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Polar orthogonal representations of real reductive algebraic groups

We prove that a polar orthogonal representation of a real reductive algebraic group has the same closed orbits as the isotropy representation of a pseudo-Riemannian symmetric space. We also develop a partial structural theory of polar orthogonal representations of real reductive algebraic groups which slightly generalizes some results of the structural theory of real reductive Lie algebras.


Introduction
A representation of a complex reductive algebraic group G on a finite-dimensional complex vector space V is called polar if there exists a subspace c ⊂ V consisting of semisimple elements such that dim c = dim V //G (the categorical quotient), and for a dense subset of c, the tangent spaces to the orbits are parallel [DK85]; then it turns out that every closed orbit of G meets c (Prop. 2.2, loc. cit.). The class of polar representations was introduced and studied by Dadok and Kac in [DK85], and it is very important in invariant theory because it includes the adjoint actions, the representations associated to symmetric spaces studied by Kostant and Rallis [KR71] as well as, more generally, the representations associated to automorphisms of finite order (θ-groups) introduced by Vinberg [Vin76] (see also [Kac80]). At present, there is no complete classification of polar representations although the paper [DK85] contains very important partial results.
A complex (resp. real) representation admitting a complex-valued (resp. real-valued) invariant non-degenerate symmetric bilinear form is called orthogonal. It is well known that a complex orthogonal representation admits a real form invariant under a maximal compact subgroup. Consider in particular the complex polar orthogonal representations and the class of compact real forms they originate. Since the complex reductive algebraic groups are exactly the complexifications of the compact Lie groups, one can equivalently define directly the concept of a real polar representation of a compact Lie group in the differentialgeometric 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. 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 generalizes 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 homomorphism 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 São 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 Università 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.

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 dim Gv ≥ dim Gx 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. It does not contain nilpotents, and is finitely generated by a theorem of Hilbert, so it is the coordinate ring of an affine algebraic variety denoted by V //G and called the categorical quotient of V by G. The embedding C[V ] G → C[V ] induces a surjective morphism of affine algebraic varieties π : V → V //G. Every fiber of π contains a unique closed orbit.
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 Then c v consists entirely of semisimple elements, and the isotropy subalgebras satisfy g 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 : U → GL(W ) [Sch80,Prop. 5.7]. The group U must be the fixed point group G θ 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 automorphismθ of V , the equationθ(g · v) = θ(g) ·θ(v) holds for g ∈ G and v ∈ V , and an invariant form ·, · 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 andσ is a real structure on V satisfyingσ(g · v) = σ(g) ·σ(v) for g ∈ G, v ∈ V . The fixed point subgroup G σ is a (not necessarily connected) real reductive algebraic group, and τ of course restricts to a representation of G σ → GL(Vσ), where Vσ is the fixed point set ofσ 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 ·, · and a real form (σ,σ) is given, then ·, · is said to be defined over R with respect toσ and (σ,σ) is called an orthogonal real form if ·, · is real-valued on Vσ. Note that the latter condition is equivalent to having σx,σy = x, y 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 ·, · is real-valued and negative-definite on Vθ. Note that (θ,θ) is a Cartan pair of τ with respect to ·, · if and only if (θ, −θ) is a Cartan pair of τ with respect to − ·, · . 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.
Proof. It is well known that there exists a Cartan involution θ of G such that θσ = σθ. Let U = G θ be the associated maximal compact subgroup of G. Consider the realification V r of V , and denote the invariant complex structure on V r by J so that V = (V r , J). Note thatσ τ (g)σ −1 = τ (σ(g)), Jτ (g)J −1 = τ (g) and JσJ −1 = −σ for g ∈ G. Let G * be the subgroup of GL(V r ) generated by τ (G),σ and J. Then G * contains τ (G) as a normal subgroup of finite index. Due to θσ = σθ, we have also thatσ normalizes τ (U). Let U * be the subgroup of G * generated by τ (U),σ and J. Then U * is a compact subgroup of G * , so we can find an U * -invariant positive-definite real inner product on V r which we denote by "·". Set (x, y) = x · y + i(x · Jy) for x, y ∈ V r . Then it is easily checked that (·, ·) is an U-invariant positive-definite Hermitian form on (V r , J) = V which is real-valued on Vσ. In particular, iu acts on V by Hermitian endomorphisms. Next, define a conjugate-linear automorphismθ of V by setting (1) (x,θy) = − x, y for x, y ∈ V . Then (2) (x,θ(gy)) = − x, gy = − g −1 x, y = (g −1 x,θy) = (x, θ(g)θ(y)), implying thatθσ =σθ. We also have that Note that λ j (x, x) = (θx,θx) > 0 if x ∈ V j \ {0}, so we also have λ j > 0. If we change (·, ·) by a factor of λ 1/2 j on V j × V j , as we do,θ is changed by a factor of λ  Proof. Suppose that (θ,θ) and (θ,θ ′ ) are two Cartan pairs of τ . Define h(x, y) = − x,θy and h ′ (x, y) = − x,θ ′ y 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 = ⊕ j V j such thatθ ′ = λ jθ on V j , where λ j > 0 and the λ j 's are pairwise distinct. Using (θ ′ ) 2 =θ 2 = 1, we finally see thatθ ′ =θ. Proof. Let (θ 1 ,θ 1 ) and (θ 2 ,θ 2 ) be two Cartan pairs of τ . It is known that there exists g ∈ G such that θ 2 = Inn g θ 1 Inn −1 g . Of course, (Inn g θ 1 Inn −1 g , gθ 1 g −1 ) is also a Cartan pair. Proposition 3 implies thatθ 2 = gθ 1 g −1 . Further, if both θ 1 and θ 2 commute with σ, it is known that g can be taken in the identity component of G σ .

