Classification of invariant AHS--structures on semisimple locally symmetric spaces
aa r X i v : . [ m a t h . DG ] J a n CLASSIFICATION OF INVARIANT AHS–STRUCTURES ONSEMISIMPLE LOCALLY SYMMETRIC SPACES
JAN GREGOROVIˇC
Abstract.
In this article, we discuss which semisimple locally symmetricspaces admit an AHS–structure invariant to local symmetries. We classifythem for all types of AHS–structures and determine possible equivalence classesof such AHS–structures. Introduction
One can describe (locally) symmetric spaces employing the language of Cartangeometries. This allows us to use a functorial construction of invariant geometricstructures. The construction is called extension and its application to symmetricspaces is described in [4]. We will review basic facts of the construction and thenapply this construction in the case of almost hermitian symmetric (shortly AHS-) structures. This will allow us to obtain full classification of these structureson semisimple locally symmetric spaces. The AHS–structures are large class ofgeometric structures, which can be described as one graded parabolic geometriesand cover the cases of projective, conformal and many other interesting geometricstructures. We will use the classification and description of AHS–structures as it issummarized in [3].1.1.
Locally symmetric spaces.
Firstly, we review a description of locally sym-metric spaces using the language of Cartan geometries. This will use the fact thatthe Cartan geometries preserves many properties of the model geometries, whichare in this case symmetric spaces. The description of symmetric spaces using Car-tan geometries is summarized in [4]. This allows us to use the following definitionof locally symmetric spaces.
Definition.
A locally symmetric space is a locally flat Cartan geometry ( p : G →
M, ω ) of type (
K, H, h ), where • M is a smooth, connected manifold • K is a Lie group with Lie subgroup H , K/H connected, h ∈ H such that h = id K and H is open in the centralizer of h in K • the maximal normal subgroup of K contained in H is trivial • G is a principal H -bundle over M • ω is a Cartan connection, i.e. a k -valued one form on G , which is H -equivariant, reproduces the fundamental vector fields and provides isomor-phism T G = G × k .We say that a locally symmetric space of type ( K, H, h ) is semisimple if K is semisimple, and we say that the homogeneous model ( K → K/H, ω K ) is asymmetric space, where ω K is the Maurer-Cartan form. Mathematics Subject Classification.
Key words and phrases. semisimple locally symmetric spaces, invariant geometric structures,almost hermitian symmetric structures, one graded parabolic geometries, classification.
Since the Cartan geometry is locally flat, the geometry is locally isomorphic to anopen subset of the homogeneous model ( K → K/H, ω K ). In particular, there is anatlas of M with images in K/H such, that the transition maps are restriction of leftmultiplication of elements of K . This allows us to locally define local symmetriesby setting S kH lH := khk − lH . These are local automorphisms of the Cartangeometry, which obviously does not depend on the chosen map from the atlas.Consequently, one can classify up to local equivalence the locally symmetricspaces in the following manner: Proposition 1.
The locally symmetric space of the type ( K, H, h ) is up to localequivalence determined by the pair ( k , h ) .Proof. This follows from the fact that K → K/H is locally equivalent to its simplyconnected covering. (cid:3)
We note that in [2], the equivalence class corresponding to pair ( k , h ) is called agerm of symmetric spaces. As we will see later, for our area of interest, the exacttype ( K, H, h ) plays a important role, but for many question, we will restrict to thepair ( k , h ).The classification of pairs ( k , h ), which are also called symmetric Lie algebras,was in the semisimple case presented in [1] and here we recall the main classificationtheorem: Proposition 2.
Each semisimple symmetric Lie algebra is a finite sum of sym-metric Lie algebras of the following two types (which are called simple): • ( h ⊕ h , h ) (we denote them simply by h ), where h is a simple Lie algebraand the second h is the diagonal in h ⊕ h . • ( k , h ) , where k is a simple Lie algebra. The pairs ( k , h ) of all possible casescan be found in the table in [1] . Extension of Cartan geometries.
Here, we summarize the functorial con-struction called extension, which will allow us to construct every Cartan geometryon locally symmetric space invariant to local symmetries.
Definition.
Let P be a Lie subgroup of a Lie group G and let H be a Lie subgroupof a Lie group K . We say, that pair a ( i, α ) is an extension of ( K, H ) to (
G, P ) ifit satisfies: • i : H → P is a Lie group homomorphism • α : k → g is a linear mapping extending T e i : h → p • α induces a vector space isomorphism of k / h and g / p • Ad ( i ( h )) ◦ α = α ◦ Ad ( h ) for all h ∈ H i.e. α is a homomorphism of therepresentations Ad ( H ) and Ad ( i ( H )) | Im( α ) Having an extension ( i, α ) of Cartan geometry ( p : G →
M, ω ) of type (
K, H ),one forms bundle
G × i P → M and defines one form ω α by ω α | G× i i ( H ) = α ◦ ω and ω α | i − ( P ) × i P = ω P and extending it by right P action to be equivariant. Thegeneral theory of [3] and results in [4] allows us to formulate the following theorem: Theorem 1.
