Classification of pseudo-Riemannian symmetric ...

Classification of pseudo-Riemannian symmetric spaces of quaternionic Kähler type Dmitri V. Alekseevsky Dept. of Mathematics, University of Hull, Cottingham Road, Hull, HU6 7RX, UK [email protected]

Vicente Cortés ´ Cartan de Mathématiques, Université Henri Poincaré - Nancy 1, B.P. 239, Institut Elie F-54506 Vandœuvre-lès-Nancy Cedex, France [email protected]

March 29, 2004 Dedicated to Arkadij L’vovich Onishchik on the occassion of his seventeenth birthday.

Abstract A pseudo-Riemannian symmetric space M = G/H of dimension 4n > 4 is called quaternionic Kähler if its holonomy group Hol is a subgroup of Sp1 · Spp,q , where p + q = n, and it is called para-quaternionic Kähler if Hol ⊂ Sp1 (R) · Spn (R). We study the structure of pseudo-Riemannian quaternionic Kähler and para-quaternionic Kähler symmetric spaces with non-zero scalar curvature as homogeneous spaces. In particular, it is proven that they have simple isometry group G. The classification of such spaces is reduced to the description of appropriate involutive automorphisms of complex simple Lie algebras, which is given in terms of Kac diagrams.

This work was supported by the ‘Schwerpunktprogramm Stringtheorie’ of the Deutsche Forschungsgemeinschaft.

Contents 1 Introduction

1

2 Definitions, basic facts and statement of the classification problem

6

2.1

²-quaternionic Kähler manifolds . . . . . . . . . . . . . . . . . . . . . . . .

6

2.2

²-quaternionic Kähler symmetric spaces . . . . . . . . . . . . . . . . . . . .

8

3 Simplicity of the isometry group of ²-quaternionic K¨ ahler symmetric spaces with non-zero scalar curvature 9 4 ²-quaternionic K¨ ahler symmetric spaces 4.1

Reduction of the classification of ²-quaternionic Kähler symmetric spaces to an algebraic problem about involutive automorphisms . . . . . . . . . . 12

5 Classification of pseudo-quaternionic K¨ ahler symmetric spaces 5.1

5.2

1

12

14

Classification of pseudo-quaternionic Kähler symmetric spaces of inner type 14 5.1.1

Involutive inner automorphisms of complex simple Lie algebras . . . 14

5.1.2

Inner real forms of pseudo-quaternionic type . . . . . . . . . . . . . 22

Classification of pseudo-quaternionic Kähler symmetric spaces of outer type 25 5.2.1

Involutive outer automorphisms of complex simple Lie algebras

. . 25

5.2.2

Outer real forms of pseudo-quaternionic type . . . . . . . . . . . . . 28

Introduction

A quaternionic structure in a 4n-dimensional real vector space V is a three-dimensional linear Lie algebra Q ⊂ End(V ) with a basis J1 , J2 , J3 which satisfies the quaternionic relations J12 = J22 = J32 = −Id , J3 = J1 J2 = −J3 J2 . (1.1) Then H = R Id + Q is an associative algebra, isomorphic to the algebra H of quaternions and V has the structure of a left H-module. A 4n-dimensional Riemannian manifold (M, g), for n > 1, is called a quaternionic K¨ ahler manifold if it admits a parallel field M 3 x 7→ Qx ⊂ so(Tx M ) of skew-symmetric quaternionic structures. This is equivalent to the condition that the holonomy group Hol of (M, g) is a subgroup of Sp1 · Spn . Such a field Q is called a parallel quaternionic structure on (M, g). The generalization to pseudo-Riemannian case is straightforward: 1

A pseudo-Riemannian 4n-dimensional manifold (M, g), n > 1, is called a pseudoquaternionic Kähler manifold if it admits a parallel quaternionic structure Q ⊂ so(T M ) or, equivalently, if its holonomy group Hol ⊂ Sp1 · Spp,q , p + q = n. Similar to the quaternionic structure, one can define the notion of para-quaternionic structure Q in 4n-dimensional vector space V as three dimensional linear Lie algebra Q with a basis J1 , J2 , J3 which satisfies the para-quaternionic relations J12 = −Id ,

J22 = J32 = Id ,

J3 = J1 J2 = −J2 J1 .

(1.2)

Then the associative algebra H = R Id + Q is isomorphic to the algebra H0 ∼ = gl2 (R) of para-quaternions (or split quaternions). A pseudo-Riemannian 4n-dimensional manifold (M, g), n > 1, with the holonomy group Hol ⊂ Sp1 (R) · Spn (R) is called a para-quaternionic Kähler manifold. It implies that (M, g) admits a parallel field Q ⊂ so(T M ) of skew-symmetric para-quaternionic structures and that the metric g has signature (2n, 2n). Pseudo-quaternionic Kähler and para-quaternionic Kähler manifolds have many common features. To unify the notions, we define the notion of an ²-quaternionic Kähler manifold, for ² = ±1, where a −1-quaternionic Kähler manifold is a pseudo-quaternionic Kähler manifold and 1-quaternionic Kähler manifold is a para-quaternionic Kähler manifold. We will denote also the linear algebra spn (R) of the (reducible) linear group Spn (R) ⊂ SO2n,2n by sp1n and the linear Lie algebra spp,q of the group Spp,q ⊂ SO4p,4q by sp−1 n . Any ²-quaternionic Kähler 4n-dimensional manifold (M, g, Q) for n > 1 is an Einstein manifold. More precisely, the curvature tensor R has a decomposition R = νR0 + W

