Classification of reductive real spherical pairs I. The simple case
Friedrich Knop, Bernhard Krötz, Tobias Pecher, Henrik Schlichtkrull
aa r X i v : . [ m a t h . R T ] M a y CLASSIFICATION OF REDUCTIVE REAL SPHERICAL PAIRSI. THE SIMPLE CASE
FRIEDRICH KNOP
FAU Erlangen-N¨urnberg, Department MathematikCauerstr. 11, D-91058 Erlangen, Germany
BERNHARD KR ¨OTZ
Universit¨at Paderborn, Institut f¨ur MathematikWarburger Straße 100, D-33098 Paderborn, Germany
TOBIAS PECHER
Universit¨at Paderborn, Institut f¨ur MathematikWarburger Straße 100, D-33098 Paderborn, Germany
HENRIK SCHLICHTKRULL
University of Copenhagen, Department of MathematicsUniversitetsparken 5, DK-2100 Copenhagen Ø, Denmark
Abstract.
This paper gives a classification of all pairs ( g , h ) with g a simple real Lie algebraand h ⊂ g a reductive subalgebra for which there exists a minimal parabolic subalgebra p ⊂ g such that g = h + p as vector sum. E-mail addresses : [email protected], [email protected], [email protected],[email protected] . Date : October 24, 2017.2000
Mathematics Subject Classification. . Introduction
Spherical pairs.
We recall that a pair ( g C , h C ) consisting of a complex reductive Liealgebra g C and a complex subalgebra h C thereof is called spherical provided there exists aBorel subalgebra b C ⊂ g C such that g C = h C + b C as a sum of vector spaces (not necessarilydirect). In particular, this is the case for symmetric pairs, that is, when h C consists of theelements fixed by an involution of g C .Complex spherical pairs with h C reductive were classified by Kr¨amer [26] for g C simpleand for g C semisimple by Brion [8] and Mikityuk [31].The objective of this paper is to obtain the appropriate real version of the classification ofKr¨amer. To be more precise, let g be a real reductive Lie algebra and h ⊂ g a subalgebra.We call h real spherical provided there exists a minimal parabolic subalgebra p ⊂ g such that g = h + p . Being in this situation we call ( g , h ) a real spherical pair . The pair is said to betrivial if h = g .We say that ( g , h ) is absolutely spherical if the complexified pair ( g C , h C ) is spherical. It iseasy to see (cf. Lemma 2.1) that then ( g , h ) is real spherical. In particular, all real symmetricpairs ( g , h ) are absolutely spherical, since the involution of g that defines h extends to aninvolution of g C . The real symmetric pairs were classified by Berger [4]. It is not difficult toclassify also the non-symmetric absolutely spherical pairs with h reductive; this is done inTable 8 at the end of the paper.1.2. Main result.
Assume that g is simple and non-compact. The main result of this paperis a classification of all reductive subalgebras of g which are real spherical. The followingTable 1 presents the most important outcome. It contains all the real spherical pairs whichare not absolutely spherical, up to isomorphism (and a few more, see Remark 1.2).Formally the classification is given in the following theorem, which refers to a number oftables in addition to Table 1. These tables are collected at the end of the paper, except forthe above-mentioned list of Berger. Theorem 1.1.
Let ( g , h ) be a non-trivial real spherical pair for which g is simple and h analgebraic and reductive subalgebra. Then at least one of the following statements holds:(i) g is compact,(ii) ( g , h ) is symmetric and listed by Berger (see [4, Tableaux II] ),(iii) ( g , h ) is absolutely spherical, but non-symmetric (see Tables 6, 7, and 8),(iv) ( g , h ) is isomorphic to some pair in Table 1.Conversely, all pairs mentioned in (i)–(iv) are real spherical. Remark 1.2.
1. We use Berger’s notation for the exceptional real Lie algebras. See Section 2.2.2. There is some overlap between ( iii ) and ( iv ), as it appeared more useful to include acouple of absolutely spherical cases in Table 1. This holds for case (1) which is absolutelyspherical unless p + q = p + q . Moreover, case (2) is absolutely spherical when q = n , case(8) is absolutely spherical when p + q is odd, and case (9) is absolutely spherical if q = n and f = u (1). h (1) su ( p + p , q + q ) ∗ su ( p , q ) + su ( p , q ) ( p , q ) = ( q , p )(2) su ( n, su ( n − q, sp ( q ) + f f ⊂ u (1) 1 ≤ q ≤ n (3) sl ( n, H ) sl ( n − , H ) + f f ⊂ C n ≥ sl ( n, H ) ∗ sl ( n, C ) n odd(5) sp ( p, q ) ∗ su ( p, q ) p = q (6) sp ( p, q ) ∗ sp ( p − , q ) p, q ≥ so (2 p, q ) ∗ su ( p, q ) p = q (8) so (2 p + 1 , q ) ∗ su ( p, q ) p = q − , q (9) so ( n, so ( n − q,
1) + su ( q ) + f f ⊂ u (1) 2 ≤ q ≤ n (10) so ( n, so ( n − q,
1) + sp ( q ) + f f ⊂ sp (1) 2 ≤ q ≤ n (11) so ( n, so ( n − ,
1) + spin (9) n ≥ so ( n, q ) so ( n − , q ) + G n ≥ , q = 1 , so ( n, q ) so ( n − , q ) + spin (7) n ≥ , q = 1 , , so (6 , so (2 ,
0) + G (15) so (7 , so (3 ,
0) + spin (4 , so ∗ (2 n ) ∗ so ∗ (2 n − n ≥ so ∗ (10) ∗ spin (6 ,
1) or ∗ spin (5 , E sl (3 , H ) + f f ⊂ u (1)(19) E ∗ E or ∗ E (20) F sp (2 ,
1) + f f ⊂ u (1)Table 13. The tables contain redundancies for small values of the parameters. These are mostlyresolved by restricting g to su ( p, q ) sl ( n, H ) sp ( p, q ) so ( p, q ) so ∗ (2 n ) p + q ≥ n ≥ p + q ≥ p + q ≥ n ≥ p ≥ q ≥ ∗ in front of h (with the exception of (2) and (9)with f = 0 and n = 2 q ). See Lemma 9.1.5. For simple Lie algebras g of split rank one the real spherical pairs were previouslydescribed in [16], and a more explicit classification was later given in [17].1.3. Method of proof.
Our starting point is the following theorem which we prove inSections 4–7, by making use of Dynkin’s classification of the maximal subalgebras in acomplex simple Lie algebra.
Theorem 1.3.
Let ( g , h ) be a real spherical pair for which g is simple and non-compact, and h is a maximal reductive subalgebra. Then ( g , h ) is absolutely spherical. Using Kr¨amer’s list [26] we then also obtain the following lemma.
Lemma 1.4.
Let g be a non-compact simple real Lie algebra without complex structure and h ( g be a maximal reductive subalgebra which is spherical. Then either h is a symmetricsubalgebra of g or a real form of sl (3 , C ) ⊂ G C or G C ⊂ so (7 , C ) . n order to complete the classification we use the following criterion, see Proposition 2.9and Corollary 2.10: If ( g , h ) is real spherical with h reductive and algebraic there exists aparabolic subalgebra q ⊃ p and a Levi decomposition q = l + u , such that for every reductiveand algebraic subalgebra h ′ ⊂ h , which is also spherical in g , one has(1.1) h = h ′ + ( l ∩ h ) . In other words (1.1) provides a factorization in the sense of Onishchik. It is not too hardto determine all l ∩ h for maximal h (see Tables 4 and 5). This allows us to conclude theclassification by means of Onishchik’s list [32] of factorizations of complex simple Lie algebras(see Proposition 2.5).1.4. Motivation.
This paper serves as the starting point for a follow up second part whichclassifies all real spherical reductive subalgebras of semisimple Lie algebras (see [23]). Withthese classifications one obtains an invaluable source of examples of real spherical pairs.Our main motivation for studying these pairs is that they provide a class of homogeneousspaces Z = G/H , which appears to be natural for the purpose of developing harmonicanalysis. Here G is a reductive Lie group and H a closed subgroup. The class includesthe reductive group G itself, when considered as a homogeneous space for the two-sidedaction. In this case the establishment of harmonic analysis is the fundamental achievementof Harish-Chandra [13]. More generally a theory of harmonic analysis has been developed forsymmetric spaces Z = G/H (see [10] and [3]). A common geometric property of these spacesis that the minimal parabolic subgroups of G have open orbits on Z , a feature which playsan important role in the cited works. This property of the pair ( G, H ) is equivalent thatthe pair of their Lie algebras is real spherical. Recent developments reveal that a furthergeneralization of harmonic analysis to real spherical spaces is feasible, see [28], [20], theoverview article [29], and [22].
Acknowledgment : It is our pleasure to thank the two referees for plenty of useful suggestions.They resulted in a significant improvement of the initially submitted manuscript.2.
Generalities
Real spherical pairs.
In the sequel g will always refer to a real reductive Lie algebraand h ⊂ g will be an algebraic subalgebra. The Lie algebra h is called real spherical providedthere exists a minimal parabolic subalgebra p such that g = h + p . The pair ( g , h ) is then referred to as a real spherical pair .Let θ be a Cartan involution of g , and let g = k + s denote the corresponding Cartan de-composition. Given a minimal parabolic subalgebra p we select a maximal abelian subspace a of s , which is contained in p , and write m for the centralizer of a in k . Then p = m + a + n ,where n is the unipotent radical of p . Moreover dim( g / p ) = dim n , and hence this gives usthe dimension bound for a real spherical subalgebra h ⊂ g :(2.1) dim h ≥ dim n = dim( g / k ) − rank R g . We note that dim( g / k ) and rank R g are both listed in Table V of [15, Ch. X, p. 518]. Furtherwe record the obvious but nevertheless sometimes useful rank inequality (2.2) rank R g ≥ rank R h . pair ( g , h ) of a complex Lie algebra and a complex subalgebra is called complex spherical or just spherical if it is real spherical when regarded as a pair of real Lie algebras. Note thatin this case the minimal parabolic subalgebras of g are precisely the Borel subalgebras.Given a pair ( g C , h C ) of a complex Lie algebra and a subalgebra, a real form of it is a pair( g , h ) of a real Lie algebra and a subalgebra such that g and h are real forms of g C and h C ,respectively. We recall from the introduction that the real form ( g , h ) is called absolutelyspherical when ( g C , h C ) is spherical. The following is easily observed (see [22, Lemma 2.1]). Lemma 2.1.
All absolutely spherical pairs ( g , h ) are real spherical. We recall also that a pair ( g , h ) is called symmetric in case there exists an involution of g for which h is the set of fixed elements, and that all such pairs are absolutely spherical.Conversely we have the following result. Lemma 2.2.
Let ( g , h ) be a real form of a complex symmetric pair ( g C , h C ) with g semisimple.Let σ be the involution of g C with fix point algebra h C . Then σ preserves g . In particular, ( g , h ) is symmetric.Proof. Let q ⊂ g be the orthogonal complement of h with respect to the Cartan-Killing formof g . Then q C is the orthogonal complement of h C in g C with respect to the Cartan-Killingform of g C . On the other hand q C is the − σ . The assertion follows. (cid:3) Fix g and let G C be a linear complex algebraic group with Lie algebra g C = g ⊗ R C . Wedenote by G the connected Lie subgroup of G C with Lie algebra g . For any Lie subalgebra l ⊂ g we denote by the corresponding upper case Latin letter L ⊂ G the associated connectedLie subgroup, unless it is indicated otherwise.Let P ⊂ G be a minimal parabolic subgroup. Then Z := G/H is called a real sphericalspace provided that ( g , h ) is real spherical, which means that there is an open P -orbit on Z .In the sequel we write P = M AN for the decomposition of P which corresponds to thepreviously introduced decomposition p = m + a + n of its Lie algebra, where the connectedgroups A and N are defined through the convention above, and the possibly non-connectedgroup M is defined as the centralizer of a in K ,2.2. Notation for classical and exceptional groups. If g C is classical, then G C will bethe corresponding classical group, i.e. G C = SL( n, C ) , SO( n, C ) , Sp( n, C ). To avoid confusionlet us stress that we use the notation Sp( n, R ), Sp( n, C ) to indicate that the underlyingclassical vector space is R n , C n . Further Sp( n ) denotes the compact real form of Sp( n, C )and likewise the underlying vector space for Sp( p, q ) is C p +2 q .By SL( n, H ) ⊂ SL(2 n, C ) and SO ∗ (2 n ) ⊂ SO(2 n, C ) we denote the subgroups of elements g which satisfy gJ = J ¯ g, J = (cid:18) I n − I n (cid:19) where I n denotes the identity matrix of size n . Another standard notation for SL( n, H ) isSU ∗ (2 n ).We denote by O( p, q ) the indefinite orthogonal group on R p + q . The identity componentof O( p, q ) is denoted by SO ( p, q ). or exceptional Lie algebras we use the notation of Berger, [4, p. 117], and write E C , E C etc. for the complex simple Lie algebras of type E , E etc., and E , E etc. for the corre-sponding compact real forms. For the non-compact real forms we write E , E , E , E for E I , E II , E III , E IV E , E , E for E V , E VI , E VII E , E for E VIII , E IX F , F for F I , F IIand finally G for G, the unique non-compact real form of G C . By slight abuse of notation wedenote the simply connected Lie groups with exceptional Lie algebras by the same symbols.2.3. Factorizations of reductive groups.
Let h be a reductive Lie algebra. Then a triple( h , h , h ) is called a factorization of h if h and h are reductive subalgebras of h and(2.3) h = h + h . It is called trivial if one of the factors equals h . Recall that a reductive subalgebra of h is asubalgebra for which ad h is completely reducible.Likewise if H is a connected reductive group and H and H are connected reductivesubgroups of H , then we call ( H, H , H ) a factorization of H provided that(2.4) H = H H . Proposition 2.3 (Onishchik [33]) . Let H be a connected reductive group and H , H reduc-tive subgroups of H . Then the following are equivalent:(i) ( h , h , h ) is a factorization of h .(ii) ( H, H , H ) is a factorization of H .(iii) H xH ⊂ H is open for some x ∈ H .Proof. We refer to [1, Prop. 4.4], for the equivalence of ( i ) and ( ii ). It is obvious that (2.4)implies H xH = H for all x , and hence in particular ( ii ) implies ( iii ).Assume ( iii ), then ( i ) is valid for the pair of h and Ad( x ) h . Hence ( ii ) holds for the pairof H and xH x − . This implies H = H xH and thus H × H acts transitively on H , thatis, ( ii ) holds for H , H . (cid:3) As a consequence we obtain the following result. Here we call a subalgebra of g compactif it generates a compact subgroup in the adjoint group of g . Lemma 2.4.