Let ( i, α ) be an extension of ( K, H ) to ( G, P ) and ( p : G →
M, ω ) aCartan geometry of type ( K, H ) . Then the pair ( G × i P → M, ω α ) defined above is aCartan geometry of type ( G, P ) and any local automorphism of the Cartan geometry ( p : G →
M, ω ) is a local automorphism of the Cartan geometry ( G × i P → M, ω α ) .Any Cartan geometry of type ( G, P ) on locally symmetric space ( p : G →
M, ω ) of type ( K, H, h ) invariant to local symmetries is constructed using an extension of ( K, H ) to ( G, P ) . LASSIFICATION OF INVARIANT AHS–STRUCTURES ON SEMISIMPLE LOCALLY SYMMETRIC SPACES3
Proof.
In [3] and [4], it is shown that the first claim is indeed true. The secondclaim is there proved globally. This means that generally any Cartan geometry oftype (
G, P ) invariant to local symmetries is only locally isomorphic to an extensionof (
K, H ) to (
G, P ). But since the group K acts locally transitively and by localsymmetries, the second claim also holds. (cid:3) Thus in particular, the group H plays a role for the existence of the extension.However, we get the following corollary of proposition 1, because the covering is infact extension to ( K, H ) and we can compose extensions.
Corollary 1.
There is extension of ( K, H ) to ( G, P ) only if there is extension ofthe simply connected covering of ( K, H ) to ( G, P ) . Holomorphic Cartan geometries.
Finally, we add few remarks about holo-morphic Cartan geometries from [3], which are Cartan geometries (
G →
M, ω ) oftype (
G, P ), where (
G, P ) are complex Lie groups,
G → M is a holomorphic princi-pal P -bundle and ω is a holomorphic Cartan connection. The curvature of Cartangeometry of type ( G, P ) splits as a real two form to κ = κ (2 , + κ (1 , + κ (0 , ac-cording to linearity and anti-linearity with respect to the complex structure. Thereis the following characterization of holomorphic Cartan geometries of type ( G, P ): Proposition 3.
Let ( G →
M, ω ) be Cartan geometry of type ( G, P ) , where ( G, P ) are complex Lie groups. Then it is holomorphic Cartan geometry of type ( G, P ) ifand only if κ = κ (2 , i.e. κ is complex bilinear. The easy consequence of the formula for the curvature of an extension is:
Corollary 2.
Extension of locally symmetric space of type ( K, H, h ) to Cartangeometry of type ( G, P ) , where ( G, P ) are complex Lie groups, is a holomorphicCartan geometry if and only if ( K, H ) are complex Lie groups and α is complexlinear. If this is the case, there is always a non equivalent extension given bycomplex conjugation. AHS–structures.
Now, we summarize basic results on AHS–structures andone graded parabolic geometries from [3], where one can find more details.Let G be a semisimple Lie group and P parabolic subgroup corresponding to thegrading g = g − ⊕ g ⊕ g , where we assume that there is no nonzero ideal of g in g . Further, we denote G the subgroup of grading preserving elements of P . Thestructure of one gradings is the following according to [3]: Proposition 4.
Let g = g − ⊕ g ⊕ g be a one graded Lie algebra. Then • g is a sum of one graded simple Lie algebras g ( j ) ; • the decomposition of g -module g − to irreducible components is given by g − = ⊕ j g ( j ) − ; • the only non isomorphic one gradings of simple complex and real Lie alge-bras are those in the table 1; • there is equivalence of categories between AHS–structures of type ( G, P ) and regular normal one graded parabolic geometries of type ( G, P ) . We use the following symbols and abbreviations in the table 1: g is simple Lie algebra with one grading g = g − + g + g g − is given as a representation of ad ( g ) on g − in terms of fundamental rep-resentations. JAN GREGOROVIˇC g g g − sl ( n, R ) sl ( p, R ) + sl ( q, R ) + R λ p − ⊗ λ sl ( n, C ) sl ( p, C ) + sl ( q, C ) + C λ p − ⊗ λ sl ( n, H ) sl ( p, H ) + sl ( q, H ) + R λ p − ⊗ λ su ( n, n ) sl ( n, C ) + R λ + ¯ λ sp (2 n, R ) sl ( n, R ) + R λ sp (2 n, C ) sl ( n, C ) + C λ sp ( n, n ) sl ( n, H ) + R λ so ( p + 1 , q + 1) so ( p, q ) + R λ so ( n + 2 , C ) so ( n, C ) + C λ so ( p, p ) sl ( n, R ) + R λ so (2 n, C ) sl ( n, C ) + C λ so ⋆ (4 n ) sl ( n, H ) + R λ e ( EI ) so (5 ,
5) + R λ e ( EIV ) so (9 ,
1) + R λ e C ( E ) so (10 , C ) + C λ e ( EV ) e + R λ e ( EV II ) e + R λ e C ( E ) e C + C λ Table 1.