(1.3)

where R0 is the curvature tensor of the standard metric of the projective space H² P n over the algebra of quaternions H−1 = H or para-quaternionic H1 = H0 , ν is a constant, proportional to the scalar curvature and W is a traceless algebraic curvature tensor of type sp²n . We say that a 4-dimensional pseudo-Riemannian manifold (M, g) of signature (4, 0) (respectively, (2, 2)) is quaternionic Kähler (respectively, para-quaternionic Kähler) if its curvature tensor has the form (1.3). In other words, this means that (M, g) is an anti-selfdual Einstein manifold. An ²-quaternionic Kähler manifold (M, g, Q) is called an ²-quaternionic Kähler symmetric space if (M, g) is a pseudo-Riemannian symmetric space. In this case, we can identify M with a homogeneous space M = G/K , where G is the connected group of isometries, generated by transvections, and K is the stabilizer of a point o ∈ M . Quaternionic Kähler symmetric space were classified by J. Wolf [W]. The compact quaternionic Kähler symmetric space (called Wolf spaces) are in one-to-one correspondence with compact simple Lie groups G with trivial center. The Wolf space associated with the group G is given by M = G/N (sµ ) where N (sµ ) is the normalizer in G of the three dimensional subalgebra sµ associated with the highest root µ of the complex Lie algebra gC corresponding to G. 2

The classification of pseudo-quaternionic Kähler symmetric spaces with zero scalar curvature (i.e. pseudo-hyperKähler symmetric spaces) was obtained in [AC]. These spaces M are in one-to-one correspondence with equivalence classes of some quartic polynomials. They have nilpotent group G generated by transvections and commutative holonomy group K and can be identified with the coset space M = G/K. A similar classification of para-quaternionic Kähler symmetric spaces with zero scalar curvature was given in [ABCV]. The main aim of the paper is to describe the structure of simply connected ²-quaternionic (i.e. pseudo-quaternionic and para-quaternionic) Kähler symmetric spaces with non-zero scalar curvature and classify pseudo-quaternionic Kähler symmetric spaces. The problem is reduced to a classification of symmetric decompositions g = k + m of simple real Lie algebras g of ²-quaternionic type. This means that the stability subalgebra k contains a three-dimensional ideal s such that ads |m is an ²-quaternionic (i.e. quaternionic or para-quaternionic) structure on m. More precisely, let (M = G/K, g, Q) be a simply connected ²-quaternionic Kähler symmetric space with non-zero scalar curvature and g = k + m the corresponding symmetric decomposition of the Lie algebra g . We identify m with the tangent space To M at the canonical base point o = eK. Then the isotropy representation of K is identified with the adjoint representation AdK |m and the isotropy group AdK |m coincides with the holonomy group Hol. If the scalar curvature is not zero, the holonomy algebra hol = adk |m contains the quaternionic structure Qo ⊂ so(m). This implies that the symmetric decomposition is of ²-quaternionic type. In Section 2, we give the main definitions and formulate the classification problem. We prove that any ²-quaternionic Kähler symmetric space M = G/K has simple isometry group G in Section 3. Let hC be a Cartan subalgebra of a complex simple Lie algebra gC , Π a system of simple roots of the root system R of (gC , hC ) and µ ∈ R the highest root. We denote by Hµ the associated vector from hC , defined by α(Hµ ) = 2 hα,µi and by sµ = span{E−µ , Hµ , Eµ } the hµ,µi three-dimensional regular subalgebra of gC spanned by Hµ and the root vectors E±µ . The adjoint operator adHµ has eigenvalues k = ±2, ±1, 0 and the corresponding eigenspaces gk give a depth two gradation gC = g−2 + g−1 + g0 + g1 + g2 with g±2 = CE±2 . In Section 4, we reduce the problem of classification of ²-quaternionic Kähler symmetric spaces M = G/K, where G is a real form of a complex simple Lie group GC to the classification of involutive automorphisms θ of the complex simple Lie algebra gC = Lie(GC ) which preserve the Cartan subalgebra hC and satisfy the following condition: 1. θ(Hµ ) = −Hµ for ² = 1 and 2. θ = Id on sµ for ² = −1.

3

In Section 5, we give a classification of all involutive automorphisms θ of gC which preserve hC and act trivially on sµ in terms of involutive Kac diagrams. ˜ be the extended Dynkin diagram of the complex simple Lie algebra gC which Let Γ represents the extended simple root system ˜ = {α0 = −µ, α1 , · · · , α` } Π P` and µ = i=1 ni αi the decomposition of the highest root µ into a linear combination of simple roots. An involutive Kac diagram (of inner type) is defined as the extended ˜ with nodes αi labeled by non-negative integers pi such that Dynkin diagram Γ ` X

ni pi = 2,

i=0

→ − ˜ − where n0 = 1. We will denote an involutive Kac diagram as (Γ, p ), where → p = (p0 , · · · , p` ) is the vector of labels. − ˜ → An involutive Kac diagram (Γ, p ) defines an (inner) involutive automorphism θ = − → C θ( p ) of g with the ±1-eigenspace decomposition X X gC = gC+ + gC− = (hC + gα ) + gα α∈R0

where Ra = {α =

P` i=1

mi αi |

P` i=1

α∈R1

mi pi ≡ a (mod 2)}, for a = 0, 1.

The involution θ commutes with the standard antilinear involution τ of gC , which → preserves hC and defines a compact real form of gC . The composition θ ◦ τ =: σ = σ(− p) − → C is an antilinear involution. It defines a non-compact real form g( p ) of g , if θ 6= Id. → → The map − p 7→ g(− p ) gives a bijection between involutive Kac diagrams (of inner type) of a simple complex Lie algebra gC (up to an isomorphism of labelled graphs) and real forms of inner type of gC (up to a conjugation by an automorphism of gC ) [K, GOV]. → ˜ − Let (Γ, p ) be an involutive Kac diagram. We denote by Π the subset of the system 0

Π of simple roots which consists of roots αi with label pi = 0. Simply connected pseudoquaternionic Kähler symmetric spaces M = G/K where G is the simply connected Lie → group with the Lie algebra g(− p ) are described by the following theorem. Theorem 1 There is a bijection between connected components of Π0 which contain a long root α of Π (up to the action of automorphisms of (gC , gC+ ) on the simple summands of gC+ ) and pseudo-quaternionic Kähler symmetric spaces M = G/K of the group G (up to homothety). If α ∈ Π0 is a long root which belongs to a connected component Π00 of Π0 , then the pseudo-quaternionic symmetric space associated with Π00 is given by M (Π00 ) = G/K, where K = NG (sα ) is the normalizer of the tree-dimensional subalgebra sα of gC associated with α. We give a similar description of pseudo-quaternionic Kähler symmetric spaces M = G/K of a simple real Lie group G associated with an outer involutive automorphism 4