The classification
LetĜ R /G R be a semisimple pseudo-Riemannian symmetric space. HereĜ R is a connected real semisimple Lie group,τ is a non-trivial involutive automorphism ofĜ R and G R is an open subgroup of the fixed point group ofτ . The automorphismτ induces an automorphism of the Lie algebraĝ R ofĜ R which we denote by the same letter. Letĝ R = g R + V R be the decomposition into ±1-eigenspaces ofτ . Of course, g R is the Lie algebra of G R . The restriction of the Killing form ofĝ R to V R × V R is Ad G R -invariant and non-degenerate, so it induces aĜ R -invariant pseudo-Riemannian metric onĜ R /G R . The adjoint action of G R on V R is equivalent to the isotropy representation ofĜ R /G R at the base-point.
Next, extendτ complex-linearly to an automorphism of the complexificationĝ = (ĝ R ) c denoted by the same letter and consider the corresponding decompositionĝ = g + V into ±1-eigenspaces. LetĜ be the simply-connected complex Lie group with Lie algebraĝ, viewτ as an involution ofĜ, and let G be the fixed point group ofτ inĜ. 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 ofĝ to V . By passing fromĜ R to a finite covering if necessary, we may assume thatĜ R embeds intoĜ and G R embeds into G, so we can view the adjoint action of G R on V R 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 G R the identity component of its real points.
The closed orbits of G on V 2 coincide with those of G v 1 , where v 1 is any semisimple vector of V 1 . In particular, V 1 and V 2 are inequivalent representations. (d) Fix a Cartan subspace c = c 1 ⊕ c 2 , let h 1 (resp. h 2 ) denote the centralizer of c 2 (resp. c 1 ) in g, and denote by H i the connected subgroup of G corresponding to h i . Then the closed orbits of τ coincide with those ofτ : 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 1 + v 2 ∈ c 1 ⊕ c 2 for τ . Then h 1 = g v 2 , h 2 = g v 1 and (c) implies that h 1 · v 1 = g · v 1 and h 2 · v 2 = g · v 2 (note that v i is semisimple for τ i since c i is a Cartan subspace). This implies that g = h 1 + h 2 , so G = H 1 · H 2 = H 2 · H 1 by connectedness of G. For any u 1 + u 2 ∈ c 1 ⊕ c 2 , we now have that G(u 1 + u 2 ) ⊂ (H 1 × H 2 )(u 1 + u 2 ). Since g = h 1 + h 2 , and G(u 1 + u 2 ), H 1 u 1 × H 2 u 2 are closed and connected, it follows that the two orbits coincide.
are polar and the closed orbits of ρ coincide with those ofρ : Proof. The complexification τ = ρ c : G → GL(V ) is polar and each τ i = ρ c i is orthogonalizable, hence stable. By Lemma 5, the closed orbits of τ coincide with those of τ : As usual, suppose that ρ is defined by (σ,σ). Now c can be taken to beσ-stable due to Lemma 13 below. In this case, H i is σ-stable; set H ′ i to be subgroup of G R given by the identity component of (H i ) σ . It is clear that the groups H ′ i have the desired properties. Given a representation ρ : . The proof of the following lemma is simple and we omit it.
Lemma 7. Let ρ : G R → GL(V R ) be orthogonalizable. Then there exists an irreducible decomposition 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.
Proof. The assertion about ρ immediately follows from that about τ and the facts that G R v is closed if and only if Gv is closed [Bir71] and dim R G R v = dim Gv for v ∈ V R . 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 dim R Uv = dim R U ′ v, so the result follows. In the general case, fix a U-invariant positive-definite Hermitian form (·, ·) and choose v 1 ∈ Gv of minimal length [DK85,p.508]. Of course, Gv 1 = Gv and v 1 is also of 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, dim c = dim V //G = dim V //G ′ , where the last equality follows from the fact that τ u and τ u | 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 .
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 : G R → GL(V R ) be a polar orthogonal representation. Let τ = ρ c : G → GL(V ), and suppose that ρ is given by (σ,σ) so that G R is the identity component of G σ . Let (θ,θ) be a Cartan pair as in Proposition 2, U = G θ , W = Vθ, and τ u : U → GL(W ) the associated real form. Then τ u is a polar representation of a compact Lie group. By Dadok's theorem quoted in the introduction and the results in [EH99,Ber01], τ u is either a Riemannian s-representation or one of the exceptions listed in those papers. We need the following fundamental lemma.
Lemma 9. If τ u is an irreducible Riemannian s-representation, then ρ is a pseudo-Riemannian s-representation.