Let g be a semisimple Lie algebra without compact ideals. Then every factor-ization of g by a reductive and a compact subalgebra is trivial.Proof. Let g = h + h be as assumed. Since h is a reductive subalgebra there exists aCartan involution which leaves it invariant. Let g = k + s denote the corresponding Cartandecomposition, and note that k = [ s , s ] since g has no compact ideals. Without loss ofgenerality we may assume that h is a maximal compact subalgebra, hence conjugate to k .It then follows from Proposition 2.3 that g = h + k . Hence s ⊂ h . Then g = [ s , s ] + s = h and the factorization is trivial. (cid:3) Factorizations of simple complex Lie algebras were classified in [32] as follows. roposition 2.5. (Onishchik) Let g be a complex simple Lie algebra and let g = h + h where h and h are proper reductive complex subalgebras of g . Then, up to interchanging h and h , the triple ( g , h , h ) is isomorphic to a triple in Table 2, where line by line, z ⊂ C and f ⊂ sp (1 , C ) . g h h h ∩ h (1) sl (2 n, C ) sl (2 n − , C ) + z sp ( n, C ) sp ( n − , C ) + z n ≥ so (2 n, C ) so (2 n − , C ) sl ( n, C ) + z sl ( n − , C ) + z n ≥ so (4 n, C ) so (4 n − , C ) sp ( n, C ) + f sp ( n − , C ) + f n ≥ so (7 , C ) so (5 , C ) + z G C sl (2 , C ) + z (5) so (7 , C ) so (6 , C ) G C sl (3 , C )(6) so (8 , C ) so (5 , C ) + f spin (7 , C ) sl (2 , C ) + f (7) so (8 , C ) so (6 , C ) + z spin (7 , C ) sl (3 , C ) + z (8) so (8 , C ) so (7 , C ) spin (7 , C ) G C (9) so (8 , C ) spin (7 , C ) + spin (7 , C ) − G C (10) so (16 , C ) so (15 , C ) spin (9 , C ) spin (7 , C )Table 2 Remark 2.6. (i)
The spin representation embeds spin (7 , C ) into so (8 , C ) and there are two conjugacyclasses of this subalgebra. In Table 2 (9) the subscripts indicate that this factorizationinvolves both conjugacy classes. (ii) In all cases h is given up to conjugation in g . Once h is fixed, there is only oneAd( H )-conjugacy class of h in g for which the factorization is valid, except where h = spin (7 , C ) is indicated without subscript. In those cases there are exactly twosuch conjugacy classes, provided by spin (7 , C ) ± . (iii) Observe that symplectic or exceptional Lie algebras do not admit factorizations.2.4.
Towers of spherical subgroups.
Let Z = G/H be a real spherical space and P ⊂ G a minimal parabolic subgroup such that P H is open in G . Then the local structure theoremof [21] asserts that there is a parabolic subgroup Q ⊃ P with Levi decomposition Q = L ⋉ U such that: (i) P H = QH . (ii) Q ∩ H = L ∩ H . (iii) L n ⊂ L ∩ H .Here L n ⊂ L is the normal subgroup with Lie algebra l n , the sum of all non-compact simpleideals of l . We refer to Q and its Levi part L as being adapted to Z and P , taking it forgranted that P H is open.
Remark 2.7.
In the special case where Z is complex spherical note that l n = [ l , l ]. Lemma 2.8.
Let H ⊂ G be reductive and real spherical, and let Q = LU be adapted to G/H and P . Then L ∩ H is reductive and contains P ∩ H as a minimal parabolic subgroup.Proof. It follows from ( iii ) above that l n is a semisimple ideal in l ∩ h . As the quotientconsists of abelian or compact factors, l ∩ h is reductive. Since P ∩ L is a minimal parabolic ubgroup in L , it also follows from ( iii ) that P ∩ L ∩ H is a minimal parabolic subgroup in L ∩ H . Since P ⊂ Q it follows from ( ii ) that P ∩ H = P ∩ L ∩ H . (cid:3) Proposition 2.9.
Let H ′ ⊂ H ⊂ G be subgroups such that H is reductive and G/H is realspherical, and let Q = LU be adapted to G/H and P . Then G/H ′ is real spherical if andonly if H/H ′ is real spherical for the action of L ∩ H , that is, it admits an open orbit for theminimal parabolic subgroup P ∩ H (cf. Lemma 2.8).Proof. Assume
G/H ′ is real spherical. Then, by density of the union of the open orbits, forsome x ∈ G the set P xH ′ is open in G and intersects non-trivially with the open set P H .It follows that
P yH ′ is open in G for some y ∈ H . Then the intersection ( P yH ′ ) ∩ H =( P ∩ H ) yH ′ is open in H .Conversely, it is clear that if ( P yH ′ ) ∩ H is open in H for some y ∈ H , then P yH ′ is openin P H and hence in G . (cid:3) Corollary 2.10.
Let H ′ ⊂ H ⊂ G be reductive subgroups and let Q be as above. If Z ′ = G/H ′ is real spherical then ( H, H ′ , L ∩ H ) is a factorization of H , that is, (2.5) H = H ′ ( L ∩ H ) . Conversely, if Q = P then (2.5) implies that Z ′ is real spherical.Proof. It follows from Proposition 2.9 that ( L ∩ H ) xH ′ is open in H for some x ∈ H . Then(2.5) follows from Proposition 2.3. Conversely, (2.5) implies that H ′ Q = HQ , and hence Z ′ is spherical if Q = P . (cid:3) We recall from [19, Prop. 9.1] the following consistency relation of adapted parabolics.
Lemma 2.11.
Let Z = G/H be a real form of a complex spherical space Z C = G C /H C . Let P ⊂ G be a minimal parabolic subgroup and B C ⊂ G C a Borel subgroup such that B C ⊂ P C and B C H C ⊂ G C open. Let Q C ⊃ B C be the Z C -adapted parabolic subgroup of G C and Q ⊃ P the Z -adapted parabolic subgroup of G . Then Q C = Q C M C . Remark 2.12.
Suppose that ( g , h ) is absolutely spherical with h self-normalizing. Let H C be the normalizer of h C in G C . Note that H C is a self-normalizing spherical subgroup of G C .In view of [18, Cor. 7.2] this implies that Z C = G C /H C admits a wonderful compactificationand as such is endowed with a Luna diagram, see [7].The Luna diagram consists of the Dynkin diagram of g C with additional information. Inparticular the roots corresponding to the adapted Levi L C ⊂ Q C are the uncircled elementsin the Luna diagram where “uncircled” means no circle around, above, or below a vertex inthe underlying Dynkin diagram. Combining this information with the Satake diagram of g then gives us the structure of L via Lemma 2.11.In view of Remark 2.7 we have(2.6) [Lie( L C ) , Lie( L C )] ⊂ h C and in particular(2.7) [ l , l ] ⊂ h in case Q C = Q C . .5. The case where g is a quasi-split real form of g C . Recall that g is called quasi-splitif the complexification p C of p is a Borel subalgebra of g C . An equivalent way of saying thisis that m is abelian. The following is clear. Lemma 2.13.
Let ( g , h ) be a real form of ( g C , h C ) and assume that g is quasi-split. Then ( g , h ) is real spherical if and only if ( g C , h C ) is spherical. Two technical Lemmas.
We conclude this section with two lemmas which are re-peatedly used in the classification later on. The first is variant of Schur’s Lemma.
Lemma 2.14.
Let V be a finite dimensional complex vector space endowed with a non-degenerate Hermitian form b . Further let G ⊂ GL( V ) be a subgroup which acts irreduciblyon V and leaves b invariant. Then any other G -invariant Hermitian form b ′ on V is a realmultiple of b . In particular if b ′ = 0 , then b and b ′ have the same signature ( p, q ) up to order.Proof. Since b is non-degenerate we find a unique T ∈ End C ( V ) such that b ′ ( v, w ) = b ( T v, w )for all v, w ∈ V . The G -invariance of both b and b ′ and the uniqueness of T then impliesthat gT g − = T for all g ∈ G . Since G acts irreducibly on V , Schur’s Lemma implies that T = λ · id V for some λ ∈ C . Since both b and b ′ are Hermitian the scalar λ needs to bereal. (cid:3) Lemma 2.15.
Let X be a real algebraic variety acted upon by a real algebraic group H .Further let f , . . . , f k be H -invariant rational functions on X . Let U ⊂ X be their commonset of definition. Consider F : U → R k , x ( f ( x ) , . . . , f k ( x )) and assume that V := { x ∈ U | rank dF ( x ) ≥ k } 6 = ∅ . Then max x ∈ X dim R Hx ≤ dim X − k . Proof.
Note that V is by assumption Zariski open in X . Hence generic H -orbits of maximaldimension meet V . Since level sets in V under F have codimension k , the assertion follows. (cid:3) Functions f , . . . , f k as above which meet the requirement V = ∅ will in the sequel becalled independent .3. The Dynkin scheme of maximal reductive subgroups of classical groups
Let G C be a complex classical group and let V be the standard representation spaceattached to G C , i.e. V = C n for G C = SL( n, C ) or SO( n, C ), and V = C n for G C = Sp( n, C ).According to Dynkin [11], there are three possible types of a connected maximal complexreductive subgroup H C of G C .3.1. Type I: The action of H C on V is reducible. Up to conjugation H C is one of thefollowing subgroups, which are all symmetric:3.1.1. G C = SL( n, C ) . Here H C = S (GL( n , C ) × GL( n , C )), n = n + n , n i > .1.2. G C = SO( n, C ) . Either H C = SO( n , C ) × SO( n , C ) with n = n + n , n i > n is even and H C = GL( n/ , C ). In the first case, the defining bilinear form on G C restrictsnon-trivially to the factors C n = V = V + V = C n ⊕ C n . In the second case V = V ⊕ V ∗ for V the standard representation of GL( n/ , C ) and both factors V and V ∗ are isotropic.3.1.3. G C = Sp( n, C ) . Here H C = Sp( n , C ) × Sp( n , C ) with n = n + n , n i >
0, or H C = GL( n, C ). In the first case, the defining bilinear form on G C restricts non-trivially tothe factors C n = V = V + V = C n ⊕ C n . In the second case V = V ⊕ V ∗ for V thestandard representation of GL( n, C ) and both factors V and V ∗ are Lagrangian.3.2. Type II: The action of H C on V is irreducible, but h C is not simple. G C = SL( n, C ) . Here H C = SL( r, C ) ⊗ SL( s, C ) and C n = C r ⊗ C s with rs = n and2 ≤ r ≤ s .3.2.2. G C = SO( n, C ) . Here there are two possibilities. The first is H C = SO( r, C ) ⊗ SO( s, C )acting on C n = C r ⊗ C s with n = rs , 3 ≤ r ≤ s , and r, s = 4. The second case is H C = Sp( r, C ) ⊗ Sp( s, C ) acting on C n = C r ⊗ C s with n = 4 rs and 1 ≤ r ≤ s .3.2.3. G C = Sp( n, C ) . Here H C = Sp( r, C ) ⊗ SO( s, C ) and C n = C r ⊗ C s with n = rs and r ≥ s ≥
3. Moreover it is requested that s = 4 unless if r = 1.3.3. Type III: The action of H C on V is irreducible and h C is simple. For this typethe different cases are listed in [11, Thm. 1.5]. However, we do not need this list.3.4.
Dynkin types in G . Let H ⊂ G be a maximal connected reductive subgroup. Notethat this implies that h is a maximal reductive subalgebra in g . To begin with we recall thefollowing result: Proposition 3.1. (Komrakov [24], [25])
Let g be a real simple Lie algebra and h a maximalreductive subalgebra. If h C is not maximal reductive in g C , then the pair ( g , h ) appears in thefollowing list:(i) ( sp (4 n, R ) , so (1 , ⊕ sp ( n, R )) , n ≥ (ii) ( sp ( p + 3 q, p + q ) , so (1 , ⊕ sp ( p, q )) , p + q ≥ (iii) ( so ∗ (8 n ) , so (1 , ⊕ so ∗ (2 n )) , n ≥ (iv) ( so ( p + 3 q, q + 3 p ) , so (1 , ⊕ so ( p, q )) , p + q ≥ (v) ( so (6 , , so (1 , ⊕ so (1 , (vi) ( so (165 , , so (1 , (vii) ( so (234 , , so (3 , (viii) ( E , G C ⊕ su (2)) (ix) ( E , G C ⊕ su (1 , Remark 3.2.
The particular embeddings of h into g in Proposition 3.1 are described in [24].For this article the particular embeddings are not needed as only dim h enters in the proofof Corollary 3.3 below. Corollary 3.3.
Let g be a real simple Lie algebra and h a real spherical maximal reductivesubalgebra. Then h C is maximal reductive in g C . roof. Recall the dimension bound dim h ≥ dim n from (2.1). Now for ( i ) we note thatdim n = (4 n ) whereas dim h = 6 + 2 n + n . For ( ii ) we use the dimension bound (6.1),for ( iii ) the dimension bound (5.2), for ( iv )–( vii ) the dimension bound (5.1), and finally weexclude ( viii ) and ( ix ) via the dimension bound (7.5): dim n ( E ) ≥ dim n ( E ) = 108. (cid:3) Definition 3.4.
Let G be a real classical group. We say that a maximal connected reductivesubgroup H is of type I, II, or III , provided H C is maximal reductive and of that type in G C . Remark 3.5.
Suppose that V is the complexification of a real vector space V R and that H ⊂ G ⊂ GL( V R ) with H C ⊂ G C maximal reductive. Suppose that there exists a complexstructure J R on V R such that H is a complex subgroup of GL( V R , J R ). Then H is of type I.Indeed H C ≃ H × H and the action of H C on V ≃ ( V R , J R ) ⊕ ( V R , − J R ) is reducible.3.5. Bilinear forms on prehomogeneous vector spaces.
Let G be an algebraic groupover C and ρ be a finite-dimensional representation of G on a complex vector space V . Thetriplet ( G, ρ, V ) is called a prehomogeneous vector space , if G × GL(1 , C ) has a Zariski openorbit in V .Let now V be an irreducible representation of a reductive group G with center at mostone-dimensional, and let G ′ denote the semisimple part of G . A necessary condition for( G, ρ, V ) to be prehomogeneous is that G ′ satisfies(3.1) dim( G ′ ) ≥ dim( V ) − . Two triplets ( G , ρ , V ) and ( G , ρ , V ) are said to be equivalent if there is a linear iso-morphism ψ : V → V such that b ψ ( ρ ( G )) = ρ ( G ) under the induced map b ψ : GL( V ) → GL( V ).With respect to this notion, ( G, ρ, V ) and (
H, ρ ◦ τ, V ) are equivalent whenever τ : H → G is a surjective homomorphism. In particular, ( G, ρ, V ) is always equivalent to (
G, ρ ∗ , V ∗ )where ρ ∗ is dual to ρ . Proposition 3.6.
Let ( ρ, V ) be an irreducible representation of a simple group G . Thetriplet ( G, ρ, V ) satisfies (3.1) if and only if it is equivalent to a triplet listed in Table 3 andit gives rise to a prehomogeneous vector space if and only if it is marked in the column ‘preh’. The table identifies the representation ρ by its highest weight (expanded in fundamentalweights using the Bourbaki numbering [6, Ch. 6, Planches I–X]) and dimension. Proof.