θ ∈ ρAdG of the complex simple Lie algebra gC , where ρ is the involutive automorphism associated with a non-trivial involution r of the Dynkin diagram and a system of canonical generators. The outer involutions are described in terms of the extended Dynkin diagram ˜ ρ of the ρ-root system, see [GOV]. In this case, the Kac diagrams, which determine an Γ outer involutive automorphism, are completely determined by a choice of a simple ρ-root βi with the label ni = 2. The final list of simply connected pseudo-quaternionic Kähler symmetric spaces is the following: Theorem 2 Let M = G/K be a simply connected symmetric pseudo-quaternionic K¨ ahler manifold. Then M is one of the following symmetric pseudo-quaternionic Kähler manifolds. A) SU(p + 2, q) , S(U(2) × U(p, q))

SL(n + 1, H) , S(GL(1, H) × GL(n, H))

BD)

SO∗ (2l + 4) , SO∗ (4) × SO∗ (2l)

SO0 (p + 4, q) , SO(4) × SO0 (p, q) C)

Sp(p + 1, q) , Sp(1) × Sp(p, q) E6)

E6(−78) , SU(2)SU(6)

E6(2) , SU(2)SU(6)

E6(2) , SU(2)SU(2, 4)

E6(6) , Sp(1)SL(3, H)

E6(−14) , SU(2)SU(2, 4)

E6(−26) , Sp(1)SL(3, H)

E7) E7(−5) E7(−5) E7(7) E7(−25) E7(−133) , , , , , ∗ SU(2)Spin(12) SU(2)Spin(12) SU(2)Spin0 (4, 8) SU(2)SO (12) SU(2)SO∗ (12) E8)

F4)

G2)

E8(−248) , SU(2)E7(133) F4(−52) , Sp(1)Sp(3)

E8(−24) , SU(2)E7(133) F4(4) , Sp(1)Sp(3)

E8(−24) , SU(2)E7(−5) F4(4) , Sp(1)Sp(1, 2)

G2(−14) , SO(4)

G2(2) . SO(4) 5

E8(8) , SU(2)E7(−5) F4(−20) , Sp(2)Sp(1, 2)

2

Definitions, basic facts and statement of the classification problem

2.1

²-quaternionic K¨ ahler manifolds

Definition 1 1. An ²-quaternionic structure (² = ±1) on a pseudo-Euclidean vector space V is a three-dimensional Lie subalgebra Q = Q² ⊂ so(V ) which admits a basis (J1 , J2 , J3 ) satisfying the ²-quaternionic relations, i.e. J3 = J1 J2 , J12 = −Id and J22 = J32 = ²Id. In both cases we will use the notation Jα2 = ²α Id and we denote by H² = R Id+Q² the associative algebra isomorphic to the algebra of quaternions for ² = −1 and paraquaternionions for ² = 1. We also denote by sp²n = Nso(V ) (Q² ) the normalizer of Q² in the pseudo-orthogonal Lie algebra so(V ), which is isomorphic to spn (R) = sp(R2n ) for ² = 1 and to spp,q for ² = −1, where 4n = dim V = 4p + 4q. 2. A pseudo-Riemannian manifold (M, g) of signature (k, l) is called strongly oriented if it is endowed with a reduction of the bundle of orthonormal frames to the connected component SO0 (k, l) ⊂ SO(k, l). 3. An ²-quaternionic Kähler manifold is a strongly oriented pseudo-Riemannian manifold (M, g) endowed with a subbundle Q ⊂ End(T M ) invariant under parallel transport such that Qp is an ²-quaternionic structure on (Tp M, gp ) (p ∈ M ) which preserves the curvature tensor of (M, g). 4. An ²-quaternionic Kähler manifold is called para-quaternionic Kähler if ² = +1 and pseudo-quaternionic Kähler if ² = −1. Remarks: 1. The dimension of an ²-quaternionic Kähler manifold is always divisible by 4: dim M = 4n. The signature of the metric is (2n, 2n) for para-quaternionic Kähler manifolds (as we shall prove below) and is (4p, 4q) for pseudo-quaternionic Kähler manifolds. 2. If n > 1 the condition that Q preserves the curvature tensor is automatically satisfied. 3. The assumption that (M, g) is strongly oriented is automatically fulfilled for pseudoquaternionic Kähler manifolds. Proposition 1 A pseudo-Riemannian manifold (M, g) admits a subbundle Q ⊂ End(T M ) such that (M, g, Q) is an ²-quaternionic Kähler manifold if and only if the holonomy group Hol Hol

⊂ Sp(1) · Sp(p, q) ⊂ SO(V ) , V = (C2 ⊗ C2n )ρ = Hn (² = −1) or (2.1) ⊂ Sp(R2 ) · Sp(R2n ) ⊂ SO(V ) , V = R2 ⊗ R2n (² = +1) , (2.2)

where ρ = jH ⊗ jE is the standard Sp(1) · Sp(p, q)-invariant real structure of C2 ⊗ C2n , which is the product of the invariant quaternionic structures jH and jE of the factors H = C2 = H and E = C2n = Hn . 6