Proof. By assumption,û = u + W admits a real Lie algebra structure extending that of u such that [HZ96,p.182] for X, Y ∈ u and ad X (Z) = [X, Z] for X ∈ u and Z ∈û. Denote the Killing form ofû by β; note that it is nondegenerate asû is semisimple. Also, it turns out that ·, · u is the restriction of β to u. Now, since β| W ×W and ·, · | W ×W are both positive-definite real-valued symmetric bilinear forms which are u-invariant, and τ u is irreducible, there exists λ > 0 such that β(x, y) = λ x, y for x, y ∈ W . By C-bilinearity, β c (x, y) = λ x, y for x, y ∈ V , where β c is the Killing form ofû c = g + V . It suffices to prove thatĝ R : where we used in the third equality that for all Z ∈ g and x, y ∈ V . Hence σ[x, y] = [σx,σy], proving the claim. Of course, ρ is now the isotropy representation of the pseudo-Riemannian symmetric spaceĜ R /G R , wherê G R := Int(ĝ R ) and G R is the connected subgroup associated to g R .
Proof of Theorem 1. In view of Lemmas 6 and 7, it is enough to consider the following two cases: (a) ρ is irreducible.
(a.2) Suppose now that ρ is irreducible but not absolutely irreducible. Then V R admits an invariant complex structure.
(a.2.1) If τ u is irreducible, then W admits an U-invariant complex structure, and by Lemma 9 we have only to consider the cases in which it is not an s-representation. According to [EH99], those are We do only the first and third cases, the others being similar in spirit. In the first case, where G * 2 is the automorphism group of the split octonions and ρ is the real tensor product of the standard representation of SO(2) and the 7-dimensional representation of G * 2 since V R admits an invariant complex structure. Now G * 2 ⊂ SO 0 (3, 4) and there exists an obvious pseudo-Riemannian s-representation ρ ′ of SO(2) × SO 0 (3, 4) on R 2 ⊗ R 7 . 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 representationρ with respect to G R must be equivalent to ρ * because ρ ⊕ρ = (τ u ) c . The only possibility is that ρ equals (SU(r, p − r) × SU(s, q − s), (C p ⊗ C q ) r ), which has the same closed orbits as the s-representation of the pseudo-Riemannian symmetric space 1 and W c 2 are inequivalent by polarity (Lemma 5(c)) and W c 2 must be the conjugate representation to W c 1 with respect to G R . Denote and τ 2 (g) = τ 1 (σ(g)) =στ 1 (g)σ for g ∈ G. Since σ commutes with θ, we can view σ as an automorphism of U. Suppose first that τ u is splitting, that is U = U 1 × U 2 and τ u is the outer direct product of (τ u ) 1 | U 1 and (τ u ) 2 | U 2 . On the level of Lie algebras, (8) implies that u 1 = ker(τ u ) 2 = σ(ker(τ u ) 1 ) = σ(u 2 ). Now we can assume that u 1 = u 2 , (τ u ) 1 = (τ u ) 2 , g = g 1 ⊕ḡ 1 , where g 1 = u c 1 andḡ 1 is the conjugate Lie algebra of g 1 , and σ : g → g is given by σ(Z ′ ,Z ′′ ) = (Z ′′ ,Z ′ ). Moreover, and ρ is just the realification of the complexification of the real polar absolutely irreducible representation (τ u ) 1 | U 1 : U 1 → GL(W 1 ). If (τ u ) 1 | U 1 is an s-representation, this means that ρ is the s-representation of a complex symmetric space viewed as a real representation. The only other possibility is that (τ u ) 1 | U 1 equals (SO(3) × Spin(7), R 3 ⊗ R 8 ). In this case, ρ : SO(3, C) r × Spin(7, C) r → GL((C 3 ⊗ C 8 ) r ) has the same closed orbits as SO(3, C) r × SO(8, C) r → GL((C 3 ⊗ C 8 ) r ), so we are done. Suppose now that τ u is not splitting. Then U = U 1 ×U 0 ×U 2 , where U 2 (resp. U 1 ) coincides with ker(τ u ) 1 (resp. ker(τ u ) 2 ) up to some discrete part. Since σ(ker(τ u ) 1 ) = ker(τ u ) 2 , the automorphism σ : U → U must restrict to isomorphisms U 1 → U 2 and U 0 → U 0 . It follows that U 0 is essential for (τ u ) 1 if and only if it is essential for (τ u ) 2 . Therefore τ u is not almost splitting in the sense of [GT00, p.58]; we use the classification given there: due to the facts that the (τ u ) i admits no invariant complex structure and dim W 1 = dim W 2 , we need only to consider the case in which U 0 = Spin(8), U 1 = U 2 = {1}, and W 1 , W 2 are two 8-dimensional inequivalent representations of Spin(8). Referring to [Oni04, Table 5, p.80], G R must be either Spin 0 (3, 5) or Spin 0 (1, 7), and ρ must be the realification of an 8-dimensional complex representation of G R 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), C 8 + ⊕ C 8 − ) (where C 8 ± denote the half-spin representations) with compact real form (Spin(8), R 8 + ⊕ R 8 − ) having the same orbits as (SO(8) × SO(8), R 8 ⊕ R 8 ), it follows that ρ has the same closed orbits as the standard action of (SO(8, C)) r on (C 8 ) 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 0 ⊕ ρ * 0 , where ρ 0 is absolutely irreducible and non-orthogonalizable. Then ρ c = ρ c 0 ⊕ (ρ * 0 ) c is polar and ρ c 0 is irreducible. By polarity, In other words, W admits a U-invariant complex structure J and ρ 0 is just a real form of the holomorphic extension of τ u : U → GL(W, J) to a representation of G on (W, J). By Lemma 9, it suffices to consider the case in which τ u is not an s-representation, namely, given in (7). We do only the case (SU(n), (Λ 2 C n ) r ) for n odd, the others being similar in spirit. Since n is odd, [Oni04, Table 5] gives that G R = SL(n, R) and ρ 0 is the representation on Λ 2 R n . Now ρ = ρ 0 ⊕ ρ * 0 has the same closed orbits as (GL + (n, R), Λ 2 R n ⊕ (Λ 2 R n ) * ), which turns out to be the s-representation of the pseudo-Riemannian symmetric space SO(n, n)/GL + (n, R) [Ber57, Tableau II].