Types of AHS–structures2.
Construction of AHS–structures on locally symmetric spaces
Now we investigate, how the extensions ( i, α ) of a symmetric space (
K, H, h ) to aAHS–structure of type (
G, P ) can look like up to equivalence. The following lemmacharacterizes, when two extension leads to locally equivalent geometries, look in [4]for proof.
Lemma 1.
Let ( i, α ) and (ˆ i, ˆ α ) be two extension of ( K, H ) to ( G, P ) . Then theextended geometries are locally equivalent if and only if there are p ∈ P and aLie algebra automorphism σ of K preserving H such, that ˆ i ( σ ( h )) = p i ( h ) p − and ˆ α = Ad p − ◦ α ◦ T σ . The first result highly restricts possible AHS–structures on locally symmetricspaces.
Theorem 2.
There is an invariant AHS–structure of type ( G, P ) on a locally sym-metric space of type ( K, H, h ) if and only if there is an invariant G -structure onthe locally symmetric space. The theorem is the direct consequence of the following proposition and Theorem1, because Ad ( G ) is a subgroup of Gl ( g − ). Proposition 5.
An invariant AHS–structure of type ( G, P ) on a locally symmetricspace ( K, H, h ) is equivalent to an extension ( i, α ) of ( K, H ) to ( G, P ) such that • i ( h ) = g ; • i ( H ) ⊂ G ; • α is given for X in the − eigenspace of Ad ( h ) as follows: LASSIFICATION OF INVARIANT AHS–STRUCTURES ON SEMISIMPLE LOCALLY SYMMETRIC SPACES5 α ( X ) − is an arbitrary isomorphism of the adjoint representations Ad ( H ) and Ad ( i ( H )) , α ( X ) = 0 , and α ( X ) is an arbitrary morphismof the adjoint representations.Proof. Let ( i, α ) be extension of (
K, H, h ) to AHS–structure of type (
G, P ), then i ( h ) = g exp( Z ) for g ∈ G and Z ∈ g , and e = i ( h ) = g exp( Z ) g exp( Z ) = g exp( Ad ( g ) Z ) exp( Z ) = g exp( − Z + Z ) = g so we have to assume, that there is g ∈ G , g = e . Now, we change the extensionby conjugation by exp( Z ) ∈ P and get equivalent extension according to the abovelemma. Then i ( h ) = exp( Z ) g exp( Z ) exp( − Z ) = g exp( Ad ( g )( Z )) exp( Z ) = g , thus we can assume that i ( h ) = g without loss of generality.Since h commutes with H , the equality g p exp( Y ) = p exp( Y ) g has to be satisfied for all p exp( Y ) ∈ i ( H ). Thus Y = 0 for all p exp( Y ) ∈ i ( H )and i ( H ) ⊂ G .Now α ( Ad ( h ) X ) = Ad ( i ( h )) α ( X ) for X in the − Ad ( h ), thus − α ( X ) = Ad ( g ) α ( X ) . Let us decompose α ( X ) = α ( X ) − + α ( X ) + α ( X ) according to the grading of g .Then the comparison of both sides provides us restriction α ( X ) = 0 . (cid:3) Now we solve, when two AHS–structures on a locally symmetric space are locallyequivalent.
Theorem 3.
On a locally symmetric space of type ( K, H, h ) , there is a bijectionbetween: • equivalence classes of invariant AHS–structure of type ( G, P ) (up to outerautomorphisms of the Lie group Ad ( H ) induced by automorphisms of K ). • pairs consisting of a conjugacy class of inclusions i of H to G and a classof elements of the centralizer of i ( H ) in Gl ( g − ) contained in Gl ( g − ) /G . The theorem is a simple consequence the following technical proposition andLemma 1.
Proposition 6.