Proof: It is clear that a pseudo-Riemannian manifold with holonomy as above is ²-quaternionic Kähler. We prove the converse statement. Let (M, g, Q) be an ²-quaternionic Kähler manifold. Then the holonomy group belongs to the normalizer of Q in the connected pseudo-orthogonal group SO0 (k, l) which is Sp(1) · Sp(p, q) (k = 4p, l = 4q) if ² = −1 and Sp(R2 ) · Sp(R2n ) if ² = +1. To see this in the second case, we first extend the representation of Q by Id to a representation of Q ⊕ R Id ∼ = R(2), the two-by-two matrix algebra. Since any representation of R(2) without trivial subrepresentation is a direct sum of standard representations, the holonomy representation can be factorized as R2 ⊗ E 0 , where E 0 is a real vector space. The metric g is a Q-invariant and Hol -invariant element of the Q-module S 2 (R2 ⊗ E 0 )∗ = S 2 R2 ⊗ S 2 (E 0 )∗ + ∧2 R2 ⊗ ∧2 (E 0 )∗ . Since S 2 R2 has no non-trivial Q-invariants, g can be written as g = ωH 0 ⊗ ωE 0 , where ωH 0 and ωE 0 are Q-invariant and Hol -invariant symplectic forms on the factors H 0 = R2 and E 0 = R2n , respectively. Now it is clear that the group of pseudo-orthogonal transformations which normalize Q is of the form Sp(R2 ) ⊗ Sp(R2n ) ⊂ GL(R2 ⊗ R2n ). Theorem 3 Any ²-quaternionic Kähler manifold M is Einstein and its curvature tensor R admits the following decomposition R = νR0 + W , where ν =

(2.3)

scal 4n(n+2)

is the reduced scalar curvature, Ã ! X 1 1X ²α hX, Jα Y iJα + X ∧Y − ² α Jα X ∧ Jα Y , R0 (X, Y ) := − 2 α 4 α

X, Y ∈ Tp M ,

is the curvature tensor of the ²-quaternionic projective space H² P n of the same signature as M and W is an algebraic curvature tensor of type sp²n . Proof: One can easily prove that trJα R(·, ·) = 2nνg(Jα ·, ·)

(2.4) ²

+ sp²n

(see [Be] for the quaternionic case) and that R0 is a curvature tensor of type Q with the reduced scalar curvature 1. Then the curvature tensor W = R − νR0 has zero reduced scalar curvature and the above formula shows that it has type sp²n , i.e. W (X, Y ) ∈ sp²n .

Corollary 1 Let M be an ²-quaternionic Kähler manifold with holonomy algebra hol and ν 6= 0. Then hol ⊃ sp²1 . Proof: Using the decomposition (2.3) of the curvature tensor and the formula 4n

Jα =

1X ²i Jα ei ∧ ei , 2 i=1 7

(2.5)

where hei , ej i = ²i δij , ²i = ±1, we calculate hol 3 R(Jα ) = R0 (Jα ) = nνJα . Theorem 4 An ²-quaternionic Kähler manifold (M, g, Q) with ν 6= 0 is locally indecomposable, i.e. is not locally a product of two pseudo-Riemannian manifolds of positive dimension. Proof: It is sufficient to show that the there is no hol-invariant proper subspace 0 6= E ⊂ V = Tp M (p ∈ M ) such that V = E ⊕ E ⊥ . This follows from the following formula νg(X, X)g(Y, Y ) = −g(R(X, J1 X)Y, J1 Y ) + ²g(R(X, J1 X)J2 Y, J3 Y ) .

(2.6)

In fact, it is sufficient to choose X ∈ E and Y ∈ E ⊥ such that g(X, X)g(Y, Y ) 6= 0. Such vectors exist, since the scalar product is non-degenerate on E and E ⊥ . Then the left-hand side of (2.6) is non-zero, since ν 6= 0. We claim that the right-hand side of (2.6) is zero. The subspaces E, E ⊥ ⊂ V are Q-invariant, by Corollary 1. In particular, we have that X, J1 X ∈ E and Y, J1 Y, J2 Y, J3 Y ∈ E ⊥ . The right-hand side of (2.6) can now be computed using the Bianchi identity: −g(R(X, J1 X)Y, J1 Y ) = g(R(J1 X, Y )X, J1 Y ) + g(R(Y, X)J1 X, J1 Y ) = 0 , by the hol-invariance of E. Similarly, one shows that g(R(X, J1 X)J2 Y, J3 Y ) = 0. This proves that (M, g) is locally indecomposable. Now it remains to prove the formula (2.6). Let us decompose the curvature operators, according to Theorem 3: R(X, Y ) = −

1X ρα (X, Y )Jα + W (X, Y ) , 2

X, Y ∈ V ,

where W (X, Y ) commutes with Q and ρα (·, Jα ·) = νg. This implies, in particular, that νg(X, X)g(Y, Y ) = = = =

ρ1 (X, J1 X)g(Y, Y ) = −²ρ1 (X, J1 X)g(J2 Y, J2 Y ) −²g([R(X, J1 X), J3 ]Y, J2 Y ) −²g(R(X, J1 X)J3 Y, J2 Y ) + ²g(J3 R(X, J1 X)Y, J2 Y ) ²g(R(X, J1 X)J2 Y, J3 Y ) − g(R(X, J1 X)Y, J1 Y ) .

Here we have used that ²1 = −1 and ²2 = e3 = ².

2.2

²-quaternionic K¨ ahler symmetric spaces

Definition 2 An ²-quaternionic Kähler symmetric space is an ²-quaternionic Kähler manifold (M, g) which is a pseudo-Riemannian symmetric space, that is for any point x ∈ M there is an isometry Sx of M (called the central symmetry) which fixes x and has the differential d(Sx )|x = −Id. 8

The connected component G of the group generated by central symmetries acts transitively on M and it is called the group generated by transvections. If K is the stabilizer of a point o ∈ M in this group, we can identify M with the quotient M = G/K. The conjugation with respect to the central symmetry So of a point o ∈ M defines an involutive automorphism σ of the group G and its Lie algebra g. The eigenspace decomposition g = k + m of g with respect to σ is a symmetric decomposition. The +1eigenspace k is the Lie algebra of the stability subgroup K of the point o. We identify M with the quotient space G/K and the AdK -invariant subspace m with the tangent space To M = TeK G/K. Then the isotropy representation of K in To M is identified with the adjoint representation AdK |m . It is known that the isotropy group AdK |m coincides with the holonomy group of M . We will always identify an ²-quaternionic Kähler symmetric space M with the homogeneous space M = G/K where G is the group generated by transvections and K is the stabilizer of a point o and we will assume that the manifold M is simply connected. For the rest of this paper, we will study the following classification problem. Problem: Classify ²-quaternionic Kähler symmetric spaces.