(b.2.2) Suppose τ u is not splitting. Then it is not almost splitting by the same argument as in case (a.2.2). Owing to the fact that (τ u ) i admits an invariant complex structure for i = 1, 2, we see from [GT00,p.59] that this case is not possible.

Isoparametric submanifolds
Let V R be a finite-dimensional real vector space equipped with a non-degenerate symmetric bilinear form ·, · . A submanifold M of V R is called a pseudo-Riemannian submanifold if the restrictions of ·, · to the tangent spaces of M are always nondegenerate. If M is a pseudo-Riemannian submanifold, the canonical flat connection D in V R induces the Levi-Cività connection ∇ in M, the second fundamental form B of M, and the connection ∇ ⊥ in the normal bundle νM of M in the usual way. Namely, and D X ξ = −A ξ X + ∇ ⊥ X ξ, where X and Y are sections of T M and ξ is a section of νM, and the Weingarten operator For each p ∈ M, the map A ξ | p : T p M → T p M is a symmetric endomorphism with respect to the induced inner product in T p M. Note that in the case in which this induced inner product is positive-definite, the Weingarten operator is automatically diagonalizable over R, whereas in the general case it may happen that A ξ | p is not diagonalizable, not even over C.
A properly embedded pseudo-Riemannian submanifold M of V R will be herein called isoparametric if the following two conditions are satisfied: (a) the normal connection is flat; (b) the Weingarten operator along a locally defined parallel normal vector field is diagonalizable over C with constant eigenvalues. Isoparametric submanifolds of Euclidean spaces are very important in submanifold geometry and share a very rich history and an extensive literature, see [Ter85,Tho00,BCO03] and the references therein. On the other hand, isoparametric submanifolds of indefinite space forms are not as common, but have already been considered before with different definitions, see e.g. [Hah84,Mag85]. Herein we consider a stronger definition which in our opinion seems more natural in view of Theorems 1 and 11.
In this section, we will consider homogeneous isoparametric submanifolds. We start with the following lemma.
Lemma 10. Let τ : G → O(V, ·, · ) be a complex polar orthogonal representation of a complex reductive algebraic group.

and the equality holds if and only if v is regular.
(b) If c ⊂ V is a Cartan subspace, then c, g · c = 0. In particular, the restrictions of ·, · to c and g · v for regular v are nondegenerate.
by G-invariance of ·, · , proving the inclusion. If v is regular, and this shows that c v is the orthocomplement of g · v in V . (b) Follows from (a).
Before stating the next theorem, a couple of remarks are in order. Let G R be a connected real form of a connected complex reductive algebraic group G, let ρ : G R → GL(V R ) be an arbitrary real representation, and let τ : Suppose now that ρ is orthogonalizable; then so is τ , hence τ is stable; in this case, V pr consists of semisimple elements only, and it follows from this discussion that V pr ∩ V R is an open and dense subset of V R consisting of closed G R -orbits. Suppose now, in addition, that τ is polar. Since the slice representations of τ are the complexifications of the slice representations of the real form τ u : U → GL(W ) [Sch80, Cor. 5.9], it follows from [BCO03, Cor. 5.4.3] that V pr is precisely the set of regular points of τ .
Theorem 11. Let ρ : G R → O(V R , ·, · ) be an orthogonal representation. If ρ is polar then every orbit of ρ through a regular element v ∈ V R is isoparametric. Conversely, if ρ is irreducible and there exists a regular element v ∈ V R such that G R v is isoparametric then ρ is polar.