The cases of Table 3 were determined in [2] and the fourth column follows fromTheorem 54 in [34]. (cid:3)
Table 3 divides into two parts: Each triplet listed in the first part represents a series ofvector spaces while a triplet in the second part is only defined for a certain dimension. Wecall a triplet (
G, ρ, V ) classical or sporadic depending on whether it is equivalent to a tripletof the former or the latter type.The final column of the table is marked by 0 if there is no non-degenerate G -invariantbilinear form on V , and by 1 (resp. 2) if there exists a non-degenerate symmetric (resp. skewsymmetric) G -invariant bilinear form. Given the highest weight ω , this data is easily deter-mined by means of [6, Ch. 8, § Remark 3.7.
Let G be a simple classical group acting on V as described in the beginningof this section, and let H be a subgroup of type III. If G is a real form of SO( n, C ), resp. ρ dim( V ) preh. inv. form1. G simple adjoint dim G
12. SL( n, C ), n ≥ ω n X
03. SL( n, C ), n ≥ ω n ( n + 1) X
04. SL( n, C ), n ≥ ω n ( n − X
05. Sp( n, C ), n ≥ ω n X
26. Sp( n, C ), n ≥ ω ( n − n + 1) 17. SO( n, C ), n ≥ n = 4 ω n X
18. SL(2 , C ) 3 ω X
29. SL(6 , C ) ω X , C ) ω X , C ) ω X , C ) ω X , C ) spin 8 X , C ) spin 16 X , C ) half spin 16 X , C ) spin 32 X , C ) half spin 32 X , C ) spin 64 219. Spin(14 , C ) half spin 64 X G C ω X F C ω
26 122. E C ω X E C ω X Sp( n, C ), then H C fixes a symmetric, resp. skew symmetric bilinear form on V . On the otherhand, if G is a real form of SL( n, C ) then H cannot be maximal if it fixes a nondegeneratebilinear form, unless H C is conjugate to SO( n, C ) or (if n is even) to Sp( n , C ).It will be a consequence of the dimension bound (2.1), that in most cases a subgroup H of type III comes from a triplet in Table 3. Hence the provided information about invariantforms reduces the number of cases which must be considered for the classification of thesesubgroups.4. Maximal reductive real spherical subgroups in case G C = SL( n, C )We prove the statement in Theorem 1.3 for g C = sl ( n, C ) and h C maximal reductive(cf. Corollary 3.3).4.1. The real forms.
It suffices to consider the non-split real forms G = SU( p, q ) with p + q = n and 1 ≤ p ≤ q , and G = SL( m, H ) with n = 2 m >
2. For these groups we obtainthe following dimension bounds from the table of [15] cited below (2.1):(4.1) dim H ≥ pq − p ( G = SU( p, q )) , (4.2) dim H ≥ m − m ( G = SL( m, H )) . For later reference we record the matrix realizations of G and P . We begin with G =SU( p, q ) which we consider as the invariance group of the Hermitian form ( · , · ) p,q defined by he symmetric matrix J = I p I p I q − p . The Lie algebra is then given by su ( p, q ) = X = A B EC − A ∗ F − F ∗ − E ∗ D (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A, B, C ∈ Mat p,p ( C ) ,E, F ∈ Mat p,q − p ( C ) ,D ∈ Mat q − p,q − p ( C ) ,B ∗ , C ∗ , D ∗ = − B, − C, − D, tr( X ) = 0 . We choose the minimal parabolic such that p = A B E − A ∗ − E ∗ D ∈ su ( p, q ) (cid:12)(cid:12)(cid:12)(cid:12) A upper triangular so that P stabilizes the isotropic flag h e i ⊂ h e , e i ⊂ . . . ⊂ h e , . . . , e p i . Moreover we recordthat G/P ≃ { V ⊂ V ⊂ . . . ⊂ V p | dim C V i = i, ( V p , V p ) p,q = { }} is the variety of full isotropic p -flags in C p + q . We denote by N p,q := { [ v ] ∈ P ( C n ) | ( v, v ) p,q = 0 } the null-cone and note that there is a G -equivariant surjective map G/P → N p,q . Moreover, if P max , C denotes the maximal parabolic subgroup of G C = SL( n, C ) which stabilizes h e i , then P C ⊂ P max , C and thus we have a G C -equivariant surjection G C /P C → G C /P max , C = P ( C n ).Thus we get: Lemma 4.1.
Let G = SU( p, q ) and H ⊂ G be a real spherical subgroup. Then the followingassertions hold:(i) There exists an H -orbit on C n of real codimension at most 3.(ii) There exists an open H C -orbit on P ( C n ) , i.e. C n is a prehomogeneous vector spacefor H C .Proof. The fact that H has an open orbit on G/P implies that there is an open H -orbit in N p,q , hence the first assertion. Secondly, the fact that H has an open orbit on G/P impliesthat H C has an open orbit on G C /P C whence on G/P max , C . (cid:3) For the group G = SL( m, H ) in G C = SL(2 m, C ) we choose P ⊂ G the upper triangularmatrices. Then G/P = { V ⊂ V ⊂ . . . ⊂ V m = H m | dim H V i = i } and in particular we obtain a G -equivariant surjection G/P → P ( H m ). Hence we get: Lemma 4.2.
Let H ⊂ SL( m, H ) be a real spherical subgroup. Then H has an open orbit on P ( H m ) and an orbit on H m = C m of real codimension at most . .2. Exclusions of sphericity via the codimension bound.
The criterion in Lemma4.1 (1) is quite useful to show that many naturally occurring subgroups are not spherical.We give an application in the lemma below.
Lemma 4.3.
Let p ≥ . Then SO ( p − , q ) is not spherical in SU( p, q ) .Proof. Write C p + q = C ⊕ C p + q − and decompose vectors v = v + v accordingly. Denoteby h· , ·i the complex symmetric bilinear form on C p + q − which defines SO ( p − , q ). Thefollowing four real valued functions are H -invariant function are independent: f ( v ) := Re v f ( v ) := Im v f ( v ) := Re h v , v i f ( v ) := Im h v , v i Hence each H -orbit on C p + q has real codimension at least 4 by Lemma 2.15, and hence H is not spherical by Lemma 4.1. (cid:3) Type I maximal subgroups.
Let H ⊂ G be a maximal subgroup of type I. Then H C = S (GL( n , C ) × GL( n , C )), n i >
0, is a symmetric subgroup of G C . In view of Lemma2.2 and Berger’s list [4] we thus obtain: Lemma 4.4.
The maximal connected subgroups of Type I for G = SU( p, q ) are given, up toconjugation, by the symmetric subgroups(i) S(U( p , q ) × U( p , q )) with p + p = p and q + q = q .(ii) GL(1 , R ) SL( p, C ) if q = p . Lemma 4.5.
The maximal connected subgroups of Type I for G = SL( n, H ) are given, upto conjugation, by the symmetric subgroups:(i) S(GL( n , H ) × GL( n , H )) with n + n = n .(ii) U(1) SL( n, C ) . Type II maximal subgroups.
In this case we have C n = C r ⊗ C s with 2 ≤ r, s and H C = SL( r, C ) ⊗ SL( s, C ). In particular, dim H = r + s − The case of G = SU( p, q ) . Lemma 4.6.
Let G = SU( p, q ) with q ≥ p ≥ . Then type II real spherical subgroups H ⊂ G occur only for p = q = 2 and are given, up to conjugation, by:(i) H = SU(2) ⊗ SU(1 , ,(ii) H = SU(1 , ⊗ SU(1 , .Both cases are symmetric.Proof. In case p = 1 all maximal reductive algebras which are real spherical are symmetricby [27]. This prevents in particular type II real spherical subgroups. Henceforth we thusassume that p ≥ , ≃ SO (2 ,
4) carries the subgroups (1) and (2) to SO (2 , × SO(3) and SO (1 , × SO (1 , p + q = rs and H C = SL( r, C ) ⊗ SL( s, C ). By Remark 3.5 we can exclude that H iscomplex, and hence we may assume that H = H ⊗ H with H , H real forms of SL( r, C ) and L( s, C ). We begin with the case where exactly one H i , say H is unitary: H = SU( p , q ).Let us first exclude the case where H = SL( s, R ). Note s ≥ , R ) ≃ SU(1 ,
1) isunitary. Then the maximal compact subgroup K := SO( s, R ) of H acts irreducibly on C s , and hence V = C r ⊗ C s is irreducible for the subgroup H ⊗ K . Now C s carries apositive definite K -invariant Hermitian form, and hence H ⊗ K leaves a Hermitian form ofsignature ( p s, q s ) invariant. According to Lemma 2.14 this form needs to be proportionalto the original form coming from G = SU( p, q ) with signature ( p, q ). It follows that the K -invariant form on C s then has to be invariant under H as well. This is impossible as H is not compact. Likewise we can argue when H = SL( k, H ) which has maximal compactsubgroup K = Sp( k ) acting irreducibly on C k . Similar to that we can argue with both H i either SL( · , R ) or SL( · , H ).Finally we need to turn to the case where H = SU( p , q ) and H = SU( p , q ), with r = p + q and s = p + q . We may assume that p ≤ q and p ≥ q . Then ( p − q )( p − q ) ≤ p p + q q ≤ p q + p q . We now exploit that H ⊗ H leaves invariant on C p + q = C r ⊗ C s both the defining formof SU( p, q ) and the tensor product of the defining forms of SU( p , q ) and SU( p , q ). Bycomparing signatures (cf. Lemma 2.14) we thus obtain( p, q ) = ( p p + q q , p q + p q ) . With that the dimension bound (4.1) reads r + s − ≥ p p + q q )( q p + q p ) − ( p p + q q )or(4.3) r + s − ≥ p q ( p + q ) + 2 p q ( p + q ) − ( p p + q q ) . Now we distinguish various cases.We first assume p , q , p , q are all non-zero. If they are all 1 then we are in case (2),hence we may assume q ≥ p ≥
2. By symmetry between r and s we can assume thelatter. With ( x + y ) ≤ x + y ) our bound (4.3) implies r + s − ≥ p q r + p q s − ( p p + q q )and hence, since p q ≥ p q ≥ r + s − ≥ r + s + 12 p q r − ( p p + q q ) . As r ≥ p q r ≥ p q r = p q ( p + q ) ≥ p p + q q and reach a contradiction.Hence we may assume now that p = 0 or q = 0. By symmetry between r and s we canassume the former, that is H = SU( r ) ⊗ SU( p , q )with p + q = s . The bound (4.3) now reads:(4.4) r + s − ≥ p q r − rq . f s = 2 then p = q = 1 and (4.4) gives r + 2 ≥ r − r , from which it follows that r = 2and we are in case (1).Hence we can assume s > p q ≥
2. As q ≤ s we obtain from (4.4) that r + s > r − rs . It easily follows that s > r .Now we use that H has an orbit of real codimension at most 3 on C r ⊗ C s (see Lemma 4.1).This implies that H has an orbit of real codimension at most 3 on C r ⊗ ( C s ) ∗ = Mat r,s ( C )with the action of H given as follows: ( h , h ) · X = h Xh − . Let Herm( r, C ) denote thespace of Hermitian matrices of size r , and for k, l > I k,l = (cid:18) I k − I l (cid:19) . The map(4.5) Φ : Mat r,s ( C ) → Herm( r, C ) , X XI p ,q X ∗ is submersive and satisfies Φ( h Xh − ) = h Φ( X ) h − for h ∈ SU( r ) and h ∈ SU( p , q ).Hence there must be an SU( r )-orbit on Herm( r, C ) of real codimension at most 3 and there-fore r ≤ H = SU( r ) ⊗ SU( p , q ) with r = 2 , s > r . Set H ′ := ⊗ SU( p , q ) ⊂ H . Then since H has an open orbit on G/P , H ′ must have an orbit of codimension at most r − G/P . We candescribe a flag
F ∈
G/P as follows F : h v i ⊂ h v , v i ⊂ . . . ⊂ h v , . . . , v p i such that { v , . . . , v p } is orthonormal with respect to the standard Hermitian scalar producton V = C n . Observe in addition that all v i are isotropic and mutually orthogonal withrespect to the form ( · , · ) p,q . It is important to note that F determines the v i uniquely up toscaling with U(1).Decompose V = C s ⊕ . . . ⊕ C s into H ′ -orthogonal summands where we have two or threesummands according to r = 2 or r = 3. This gives us r projections π j : C n → C s . Likewisefor every 1 ≤ m ≤ p the π j induce projections V m C n → V m C s which will be also denotedby π j . Further, the invariant form ( · , · ) p,q induces an invariant form on V m C n , denoted bythe same symbol.We define functions g mjk on G/P for 1 ≤ m ≤ p and 1 ≤ j, k ≤ r by g mjk ( F ) := ( π j ( v ∧ . . . ∧ v m ) , π k ( v ∧ . . . ∧ v m )) p,q . Note, that for fixed m , the rational functions f mjk := Re (cid:18) g mjk g m (cid:19) and f ′ mjk := Im (cid:18) g mjk g m (cid:19) are all H ′ -invariant. Already for m = 1, we obtain r − p ≥ n > m = 2 (itwill depend on the non-trivial v -coordinate of F ), which gives a contradiction by Lemma2.15. (cid:3) .4.2. The case of G = SL( m, H ) . Lemma 4.7.
Let G = SL( m, H ) with m ≥ . The only type II real spherical subgroup H ⊂ G occurs for m = 2 and is given, up to conjugation, by: H = SU(1 , ⊗ SU(2)
This is a symmetric subgroup.Proof.
For H C = SL( r, C ) ⊗ SL( s, C ) the dimension bound (4.2) reads r + s − ≥ m − m. The equation rs = 2 m together with r, s ≥ r + s ≤ m + 2 and hence r + s ≤ m + 4.Hence 2 m − m ≤ m + 2, which implies m = 2. Then r = s = 2. The local isomorphismSL(2 , H ) ≃ SO (1 ,
5) carries H = SU(1 , ⊗ SU(2) to SO (1 , × SO(3), which is symmetric.On the other hand, H = SU(1 , ⊗ SU(1 ,
1) is excluded by the rank inequality. (cid:3)
Type III maximal reductive subgroups.
Here H C is simple and acts irreduciblyon C n . In the following we denote by Sym( m, C ) and Skew( m, C ) the space of symmetric,respectively skew-symmetric, matrices of size m .4.5.1. The case of G = SU( p, q ) . Lemma 4.8.
Let p + q ≥ and let H ⊂ SU( p, q ) be a reductive real spherical subgroup oftype III. Then, up to conjugation, H is one of the following symmetric subgroups:(i) SO ( p, q ) .(ii) SO ∗ (2 p ) if p = q .(iii) Sp( p/ , q/ if p, q are even.(iv) Sp( p, R ) if p = q .Proof. According to [27], the assertion is true for p = 1 and henceforth we assume that q ≥ p ≥
2. By the dimension bound (4.1) we have for n = p + q (4.6) dim H ≥ pq − p = 2 p ( n − p ) − p ≥ n − , where the last inequality follows since 2 ≤ p ≤ n . We recall from Lemma 4.1 that V = C n is a prehomogeneous vector space for H C , and since V is irreducible and H C is simple, wecan apply Proposition 3.6 and Table 3 as explained in Remark 3.7. • H C = SL( m, C ) acting on V = Skew( m, C ), m ≥
5. Here n = m ( m −
1) and dim H = m −
1. Hence by (4.6) we obtain m − m − ≤ m ≥ • H C = SL( m, C ) acting on V = Sym( m, C ), m ≥
3. Here n = m ( m + 1) and we get m + 2 m − ≤
0, which is excluded with m ≥ • H C = SO( n, C ) acting on V = C n . This leads to (1) and (2). • H C = Sp( m, C ) acting on C n = C m . This leads to (3) and (4). • The sporadic prehomogeneous vector spaces.