For a symmetric space ( K, H, h ) , there is a bijection between: • extensions of ( K, H ) to AHS–structure of type ( G, P ) such, that i ( h ) = g • couples β, b , where β is a frame of the − eigenspace g − of Ad ( h ) such,that the inclusion i β : H → Gl ( g − ) induced by the frame β is contained in G , and b is an endomorphism of g commuting with i β ( Ad ( H )) .Two frames of the − eigenspace of Ad ( h ) determine the same homomorphism i : H → G if and only if the transition map between them commutes with i ( H ) .Two frames of the − eigenspace of Ad ( h ) determine equivalent AHS–structuresof type ( G, P ) if and only if the transition map between them is composition of ele-ments of P (in fact G ) and outer automorphisms of the Lie group Ad ( H ) inducedby automorphisms of K .Proof. The first two claims immediately follows from the previous theorem andlemma, and the same holds for the third claim in all cases except (H–)projectivestructures, because in these cases the b -part of the extension plays no role in thequestion of equivalence.In the case of (H–)projective structures, for a fixed frame β , the b -part is de-termined from the normality conditions using formula and notation from [4] on the JAN GREGOROVIˇC curvature κ :0 = ( ∂ ∗ κ )([ e, e ])( a i X i + p ) = X i [ Z i , [ α ( a j X j ) , α ( X i )] − α ([ a j X j , X i ])]= X i,j a j ([ Z i , [ X j + b ( X j ) , X i + b ( X i )] − α ([ X j , X i ])])= X i,j a j (( b ji − nb ij ) Z i − [ Z i , [ α ([ X j , X i ])])where X i is vector in g − with 1 on i-th row and rest 0, Z i ∈ g is covector with 1on i-th column and rest 0 and b ( X j ) = b ij Z i . So we get system of linear equationsand we know, there always has to be at least one solution. The homogeneous partsof the equations are (1 − n ) b ii = 0, b ji − nb ij = 0 and b ij − nb ji = 0. Clearly, b = 0is the only solution of the homogeneous part. Thus there is unique b and the lastclaim follows. (cid:3) Since the classification of the semisimple symmetric spaces is known, we canclassify all AHS–structures on them. Moreover, a simple consequence of the classi-fication of semisimple locally symmetric spaces is, that the adjoint representationof H on the − k / h of Ad ( h ), which is complementary to h in k , iscompletely reducible. Thus: Proposition 7.
Let ( K, H, h ) be semisimple symmetric space. Then Z Gl ( k / h ) ( Ad ( H )) is product of centralizers Z Gl ( k i / h i ) ( Ad ( H i )) of simple factors ( K i , H i , h i ) of ( K, H, h ) . The possible centralizers are described in the case of simple symmetric spaces in[2]. The only possibilities are R , C , R × R and C × C .Finally, there is a large class of AHS–structures, where the question of equiva-lence is trivial. This is the result of [5] summarized in the following proposition. Proposition 8.
The AHS–structures invariant to local symmetries are torsion-freeand if the only component of the harmonic curvature is torsion, then the AHS–structures are unique up to equivalence (because they are locally flat).
So firstly we investigate the structures, which allow non-trivial curvature andthen the rest. 3.
Non-flat invariant AHS–structures
Projective and H–projective structures.
The (H–)projective structurescorrespond to the following grading of g = sl ( n + 1 , K ) , where K = R for projectiveor C for H–projective structures: (cid:18) a ZX A (cid:19) , where A ∈ gl ( n, K ), a = − tr ( A ), X ∈ K n and Z ∈ ( K n ) ∗ .The corresponding effective homogeneous model has G = P Gl ( n + 1 , K ) and g = (cid:18) − E (cid:19) . Let (
K, H, h ) be a (complex) symmetric space (see [2] for details about complexsymmetric spaces). Then the choice of any frame β of the − Ad ( h )provides i β : Ad ( H ) → Gl ( n, K ) = G , and it is obvious, that all frames β provideequivalent extensions. Thus: LASSIFICATION OF INVARIANT AHS–STRUCTURES ON SEMISIMPLE LOCALLY SYMMETRIC SPACES7
Proposition 9.
There is (up to equivalence) unique projective structure on anysemisimple locally symmetric space. There is a H–projective structure on any com-plex semisimple locally symmetric space. The non-equivalent (up to outer auto-morphisms of the Lie group Ad ( H ) induced by automorphisms of K ) H–projectivestructures are given by corollary 2. Conformal structures.
The (complex) conformal structures correspond tothe following grading of g = so ( p + 1 , q + 1) or g = so ( n + 2 , C ): a Z X A − I p,q Z T − X T I p,q − a , where A ∈ so ( p, q ), a ∈ R , X ∈ R p + q , Z ∈ ( R p + q ) ∗ and is diagonal matrix I p,q with ± p, q ), or A ∈ so ( n, C ), a ∈ C , X ∈ C n , Z ∈ ( C n ) ∗ and I p,q is identity matrix.The corresponding effective model has G = P O ( p + 1 , q + 1) or G = P O ( n + 2 , C )and g = − E
00 0 − . On a semisimple symmetric space (
K, H, h ), we can restrict the Killing form B : k ⊗ k → R to the H -invariant complement k / h of h . This defines a non-degenerate Ad ( H )-invariant symmetric bilinear form on T e K/H , which defines aninclusion i : H → O ( n − p, p ), where p = dim( C ) − dim( C ∩ H ) and C is themaximal compact subgroup of K .Moreover, the Killing form B provides an Ad ( H )-invariant bijection gl ( k / h ) → ( k / h × k / h ) ∗ , X B ( X · , · ) . In particular, B ( X · , · ) is a non-degenerate Ad ( H )-invariant symmetric bilinear formon T e K/H if and only if X is an element of the centralizer of i ( H ) in Gl ( g − )contained in Gl ( g − ) /O ( n − p, p ). Thus: Proposition 10.