Proof. Suppose ρ is polar and v ∈ V R is regular. Then c v = (g · v) ⊥ is a Cartan subspace of τ = ρ c defined over R. Denote the set of real points of c v by (c v ) R and let M = G R v. Then the normal space ν v M = (c v ) R . Since g v · c v = 0 [DK85, Lem. 2.1(iii)], any ξ ∈ ν v M extends to a locally defined equivariant normal vector fieldξ along M given byξ(gv) = gξ for g ∈ (G R ) • (the connected component of the identity). For X ∈ g R , we have that ∇ ⊥ X·vξ is the orthogonal projection in ν v M of d dt t=0 (exp tX)ξ = X ·ξ ∈ g R ·ξ. Since g R ·ξ ⊂ g R ·v, it follows that ∇ ⊥ X·vξ = 0. This proves that a locally defined equivariant normal vector field along M is parallel. By taking a basis of ν v M, we get a locally defined parallel normal frame along ν v M, which implies that ν v M is flat. It is clear that the eigenvalues of the Weingarten operator along an equivariant normal vector field are constant, and that operator is diagonalizable over C by Example 12 below. Hence M is isoparametric.
Conversely, suppose ρ is irreducible and there exists a regular element v ∈ V R such that M = G R v is isoparametric. Irreducibility of ρ yields that M is full in V R , that is, not contained in a proper affine subspace. We first claim that a locally defined parallel normal vector fieldξ along M is equivariant. Let U be a neighborhood of v in M whereξ is defined, and letξ(v) = ξ. Suppose that g(t) is a continuous curve in G R satisfying g(0) = 1 and g(t)v ∈ U. Consider the continuous curve ξ(t) = g(t) −1ξ (g(t)v) in ν v M. By the isoparametric condition and the fact that the action of G R is orthogonal, we have that A ξ(t) and A ξ have the same complex eigenvalues. By connectedness of the domain interval of g(t) and the facts that they are diagonalizable over C and commute, we get that A ξ(t) = A ξ for all t. Fullness of M implies the injectivity of the map ξ → A ξ , so ξ(t) = ξ for all t. This proves the claim.
Since the locally defined equivariant normal vector fields are parallel with respect to the normal connection, where ξ ∈ ν v M and X ∈ g R . This proves that we get that dim c v = dim V //G and hence τ = ρ c (resp. ρ) is polar.
Example 12. Let τ : G → O(V, ·, · ) be a complex polar orthogonal representation and fix an orthogonal real form ρ : G R → O(V R , ·, · ) defined by (σ,σ). In this example, we compute the Weingarten operator of an orbit M = G R v for a regular v ∈ V R = Vσ. Let c be aθ-andσ-stable Cartan subspace of τ and consider the corresponding root space decomposition g = m + α∈Ag α (see subsection 5.2 for the notation and terminology used in this example). By Proposition 14 below, we may assume that v ∈ cσ. Let ξ be a vector normal to M at v in V R . Then also ξ ∈ cσ.
If α is a noncomplex root,g α is σ-stable. We have (the superscript "⊤" denotes the tangential component to the orbit) where X α ∈g σ α , and is a real eigenvalue and X α ·v α (resp. i(X α ·v α )) is the associated eigenvector if α is real (resp. imaginary).
If α is a complex root,g α is not σ-stable and (g α ⊕g |σα| ) σ is spanned by X α + σX α and i(X α − σX α ) for X α ∈g θ α . The associated subspace of T v M is spanned by is not real and the matrix of A ξ in the basis (9) is given by which is of course diagonalizable over C.

Structural theory of polar representations of real reductive algebraic groups
Consider a semisimple pseudo-Riemannian symmetric spaceĜ R /G R and its complexifi-cationĜ/G as in the first two paragraphs of section 3. Letσ denote the conjugation ofĜ overĜ R . We can choose a Cartan involutionθ ofĜ R that commutes withτ onĜ R . Sincê G is simply-connected, we can extendθ anti-holomorphically to a Cartan involution ofĜ which will be denoted by the same letter. Note thatθ commutes withτ andσ onĜ. Set θ (resp.θ) to be the restriction ofθ to G (resp. V ), and set σ (resp.σ) to be the restriction ofσ to G (resp. V ). ThenÛ =Ĝθ (resp. U = G θ ) is a compact real form ofĜ (resp. G). Write W = Vθ. Now we have the combined decomposition In this context, an element v ∈ V R is called semisimple if ad v is a semisimple endomorphism ofĝ, and a Cartan subspace ofĝ R is a maximal Abelian subspace of V R consisting of semisimple elements. It is known that the Ad G R -orbit of v ∈ V R is closed if and only if v is semisimple [BH62,Cor. 10.3]; every semisimple element of V R belongs to some Cartan subspace; every Cartan subspace of V R is Ad (G R ) • -conjugate to aθ-stable Cartan subspace; there exist finitely many Ad (G R ) • -conjugacy classes ofθ-stable Cartan subspaces in V R ; two suchθ-stable Cartan subspaces are Ad (K R ) • -conjugate if and only they are Ad (G R ) • -conjugate if and only they are Ad G -conjugate [HHNO99].
Throughout this section, we let τ : G → GL(V ) be a complex polar representation of a connected complex reductive algebraic group, consider a real form ρ : G R → GL(V R ) defined by (σ,σ), where G R is the identity component of G σ , and prove a collection of results for ρ similar to those stated in the previous paragraph for an s-representation. The first three results do not require that τ and ρ be orthogonalizable.

General facts about Cartan subspaces. A Cartan subspace of ρ is a subspace of
Vσ which is theσ-fixed point vector space of aσ-stable Cartan subspace of τ .