Since we assume that p ≥
2, the dimensionbound gives no possibilities. (cid:3) .5.2. The case of G = SL( m, H ) . Lemma 4.9.
Let H ⊂ SL( m, H ) for m ≥ be a real spherical subgroup of type III. Then H is conjugate to one of the following symmetric subgroups:(i) SO ∗ (2 m ) (ii) Sp( p, q ) , p + q = m .Proof. Let V = C m = C n . By (4.2) a spherical subgroup H satisfiesdim H ≥ m − m > m = n = dim C V, as m >
2. Since H C acts via ρ irreducibly on V , it follows from Lemma 4.2 that the triplet( H C , ρ, V ) appears among the even-dimensional cases in Proposition 3.6. In particular wedo not have to consider the odd dimensional cases (10) and (22) from Table 3. Further, viaRemark 3.7, we can eliminate the cases (1), (6), (8), (9), (12) - (14), (16) - (18), (20), (21)and (23) from Table 3. Since H has to be proper, case (2) is excluded as well. This leavesus with the following possibilities: • H C = SL( k, C ) , acting on V = Skew( k, C ) with k ≥
5. The dimension bound for H reads(4.7) k − ≥ m − m. Since 2 m = k ( k −
1) and k ≥
5, we have m ≥ k ≥
5. Furthermore, k = 4 m + k and by(4.7) we get the contradiction4 m + k − ≥ m ( m − ≥ m. • H C = SL( k, C ) , acting on V = Sym( k, C ) with k ≥
3. Here 2 m = k ( k + 1). Since k ≥ m ≥ k ≥
3. Furthermore, k = 4 m − k and by (4.7) we get the contradiction4 m − k − ≥ m ( m − ≥ m. • H C = SO(2 m, C ) acting on V = C m . The real form H = SO ∗ (2 m ) gives case (1) of thelemma. The real form H = SO ( p, q ), p + q = 2 m cannot occur, since its maximal compactsubgroup SO( p ) × SO( q ) must be conjugate to a subgroup of K = Sp( m ) ⊂ SU( m, m ) fromwhich we conclude p = q = m . But then, rank R ( H ) = m > m − R ( G ). • H C = Sp( m, C ) acting on V = C m . The real form H = Sp( p, q ) with p + q = m gives case(2) of the lemma. The real form H = Sp( m, R ) does not occur, since its real rank equals m which is greater than rank R ( G ) = m − • H C = SL( k, C ) acting on V = V C k , k = 7 ,
8. It is easy to see that for k = 8 thedimension bound is violated, while for k = 7 the dimension of V is odd. • H C = Spin( k, C ) acting on a half spin representation, k = 10 ,
14. The representationspaces are C and C respectively. The dimension bound for H reads k ( k − ≥ m ( m − k ≥ m . (cid:3) This concludes the proof of Theorem 1.3 for G C = SL( n, C ). . Maximal reductive real spherical subgroups for the orthogonal groups
We prove the statement in Theorem 1.3 for g C = so ( n, C ), assuming n ≥ h C is maximal reductive (cf. Corollary 3.3).5.1. The real forms.
Let G C = SO( n, C ). Our focus is on the real forms G = SO ( p, q )with p + q = n and p ≤ q , and G = SO ∗ (2 m ) with n = 2 m .Note that so ∗ (6) ≃ su (1 ,
3) was already treated in Lemmas 4.4, 4.6, and 4.8. Furthermore, so ∗ (8) ≃ so (2 ,
6) will be treated below through the general case of SO ( p, q ). We may thusassume m ≥ ∗ (2 m ).The dimension bounds obtained from (2.1) and the cited table of [15] read:(5.1) dim H ≥ pq − p ( G = SO ( p, q )) , (5.2) dim H ≥ m − m ( G = SO ∗ (2 m )) . For further reference we record the matrix realizations of G and P . We begin with G =SO ( p, q ) which we consider as the invariance group of the symmetric form h· , ·i p,q definedby I p I p I q − p . Accordingly we obtain for the Lie algebra so ( p, q ) = A B EC − A T F − F T − E T D (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A, B, C ∈ Mat p,p ( R ) ,E, F ∈ Mat p,q − p ( R ) ,D ∈ Mat q − p,q − p ( C ) ,B T , C T , D T = − B, − C, − D . We choose the minimal parabolic such that p = A B E − A T − E T D ∈ so ( p, q ) | A upper triangular so that P stabilizes the isotropic real flag h e i ⊂ h e , e i ⊂ . . . ⊂ h e , . . . , e p i in R n . Moreover G/P ≃ { V ⊂ . . . ⊂ V p | V p ⊂ R n , dim R V i = i, h V p , V p i p,q = { }} is the variety of full isotropic p -flags in R p + q with respect to the symmetric bilinear form h· , ·i p,q .Let us denote N R p,q := { [ v ] ∈ P ( R n ) | h v, v i p,q = 0 } and note that there is a G -equivariant surjective map G/P → N R p,q . Hence we obtain thefollowing lemma. Lemma 5.1.
Let G = SO ( p, q ) and H ⊂ G a real spherical subgroup. Then there exists an H -orbit on R n of codimension at most 2.Proof. The fact that H has an open orbit on G/P implies that there is an open H -orbit in N R p,q . (cid:3) e continue to recall a few structural facts for the group SO ∗ (2 m ). We identify H m with C m via H m = C m ⊕ j C m . Denote by h h the conjugation on H m . The group SO ∗ (2 m )consists of the right H -linear transformations on H m which preserve the H -valued form φ ( h, h ′ ) = h jh ′ + . . . + h m jh ′ m ( h i , h ′ i ∈ H ) . Observe that φ is a so-called skew-Hermitian form, i.e. it is sesquilinear and skew. Werecall that sesquilinear means φ ( hx, h ′ x ′ ) = xφ ( h, h ′ ) x ′ for all h, h ′ ∈ H m , x, x ′ ∈ H , and skew refers to φ ( h, h ′ ) = − φ ( h ′ , h ).Denote the C -part of φ ( h, h ′ ) by ( h, h ′ ) m,m ∈ C and the j C -part by h h, h ′ i ∈ C . If wewrite elements h ∈ H m as h = x + jy with x, y ∈ C m , then( h, h ′ ) m,m = y T x ′ − x T y ′ and h h, h ′ i = x T x ′ + y T y ′ . Notice that i ( · , · ) m,m is a Hermitian form of signature ( m, m ). In particular, if we viewSO ∗ (2 m ) as a subgroup of SL(2 m, C ), then SO ∗ (2 m ) = SO(2 m, C ) ∩ SU( m, m ).The minimal flag variety is given by isotropic right H -flags(5.3) G/P = { V ⊂ . . . ⊂ V [ m/ ⊂ H m | φ ( V i , V i ) = { } , dim H V i = i } . Remark 5.2.
Observe that the sesquilinear form φ is uniquely determined by its C -partor j C -part. Hence an H -subspace V i ⊂ H m is isotropic if and only if it is isotropic for h· , ·i (or ( · , · ) m,m ). Recall that G C /B C is the variety of h· , ·i -isotropic (left) C -flags in C m = H m .Hence the right hand side of (5.3) embeds totally real into the quotient of G C /B C consistingof even-dimensional isotropic complex flags. A simple dimension count then shows equalityin (5.3).5.2. Type I maximal subgroups.
Let H ⊂ G be a maximal subgroup of type I. Then H C = SO( n , C ) × SO( n , C ), n i > n = n + n , or H C = GL( n/ , C ) for n even. Inboth cases H C is a symmetric subgroup of G C . Hence with Lemma 2.2 and Berger’s list [4]we obtain: Lemma 5.3.
Let H ⊂ SO ∗ (2 m ) be a subgroup of type I. Then H is symmetric, and up toconjugation it equals one of the following groups:(i) SO ∗ (2 m ) × SO ∗ (2 m ) with m + m = m , m , m > ,(ii) SO( m, C ) ,(iii) GL( m/ , H ) for m even,(iv) U( k, l ) with k + l = m . Lemma 5.4.
Let H ⊂ SO ( p, q ) be a subgroup of type I. Then H is symmetric, and up toconjugation it equals one of the following groups:(i) SO ( p , q ) × SO ( p , q ) with p + p = p and q + q = q ,(ii) SO( p, C ) for p = q ,(iii) GL( p, R ) with p = q .(iv) U( p/ , q/ for p, q even. .3. Type II maximal reductive subgroups.
Here we suppose that H C is a maximalreductive subgroup of G C = SO( n, C ), n ≥ • H C = SO( r, C ) ⊗ SO( s, C ) with rs = n , 3 ≤ r ≤ s , and r, s = 4. • H C = Sp( r, C ) ⊗ Sp( s, C ) with 4 rs = n and 1 ≤ r ≤ s .5.3.1. The case of G = SO ∗ (2 m ) . Lemma 5.5.
Let G = SO ∗ (2 m ) for m ≥ . Then there exist no real spherical subgroups H ⊂ G of type II.Proof. When m ≥ (cid:3) The case of G = SO ( p, q ) . Lemma 5.6.
Let G = SO ( p, q ) with p + q ≥ . Then type II real spherical subgroups H ⊂ G occur only for p = q = 4 and are given, up to conjugation, by:(i) H = Sp(1 , R ) ⊗ Sp(2 , R ) ,(ii) H = Sp(1) ⊗ Sp(1 , .Both cases are symmetric.Proof. We first prove that the groups listed under (1) and (2) are symmetric and hence realspherical. Write H = H ⊗ H and C = C ⊗ C . The symplectic forms Ω i on C i defined by H i give rise to the SO (4 , h· , ·i = Ω ⊗ Ω on C . Write J and J for the matrices defining Ω and Ω . Then B = Ω ( J · , · ) ⊗ Ω ( J · , · ) defines a symmetricbilinear form on C and we write g g t for the corresponding transpose on matrices. Thenthe assignment g ( J ⊗ J ) g − t ( J ⊗ J ) defines an involution on G = SO (4 ,
4) with fixedgroup H . Hence H is symmetric (and outer isomorphic to SO (2 , × SO (2 , × SO (1 , H = H ⊗ H be a type II subgroup. We consider first the case where each H i C issymplectic. We start with H = Sp( r, R ) ⊗ Sp( s, R ). The invariant Hermitian form on eachfactor gives an invariant Hermitian form on the tensor product with signature (2 rs, rs )which then must be equal to ( p, q ). The dimension bound (5.1) becomes r (2 r + 1) + s (2 s + 1) ≥ r s − rs. For r ≥ s ≥ r (2 r + 1) ≤ r ≤ r s and s (2 s + 1) ≤ r s . It follows that4 r s − rs ≤ r s which easily implies rs <
3. Since 4 rs = n ≥ r = 1and s = 2. These data produce the first symmetric subgroup mentioned in the lemma.For H = Sp( r, R ) ⊗ Sp( p , q ) we obtain the same signature condition p = q = 2 rs asbefore and hence H = Sp(1 , R ) ⊗ Sp(1 , G C , which can be excluded with Berger’s list for G = SO (4 , H = Sp( p , q ) ⊗ Sp( p , q ) with r = p + q and s = p + q is treatedanalogously as Lemma 4.6. We can assume p ≤ q and p ≥ q . The group H leavesinvariant a Hermitian form of signature (4( p p + q q ) , p q + p q )), which must thenequal ( p, q ) by Lemma 2.14. Then the dimension bound r (2 r + 1) + s (2 s + 1) ≥ p q ( p + q ) + 16 p q ( p + q ) − p p + q q )leads to the absurd unless p = 0 and H = Sp( r ) ⊗ Sp( p , q ) with r ≤ s . Using a matrixsubmersion as (4.5) we obtain with Lemma 5.1 that r = 1, hence H = Sp(1) ⊗ Sp( p , q )and G = SO (4 p , q ). Set H ′ := Sp( p , q ) ⊂ H . Then H ′ is of codimension 3 in H and hus H ′ admits an orbit of codimension 3 on G/P . We parameterize flags
F ∈
G/P as inLemma 4.6. Let V = R p +4 q ≃ C p +2 q . First we note that there are three independentreal symplectic forms which are invariant under H ′ . In fact, if Ω is the complex symplecticform on C p +2 q which defines H ′ , then Ω = Re Ω, and Ω = Im Ω give two independentsymplectic forms. A third form is given by Ω = Im( · , · ) p , q . Concretely, the Ω i are givenas follows: Out of the standard symplectic forms J i on R J = − − J = − − J = − − we build the forms J i = diag( J i ) ∈ Mat p + q ( R ). Then Ω i ( · , · ) = h J i · , ·i p,q .This gives us two independent rational invariants f i ( F ) := Ω i ( v ∧ v )Ω ( v ∧ v ) ( i = 1 , . Further invariants we obtain via g j,k ( F ) = (Ω j ∧ Ω k )( v ∧ v ∧ v ∧ v )(Ω ∧ Ω )( v ∧ v ∧ v ∧ v ) (1 ≤ j ≤ k ≤ , ( j, k ) = (3 , . Clearly each g j,k is independent to { f , f } as the f i only depend on the first two coordinates v , v . Moreover, if p > V V . Thus for p > H ′ -invariants on G/P contradicting the fact that the generic H ′ -orbit is of codimension at most 3 (cf. Lemma2.15). This leaves us to investigate the case with p = 1. Now if q = 1, the g j,k are alldependent and H is the second symmetric subgroup mentioned in the lemma. If q > f i , g jk . To verify thatwe may restrict ourselves to the case q = 2. We fix the first two coordinates of F to be v = e , v = e . For λ, µ, ν, ǫ, δ ∈ R we consider˜ v = e + λe + µe + νe + ǫe ˜ v = e − µe + δe . Then { v , v , ˜ v , ˜ v } is a set of mutually orthogonal vectors with respect to h· , ·i p,q . Moreover˜ v , resp. ˜ v , is isotropic provided that 2( µ + λν ) + ǫ = 0, resp. − µ + δ = 0. We choose nowthe parameters such that both ˜ v and ˜ v are isotropic. Let v , v be unit vectors obtainedfrom ˜ v , ˜ v .This then gives us isotropic flags F = {h e i ⊂ h e , e i ⊂ h e , e , v i ⊂ h e , e , v , v i} . Then g ( F ) = Ω ( v , v )Ω ( v , v )Ω ( v , v )Ω ( v , v ) and g ( F ) = Ω ( v , v )Ω ( v , v )Ω ( v , v )Ω ( v , v )and in particular g ( F ) = λµµλ − ν + ǫδ and g ( F ) = − νµλ − ν + ǫδ . It follows that { f , f , g , g } are independent and hence H is not real spherical for q > ext we look at the case where H = H ⊗ H with both complexifications orthogonal.We begin with both H i = SO ∗ (2 m i ) quaternionic, and m i >
2. The invariant Hermitianform on each factor gives an invariant Hermitian form on the tensor product with signature(2 m m , m m ) which then must be equal to ( p, q ) by Lemma 2.14. Then the dimensionbound 2 m − m + 2 m − m ≥ m m − m m is easily seen to be violated. Similarly if H = SO ∗ (2 m ) and H = SO ( p , q ) with p + q = s , then p = q = m ( p + q ) by a signature argument, and exactly the same bound as aboveresults.This reduces to the final case where H = SO ( p , q ) ⊗ SO ( p , q ), which is treatedsimilarly as the previous case of H = Sp( p , q ) ⊗ Sp( p , q ). Comparing signatures wefind that p = p p + q q and q = p q + p q , and the dimension bound then implies H = SO( r ) ⊗ SO ( p , q ) with r ≤ s . By applying a matrix submersion as (4.5) we obtainwith Lemma 5.1 that H must have an orbit on Sym( r, R ) of codimension 2. This contradictsthat r ≥ (cid:3) Type III maximal subgroups.