3

Simplicity of the isometry group of ²-quaternionic K¨ ahler symmetric spaces with non-zero scalar curvature

Lemma 1 Let M = G/K be an ²-quaternionic Kähler symmetric space and g = k + m the corresponding symmetric decomposition. If M has non-zero scalar curvature then adk |m ⊃ sp²1 . Proof: Since the holonomy Lie algebra hol coincides with the isotropy Lie algebra adk |m the lemma follows from Corollary 1. Corollary 2 Let M = G/K be an ²-quaternionic Kähler symmetric space and g = k+m the corresponding symmetric decomposition. Then k = sp²1 ⊕ k0 ,

k0 = Zg (sp²1 ) ,

[sp²1 , g] = sp²1 + m

and

adk0 |m ⊂ sp²n .

(3.1)

Theorem 5 Let M = G/K be an ²-quaternionic Kähler symmetric space with effective action of the Lie group G and reduced scalar curvature ν 6= 0. Then G is simple. Proof: We first prove that G is semisimple. Let g = s + r be a Levi-Malcev decomposition such that s ⊃ sp²1 . Then we have the following decompositions s r ² Zs (sp1 ) Zr (sp²1 )

= = = =

Zs (sp²1 ) + [sp²1 , s] , Zr (sp²1 ) + [sp²1 , r] , k0 ∩ s =: k0s , [sp²1 , s] = sp²1 + ms , ms := m ∩ [sp²1 , s] = m ∩ s , k ∩ r =: kr , [sp²1 , r] = m ∩ r =: mr . 9

(3.2) (3.3) (3.4) (3.5)

So putting ks := sp²1 ⊕ k0s = k ∩ s we have s = ks + ms ,

r = kr + mr ,

k = ks + kr = (sp²1 ⊕ k0s ) + kr

and m = ms + mr . (3.6)

Since mr = [sp²1 , r] ⊂ nilrad(g) =: n the operators adX are nilpotent for all X ∈ mr . For X ∈ mr and Y ∈ m we have that a := [X, Y ] ∈ k ∩ n ⊂ k ∩ r = kr = Zr (sp²1 ) and hence [ada |m , Jα ] = 0. This implies tr (Jα ◦ ada |m ) = 0 since Jα ◦ ada |m is the product of two commuting endomorphisms one of which is nilpotent. To finish the proof of the semisimplicity we use the next lemma. Lemma 2

Any ²-quaternionic Kähler symmetric space of dimension 4n satisfies ¡ ¢ tr Jα ◦ ad[X,Y ] |m = −2nνhJα X, Y i for all X, Y ∈ m .

Proof: This follows from (2.4) and Theorem 3 using the formula R(X, Y ) = −ad[X,Y ] |m

for all X, Y ∈ m

for the curvature tensor of a symmetric space. The lemma applied to X ∈ mr now shows that ¡ ¢ 0 = tr Jα ◦ ad[X,Y ] |m = −2nνhJα X, Y i for all Y ∈ m and hence mr = 0. Here we have again used that ν 6= 0. Finally r = kr ⊂ k has to vanish by effectivity. This finishes the proof of the semisimplicity of G. It remains to prove that G is simple. For this we need some facts about semisimple graded Lie algebras. Let gC be a complex semisimple Lie algebra and gC = gC−d + · · · + gC−1 + gC0 + gC1 + · · · + gCd a grading of depth d. The endomorphism A of gC defined by A|gCk = k Id is a derivation. Since all derivations of a semisimple Lie algebra are of inner type we have A = adD for some D ∈ gC . The element D determines the grading and is called the grading element. Lemma 3 (i) Let gC be a complex simple Lie algebra, µ ∈ (hC )∗ a long root with respect to a Cartan subalgebra hC ⊂ gC and Hµ ∈ hC defined by the condition α(Hµ ) = 2 hα,µi hµ,µi for all α ∈ (hC )∗ . Then Hµ is the grading element of a depth two grading of gC with gC±2 = CE±µ , where E±µ are the corresponding root vectors. Moreover, sµ := span{E−µ , Hµ , Eµ } is the three-dimensional subalgebra associated with µ and gC0 = CHµ ⊕ ZgC (sµ ). Conversely, any depth two grading of gC with dim gC±2 = 1 is of this form. 10