Proof. Owing to the remarks preceding Theorem 11, the set V pr ∩ V R is a nonempty open subset of V R = Vσ consisting of regular elements of τ ; it suffices to take c v where v lies therein.
We will use the following notion in the proof of the next proposition. The rank of τ is defined to be the difference dim c − dim c g , where c ⊂ V is a Cartan subspace and c g denotes the subspace of G-fixed points in c.
Proposition 14. Given a semisimple x ∈ Vσ, there exists a Cartan subspace of Vσ which contains x.
Proof. Note that for a regular x ∈ Vσ, one can simply take c = c x . In the general case, we proceed by induction on the rank of τ . Since x is semisimple, there exists a Cartan subspace c ′ such that x ∈ c ′ . If x ∈ (c ′ ) g , then x belongs to anyσ-stable Cartan subspace of τ . Suppose now x ∈ (c ′ ) g . Then the slice representation (G x , N x ) is polar with rank stricly lower than τ , and c ′ ⊂ N x is a Cartan subspace of (G x , N x ) [DK85, Thm. 2.4]. Without loss of generality, x is a minimal vector with respect to some U-invariant positive-definite Hermitian form (·, ·) which is real-valued on Vσ, and N x is the orthocomplement of g · x with respect to (·, ·) [DK85, Rmk. 1.4]. Since x ∈ Vσ, it follows that G x is σ-stable, N x is σ-stable and (G x , N x ) is defined over R with respect to (σ,σ). By the induction hypothesis, there exists aσ-stable Cartan subspace c ⊂ N x such that x ∈ c. Now c, c ′ are two Cartan subspaces of (G x , N x ), so there exists g ∈ G x such that g · c ′ = c. It follows that c is a Cartan subspace of τ .
Theorem 15. There exist only finitely many G R -conjugacy classes of Cartan subspaces of V R .
Proof. According to the remarks preceding Theorem 11, the set of regular points of τ is a Zariski-open and dense subset V pr of V . By a theorem of Whitney [Whi57], V pr ∩ V R has finitely many connected components.
Suppose now that cσ is a Cartan subspace of V R . Consider the map it is easily seen to be a smooth submersion at v if v is a regular point of τ . It follows that But the sets G R · (cσ ∩ V pr ) for varying cσ obviously cover V pr ∩ V R . Any two of them are not disjoint if and only if the corresponding Cartan subspaces are conjugate, in which case the sets coincide. The result follows.
Consider the categorical quotient map π : V → V //G. Since G, V , and the action of G on V are defined over R, so is the variety V //G; denote its set of real points by (V //G) R . By a theorem of Tarski and Seidenberg, π(V R ) is a real semialgebraic subset of (V //G) R . Recall that π(V pr ∩ V R ) is an open and dense subset of π(V R ). We propose the following conjecture (compare [Rot71]).
Conjecture 16. The map π sets up a one-to-one correspondence between the G R -conjugacy classes of Cartan subspaces of V R and the connected components of the stratum π(V pr ∩ V R ).
Henceforth we assume that ρ (and hence τ ) is orthogonal with respect to ·, · .
The following construction of η is standard (compare [Oni04, §3, Prop. 6]). Set ω = σµ. We can view ω as a complex linear automorphism of g. Consider the decomposition into the center and semisimple factor g = z ⊕ g ss . Let β be the Killing form of g ss . We can extend β to an ad-invariant symmetric bilinear form on g, denoted by the same letter, which is realvalued on g µ , g σ and negative-definite on g µ . Then one easily sees that ω is Hermitian with respect to the positive-definite Hermitian form B µ (X, Y ) = −β(X, µY ), where X, Y ∈ g. It follows that ω 2 is Hermitian and positive-definite, and hence belongs to a one-parameter family of Hermitian and positive-definite automorphisms of g. Therefore there exists a unique Hermitian, positive-definite automorphism ϕ of g such that ϕ 4 = ω 2 . Since ϕ| gss belongs to a one-parameter group of automorphisms of g, we have that ϕ| gss is inner, that is, equals Ad h for some h ∈ G ss . Set η = Inn h µ Inn −1 h . Then η is a Cartan involution of G. Also, on the Lie algebra level, µωµ = ω −1 , so µω 2 µ = ω −2 and µϕµ = ϕ −1 . Of course, ωω 2 ω −1 = ω 2 , so ωϕω −1 = ϕ and ωϕ 2 ω −1 = ϕ 2 . Now we have so ϕ 4 = ω 2 implies that ησ = ση on g, and also on G.
Since the irreducible summands of V must be pairwise inequivalent by polarity, each one of them isω-invariant. Let V 0 be an irreducible summand of V and suppose that the action of z on V 0 is given by a linear functional Λ : z → C. Equation (11) implies that Λ(X) = Λ(ωX) for X ∈ z. Now, if X ∈ z and v ∈ V 0 , we have and if X ∈ g ss , = dτ (ω 2 (X)).
5.2. Roots and co-roots. In the rest of the paper, we assume that ρ is orthogonal with respect to ·, · and a Cartan pair (θ,θ) commuting with (σ,σ) has been fixed according to Proposition 2. We also recall the Hermitian form (·, ·) that was introduced in that proposition and satisfies equation (1).