We assume that H C is simple and acts irreduciblyon V .5.4.1. The case G = SO ( p, q ) . Lemma 5.7.
Let G = SO ( p, q ) for ≤ p ≤ q and p + q ≥ . Then the only real sphericalsubgroups H ⊂ G of type III are given, up to isomorphism, by(i) h = G in g = so (3 , .(ii) h = spin (3 , in g = so (4 , (two conjugacy classes swapped by an outer automor-phism of g ).These pairs are absolutely spherical and the second one is symmetric. Note that although the pair of Lie algebras ( g , h ) in ( ii ) is symmetric, this is not the casefor the space G/H , since the corresponding involution does not lift to G . Nevertheless G/H is real spherical since the existence of an open P -orbit is a property of the Lie algebras. Proof.
For p = 1 it follows from [27] that there are no such spherical subgroups. The case so (2 ,
3) is quasi-split and features no type III subalgebras according to Kr¨amer. Hence wemay assume here 2 ≤ p ≤ q , q ≥ p + q >
5. Then( pq − p ) − ( p + q ) = ( p − q −
3) + p + q − ≥ . Hence the dimension bound (5.1) implies(5.4) dim H ≥ pq − p ≥ p + q = dim V. In particular, we can apply Proposition 3.6 and Remark 3.7. We observe also that pq − p >p + q if p + q > • H C , adjoint representation. Then dim H = dim V , which is excluded unless dim H = 6,by the strictness of (5.4). Then H = SO (3 ,
1) and H C is not simple. • H C = SO( m, C ) acting on V = C m . This is possible, but then we would have H C = G C . H C = Sp( m, C ) acting on V C m for m ≥
3. Here n = 2 m − m − p + q . Thedimension bound (5.1) then gives 2 m + m ≥ pq − p = p (2 m − m − − p ). Already for p = 2 this implies m ≤
2, and hence there are no solutions. • H C = Spin( m, C ) acting on a spin representation, m = 7 ,
9. Since the representationspaces are C and C respectively, the dimension bound leaves the following possibilities: • G = SO (2 , (3 ,
5) or SO (4 ,
4) if m = 7, • G = SO (2 ,
14) or SO (3 ,
13) if m = 9.It follows from the signature laws of the spin representations [14, Theorems 13.1 and 13.8]that only symmetric signatures (i.e. of the form ( p, p )) can occur. Hence only G = SO (4 , h = spin (3 ,
4) := spin (7 , C ) ∩ so (4 , spin (7 , C ) is symmetric in so (8 , C ). • H C of exceptional type. The case H C = G C is possible; with h = G , the pair ( so (3 , , h )is absolutely spherical (see Table 8). In view of Table 3 we are left with H C = F C . Thendim V = 26 and the dimension bound implies that G = SO (2 , H C with rank ≤ F . Its representation space V = { X ∈ Herm(3 , O ) C : T rX = 0 } . According to (2.2) in [9], the space V carries an invariant symmetric bilinear form withsignature (10 , , (cid:3) The case G = SO ∗ (2 m ) . Lemma 5.8.
Let G = SO ∗ (2 m ) for m ≥ . Then there exists no real spherical subgroup H ⊂ G of type III.Proof. By assumption m ≥ m − m = m · (2 m − > m = dim( V ) . Hence, if H is spherical it follows from (5.2) that dim H > dim V . In particular, V is thena representation from Proposition 3.6, to which also Remark 3.7 applies. • H C simple, adjoint representation. Since dim H = dim V , this is impossible by thestrictness of the dimension bound. • H C = SO( k, C ) acting on V = C k . This is possible for k = 2 m , but then we would have H C = G C . • H C = Spin( k, C ) acting on a spin representation for k = 7 ,
9. Note that k = 7 is excludedsince m ≥
5. For Spin(9 , C ) on C we have m = 8 and dim H = 36 <
52, so H does notsatisfy the dimension bound. • H C = Sp( k, C ) acting on V C k . Here 2 m = 2 k − k −
1. Since k ≥
3, we have m ≥ H = 2 k + k = 2 m + 2 k + 1 ≥ m ( m −
32 )and k ≤ m that 4 m + 1 ≥ m , which is impossible. H C of exceptional type. Only for F C is dim V even and not excluded by Table 3. Then m = 13 and (5.2) is invalid. (cid:3) This concludes the proof of Theorem 1.3 for G C = SO( n, C ).6. Maximal reductive real spherical subgroups for the symplectic groups
We only consider the real forms G = Sp( p, q ) of G C = Sp( n, C ), p + q = n and 0 < p ≤ q ,as the real form Sp( n, R ) is split. Then dim( G/K ) = 4 pq and rank R G = p , so that by (2.1)(6.1) dim H ≥ pq − p. About
Sp( p, q ) . For later reference we record some structural facts for the groupSp( p, q ). As before we identify H n with C n and denote by h h the conjugation on H n . The group Sp( p, q ) consists of the right H -linear transformations on H n which preservethe Hermitian H -valued form φ ( h, h ′ ) = h h ′ + . . . + h p h ′ p − h p +1 h ′ p +1 − . . . − h n h ′ n . Similar to the SO ∗ (2 m )-case the C -part of φ yields a Hermitian form ( · , · ) p, q and the j C -part a symplectic form h· , ·i , both being kept invariant under Sp( p, q ). In particular, ifwe view Sp( p, q ) as a subgroup of SL(2 n, C ), then Sp( p, q ) = Sp( n, C ) ∩ SU(2 p, q ).The minimal flag variety is given by the isotropic right H -flags:(6.2) G/P = { V ⊂ . . . ⊂ V p ⊂ H n | dim H V i = i, φ ( V i , V i ) = { }} . Lemma 6.1.
Let H ⊂ Sp( p, q ) be a real spherical subgroup. Then H admits an orbit on P ( H n ) of real codimension at most and an orbit on H n = C p +2 q of real codimension atmost .Proof. Let L be the variety of φ -isotropic H -lines. According to (6.2) L is a G -quotient of G/P and hence a real spherical subgroup H ⊂ G must admit an open orbit on L . Observethat a line v H is isotropic if and only if the real valued function v φ ( v, v ) vanishes. Fromthat the assertion follows. (cid:3) Type I maximal subgroups.
Let H ⊂ G be a maximal subgroup of type I. Then H C = Sp( r, C ) × Sp( s, C )), r, s > n = r + s or H C = GL( n, C ). In both cases H C is asymmetric subgroup of G C . Hence with Lemma 2.2 and Berger’s list [4] we obtain: Lemma 6.2.
Let H ⊂ Sp( p, q ) be a subgroup of type I. Then H is symmetric and up toconjugation one of the following:(i) Sp( p , q ) × Sp( p , q ) with p + p = p and q + q = q ,(ii) Sp( p, C ) if p = q ,(iii) U( p, q ) ,(iv) GL( p, H ) for p = q . Type II maximal subgroups.
In this situation we have H C = Sp( r, C ) ⊗ SO( s, C )with s ≥ s = 4 or ( r, s ) = (1 , Lemma 6.3.
There are no real spherical subgroups H ⊂ Sp( p, q ) of type II. roof. We first claim that the real forms H = Sp( r, R ) ⊗ SO ( p , q ) with p + q = s and H = Sp( r, R ) ⊗ SO ∗ (2 m ) with 2 m = s are not possible. Note that Sp( r, R ) ⊂ SU( r, r ),SO ( p , q ) ⊂ SU( p , q ), SO ∗ (2 m ) ⊂ SU( m, m ). It follows that both H and H leave aHermitian form on C n = C r ⊗ C s invariant which is of type ( rs, rs ) and hence p = q = rs by Lemma 2.14. The dimension bound (6.1) then gives2 r + r + 12 s ( s − ≥ p − p with rs = 2 p . This has no solutions since for r ≥ s ≥ r + r + 12 s ≤ r s + 19 r s + 12 r s = 56 r s ≤ r s − rs and thus neither H nor H can be spherical.The case where H = Sp( p , q ) ⊗ SO ∗ (2 m ) is similar, as H leaves a Hermitian form ofequal parity invariant.This leaves us with the last case where H = Sp( p , q ) ⊗ SO ( p , q ). It requires a moredetailed investigation. We request p ≤ q and q ≤ p . Then p p + q q ≤ p q + p q and(2 p p + 2 q q , p q + 2 p q ) = (2 p, q )as H leaves invariant a Hermitian form of this signature (cf. Lemma 2.14). Inserting thatin the dimension bound (6.1) gives2 r + r + 12 s ( s − ≥ p q ( p + q ) + 4 p q ( p + q ) − ( p p + q q ) . As in (4.3) we deduce that one factor must be compact. Suppose first that q = 0 hence H = Sp( p , q ) ⊗ SO( s ) with s = p . The dimension bound in this case is:2 r + r + 12 s ( s − ≥ q p s − p s . There are no solutions for 2 r ≤ s . For s ≤ r , a matrix computation (use an analogue ofthe map (4.5)) combined with Lemma 6.1 yields that SO( s ) needs to have an orbit of realcodimension at most 5 on Herm( s, C ). The orbits of maximal dimension are in Sym( s, R ),and they have codimension s in this space, hence s ( s + 1) in Herm( s, C ). It follows that no s ≥ p = 0. Then r = q and H = Sp( r ) ⊗ SO ( p , q )and the dimension inequality becomes:2 r + r + 12 s ( s − ≥ q p r − p r There is no solution if 3 ≤ s ≤ r so we may assume that s > r . With the matrixcomputations similar to the SU( p, q )-case (see (4.5)) combined with Lemma 6.1 this reducesmatters to study Sp( r )-orbits on Herm(2 r, C ) with codimension at most 5. This implies r = 1, i.e. H = Sp(1) ⊗ SO ( p, q ) with p + q ≥
3. We now proceed as in Lemma 4.6:Consider H ′ := ⊗ SO ( p, q ) ⊂ H . Then H ′ is required to have an orbit on G/P ofcodimension at most 3. We now produce many H ′ -invariant functions on G/P . First wedecompose V = C n = C p + q ⊕ C p + q into H ′ -orthogonal summands and write p i : V → C p + q ,1 ≤ i ≤ H ′ -equivariant projections. Now given a flag F = { V ⊂ . . . ⊂ V p } , wechoose an orthonormal basis v , . . . , v p of V p with respect to the standard Hermitian inner roduct on C n such that V i is spanned by v , . . . , v i . Denote by ( · , · ) p,q the complex bilinearform on C p + q which is invariant for H ′ .Then for 1 ≤ m ≤ p and 1 ≤ j, k ≤ g mjk ( F ) := ( p j ( v ∧ . . . ∧ v m ) , p k ( v ∧ . . . ∧ v m )) p,q . Similarly as before the rational functions f mjk := Re (cid:18) g mjk g m (cid:19) and f ′ mjk := Im (cid:18) g mjk g m (cid:19) are all H ′ -invariant. Already for m = 1 we obtain 4 independent invariants this way, andthus H ′ cannot have an orbit of codimension 3 by Lemma 2.15. (cid:3) Type III maximal subgroups.Lemma 6.4.
Let G = Sp( p, q ) . Then there exist no real spherical subgroups of type III.Proof. We may assume that 1 < p as it is known for p = 1 by [27]. Then 2 ≤ p ≤ q implies3 p + 2 q ≤ q < q ≤ pq and hence4 pq − p > dim V = 2 p + 2 q. Hence we get from (6.1) the strict inequality dim H C > dim V , and again we can use Propo-sition 3.6 and Remark 3.7. We are thus left with testing some sporadic cases, and it is easyto see that they never satisfy the dimension bound. (cid:3) This concludes the proof of Theorem 1.3 for G C = Sp( n, C ).7. The maximal real spherical subalgebras of the exceptional Lie algebras
Here g is such that g C is exceptional simple. We assume that g is not compact. Lemma 7.1.
Let g be a non-complex exceptional non-compact simple real Lie algebra and h a real spherical maximal reductive subalgebra. Then,(i) If g = G , then h is symmetric.(ii) If g = G , then h is symmetric or conjugate to either h = su (2 , or h = sl (3 , R ) which are both absolutely spherical but not symmetric.Proof. Recall from Corollary 3.3 that h C is maximal reductive in g C . If g is quasi-split, thenthe lemma follows from Lemma 2.13 combined with the work of Kr¨amer [26] (see Table 6).In particular G , the only non-compact real form of G C , is split, and thus the assertion (2)is obtained with Table 8.From now on we assume that g is not quasi-split. For g C = F C the only non-split real form F has real rank one, and for that the result is given in [27, Lemma 6.2]. This leaves us toconsider for g only the real forms E , E of E C , E , E of E C and E of E C .We follow [31]. According to Dynkin [12], a subalgebra h C of g C is called regular , if it isnormalized by a Cartan subalgebra of g C . On the other hand, h C is called an S-subalgebra of g C if it is not contained in any proper regular subalgebra of g C .Let h C be a maximal reductive subalgebra of g C . Then it is either regular or an S-subalgebra. According to [12, Theorem 14.1], the pairs ( g C : h C ), where h C is non-symmetric nd a maximal S-subalgebra of E C , E C or E C , are given by:( E C : A , G C , A ⊕ G C ) , ( E C : A , A ⊕ A , A , G C ⊕ C , A ⊕ F C , A ⊕ G C ) , ( E C : A , A ⊕ A , B , G C ⊕ F C ) , and by [12, Theorem 5.5] (together with the correction on p. 311 of the selected works) thepairs ( g C : h C ), where h C is non-symmetric, semisimple, and a maximal regular subalgebra,are given by: ( E C : A ⊕ A ⊕ A ) , ( E C : A ⊕ A ) , ( E C : A , A ⊕ A , A ⊕ E C ) . Note that a maximal reductive subalgebra of g C is either a semisimple maximal subalgebraor a maximal Levi subalgebra of g C . From the Dynkin diagram of g we can read the maximalLevi subalgebras and deduce that they are either symmetric or are contained in a semisimplemaximal regular subalgebra listed above (see [5, Table on p. 219]). Hence the two liststogether consist in fact of all maximal reductive subalgebras which are not symmetric.Next we record the dimension bounds obtained from (2.1):dim H ≥
30 ( g = E )(7.1) dim H ≥
24 ( g = E )(7.2) dim H ≥
60 ( g = E )(7.3) dim H ≥
51 ( g = E )(7.4) dim H ≥
108 ( g = E )(7.5)Going through the lists of maximal regular- and S-subalgebras of g C , we see that onlythe following two pairs ( G, H C ) satisfy the bound and thus may correspond to real sphericalpairs: ( E , A ⊕ A ⊕ A ) and ( E , A ⊕ F C ) . We claim that G = E and a real form in G of H C = SL(3 , C ) × SL(3 , C ) × SL(3 , C ) cannotcorrespond to a real spherical pair. By inspecting the Satake diagram of E we see thatthe minimal parabolic P of G is contained in the maximal parabolic P max , C of G C , whichis related to the 27-dimensional fundamental representation of G C . This representation isprehomogeneous, see case (22) in Table 3 of Proposition 3.6. If the pair was spherical then C would thus become a prehomogeneous vector space for H C . As dim H C = 24 <
26 thisis excluded.This leaves us with the case ( E , A ⊕ F C ). An inspection of the Satake diagram of G = E shows that the minimal parabolic P of G is contained in the maximal parabolic P max , C of G C , which is related to the 56-dimensional fundamental representation of G C . Again this isprehomogeneous, see case (23) in Table 3. If the pair was spherical then C would thus bea prehomogeneous vector space for H C = SL(2 , C ) × F C . In particular, every irreducible H C -submodule of C is then prehomogeneous. We recall from [34, Thm. 54], that F C does notadmit a prehomogeneous vector space in the generalized sense: for no non-trivial irreducible epresentation V of F C and for no n ∈ N does F C × GL( n, C ) admit an open orbit on V ⊗ C n .In particular, C cannot be prehomogeneous for H C . (cid:3) The lemmas in Sections 4-7 together with the list of Kr¨amer [26] finally conclude theproofs of Theorem 1.3 and Lemma 1.4.8.