(ii) Let D be the grading element of a depth two grading of a complex semisimple Lie algebra gC such that dim gC±2 = 1, [g−2 , g2 ] = CD and gC0 does not contain any non-trivial ideal of gC (“effectivity assumption”). Then gC is simple and D = Hµ . Proof: (i) It is well known that Hµ defines a depth two grading with stated properties. We prove only the converse statement. Any grading of a complex semisimple Lie algebra is defined by a choice of non-negative labels mi for each simple root αi such that deg(Eαi ) = P mi and deg(h) = 0. If µ = ni αi is the decomposition of the highest root P with respect to simple roots, then the depth of the grading is given by d = deg(Eµ ) = ni mi . In the case d = 2 there are only three possibilities. (a) mk = 1, nk = 2, (b) mp = mq = 1, np = nq = 1 and (c) mk = 2, nk = 1, where all other mi = 0. The condition dim gC±2 P = 1 implies that deg(Eµ−αi ) < 2 for all αi for which µ − αi is a root. We can write µ = ki πi as linear combination of fundamental weights. Then µ − αi is a root if and only if ki > 0. In fact, from hπj , αi i = 21 hαi , αi iδij we obtain X 1 hµ, αi i = kj hπj , αi i = hαi , αi iki > 0 if and only if ki > 0 . 2 This implies that mi > 0 if ki > 0. With exception of the A series the highest root is always proportional to a fundamental weight πk with nk = 2 so we are in case (a). For Al we have µ = π1 + πl with all ni = 1 and hence m1 = ml = 1 and we are in case (b). The case (c) never occurs. In all cases we obtain a unique labeling which gives the standard grading by Hµ . (ii) Let hC be a Cartan subalgebra of gC0 . It contains D and is a Cartan subalgebra of gC . Since [hC , gC±2 ] = gC±2 , the complex lines gC±2 are root spaces of some simple ideal g˜C ⊂ gC . Hence [g−2 , g2 ] = CD ⊂ g˜C . This implies that all other ideals of gC belong to gC0 and hence vanish by effectivity. Lemma 4 Let M = G/K be an ²-quaternionic Kähler symmetric space as above and h ∈ sp²1 such that the corresponding isotropy operator I = adh |m satisfies I 2 = Id or I 2 = −Id. Then D = h or D = ih, respectively, is the grading element of a depth two grading gC = gC−2 + gC−1 + gC0 + gC1 + gC2 , (3.7) of the complexified Lie algebra gC such that dim gC±2 = 1, [g−2 , g2 ] = CD, kC = gCev = gC−2 + gC0 + gC2 and mC = gCodd = gC−1 + gC1 . In the case D = h the grading is defined over R, i.e. on g. In both cases sp1 (C) = sp²1 ⊗ C = gC−2 + CD + gC2 . This lemma shows that the grading associated with h ∈ sp²1 satisfies the assumptions of Lemma 3 (ii). So we obtain that G is simple. This finishes the proof of Theorem 5. Corollary 3 Under the assumptions of the previous lemma, there exists a Cartan subalgebra hC ⊂ gC0 of gC and a long root µ of (gC , hC ) such that D = Hµ and gC±2 = CE±µ . Proof: This follows from Lemma 3 (i). 11

4 4.1

²-quaternionic K¨ ahler symmetric spaces Reduction of the classification of ²-quaternionic K¨ ahler symmetric spaces to an algebraic problem about involutive automorphisms

Now we reduce the classification problem for ²-quaternionic Kähler symmetric spaces to the description of involutive automorphisms of a complex simple Lie algebra gC which preserve the three-dimensional subalgebra sµ ⊂ gC associated with a highest root µ. Let M = G/K be an ²-quaternionic Kähler symmetric space and g = k + m = (sp²1 ⊕ k0 ) + m the corresponding symmetric decomposition. By Lemma 4 any element h ∈ sp²1 such that the corresponding isotropy operator I = adh |m satisfies I 2 = Id or I 2 = −Id defines a depth two grading of gC with grading element D = h or ih. Thanks to Corollary 3, we can identify D = Hµ , gC±2 = CE±µ and sp²1 ⊗ C = sµ . The antilinear involution (i.e. involutive automorphism) σ of the Lie algebra gC defined by complex conjugation with respect to g preserves sµ and σHµ = Hµ if Hµ = h ∈ sp²1 and σHµ = −Hµ if Hµ = ih ∈ isp²1 . This implies that σgCj = gCj or σgCj = gC−j , respectively. In both cases σ preserves gCev = kC and gCodd = mC . Conversely, any antilinear involution σ of gC , which preserves sµ defines an ²-quaternionic Kähler symmetric space. We denote by Aut(gC , sµ ) the group of automorphisms which preserves sµ . Proposition 2 The above correspondence M = G/K 7→ σ induces a bijection between simply connected ²-quaternionic Kähler symmetric spaces (up to homothety) and antilinear involutions σ of gC (up to conjugation by elements of Aut(gC , sµ )) which preserve the threedimensional subalgebra sµ ⊂ gC associated with a long root µ and have the fixed point set sσµ ∼ = sp²1 . Proof: We prove that any antilinear involution σ as above defines an ²-quaternionic Kähler symmetric space. By Lemma 3 (i) there exists a depth two grading gC = gC−2 + gC−1 + gC0 + gC1 + gC2 defined by Hµ such that sµ = gC−2 + CHµ + gC2 . Since σ is an automorphism of sµ we may assume, without loss of generality, that σHµ = ±Hµ . This implies that σ preserves kC := gCev = sµ ⊕ ZgC (sµ ), (k0 )C := ZgC (sµ ) and mC := gCodd . We have sσµ = sp²1 for ² = ±1. We denote by g, k, k0 and m the fixed point sets of σ on gC , kC , (k0 )C and mC , respectively, and by M = G/K the simply connected symmetric space associated with the symmetric decomposition g = k + m. Here G is the simply connected Lie group with Lie algebra g and K ⊂ G is the connected Lie subgroup generated by k. We will prove that M is an ²-quaternionic Kähler symmetric space. The Killing form of g induces a k-invariant C scalar product g on m. Up to scale, g is the restriction of the Killing form B g |mC =: g C to mC . The latter is invariant under sµ . This shows that the pseudo-Riemannian metric g is invariant under sp²1 . Since adHµ |mC has eigenvalues ±1, we can identify mC = C2 ⊗ C2n such that sµ ∼ = sp1 (C) acts in the standard way on the first factor C2 Then it is clear the real subalgebra sp²1 ⊂ End(mC ) has a basis (J1 , J2 , J3 ) which satisfies the ²-quaternion relations. Moreover sp²1 preserves the real form m of mC and hence defines on m an ²-quaternionic structure 12