For any semisimple locally symmetric space of type ( K, H, h ) holds: • The simple factors are orthogonal to each other with respect to any invariantmetric and the only non-degenerate invariant symmetric bilinear form on asimple factor is a real multiple (a complex multiple if B is complex linear)of the Killing form B . • There is unique (up to conjugacy) Lie group homomorphism i : H → G = CO ( p, q ) if and only if there are multiples of the Killing forms of the simplefactors such that the resulting form has signature ( p, q ) . There is bijectionbetween the conjugacy classes of conformal structures on the locally sym-metric space and ( S ) r , where r is the number of the simple factors withcomplex linear Killing form. • There is a complex conformal structure if and only if there is a complexstructure on ( K, H, h ) . The non-equivalent (up to outer automorphisms ofthe Lie group Ad ( H ) induced by automorphisms of K ) complex conformalstructures are given by corollary 2.Proof. The first claim clearly follows from description of elements of the centralizerof i ( H ) in Gl ( g − ) contained in Gl ( g − ) /O ( n − p, p ). It follows from [2], that onlythose in claim are possible. Second claim clearly follows, because we are taking Gl ( g − ) /CO ( p, q ) instead of Gl ( g − ) /O ( p, q ). The third claim then clearly followsfrom Lemma 2. (cid:3) JAN GREGOROVIˇC
Quaternionic structures.
The quaternionic structures correspond to thefollowing grading of g = sl ( n + 1 , H ): (cid:18) a ZX A (cid:19) , where A ∈ gl ( n, H ), a ∈ H , Re( a ) + Re( tr ( A )) = 0, X ∈ H p + q and Z ∈ ( H p + q ) ∗ .The corresponding effective model has G = P Gl ( n + 1 , H ) and g = (cid:18) − E (cid:19) . We will assume that n >
1, because sl (2 , H ) ∼ = so (5 ,
1) and the parabolic geom-etry is in fact conformal geometry, which we already discussed.
Example.
Quaternionic structure on ( so ∗ (2 n + 2) , so ∗ (2) ⊕ so ∗ (2 n )).If we look in classification of simple symmetric spaces, SO ∗ (2 n ) acts by a quater-nionic representation i.e. there is i : SO ∗ (2 n ) → Gl ( n, H ). Further SO ∗ (2) acts bymultiples of − k ∈ Sp (1) from left i.e. SO ∗ (2) × SO ∗ (2 n ) sits uniquely up to conju-gation in G := P ( Sp (1) × Gl ( n, H )). In fact, we immediately get a flat quaternionicstructure on SO ∗ (2 n + 2) /SO ∗ (2) × SO ∗ (2 n ) just by inclusion of SO ∗ (2 n + 2) to P Gl ( n + 1 , H ).We show that there is a quaternionic structure on any pseudo–quaternionic–K¨ahler symmetric space and there are no quaternionic structures on other semisim-ple symmetric spaces except the previous example. Proposition 11.
There is a quaternionic structure on any pseudo–quaternionic–K¨ahler locally symmetric space and there are no quaternionic structures on othersemisimple locally symmetric spaces except ( so ∗ (2 n + 2) , so ∗ (2) ⊕ so ∗ (2 n )) . Thestructure is unique up to equivalence.Proof. Let (
K, H, h ) be a semisimple homogeneous symmetric space and assumethat the image of i is contained in Gl ( n, H ). Then the representation of Ad ( i ( H )) isof quternionic type and there is no such in the classification of semisimple symmetricspaces. The same is true in the case that the image of i is contained in the partgiven by a ∈ H . So the image of i has intersection with both parts, but thisimplies that the representation of Ad ( i ( H )) is irreducible and going through thelist of simple symmetric spaces we check that the only possibilities are pseudo-quaternionic-K¨ahler symmetric spaces (where Sp (1) × Sp ( p, q ) trivially sits in G )and the previous example. Since i ( H ) acts by irreducible representations, its imageis unique, and the centralizer of i ( H ) in Gl ( g − ) contained Gl ( g − ) /G is trivial. (cid:3) Para-quaternionic structures.
The para-quaternionic structures correspondto the following grading of g = sl ( n + 2 , R ): (cid:18) a ZX A (cid:19) , where A ∈ gl ( n, R ), a ∈ gl (2 , R ), tr ( a ) + tr ( A ) = 0, X ∈ R n ⊗ ( R ) ∗ and Z ∈ R ⊗ ( R n ) ∗ .The corresponding effective model has G = P Gl ( n + 2 , R ) and g = − − E . We will assume that n >
2, because sl (4 , R ) ∼ = so (3 ,
3) and the parabolic geom-etry is in fact conformal geometry, which we already discussed.
LASSIFICATION OF INVARIANT AHS–STRUCTURES ON SEMISIMPLE LOCALLY SYMMETRIC SPACES9
Example.