Tables for L ∩ H Let ( g , h ) be a real spherical pair and recall from Section 2.4 the parabolic subgroup Q = L ⋉ U adapted to Z = G/H and P . In view of Proposition 2.9 the Lie algebra l ∩ h isof central importance to us. In the following Tables 4-5 (classical and exceptional) we list allsymmetric pairs ( g , h ) (from Berger’s list [4]) with g not quasi-split nor compact, togetherwith the associated subalgebra l ∩ h . g h l ∩ h (1) su ( p, q ) so ( p, q ) so ( q − p ) 1 ≤ p ≤ q − sl ( n, H ) so ∗ (2 n ) u (1) n n ≥ su (2 p, q ) sp ( p, q ) sp ( q − p ) + sl (2 , C ) p ≤ p ≤ q − sl ( n, H ) sp ( n − k, k ) sp (1) n ≤ k ≤ n (5) su ( p, q ) s [ u ( p , q ) + u ( p , q )] ( s [ u ( p − q , q − p ) + u (1) p + q ] s [ u ( q − p ) + u ( q − p ) + u (1) p + p ] p ≥ q p ≤ q (6a) sl ( n, H ) s [ gl ( n − k, H ) + gl ( k, H )] s [ gl ( n − k, H ) + gl (1 , H ) k ] 0 ≤ k ≤ n (6b) sl ( n, H ) sl ( n, C ) + u (1) ( s [ gl (1 , H ) n ] s [ gl (1 , H ) [ n ] ] + u (1) n even n odd(7) so (2 p, q ) u ( p, q ) u ( q − p ) + su (1 , p ≤ p ≤ q − so ∗ (2 n ) u ( n − k, k ) su (2) n , n and k even su (2) n − + su (1 ,
1) + u (1) , n even , k odd su (2) n − + u (1) , n odd 0 ≤ k ≤ n , n ≥ so ∗ (2 n ) gl ( n , H ) su (2) n n ≥ so ( p, q ) so ( p , q ) + so ( p , q ) ( so ( p − q , q − p ) so ( q − p ) + so ( q − p ) p ≥ q p ≤ q (10a) so ∗ (2 n ) so ∗ (2 n − k ) + so ∗ (2 k ) so ∗ (2 n − k ) + u (1) k ≤ k ≤ n (10b) so ∗ (2 n ) so ( n, C ) u (1) [ n ] n ≥ sp ( p, q ) ( u ( p, q ) gl ( p, H ) , p = q u ( q − p ) + u (1) p ≤ p ≤ q (12a) sp ( p, q ) sp ( p , q ) + sp ( p , q ) ( sp ( p − q , q − p ) + sp (1) p + q sp ( q − p ) + sp ( q − p ) + sp (1) p + p p ≥ q p ≤ q (12b) sp ( p, p ) sp ( p, C ) sp (1) p Table 4The notation in Table 4 follows certain conventions: in each row where the letters appearone has p = p + p ≤ q = q + q and p + q ≤ p + q (whence p ≤ q ).Following are some remarks on how the intersections l ∩ h have been calculated. Forthis we made extensive use of [19], especially §
10. The complexification Z C = G C /H C of he symmetric space Z = G/H is (complex) spherical. We may assume that Z C admits awonderful compactification, see Remark 2.12. Its Luna diagram is a collection of variousdata of which we need only two. First, a subset S ( p ) of the set S of simple roots of G whoseelements are called the parabolic roots of Z . Second, a finite set Σ Z of characters of G , calledthe spherical roots of Z . Each spherical root is an N -linear combination of simple roots.The real structure provides us with the set S ⊂ S of compact simple roots (the blackdots in the Satake diagram). Then, as mentioned already in Remark 2.12 , the set of simpleroots of L is the union S ∪ S ( p ) . Now let Σ Z ⊂ Σ Z be the set of those spherical rootswhich lie in the span of S ∪ S ( p ) . Then [19, Cor. 10.16] implies that Z := L/L ∩ H is anabsolutely spherical variety whose Luna diagram has still S ( p ) as set of parabolic roots andΣ Z as set of spherical roots. Since L ∩ H is reductive, these two data suffice to determinethe isomorphism type of the derived subgroup ( L ∩ H ) ′ by use of tables in [7].To determine the connected center C of L ∩ H it suffices to know its dimension and itsreal rank. The local structure theorem implies that L/L ∩ H is an open subset of the doublecoset space U \ G/H where U is the unipotent radical of the adapted parabolic of Z . Fromthis we get dim L ∩ H = dim H − dim U = dim H −
12 (dim G − dim L ) . Knowing ( L ∩ H ) ′ we get dim C . For its real rank, we userank R C = rank R L − rank R Z = rank R L − dim h res A σ | σ ∈ Σ Z i Q . (see [19, Lemma 4.18]). Here A ⊂ L is a maximally split subtorus.In most cases, it is not necessary to know the embedding of l ∩ h into h but in some itdoes matter. For this, the following lemma is useful. Lemma 8.1.
Let h , h be two self-normalizing real spherical subalgebras of g with adaptedparabolic subalgebras q and q corresponding to minimal parabolic subalgebras p and p .Suppose that h , C = Ad( x ) h , C for some x ∈ G C . Then there exists an element g ∈ G C of theform g = tg with g ∈ G and t ∈ Z ( L , C ) such that Ad( g ) maps h , C , q , C , and l ∩ h onto h , C , q , C , and l ∩ h , respectively.Proof. See [19, Section 13]. (cid:3)
In particular, the lemma says that the complexification of the embedding l ∩ h ֒ → h doesnot depend on the particular real form h . This is used in part (a) of the following remark. Remark 8.2. (a) In Table 5 there is ambiguity how l ∩ h is embedded into h in some caseswhere h = h ⊕ h consists of two factors. However, with Lemma 8.1 one can derive thefollowing additional data from the table:(a ) For g = E and h C = sl (6 , C ) ⊕ sl (2 , C ) one has [ l ∩ h , l ∩ h ] ⊂ h .(a ) For g = E and h C = so (12 , C ) ⊕ sl (2 , C ) one has l ∩ h ⊂ h .(a ) For g = E and h C = so (12 , C ) ⊕ sl (2 , C ), one has [ l ∩ h , l ∩ h ] ⊂ h .To see that, we discuss the case (a ). The arguments for (a ) and (a ) are similar. Table 5shows that there are two symmetric subalgebras in g , say h ′ = h ′ ⊕ h ′ = su (4 , ⊕ su (2) and h ′′ = h ′′ ⊕ h ′′ = su (5 , ⊕ sl (2 , R ) with isomorphic complexifications. By Lemma 8.1 there isan isomorphism of g C which carries h ′′ C to h ′ C and l ′′ ∩ h ′′ to l ′ ∩ h ′ . Table 5 shows that l ∩ h is of compact type, and hence [ l ′′ ∩ h ′′ , l ′′ ∩ h ′′ ] ⊂ h ′′ by Schur’s lemma. It then follows that[ l ′ ∩ h ′ , l ′ ∩ h ′ ] ⊂ h ′ . h l ∩ h E sp (2 , so (4) E sp (3 , so (4) + so (4) E ( su (4 ,
2) + su (2) su (5 ,
1) + sl (2 , R ) u (2) + u (2) E sl (3 , H ) + su (2) so (5) + so (3) + gl (1 , R ) E so (10) + u (1) so (8 ,
2) + u (1) so ∗ (10) + u (1) u (4) E so (9 ,
1) + gl (1 , R ) spin (7) + gl (1 , R ) E F so (7 , E ( F F so (8) E ( su (6 , su (4 , so (2) E ( su (6 , sl (4 , H ) so (4) + so (4) E so (12) + su (2) so (8 ,
4) + su (2) so ∗ (12) + sl (2 , R ) su (2) E ( so (10 ,
2) + sl (2 , R ) so ∗ (12) + su (2) so (6) + so (2) + sl (2 , R ) E ( E + u (1) E + u (1) so (6 ,
2) + so (2) E E + u (1) E + gl (1 , R ) E + u (1) so (8) E ( so (12 , so ∗ (16) so (4) + so (4) E E + su (2) E + sl (2 , R ) E + su (2) so (8) F sp (2 ,
1) + su (2) so (4) + so (3) F ( so (9) so (8 , spin (7) Table 5(b) For g = F and h = sp (2 , ⊕ su (2) the algebra l ∩ h surjects onto h . To see this,let V be the 52-dimensional irreducible (adjoint) representation of F . Then we claim thatdim V l ∩ h = 1. This can be shown by branching V (with highest weight ω ) to l ∩ h by using he following chain of maximal subalgebras l ∩ h = so (4) + so (3) ⊂ l ′ = so (7) ⊂ so (8) ⊂ so (9) ⊂ F . One can do that either by hand (starting with Res F so (9) V = L ( ω ) + L ( ω )) or by help of acomputer algebra package. We used LiE , [30], with the functions resmat() to generate therestriction matrices and branch() to perform the branching.On the other hand, res gh V contains the 3-dimensional sp (3) ⊕ sl (2)-module C ⊗ S C . Thiscannot happen if the projection of l ∩ h to h were trivial.(c) In the classical case (Table 4) there are also some situations where h is not simple, andwhere it is of interest how certain factors of l ∩ h are embedded into h . These are: • In (5) u (1) p + q , resp. u (1) p + p , is diagonally embedded into h = h + h , • In (6a) gl (1 , H ) k is diagonally embedded into h = h + h , • In (10a) u (1) k is diagonally embedded into h = h + h , • In (12a) sp (1) p + q , resp. sp (1) p + p , is diagonally embedded into h = h + h .This can be verified as follows: Let σ be the involution which determines h and let θ bethe standard Cartan involution which commutes with σ . Let k be the fixed point set of θ .Then l can be chosen as the centralizer of a generic element X ∈ h ⊥ ∩ k ⊥ where ⊥ refersto the orthogonal complement with respect to the Cartan-Killing form of g . Simple matrixcomputations then verify the bulleted assertions.(d) In the last two lines of Table 5 we have l ∩ h = spin (7). That it is the spin embedding(and not so (7)) is seen in both cases from the fact that the complement in g contains thespin representation.9. The classification of reductive real spherical pairs
Now that we have classified all maximal spherical subalgebras which are reductive, we cancomplete the classification.We recall the adapted parabolic Q = L ⋉ U ⊃ P of a real spherical space. We set L H := L ∩ H and denote its Lie algebra by l h . Further we may assume that M A ⊂ L .The general strategy is as follows. Given G and a maximal reductive real spherical sub-group H we let H ′ ⊂ H be a proper reductive subgroup. According to Proposition 2.9 thespace G/H ′ is real spherical if and only if H/H ′ is a real spherical L H -variety. In particular,(9.1) h = h ′ + l h needs to hold by Corollary 2.10. By Lemma 1.4, H is symmetric in almost all cases, andhence l h is given by the tables in Section 8. By Proposition 2.5 we can then determinewhether (9.1) is valid and thus limit the number of subgroups H ′ to consider.After the following preliminary result this section will be divided into two parts: classicaland exceptional.9.1. Almost absolutely spherical pairs.
In addition to (9.1) there is a second generalfact which will be useful in the classification. Let us call h almost absolutely spherical if it isreal spherical and there exists an absolutely spherical subalgebra h of g with [ h , h ] ⊂ h ⊂ h . Lemma 9.1.