Q = sp²1 ⊂ End(m). Since the isotropy representation of the symmetric space M = G/K preserves Q, this defines on M the structure of an ²-quaternionic Kähler symmetric space (M, g, Q). It is clear that antilinear involutions σ and σ 0 = φ ◦ σ ◦ φ−1 where φ ∈ Aut(gC , sµ ) define homothetic ²-quaternionic Kähler symmetric spaces. Conversely, if g = sp²1 + k0 + m and g0 = sp²1 0 + k00 + m0 are two symmetric decompositions of ²-quaternionic type, which define homothetic ²-quaternionic Kähler symmetric spaces, then there is an isomorphism φ : g → g0 which is consistent with these decompositions. It is extended to a complex linear automorphism φ : gC → gC which preserves sµ = (sp²1 )C = (sp²1 0 )C . Next we reduce the description of antilinear involutions as in Proposition 2 to complex linear involutions. Lemma 5 Let σ be an antilinear involution of a complex simple Lie algebra gC which preserves sµ . Then there exists a compact (i.e. such that the real form (gC )τ is compact) antilinear involution τ which commutes with σ, satisfies τ Hµ = −Hµ and, in the case ² = −1, coincides with σ on sµ . Moreover, there exists a Cartan subalgebra h ⊂ (gC )τ , which is invariant under σ. Its complexification hC ⊂ gC is a Cartan subalgebra invariant under σ and τ and contains Hµ . Proof: We can assume that σHµ = ²Hµ . There exists a Cartan decomposition g := (gC )σ = k + m, where k ⊂ g is a maximal compact subalgebra, such that Hµ ∈ m if ² = +1 and iHµ ∈ k if ² = −1. In the case ² = −1 we choose k such that k ⊃ (sµ )σ = sp²1 = sp1 . We define τ by (gC )τ = k + im. Then τ commutes with σ and τ Hµ = −Hµ . This implies that τ preserves the three-dimensional subalgebra sµ = g−2 + CHµ + g2 . Moreover, in the case ² = −1, we have τ = σ on sµ . Now we can choose a Cartan subalgebra h ⊂ (gC )τ of the form h = hk + ihm , where hk = h ∩ k and hm = h ∩ m, such that Hµ ∈ hm if ² = +1 and iHµ ∈ hk if ² = −1. It has the claimed properties. Let us fix a Cartan subalgebra hC and a long root µ of the simple Lie algebra gC . We denote by τ the standard compact antilinear involution of (gC , hC ). Notice that any involutive automorphism θ of the complex Lie algebra gC which commutes with τ defines a real form g = g(θ) := (gC )σ , where σ = θ ◦ τ . The real form g(θ) is called of inner (respectively, outer) type if θ is an inner (respectively, outer) automorphism. Theorem 6 There is one-to-one correspondence between ²-quaternionic Kähler symmetric spaces M = G/K (up to homothety) and involutive automorphisms θ (up to conjugation by admissible elements of Aut(gC )) of gC which preserve hC , commute with τ and satisfy θHµ = −²Hµ and, in the case ² = −1, θ|sµ = Id, i.e. sµ ⊂ (gC )θ . (An element g ∈ Aut(gC ) is admissible if θg = g ◦ θ ◦ g −1 satisfies the same conditions as θ, i.e. θg hC = hC , [θg , τ ] = 0, etc.) Proof: It is sufficient to establish a correspondence between anti-linear involutions σ as in Proposition 2 and involutive automorphisms θ as in the statement of the theorem. Let σ be an anti-linear involution of gC which preserves the three-dimensional subalgebra sµ ⊂ gC and has the fixed point set sσµ = sp²1 . Thanks to the previous Lemma, up to conjugation 13

of σ by an element of Aut(gC ), we can assume that σ preserves the Cartan subalgebra hC , commutes with τ , satisfies σHµ = ²Hµ and, in the case ² = −1, coincides with τ on sµ . Then θ := στ is an involutive automorphism which commutes with τ , preserves hC , satisfies θHµ = −²Hµ and, in the case ² = −1, verifies θ|sµ = Id. Conversely, let θ be an involutive automorphism of gC which preserves hC , commutes with τ , satisfies θHµ = −²Hµ and, in the case ² = −1, θ|sµ = Id. Then σ := θτ is an anti-linear involution which preserves sµ and has the fixed point set sσµ = sp²1 . In fact, if ² = −1 then σ = τ on sµ , which shows that (sµ )σ = (sµ )τ = sp1 . If ² = +1 then σHµ = Hµ , which shows that (sµ )σ is non-compact, since it contains the non-compact element Hµ . This Theorem reduces the problem of classification of ²-quaternionic Kähler symmetric spaces to the problem of classification of involutive automorphisms of a simple complex Lie algebra gC which preserve the Cartan subalgebra hC and satisfy the following conditions: ² = 1: θHµ = −Hµ ² = −1: θ = Id on sµ . As we shall see, this can be done using the Kac diagram of θ.

5 5.1 5.1.1

Classification of pseudo-quaternionic K¨ ahler symmetric spaces Classification of pseudo-quaternionic K¨ ahler symmetric spaces of inner type Involutive inner automorphisms of complex simple Lie algebras

We recall the description of involutive automorphisms of inner type of a complex simple Lie algebra gC in terms of Kac diagrams. Let R be the root system of a complex simple Lie algebra gC with respect to a Cartan subalgebra hC , R+ a system of positive roots and Π = {α1 , . . . , αl } the corresponding ˜ := Π ∪ {α0 } the extended system of simple system of simple roots of R. We denote by Π roots, where α0 = −µ is the lowest root. We denote by ni the coordinates of µ with respect to Π, l X µ= ni αi , i=1

and put n0 = 1. The extended system of simple roots is represented by the extended Dynkin diagram with the labels ni over the nodes associated with αi , see the table 3 [GOV]. Definition 3 An involutive Kac diagram of a semisimple Lie algebra gC is the extended Dynkin diagram of gC labeled by non-negative integers pi over the nodes αi , which satisfy the relation l X ni pi = 2 . i=0

14

An involutive Kac diagram defines a decomposition R = R0 ∪ R1 , where Ra := {α =

l X

ki αi ∈ R |

X

ki pi ≡ a (mod 2)} ,

i=1

for a = 0, 1. Notice that R0 is a root system and that its Dynkin diagramm Π0 is obtained from the Kac diagramm by deleting all nodes which have non-zero labels. We denote by − → p = (p0 , · · · , pl ) the (l + 1)-vector with the components pi . Note that any semi-simple inner automorphism of a complex semisimple Lie algebra gC is of the form Adg , g = exp x, where x is an element of a Cartan subalgebra hC ⊂ gC . V. Kac described all x ∈ hC which define an automorphism of finite order using Kac diagrams, see [GOV]. The following theorem is a special case. Theorem 7 There exists a natural one-to-one correspondence between involutive Kac diagrams, up to automorphisms of the extended Dynkin diagram, and involutive inner automorphisms of gC , up to conjugation by automorphisms of gC . More precisely, an involutive Kac diagram with labels pi defines an involutive automorphism θ with eigenspace decomposition gC = gC+ + gC− , such that θ|gC± = ± Id, where gC+ = hC + gC− =

X

X

CEα

(5.1)

α∈R0

CEα .