For a given Cartan subspace c ⊂ V , the set of singular elements c sing ⊂ c is by definition the complement of the set of regular elements in c. If the rank of τ is not zero, it is known that c sing is a union of finitely many complex hyperplanes where A is a finite index set [DK85,Lem. 2.11]. Fix aσ-andθ-stable Cartan subspace c ⊂ V , set g α to be the centralizer of c α in g and G α to be the corresponding connected subgroup of G.
Lemma 20. Each c α meets V ±θ in a real hyperplane.
Since ·, · is positive-definite on V −θ × V −θ , the vector v α in the proof of Lemma 20 can be chosen to satisfy and then it is uniquely defined up to a sign. We select a connected component of c −θ − ∪ α∈A c −θ α once and for all, and then v α is uniquely defined (but the sign of v α will not actually matter for our purposes). The vector v α is called a (unnormalized) co-root. The associated root is the linear functional α : c → C obtained by setting for v ∈ c −θ and then considering its complex-linear extension to c. A root is called: real (resp. imaginary) if α is real-valued (resp. purely imaginary-valued) on cσ, and it is called complex otherwise. It follows from the ·, · -orthogonality of the decomposition . It follows that α in noncomplex if and only if c α isσ-invariant if and only if it isω-invariant, whereω =σθ =θσ. Recall thatθ gets replaced by its opposite by changing the sign of ·, · , so the choice of some signs above does not have intrinsic meaning, as compared to the case of an s-representation in which the sign of ·, · is fixed by the Killing form ofĝ R (see (10)).
Let m be the centralizer of c in g. Then m is σ-, θ-stable. Since m is a reductive subalgebra of g α , there exists a θ-and ad m -stable splitting g α = m ⊕g α , whereg α is a subspace, which is called a root space. Now assume α is noncomplex. Theng α can be taken ω-stable, so thatg α =g ω α ⊕g −ω α . An imaginary root α ∈ A is called noncompact imaginary ifg −ω α = 0 and compact imaginary otherwise. A real root α ∈ A is called compact real ifg ω α = 0 and noncompact real otherwise. Finally, definẽ where v ∈ c. Sinceσ takes singular orbits to singular orbits and maps hyperplanes of c to hyperplanes of c, this defines an action on A ∪ (−A ). Also,σα = α (resp.σα = −α) if and only if α is real (resp. imaginary). We can choose the root spaces so that σg α =g |σα| for all α ∈ A , where | · | : A ∪ (−A ) → A has its obvious meaning.
5.3. Cayley transforms. By Corollary 18, every closed G R -orbit in V R meets some standard Cartan subspace of V R . We want to study standard Cartan subspaces of V R , so consider aσ-andθ-stable Cartan subspace c ⊂ V . Note that and dim R cσ ∩ cθ (resp. dim R cσ ∩ c −θ ) is an invariant of the G R -conjugacy class of cσ, called the compact dimension (resp. noncompact dimension) of cσ. We call a standard Cartan subspace cσ maximally compact (resp. maximally noncompact) if its compact dimension (resp. noncompact dimension) is as large as possible. Note that the compact and noncompact dimensions of cσ are interchanged if we replace ·, · andθ by their opposites. A maximally compact or maximally noncompact standard Cartan subspace will also be called extremal. Cayley transforms are used to pass from one G R -conjugacy class of Cartan subspaces to another one, namely, to increase or decrease its compact dimension by one (and correspondingly decrease or increase its noncompact dimension by one). In general, an element g ∈ G maps aσ-andθ-stable Cartan subspace c of V to anotherσ-andθ-stable Cartan subspace if and only if σ(g)g −1 and θ(g)g −1 belong to the normalizer N G (c) of c in G, as is easily seen. Recall that the Weyl group of c is the finite group [DK85, p. 513] where Z G (c) denotes the centralizer of c in G. We will construct a special kind of Cayley transform. We first consider the case of a rank one polar orthogonal irreducible representation τ : G → O(V, ·, · ). Fix a standard Cartan subspace c which is extremal, say maximally compact. Here cσ = cσ ∩ cθ and A = {α}. Assume that α is an imaginary root. We will show how one can pass from cσ to a Cartan subspaceĉ in another G R -conjugacy class which in this case, by dimensional reasons, must be maximally noncompact, namely,ĉσ =ĉσ ∩ĉ −θ .
Since the rank is one, τ u : U → O(W, ·, · ) is a co-homogeneity one action of a compact Lie group. Let v = iv α ∈ cσ ∩ cθ. Then v, v = −1 and U(v) is a round sphere S n−1 ≈ U/U v in W . Introduce the following notation: g = k + p is the decomposition into ±1-eigenspaces of ω, k R = k σ = k θ , p R = p σ , and K R = U σ = U ∩ G R ; note that K R is a maximal compact subgroup of G R and hence it is connected since G R is so.
Claim 1. We have that α is compact imaginary if and only if K R ⊂ U is transitive on S n−1 .