Para-quaternionic structures on ( so ( k + 1 , l + 1) , so (1 , ⊕ so ( k, l )) and( so ( k + 2 , l ) , so (2) ⊕ so ( k, l )).First we notice that SO ( n + 2) /SO (2) × SO ( n ) is equivalent to the homogeneousmodel of ( G, P ). Thus SO (2) × SO ( n ) sits uniquely up to conjugation in G := P ( Gl (2 , R ) × Gl ( n, R )). Since it does not matter, which signature the matrices have,we immediately get a flat para-quaternionic structure on SO ( k +1 , l +1) /SO (1 , × SO ( k, l ) and SO ( k, l + 2) /SO (2) × SO ( k, l ) just by inclusion.We show that there is a para-quaternionic structure on any pseudo-para-quaternionic-K¨ahler symmetric space and there are no para-quaternionic structures on othersemisimple symmetric spaces except the previous examples. Proposition 12.
There is a para-quaternionic structure on any pseudo-para-quaternionic-K¨ahler locally symmetric space and there are no para-quaternionic structures onother semisimple locally symmetric spaces except ( so ( k + 1 , l + 1) , so (1 , ⊕ so ( k, l )) and ( so ( k + 2 , l ) , so (2) ⊕ so ( k, l )) . The structure is unique up to equivalence.Proof. Let (
K, H, h ) be a semisimple homogeneous symmetric space and assumethat the image of i is contained in Sl ( n, R ). Then the representation of Ad ( i ( H ))decomposes to two copies of standard representation of sl ( n, R ) and there is nosuch in the classification of semisimple symmetric spaces. The same is true in thecase that the image of i is contained in the part given by a ∈ gl (2 , R ). So theimage of i has intersection with both parts, but this implies that the representationof Ad ( i ( H )) is irreducible and going through the list of simple symmetric spaceswe check that the only possibilities are pseudo-para-quaternionic-K¨ahler symmetricspaces (where Sp (2 , R ) × Sp (2 n, R ) trivially sits in P ( Gl (2 , R ) × Gl (2 n, R ))) andthose in previous example. Since i ( H ) acts by irreducible representations, its imageis unique, and the centralizer of i ( H ) in Gl ( g − ) contained Gl ( g − ) /G is trivial. (cid:3) Flat invariant AHS–structures
Many of these symmetric spaces and structures on them are described in [2] inlanguage of algebraic geometry. We present complete list of such structures.4.1.
General Grassmannian structures.
The Grassmannian structures of type( p, q ) correspond to the following grading of g = sl ( p + q, K ), where p > , q > K = R , or p > , q > K = C , or p > , q > K = H : (cid:18) A ZX B (cid:19) , where A ∈ gl ( p, K ), B ∈ gl ( q, K ), Re( tr ( a )) + Re( tr ( A )) = 0, X ∈ K q ⊗ ( K p ) ∗ and Z ∈ K q ⊗ ( K p ) ∗ .The corresponding effective model has G = P Gl ( p + q, K ) and g = (cid:18) − E p E q (cid:19) , where E p , E q are identity matrices of sizes p × p and q × q .We show that general Grassmannian structures are only on simple symmetricspaces and list them. Proposition 13.
The only locally symmetric spaces admitting a (unique) generalGrassmannian structure are the following types of simple symmetric spaces: ( so ( a + b, c + d ) , so ( a, c ) ⊕ so ( b, d )) , K = R of type ( a + c, b + d ) ; ( sp (2( p + q ) , R ) , sp (2 p, R ) ⊕ sp (2 q, R )) , K = R of type (2 p, q ) ; ( so ( p + q, C ) , so ( p, C ) ⊕ so ( q, C )) , K = C of type ( p, q ) , holomorphic Cartangeometry; ( sp ( p + q, C ) , sp ( p, C ) ⊕ sp ( q, C )) , K = C of type ( p, q ) , holomorphic Cartangeometry; ( so ∗ (2( p + q )) , so ∗ (2 p ) ⊕ so ∗ (2 q )) , K = H of type (2 p, q ) ; ( sp ( a + b, c + d ) , sp ( a, c ) ⊕ sp ( b, d )) , K = H of type ( a + c, b + d ) ; ( su ( a + b, c + d ) , su ( a, c ) ⊕ su ( b, d )) ⊕ so (2) , K = C of type ( a + c, b + d ) .Proof. Let (
K, H, h ) be a semisimple homogeneous symmetric space and assumethat the image of i is contained in Sl ( p, K ) or Sl ( q, K ). Then the representation of Ad ( i ( H )) decomposes to several copies of standard representation of sl ( p, R ) andthere is no such in the classification of semisimple symmetric spaces. So the imageof i has intersection with both parts, but this implies that the representation of Ad ( i ( H )) is irreducible and going through the list of simple symmetric spaces wesee that the only those in the proposition act by the prescribed representations of G . (cid:3) We note that there the following types of symmetric spaces with Grassmannianstructures, which are not semisimple. The inclusion to G is given by the adjointrepresentation. R ⊕ sl ( n, R ), K = R of type ( n, n ); C ⊕ sl ( n, C ), K = C of type ( n, n ), holomorphic Cartan geometry; R ⊕ sl ( n, H ), K = H of type ( n, n ).4.2. General Lagrangean structures.