Let g be a non-compact and non-complex simple Lie algebra and h a reductivesubalgebra which is not absolutely spherical. Then ( g , h ) is almost absolutely spherical if and nly if it is isomorphic to one of the pairs in Table 1 of Theorem 1.1 which are marked byan asterisk.Proof. We use the real version of the Vinberg-Kimel’feld criterion (see [21, Prop. 3.7]): thesubalgebra h is real spherical if and only if dim V h ≤ G , forwhich there exists a P -semiinvariant vector. Observe, that it suffices to check this conditionover C .Now let h ⊂ g be an absolutely spherical subalgebra in which h is coabelian. Because h is not absolutely spherical the complexified pair ( g C , h C ) will appear (according to Kr¨amer[26]) in the following table: g C h C M α sl (2 n, C ) sl ( n, C ) + sl ( n, C ) ω , . . . , ω n − , ω n ± ǫ, ω n +1 , . . . , ω n − α n so (4 n, C ) sl (2 n, C ) ω , ω , . . . , ω n − , ω n ± ǫ α n so (2 n + 1 , C ) sl ( n, C ) ω , . . . , ω n − , ω n ± ǫ α n so ( n, C ) so ( n − , C ) ω ± ǫ, ω α so (10 , C ) spin (7 , C ) ω ± ǫ, ω , ω + ǫ, ω − ǫ α sp ( n, C ) sl ( n, C ) 2 ω , . . . , ω n − , ω n ± ǫ α n sp ( n + 1 , C ) sp ( n, C ) ω ± ǫ, ω α E C E C ω , ω , ω ± ǫ α Since H C normalizes X = G C /H C , there is a right action of T := H C /H C ∼ = C ∗ on X .Moreover, because G C /H C is (absolutely) spherical, X is spherical as G = G C × T -variety.The corresponding weights (i.e. highest weights of irreducible G -modules V containing anon-trivial H C -fixed vector) are called the extended weights of X . They are charactersof B × T , where B ⊂ P C ⊂ G C is a Borel subgroup, and form a monoid M . Now thethird column shows the generators of this monoid. Here, ǫ generates the character groupof T . The expansion of a character is given w.r.t. the fundamental weights following theBourbaki notation. The set of weights M of X as a G -variety is obtained by dropping the T -components, i.e., by setting ǫ = 0. This way we get a surjective map π : M → M . Let M P ⊂ M be the submonoid of weights whose first component (i.e. its restriction to B ) isa weight of P C . Then the Vinberg-Kimel’feld criterion implies that G/H is real spherical ifand only if the restriction of π to M P is injective.To decide injectivity one checks that in every case there is a unique simple root α of g C (given in the fourth column) with the property that the restriction of π to M∩ H α is injectivewhere H α is the hyperplane perpendicular to α . We claim that G/H is real spherical if andonly if α is a compact simple root of G . Indeed, if α is compact then M P ⊂ H α and therestriction of π is injective. Conversely, if α is non-compact then the unique fundamentalweight ω with h ω, α ∨ i = 1 is a weight of P . Moreover, by inspection of the table one seesthat there is d ≥ ω ± dǫ ∈ M . Thus the restriction of π is not injective.Finally, the lemma is proved by simply finding all real forms of ( g C , h C ), for which α is acompact root of g . For this we use Berger’s list together with Table 8. For example the firstitem yields among others the pair ( sl ( n, H ) , sl ( n, C )) which is real spherical if and only if α n is compact for sl ( n, H ), hence if and only if n is odd. (cid:3) The classical cases. roposition 9.2. Let g = su ( p, q ) , ≤ p ≤ q , and let h be a reductive subalgebra. Then ( g , h ) is real spherical if and only if either it is absolutely spherical or h is conjugate to oneof the following:(i) h = h ⊕ h = su ( p , q ) ⊕ su ( p , q ) with p + p = p, q + q = q and p + q = p + q ,but ( p , q ) = ( q , p ) , or(ii) p = 1 , q = q + q , q even and h = su (1 , q ) ⊕ sp ( q / ⊕ f with f ⊂ u (1) .Proof. Since we shall refer to Proposition 2.9 it will be convenient to replace the notation h in the above statement by h ′ , and let h instead denote a maximal reductive subalgebra with h ′ ⊂ h ⊂ g . Then h is symmetric by Lemma 1.4.We need to consider the cases (1), (3) and (5) from Table 4. For case (1) we observe that l h is compact but not h . Hence by Lemma 2.4 there is no proper real spherical subalgebra h ′ of h . We can argue similarly for (3) as symplectic algebras do not admit factorizations byProposition 2.5.This leaves us with (5), i.e. h = h ⊕ h ⊕ h = su ( p , q ) ⊕ su ( p , q ) ⊕ u (1)with p + q , p + q > p + p = p , q + q = q . Set r := p + q and s := p + q . Wemay assume that r ≤ s . Note that since p ≤ q this implies that q − p ≥ | p − q | .According to Table 4 we have(9.2) l h = s [ u ( p − q , q − p ) ⊕ u (1) r ]when p ≥ q , and when p ≤ q we have(9.3) l h = s [ u ( q − p ) ⊕ u ( q − p ) ⊕ u (1) p ] . Let us first consider the case where h ′ = [ h , h ] and start with p ≥ q . The embeddinginto h of (9.2) is such that the projection of l h to h is injective. Hence we deduce from (9.1)and Proposition 2.5 that r = p + q = 1 and h ′ = sp ( p , q ) or h ′ = sp ( p , q ) ⊕ u (1) both ofwhich are absolutely spherical according to Table 8.Next we consider the case where p ≤ q with l h given by (9.3). Note that u (1) p projectsinjectively to both factors h and h . Hence we deduce from (9.1) and Proposition 2.5 that p = p + p = 1. Without loss of generality let p = 1 and p = 0, i.e. g = su (1 , q )and h = su (1 , q ) ⊕ su ( q ) ⊕ u (1). Proposition 2.5 forces q to be even and shows that h ′ = h ⊕ sp ( q /
2) or h ′ = h ⊕ sp ( q / ⊕ u (1), both of which are real spherical. This is case( ii ).Finally let us consider the case where h ′ = [ h , h ] = h + h . According to Table 8 this isabsolutely spherical provided that r < s . For r = s , case ( i ) follows from Lemma 9.1. (cid:3) Proposition 9.3.
Let g = sl ( m, H ) for m ≥ , and let h be a reductive subalgebra. Then ( g , h ) is a real spherical pair if and only if it is absolutely spherical or, up to conjugation,(i) h = sl ( m − , H ) ⊕ f with f ⊂ C , or(ii) h = sl ( m, C ) with m odd.Proof. We need to treat the cases (4) and (6) from Table 4. Now, since symplectic algebrasdo not admit factorizations, we are left with the two cases in (6).We begin with h = s ( gl ( m , H ) ⊕ gl ( m , H )), m = m + m , m ≥ m . Set h = sl ( m , H ), h = sl ( m , H ) and h = z ( h ) = gl (1 , R ). ere we have from Table 4 l h = s [ gl ( m − m , H ) ⊕ gl (1 , H ) m ]with sl ( m − m , H ) ⊂ h and gl (1 , H ) m diagonally embedded. According to [26], [ h , h ] isabsolutely spherical if and only if m = m . If m = m , then l h does not surject to the centerof h and hence [ h , h ] is not spherical. If m >
1, then we obtain via (9.1) and Proposition2.5 that the only possible spherical subalgebra contained in h is [ h , h ].In case m = 1, m ∩ h surjects onto h = su (2) and we obtain the cases listed in (1).The second possibility for h is h = u (1) ⊕ sl ( m, C ). For that we first note that the dimensionbound excludes h ′ := u (1) ⊕ h ′ with h a proper reductive subalgebra of sl ( m, C ) to be realspherical. The cases where [ h , h ] are spherical are deduced from Lemma 9.1. (cid:3) Proposition 9.4.
Let g = so ∗ (2 m ) for m ≥ . Then a reductive subalgebra is real sphericalif and only if it is absolutely spherical or conjugate to one of the following:(i) h = so ∗ (2 m − , or(ii) m = 5 , h = spin (5 , or spin (6 , .Proof. We need to consider the cases (8) and (10) from Table 4. In case of (8) there are noproper real spherical subalgebras of g contained in h by (9.1) and Proposition 2.5. In case(10) with h = so ( m, C ) the dimension bound excludes a proper reductive subalgebra of h tobe real spherical.Finally we need to treat the case where h = h ⊕ h = so ∗ (2 m ) ⊕ so ∗ (2 m ) with m ≤ m , m + m = m . Here l h = so ∗ (2( m − m )) ⊕ so (2) m with so ∗ (2( m − m )) ⊂ so ∗ (2 m ).When m > h which are real spherical. If m = 1, then h = so (2 , R ) ⊕ so ∗ (2 m ), m ∩ h surjects onto so (2 , R ) and thus so ∗ (2 m ) is real spherical. Further factorizations are onlypossible for m = 4 which results in the real spherical subalgebras spin (6 ,
1) and spin (5 , h = so ∗ (8) ≃ so (6 , (cid:3) Proposition 9.5.
Let g = so ( p, q ) , ≤ p ≤ q , p + q ≥ . Then a reductive subalgebra is realspherical if and only if it is either absolutely spherical or conjugate to one of the following:(i) p , q , and p + q all even, p = q , and h = su ( p , q ) .(ii) p = 2 r , q = 2 s + 1 , r = s , and h = su ( r, s ) .(iii) p = 2 r + 1 , q = 2 s , r = s , and h = su ( r, s ) .(iv) p = 1 , q = q + 4 and h = so (1 , q ) + h ′ with h ′ ( so (4) ≃ so (3) × so (3) a subalgebrasuch that h ′ + diag so (3) = so (4) .(v) p = 1 , q = q + q , q ≥ , and h = so (1 , q ) ⊕ h with h ( so ( q ) a subalgebra suchthat h + so ( q −
1) = so ( q ) (see Proposition 2.5).(vi) p = 2 , q = q + 7 and h = so (2 , q ) ⊕ G .(vii) p = 2 , q = q + 8 and h = so (2 , q ) ⊕ spin (7) .(viii) p = 3 , q = q + 8 and h = so (3 , q ) ⊕ spin (7) .(ix) p = 3 , q = 6 and h = so (2) ⊕ G .(x) p = 4 , q = 7 and h = so (3) ⊕ spin (3 , .Proof. According to Lemma 5.6 and Lemma 5.7 there are no maximal spherical subalgebrasof type II or III unless g is split. Moreover, by Lemma 5.4 all maximal subalgebras of typeI are symmetric. As we may assume that g is not quasisplit, we need only to considersubalgebras of the symmetric subalgebras h from cases (7) and (9) in Table 4. e begin with the first of these, i.e. h = u ( p , q ) with p = q even and l h = u ( q − p ) ⊕ sl (2 , R ) q where u ( q − p ) ⊂ so ( q − p ) = m . In particular, u ( q − p ) surjects onto the center of h and wededuce that [ h , h ] is spherical. According to Kr¨amer this is absolutely spherical if and onlyif p + q is odd, and we obtain ( i ). From the structure of l h we deduce from Proposition 2.5and Proposition 2.9 that no other reductive proper subalgebra h ′ ⊂ h is real spherical in g .This leaves us with the case (9) from Table 4, where h = h ⊕ h = so ( p , q ) ⊕ so ( p , q )with p = p + p ≤ q = q + q , r = p + q ≤ s = p + q , and p ≤ q . In case p ≤ q wehave P = Q and(9.4) l h = so ( q − p ) ⊕ so ( q − p ) . In case p ≥ q we have(9.5) l h = so ( p − q , q − p ) ⊂ h . To start with we exclude the diagonal case where h ≃ h and h ′ ≃ h is “diagonally”embedded into h = h ⊕ h . For that we note that h ≃ h either means that h = h or( p , q ) = ( q , p ). In the latter case g is split and we are left with h = h . In particular h isnon-compact and semisimple, but l h = so ( q − p ) ⊕ so ( q − p ) is compact. Thus H = H ′ L H is not possible by Lemma 2.4.We now begin with the case of p ≤ q and (9.4). Suppose that h ⊕ h ′ is a sphericalsubalgebra of g , with h ′ ( h . Then h = h ′ + l h with l h = so ( q − p ), i.e.(9.6) h ′ + so ( q − p ) = h = so ( p , q ) . Suppose first that h is simple. Then this is a factorization of of so ( p , q ) with one factorcompact and we deduce from Lemma 2.4 that so ( p , q ) is compact as well, i.e. p = 0 or q = 0. If q = 0, then p = 0, and h ′ = h . Hence we may assume that p = 0 and then h ′ + so ( q − p ) = so ( q ) . We deduce from Proposition 2.5 that p = 1 , , v ), ( vi ), ( vii ),( viii ) for h ′ .In case h is not simple possibilities are h = so (2 ,
2) and h = so (4). Only the latter ispossible with (9.6), which in that case gives p = 1 and h ′ ≃ so (3) and leads to ( iv ).The case where h ′ ⊕ h with h ′ ( h is analogous, and leads to the same results but with r and s interchanged. This finishes the treatment of p ≤ q and (9.4).We now treat the case of p ≥ q and (9.5), where l h ⊂ h . Let h ′ := h ⊕ h ′ ⊂ h with h ′ ⊂ h be a spherical subalgebra of g . The condition is that H /H ′ is a real spherical spacefor the action of L H . In particular we must have (see Corollary 2.10)(9.7) h ′ + so ( p − q , q − p ) = h = so ( p , q )and p , q both non-zero. Hence h is non-compact and we may assume it is simple. Ac-cording to Proposition 2.5 we derive that p + q equals 1 , ,
3. Suppose first that h ′ is ofType I in h , see the four cases in Lemma 5.4. Onishchik’s list, Table 2, shows that only h ′ = u ( p , q ) can be compatible with (9.7), and then p + q = 1. Hence h = h and h ′ = h ′ is real spherical according to Kr¨amer. Further subalgebras of type h ′′ := u (1) ⊕ h ′′ with h ′′ ⊂ [ h ′ , h ′ ] a proper maximal spherical subalgebra are excluded. Indeed (9.7) andProposition 2.5 only allow h ′′ = sp ( p , q ) and we arrive at the tower g = so ( p + 1 , q ) ⊃ u ( p , q ⊃ h ′′ = sp ( p , q u (1) . ow the real spherical pair ( g , h ) = ( so ( p + 1 , q ) , u ( p , q )) has structural algebra l h = u ( | q − p | ) and we deduce that ( g , h ′′ ) is not real spherical by Proposition 2.9 and Proposition2.5. This leads us to decide whether h := [ h ′ , h ′ ] = su ( p , q ) is real spherical. Accordingto Kr¨amer, h is not absolutely spherical. Without loss of generality we may assume p = 1and q = 0 and then l h = so ( p , q − h is real spherical if and only if h ∩ m surjects onto the center of h ′ . This is the case precisely when p = q and p = q − ii )–( iii ).Finally assume that h ′ is of type II or III in h . The type II subalgebras appear onlyfor h = so (4 ,
4) (see Lemma 5.6) and are excluded by the dimension bound (5.1). Thisleaves us with the examination of the two type III cases from Lemma 5.7. We begin with h ′ = spin (4 ,
3) in h = so (4 ,
4) = so ( p , q ). Recall that p + q = 1 , , p and q have to differ by three in order for g not to be quasisplit. Hence we may assume that h = so (3), i.e. g = so (4 ,
7) and h ′ = so (3) ⊕ spin (4 , ⊂ so (3) ⊕ so (4 , h ′ is real spherical. For that we need to show that the L H = SO (1 , H /H ′ = SO (4 , / Spin(4 ,
3) is real spherical. To this end we lift to Spin(4 , α and α , and go backto SO (4 , ,
3) and SO (1 , ⊂ SO (2 ,
4) are converted to SO (4 ,
3) andSp(1 , ⊂ SU(2 , n = 2. Using the last column of the table we get SO (4 , / SO (4 ,
3) =Sp(1 , / Sp(0 , , x ).Next we move on to the case where h ′ = G and h = so (3 , p + q = 1 , , p + q = 3 is excluded by the dimension bound and the case with p + q = 1leads to absolutely spherical pairs. The case p = q = 1 is quasi-split. This leaves us with h ′ = so (2) ⊕ G in g = so (3 , H /H ′ = SO (3 , / G is spherical as L H = SO (1 , (3 , / G ∼ = SO (4 , / Spin(3 ,
4) (eighth case of Table 2) and the proof of case ( x ) above.This yields case ( ix ). (cid:3) Proposition 9.6.
Let g = sp ( p, q ) and let h be a reductive subalgebra. Then h is realspherical if and only if it is absolutely spherical or conjugate to one of the following:(i) sp ( p − , q ) ,(ii) sp ( p, q − ,(iii) su ( p, q ) with p = q .Proof. We need to consider subalgebras of the following cases from (11) and (12) in Table 4: h = sp ( p, C ) , sp ( p , q ) × sp ( p , q ) , u ( p, q ) , gl ( p, H )where q = p in the first and last cases. Since symplectic algebras admit no factorizationsby Proposition 2.5 the first case is excluded with Proposition 2.9. We can argue similarly inthe second case, except when l h surjects onto one of the factors of h = h ⊕ h . Accordingto Table 4 this happens if and only if h or h is sp (1), in which case the other factor h ′ of h is sp ( p − , q ) or sp ( p, q − m belongs to h and surjects onto sp (1) along h ′ . Hence h ′ is spherical. Further we observe that it is not absolutely spherical, but any strictly largersubalgebra is. This gives (1)-(2). or the third case, h = u ( p, q ), we note that l ∩ h is compact according to Table 4. Henceif h ′ ⊂ h satisfies (9.1) then Lemma 2.4 implies su ( p, q ) ⊂ h ′ . With Lemma 9.1 we conclude(3).Finally, in the fourth case h = gl ( p, H ) we have l h = u (1) p and hence no proper factorizationis possible. (cid:3) The exceptional cases.
