(b.2.2) Suppose τ is not splitting. Then it is not almost splitting by the same argument
u
as in case (a.2.2). Owing to the fact that (τ ) admits an invariant complex structure for
u i
i = 1, 2, we see from [GT00, p.59] that this case is not possible.
10