The Lagrangean structures correspondto the following grading of g = sp (2 n, K ), where K = R , or K = C , or g = sp ( n, n )in the case K = H : (cid:18) − A T ZX A (cid:19) , where A ∈ gl ( n, K ), , X ∈ S K n and Z ∈ ( S K n ) ∗ .The corresponding effective model has G = Sp (2 n, K ) ⋊ Z or Sp ( n, n ) ⋊ Z ,because g = (cid:18) − E n E n (cid:19) , where E n is n × n identity matrix, is not in the connected component of identityof G .We show that general Lagrangean structures are only on simple symmetric spacesand list them. Proposition 14.
The only locally symmetric spaces admitting a (unique) generalLagrangean structure are the following types of simple symmetric spaces: ( sp (2 n, C ) , sp (2 n, R )) , K = R ; sp (2 n, R ) , K = R ; sp ( n, C ) , K = C , holomorphic Cartan geometry; ( sp ( n, C ) , sp ( p, q )) , K = H ; sp ( p, q ) , K = H ; ( sp (2 n, R ) , su ( p, q ) ⊕ so (2)) , K = C ; ( sp ( p, q ) , su ( p, q ) ⊕ so (2)) , K = C .Proof. Let (
K, H, h ) be a semisimple homogeneous symmetric space and assume i ( H ) ⊂ G = Gl ( n, K ). Since the representation of i ( H ) is completely reducible,there is invariant complement to any invariant subspace simple factor of i ( H ) in S K n . Since due to structure of S K n there is at most one invariant subspacewith non-trivial action of i ( H ), the symmetric space has to be simple. Then goingthrough the list of simple symmetric spaces we see that the only those in theproposition act by the prescribed representations of G except last two examples. LASSIFICATION OF INVARIANT AHS–STRUCTURES ON SEMISIMPLE LOCALLY SYMMETRIC SPACES11
The last two examples corresponds to the maximal compact subgroup of sp ( n, C )and the signature and the real form of k plays no role. (cid:3) Of course, if we add abelian factors (which are mapped to diagonal in S K n ) tocertain types of simple symmetric spaces we obtain the following symmetric spaceswith general Lagrangean structures:( R ⊕ su ( p, q ) , so ( p, q )), K = R ;( R ⊕ sl ( p + q, R ) , so ( p, q )), K = R ;( C ⊕ sl ( n, C ) , so ( n, C )), K = C , holomorphic Cartan geometry;( R ⊕ su ( n, n ) , so ∗ (2 n )), K = H ;( R ⊕ sl ( n, H ) , so ∗ (2 n )), K = H .4.3. General spinorial structures.
The spinorial structures correspond to thefollowing grading of g = so ( n, n ) for K = R , g = so (2 n, C ) for K = C , or g = so ⋆ (4 n )for K = H : (cid:18) − A T ZX A (cid:19) , where A ∈ gl ( n, K ), , X ∈ V K n and Z ∈ ( V K n ) ∗ .The corresponding effective model has G = SO ( n, n ) ⋊ Z , SO (2 n, C ) ⋊ Z or SO ∗ (4 n ) ⋊ Z , because g = (cid:18) − E n E n (cid:19) , where E n is n × n identity matrix, is not in the connected component of identityof G .We show that general spinorial structures are only on simple symmetric spacesand list them. Proposition 15.
The only locally symmetric spaces admitting a (unique) generalspinorial structure are the following types of simple symmetric spaces: so ( p, q ) , K = R ; ( so ( n, C ) , so ( p, q )) , K = R ; so ( n, C ) , K = C , holomorphic Cartan geometry; ( so (2 n, C ) , so ∗ (2 n )) , K = H ; so ∗ (2 n ) , K = H ; ( so (2 p, q ) , su ( p, q ) ⊕ so (2)) , K = C ; ( so ∗ (2 n ) , su ( p, q ) ⊕ so (2)) , K = C .Proof. Let (
K, H, h ) be a semisimple homogeneous symmetric space and assume i ( H ) ⊂ G = Gl ( n, K ). Since the representation of i ( H ) is completely reducible,there is invariant complement to any invariant subspace simple factor of i ( H ) in V K n . Since due to structure of V K n there is at most one invariant subspacewith non-trivial action of i ( H ), the symmetric space has to be simple. Then goingthrough the list of simple symmetric spaces we see that the only those in theproposition act by the prescribed representations of G except last two examples.The last two examples corresponds to the maximal compact subgroup of so (2 n, C )and the signature and the real form of k plays no role. (cid:3) Of course, if we add abelian factors to certain types of simple symmetric spaceswe obtain the following symmetric spaces with general spinorial structures:( u ( n, n ) , sp (2 n, R )), K = R ;( gl ( n, R ) , sp (2 n, R )), K = R ;( gl (2 n, C ) , sp (2 n, C )), K = C , holomorphic Cartan geometry;( gl ( n, H ) , sp ( p, q )), K = H ;( u (2 p, q ) , sp ( p, q )), K = H . su ( p, p ) -Cartan geometries. The su ( p, p )-Cartan geometries structures cor-respond to the following grading of g = su ( p, p ): (cid:18) − A T ZX A (cid:19) , where A ∈ sl ( n, C ), and X, Z ∈ u ( n ).The corresponding effective model has G = SU ( n, n ) ⋊ Z , because g = (cid:18) − E n E n (cid:19) , where E n is n × n identity matrix, is not in the connected component of identityof G .We show that su ( p, p )-Cartan geometries are only on simple symmetric spacesand list them. Proposition 16.