In fact, here we have g α = m ⊕g α where g α = g, m = g v , and g = g v ⊕g α is θ-stable. Taking θ-fixed points, we get u = u v ⊕g θ α . Now K R is transitive on S n−1 if and only if u v + k R = u if and only if k R ⊃g θ α if and only if k ⊃g α if and only ifg −ω α = {0}. Claim 2. If K R is not transitive on S n−1 , then we can take g ∈ G such that θ(g) = g, σ(g) = g −1 and g 2 = −id ∈ W (c).
Indeed, suppose now that the rank of τ is arbitrary, let c be an arbitraryσ-andθ-stable Cartan subspace and assume there exists a noncompact imaginary root α ∈ A which we suppose fixed. Write c = c α ⊕ Cv α where v α ∈ i(cσ ∩ cθ) = c −σ ∩ c −θ is the co-root. Note that cσ = cσ α ⊕ R(iv α ), and iv α ∈ cθ. Now (g α , c ⊕ g α · c) is a rank one polar action [DK85, Th. 2.12]. Since V = c ⊕ α∈A g α · c is a ·, · -orthogonal direct sum, (g α , c ⊕ g α · c) is orthogonal with respect to the restriction of ·, · ; we restrict it to (g α , Cv α ⊕g α · v α ) to get an irreducible polar orthogonal action of rank one. Since X ∈ g α → X · v α is injective ong α , the kernel of this representation is contained in m. Let Z ∈ m. Then Z · v α = 0. If Z ·g α · v α = 0, then [Z,g α ] · v α = 0. Since [Z,g α ] ⊂g α , we get that [Z,g α ] = 0, so Z ∈ Z m (g α ). Now (g ′ α , V α ) is an effective irreducible polar orthogonal action of rank one, where we have set g ′ α = g α /Z m (g α ) and V α = Cv α ⊕g α · v α . Note that α can also be considered as a root of (g ′ α , V α ), and then it is a noncompact imaginary root, so by the previous discussion we can find g ∈ G α as above and perform a Cayley transform c α = τ (g) as follows: c = c α (c) = c α ⊕ C(gv α ).
Note thatĉσ = cσ α ⊕ R(gv α ), and gv α ∈ c −θ , so the noncompact dimension ofĉσ is one higher than that of cσ. In a completely analogous way, one can define a Cayley transform that increases the compact dimension of cσ by one by using a compact real root.
5.4. Uniqueness of extremal Cartan subspaces. The Cayley transform allows us to derive some important properties of extremal Cartan subspaces.
Proof. It suffices to treat the case of (K R , V R ∩ iW ). Let cσ be a maximally noncompact Cartan subspace. Then there are no noncompact imaginary roots, for otherwise a Cayley transform could be performed increasing the noncompact dimension of cσ. We claim that there exists v 2 ∈ cσ ∩ c −θ such that In order to prove this claim, we first remark that [DK85, Thm.2.12] g v = m + α imaggα for generic v ∈ c −ω , m + α realgα for generic v ∈ cω, and where in the last line we have used the nonexistence of noncompact imaginary roots. Select generic v 1 ∈ cσ ∩ cθ, v 2 ∈ cσ ∩ c −θ and set v = v 1 + v 2 ∈ cσ. For each α ∈ A , where at least one of the two summands on the right hand-side in not zero by the choice of v 1 , v 2 . This shows that v is regular for (G, V ). By polarity, g · v ⊕ c = V . Taking real parts in V R yields g R (v) ⊕ cσ = V R , which is the same as In particular, (k R (v 2 ) + p R (v 1 )) ⊕ cσ ∩ c −θ = V R ∩ iW. The claim will follow if we show that k R (v 2 ) ⊃ p R (v 1 ). This is to be a consequence of k · v 2 ⊃ p · v 1 , as k · v 2 and p · v 1 areσ-stable and k R (v 2 ) = (k · v 2 )σ, p R (v 1 ) = (p · v 1 )σ. Now p · v 1 is spanned byg −ω α =0 ·v 1 for α imaginary, and (1 −ω)(X α · v α ) for α complex and X α ∈g α .
This proves that p·v 1 ⊂ k·v 2 , and hence that k R (v 2 )⊕cσ ∩ c −θ = V R ∩iW . Since g · c, c = 0, we get that cσ ∩ c −θ is the ·, · -orthogonal complement of K R (v 2 ) in V R ∩ iW . Since K R is compact and ·, · is positive-definite on V R ∩ iW , it easily follows that cσ ∩ c −θ meets all the K R -orbits in V R ∩ iW . Again by g · c, c = 0, one has that cσ ∩ c −θ meets all the other K R -orbits orthogonally. This finishes the proof.
Corollary 22. Two maximally noncompact (resp. compact) Cartan subspaces cσ 1 and cσ 2 of Vσ = V R are K R -conjugate. As a consequence, there exists a unique G R -conjugacy class of maximally noncompact (resp. compact) Cartan subspaces of V R .
Proof. Again, it suffices to treat the case of maximally noncompact Cartan subspaces. By Theorem 21, we may assume that cσ 1 ∩ c −θ 1 = cσ 2 ∩ c −θ 2 . Take a generic point v 2 lying therein. Since v 2 ∈ V R ∩iW , we have that u v 2 = (k R ) v 2 +(ip R ) v 2 , and this is a decomposition into the ±1-eigenspaces of σ on u v 2 , so