For convenience we record here Cartan’s list of the ninesymmetric subgroups in the complex exceptional Lie groups of type E: G C E C E C E C H C Sp(4 , C ) SL(8 , C ) SO(16 , C ) H C SL(6 , C ) × SL(2 , C ) SO(12 , C ) × SL(2 , C ) E C × SL(2 , C ) H C SO(10 , C ) × SO(2 , C ) E C × SO(2 , C ) H C F C For the list of real symmetric subgroups we shall refer to [4].
Proposition 9.7.
Let g be a non-complex and non-compact simple exceptional Lie algebraand let h ⊂ g be reductive. Then ( g , h ) is real spherical if and only if it is absolutely sphericalor, up to conjugation,(i) g = F and h = sp (2 , ⊕ f with f ⊂ u (1) , or(ii) g = E and h = sl (3 , H ) ⊕ f with f ⊂ u (1) , or(iii) g = E and h = E or E .Proof. If g is quasi-split we apply Lemma 2.13. In particular we can then assume g C = G C .This leaves us with the E and F -cases which are not quasi-split. As before we shall use thenotation h ′ for a given candidate of a real spherical subalgebra of g . It follows from Lemma7.1 that we may assume h ′ is contained in a symmetric subalgebra which we then denote by h . We can assume the inclusion is proper.We use Table 5 for l ∩ h , where Q = LU is the adapted parabolic for Z = G/H . We notethat l ∩ h only depends on the real form g and the complexification h C (see Lemma 8.1).We start with g = F . From Table 5 we deduce that either h = so (8 ,
1) and l h = so (7),or h = sp (2 , ⊕ su (2) and l h = so (4) ⊕ so (3). Since so (9 , C ) does not admit non-trivialfactorizations, no proper reductive subalgebra of h can be spherical in the first case. In thesecond case we observe with Remark 8.2 that l h projects onto su (2), the second factor of h ,and the cases in (1) emerge.We continue with g = E . Any symmetric subalgebra of g is a real form of either h , C or h , C from the table above. However, no proper reductive subalgebra of h can satisfy thedimension bound (7.5). We assume h ′ ( h , a real form of h , C , and let Q = LU be theadapted parabolic for G/H . Then(9.8) l ∩ h + h ′ = h by (9.1). Here ( l ∩ h ) C = so (8 , C ) and the projection of this algebra to the second componentof h , C must be zero as so (8 , C ) is irreducible and of higher dimension than sl (2 , C ). Hence( l ∩ h ) C ⊂ E C . However by Proposition 2.5, E C admits no proper factorizations, and hencewe must have E C ⊂ h ′ . Thus l ∩ h ⊂ h ′ which contradicts (9.8).Next we investigate the real forms of g C = E C . According to the table above the symmetricsubalgebras in g C are h i, C , i = 1 , , e start with g = E and note that the dimension bound (7.3) excludes that h ( h .Next we consider the pair ( g , h ) and the corresponding adapted parabolic Q = LU . Assume h ( h , then (9.8) holds as before. Here ( l ∩ h ) C = sl (2 , C ) embeds in the first component so (12 , C ) of h , C (see Remark 8.2). By Proposition 2.5 there exists no proper factorizationof so (12 , C ) with sl (2 , C ) as a factor, and hence we conclude that so (12 , C ) ⊂ h C . Thus l ∩ h ⊂ h which contradicts (9.8).For the third case we note that E C has codimension 1 in h , C and does not admit properfactorizations by Proposition 2.5. Hence if h ( h then h C = E C . We deduce that h is realspherical from Lemma 9.1. This gives (3).We move on with g = E , and start with the assumption that h ( h . Here Q = P byLemma 2.11 and this implies that dim H /H ∩ L = dim G/P = 51, and hence dim H ∩ L =12. According to Proposition 2.5 the only factorizations for h are given by • ( sl (8 , C ) , sp (4 , C ) , sl (7 , C )), • ( sl (8 , C ) , sp (4 , C ) , s ( gl (1 , C ) ⊕ gl (7 , C ))),and none of these factors have dimension 12. With Proposition 2.9 we reach a contradiction.For the case of h we first recall h ∩ l = so (6) ⊕ so (2) ⊕ sl (2 , R ). It follows from Remark8.2 that h ∩ l does not surject onto the sl (2)-component of h . Hence no proper reductivesubalgebras of h can be real spherical.For h we are again left to check whether a real form of the first component E C of h , C isreal spherical. Here ( l ∩ h ) C = so (8 , C ) and thus the projection of ( l ∩ h ) C to the secondcomponent of h is trivial, and hence no real form of E C can be spherical.Finally we consider g C = E C and g = E or E . The complex symmetric subalgebras h , C , i = 1 , . . . ,
4, are given in the table.Since both h and h admit no factorizations there are no reductive real spherical subal-gebras which are contained in h or h . We move on and assume h ( h , where h is a realform of h , C in g . Write h = h ′ ⊕ h ′′ with h ′ , C = sl (6 , C ) and h ′′ , C = sl (2 , C ). From Table 5we infer that(9.9) l ∩ h = u (2) ⊕ u (2) for g = E (9.10) l ∩ h = so (5) ⊕ so (3) ⊕ gl (1 , R ) for g = E We claim that h is not spherical for g = E . Otherwise, according to Propositions 2.5, 2.9, l h would surject to a factor of h ′′ . But this is not possible by Remark 8.2.For g = E , we claim that l ∩ h surjects onto h ′′ = su (2) and in particular that h ′ = sl (3 , H )is real spherical. In order to establish that we let V ⊂ g C be the orthogonal complementof h C in g C . Note that dim C V = 40 and that V is an irreducible module for h C . Hence V = V C ⊗ C as an h , C -module. Notice that a := V ∩ z ( l ) = { } and that a is fixed under l h . In order to obtain a contradiction, assume that l h ⊂ h ′ . Then, as an l h -module, V = V C ⊕ V C . Since V l h = { } we deduce that the irreducible h ′ , C = sl (6 , C )-module V C is spherical for [ l ∩ h , l ∩ h ] C ≃ sp (2 , C ) ⊕ sp (1 , C ). But V C decomposes under sp (3 , C )into V ( ω ) ⊕ V ( ω ) and hence is not spherical for the pair ( sp (3 , C ) , sp (2 , C ) ⊕ sp (1 , C )) by[26, Tabelle 1]. This gives the desired contradiction, and hence (2).Finally we come to the case of h , C . Here it is known that h ′ , C = so (10 , C ) is a com-plex spherical subgroup by [26]. From the list in Proposition 2.5 we extract that the onlyfactorizations of so (10 , C ) are given by ( so (10 , C ) , so (9 , C ) , sl (5 , C ) + f ), f ⊂ u (1). Now for = E , E we have [ l ∩ h , l ∩ h ] C is so (6 , C ) or spin (7 , C ) and the factorization of h , C is notpossible. This concludes the proof of the proposition. (cid:3) By combining Propositions 9.2, 9.3, 9.4, 9.5, 9.6, and 9.7 we finally obtain Table 1.10.
Absolutely spherical pairs
In this section we prove Theorem 1.1. For that it only remains to classify the absolutelyspherical pairs, and we refer to [4] for the symmetric ones.10.1.
The complex cases.
We begin by determining the cases for which g has a complexstructure. Proposition 10.1.
Let g be a complex simple Lie algebra and h ⊂ g a real reductive subal-gebra. Then ( g , h ) is real spherical if and only it is absolutely spherical. This is the case ifand only if one of the following holds(i) h is a real form of g (and hence symmetric),(ii) h is a complex spherical subalgebra of g ,or h is conjugate to h ⊕ h with(iii) ( h , h ) = ( z ( h ) , [ h , h ]) , dim R h = 1 , and ( g , h ) is one of the following complexspherical pairs(a) ( sl ( n + m, C ) , sl ( n, C ) × sl ( m, C )) , < m < n (b) ( sl (2 n + 1 , C ) , sp ( n, C )) , n ≥ (c) ( so (2 n, C ) , sl ( n, C )) , n odd, n ≥ (d) ( E C , so (10 , C )) ,(iv) h = su (2) or sl (2 , R ) , and ( g , h ) = ( sp ( n + 1 , C ) , sp ( n, C )) , n ≥ .Proof. First observe that g , considered as a real Lie algebra, is quasisplit. Hence h ⊂ g is realspherical if and only if h C ⊂ g C is spherical. We may identify g C with g ⊕ g . Now accordingto [31, Section 5], the complex spherical subalgebras ˜ h of g ⊕ g are given as follows (i) ˜ h = diag( g ) (cf. [31, Prop. 5.4]). (ii) ˜ h = h ⊕ h with h i ⊂ g complex spherical. (iii) There exists a complex spherical subalgebra h ⊂ g with z ( h ) = 0, [ h , h ] complexspherical and ˜ h = [ h , h ] ⊕ z ( h ) ⊕ [ h , h ] and z ( h ) diagonally embedded (see [31],beginning of Section 5 with the notion of a principal irreducible spherical pair ). (iv) g = sp ( n + 1 , C ) and ˜ h = sp ( n, C ) ⊕ sp (1 , C ) ⊕ sp ( n, C ) with sp (1 , C ) diagonallyembedded and n ≥ h = h C these four cases correspond to the fourcases listed in the proposition. This is easily seen, with use of Kr¨amer’s list for ( iii ). (cid:3) In Table 6 at the end of this section we record the list of the non-symmetric pairs of case( ii ). The pairs in ( iii )-( iv ) are tabulated in Table 7. Remark 10.2.
Inspecting Table 6 one realizes that it has a certain structure (cf. [35, Table(12.7.2)]). In all cases but (8) and (9) there is a canonical intermediate subalgebra h C , givenin the last column, with the following properties (a)-(c).(a) The pair ( g C , h C ) is symmetric. Hence all of its real forms appear up to isomorphismin Berger’s list. b) Except for case (10), the pair ( h C , h C ) is symmetric, as well. Even though h C maynot be simple the real forms of these pairs are easily read off from Berger’s list.(c) If ( g C , h C ) is defined over R then also h C is defined over R . Indeed, h C = N g C ( h C )in cases (1), (4), and (11). Moreover, h C = N g C ( C g C ([ h C , h C ])) in cases (2), (3), and(7). For cases (5) and (6) one argues as follows: for all real forms of g C , the definingrepresentation V is defined over R . Then h C = C g C ( V h C ) is defined over R , as well.In case (10) the isomorphism so ∗ (8) ∼ = so (6 ,
2) (via triality) shows that it suffices toconsider real forms for which V is defined over R . Then the argument above works.10.2. The non-complex cases.
We recall that g carries no complex structure if and onlyif it remains simple upon complexification. In this case we say that g is absolutely simple .Assume g is non-compact and absolutely simple. Using the reasoning in Remark 10.2, weobtain all non-symmetric, absolutely spherical reductive subalgebras h . The list is given inTable 8 below. Only the last five rows, which relate to (8), (9) and (10) above, require aseparate argument.The cases involving real forms of G are handled using the following remarks. The maximalcompact subalgebra of G is su (2) + su (2). Hence su (3) G . Moreover, the invariant scalarproduct on the 7-dimensional representation of G and G has signature (7 ,
0) and (4 , , R ) or SU(2 , Tables.
Here we tabulate (up to isomorphism) all absolutely spherical non-symmetricpairs ( g , h ) with g non-compact, simple and h ⊂ g reductive:- Table 6 lists those pairs in which both g and h have a complex structure. In thistable all algebras are implied to be complex. The table is due to Kr¨amer [26].- Table 7 lists those pairs in which g but not h has a complex structure. The table isextracted from Proposition 10.1.- Table 8 lists those pairs in which g is absolutely simple. See Section 10.2. g C h C h C (1) sl ( m + n ) sl ( m ) + sl ( n ) m > n ≥ s [ gl ( m ) + gl ( n )](2) sl (2 n + 1) sp ( n ) + f n ≥ , f ⊂ C gl (2 n )(3) sp ( n ) sp ( n −
1) + C n ≥ sp ( n −
1) + sp (1)(4) so (2 n ) sl ( n ) n ≥ gl ( n )(5) so (2 n + 1) gl ( n ) n ≥ so (2 n )(6) so (9) spin (7) so (8)(7) so (10) spin (7) + C so (8) + C (8) G sl (3) − (9) so (7) G − (10) so (8) G so (7)(11) E spin (10) spin (10) + C Table 6 hsl ( n + m, C ) sl ( n, C ) + sl ( m, C ) + z z ⊂ C , dim R z = 1 0 < m < n sl (2 n + 1 , C ) sp ( n, C ) + z z ⊂ C , dim R z = 1 n ≥ so (2 n, C ) sl ( n, C ) + z z ⊂ C , dim R z = 1 n ≥ , n odd E C so (10 , C ) + z z ⊂ C , dim R z = 1 sp ( n + 1 , C ) sp ( n, C ) + f f ∈ { sp (1) , sp (1 , R ) } n ≥ g hsl ( m + n, R ) sl ( m, R ) + sl ( n, R ) m > n ≥ su ( p + p , q + q ) su ( p , q ) + su ( p , q ) p + q > p + q ≥ sl ( m + n, H ) sl ( m, H ) + sl ( n, H ) m > n ≥ sl (2 n + 1 , R ) sp ( n, R ) + f n ≥ , f ⊂ R su (2 p + 1 , q ) sp ( p, q ) + f p + q ≥ , f ⊂ i R su ( n + 1 , n ) sp ( n, R ) + f n ≥ , f ⊂ i R sp ( n, R ) sp ( n − , R ) + f n ≥ , f ∈ { R , i R } sp ( p, q ) sp ( p − , q ) + i R p, q ≥ so (2 p, q ) su ( p, q ) p ≥ q ≥ , p + q odd so ( n, n ) sl ( n, R ) n ≥ so ∗ (2 n ) su ( p, q ) n = p + q ≥ so (2 p + 1 , q ) su ( p, q ) + i R p + q ≥ so ( n + 1 , n ) sl ( n, R ) + R n ≥ so (5 , spin (4 , so (8 , spin (7 , so (5 , spin (4 ,
3) + R so (6 , spin (4 ,
3) + i R so (8 , spin (7 ,
0) + i R so (9 , spin (7 ,
0) + R so ∗ (10) spin (6 ,
1) + i R , spin (5 ,
2) + i R E so (5 , E so (6 , , so ∗ (10) E so (10) , so (8 , , so ∗ (10) E so (9 , G sl (3 , R ) , su (2 , so (4 , G so (4 , G so (5 , G so (7 , G Table 8
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