The only locally symmetric spaces admitting a (unique) su ( p, p ) -Cartan geometry are the following types of simple symmetric spaces: ( so ( n, n ) , so ( n, C )) ; ( so ∗ (2 n ) , so ( n, C )) ; ( sp ( n, n ) , sp ( n, C )) ; ( sp (2 n, R ) , sp ( n, C )) , ( sl (2 n, R ) , sl ( n, C ) ⊕ so (2)) ; ( sl ( n, H ) / sl ( n, C ) ⊕ so (2)) .Proof. Let (
K, H, h ) be a semisimple homogeneous symmetric space and assume i ( H ) ⊂ G = S ( Gl ( n, C )). Since the representation of i ( H ) is completely reducible,there is invariant complement to any invariant subspace simple factor of i ( H ) in u ( n ). Since due to structure of u ( n ) there is at most one invariant subspace withnon-trivial action of i ( H ), the symmetric space has to be simple. Then goingthrough the list of simple symmetric spaces we see that the only those in theproposition act by the prescribed representations of G except last two examples.For the last two examples sl ( n, C ) is mapped to real part of G = sl (2 n, C ) and so (2) is mapped to complex diagonal. (cid:3) Of course, if there is a su ( p, p )-Cartan geometry on u ( p, q ), which is not semisim-ple.4.5. Exceptional structures.
There are six AHS–structures corresponding to ex-ceptional Lie groups. We discuss them only briefly.For the first three G = SO (5 , G = SO (9 ,
1) or G = SO (10 , C ) and actsby spin representation. Consequently such structures can exist only on simplesymmetric spaces and going trough the list of simple symmetric spaces we obtainthe following lists:For G = SO (5 , f , so (5 , sp (4 , R ) , sp (2 , R ) ⊕ sp (2 , R )); ( sp (2 , , sp (1 , ⊕ sp (1 , sp (4) , sp (2) ⊕ sp (2)); ( sp (2 , , sp (2) ⊕ sp (2));For G = SO (9 , f , so (9)); ( f , so (8 , f , so (9)); ( sp (3 , , sp (1 , ⊕ sp (2))For G = SO (10 , C ):( f C , so (9 , C )), holomorphic Cartan geometry; ( sp (4 , C ) , sp (2 , C ) ⊕ sp (2 , C )), holo-morphic Cartan geometry; ( e , so (10) ⊕ so (2)); ( e , so (10) ⊕ so (2));The inclusions i : H → G are in above cases induced by inclusions of maximalcompact subgroups and we use the isomorphism so (5 , C ) ∼ = sp (2 , C ) (and corre-sponding isomorphism of real forms).For the second three G = E , G = E or G = E C and acts by standardrepresentation. Consequently such structures can exist only on simple symmetricspaces and going trough the list of simple symmetric spaces we obtain the followinglists, where we also add symmetric spaces with these structures coming from simplesymmetric spaces after adding abelian factor): LASSIFICATION OF INVARIANT AHS–STRUCTURES ON SEMISIMPLE LOCALLY SYMMETRIC SPACES13
For G = E :( su (8) , sp (4)); ( su (4 , , sp (2 , su (4 , , sp (4 , R )); ( sl (4 , H ) , sp (4)); ( sl (4 , H ) , sp (2 , R ⊕ e , f ); ( R ⊕ e , f );For G = E :( su (6 , , sp (3 , sl (4 , H ) , sp (3 , R ⊕ e , f ); ( R ⊕ e , f ); ( R ⊕ e , f ); ( R ⊕ e , f );For G = E C :( sl (8 , C ) , sp (4 , C ), holomorphic Cartan geometry; ( C ⊕ e C , f C ), holomorphic Car-tan geometry; ( e , e × so (2)); ( e / e × so (2));The inclusions i : H → G are in above cases induced by inclusions of maximalcompact subgroups. Acknowledgements
The author has been supported by the grant GACR 201/09/H012.
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Jan Gregoroviˇc, Department of Mathematics, Faculty of Sciences and Technology,Aarhus University, Ny Munkegade 118, 8000 Aarhus C, Denmark
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