(5.2)

α∈R1

Applying this theorem, for each simple complex Lie algebra we will now give the list of involutive automorphisms. We indicate also the corresponding non-compact Riemannian symmetric space S(θ) = G/G+ associated with the Cartan decomposition g = g+ + g− ,

(5.3)

where g = g(θ) = (gC )σ is the real form of gC associated with the antilinear involution σ = θ ◦ τ and g± = gC± ∩ g. Recall that τ is the standard compact antilinear involution associated with the Cartan subalgebra hC . Corollary 4 The non-trivial involutive automorphisms of the simple complex Lie algebras gC (up to conjugation in Aut(gC )) are defined by the following involutive Kac diagrams. For each case, the corresponding system of positive roots R0+ = R0 ∩ R+ of gC+ is indicated together with the symmetric space S(θ). Al ) System of positive roots: R+ = {²i − ej | 1 ≤ i < j ≤ l + 1}, Simple roots: αi = ²i − ²i+1 , 15

Pl

Highest root: µ = ²1 − ²l+1 = ˜ Extended Dynkin diagram Π:

i=1

→ αi , i.e. − n = (1, . . . , 1), 1 •

© αH ©© 0 HH

©© © 1 © 1 •© •· · ·

α1

HH H

1 H 1 · · · • H• αl−1 αl

α2

AIII.m) Kac diagram: 1

©©

• ©©HHH © HH ©©

HH H

1 •· · · • αm

•©

· · ·•

•

− → p (m) = (1, 0, . . . 0, 1, 0, . . . , 0), where 1 ≤ m ≤ l, | {z } m+1

R0+ = {²i − ²j | i < j ≤ m

or

j > i ≥ m + 1},

S(θ) = SU(m, n − m)/S(U(m) × U(n − m)), where n = l + 1. Bl ) System of positive roots: R+ = {²i ± ej | 1 ≤ i < j ≤ l + 1} ∪ {²i | i = 1, . . . , l}, Simple roots: αi = ²i − ²i+1 , i = 1, . . . , l − 1, αl = ²l , P − Highest root: µ = ²1 + ²2 = α1 + 2 li=2 αi , i.e. → n = (1, 1, 2, . . . , 2), ˜ Extended Dynkin diagram Π: 1 •H α0 HH•2 1 ©© •© α2 α1

2 •· · · α3

2 2 ··· • • αl−2 αl−1

2 • > αl

BI.1) Kac diagram: 1 •H

H

− → p = (1, 1, 0 . . . 0),

H• 1©©© •

•· · ·

R0+ = {²i ± ²j | 2 ≤ i < j} ∪ {²i | i = 1, . . . , l}, S(θ) = SO0 (2, 2l − 1)/SO(2) × SO(2l − 1). BI.m) Kac diagram:

16

··· •

•

• >

•H

HH • ©© © •

•· · ·

1 • αm

··· •

•

• >

− → p (m) = (0, . . . , 0, 1, 0, . . . , 0), where 2 ≤ m ≤ l, | {z } m+1

R0+

= {²i ± ²j | i < j ≤ m

or

j > i ≥ m + 1} ∪ {²i | i ≥ m + 1},

S(θ) = SO0 (2m, 2(l − m) + 1)/SO(2m) × SO(2(l − m) − 1). Cl ) System of positive roots: R+ = {²i ± ej | 1 ≤ i < j ≤ l} ∪ {2²i | i = 1, . . . , l}, Simple roots: αi = ²i − ²i+1 , i = 1, . . . , l − 1, αl = 2²l , P → − Highest root: µ = 2²1 = 2 l−1 i=1 αi + αl , i.e. n = (1, 2, . . . , 2, 1), ˜ Extended Dynkin diagram Π: 1 • α0

2 • > α1

2 •· · · α2

2 2 ··· • • < αl−2 αl−1

1 • αl

CI) Kac diagram: 1 •

• >

•· · ·

··· •

•
i ≥ m + 1} ∪ {2²i | i ≥ l},

S(θ) = Sp(m, l − m)/Sp(m) × Sp(l − m). Dl ) System of positive roots: R+ = {²i ± ²j | 1 ≤ i < j ≤ l}, Simple roots: αi = ²i − ²i+1 , i = 1, . . . , l − 1, αl = ²l−1 + ²l , P − → Highest root: µ = ²1 + ²2 = α1 + 2 l−2 i=2 αi + αl−1 + αl , i.e. n = (1, 1, 2, . . . , 2, 1, 1), ˜ Extended Dynkin diagram Π: 1 •H α0 HH•2 1 ©© •© α2 α1

1

2 •· · · α3 17

• ©©

2 2© αl−1 ··· • •H αl−3 αl−2HH1 • αl

DI.1) Kac diagram: 1 •H

HH • 1©©©

•· · ·

··· •

•

− → p = (1, 1, 0, . . . , 0),

©• © © •H HH •

R0+ = {²i ± ²j | 2 ≤ i < j}, S(θ) = SO0 (2, 2l − 2)/SO(2) × SO(2l − 2). DI.m) Kac diagram: •H

•

1 • αm

HH ©• ©©

··· • •· · · • − → p (m) = (0, . . . , 0, 1, 0, . . . , 0), where 2 ≤ m ≤ l − 2, | {z }

©© © •H HH •

m+1

R0+ = {²i ± ²j | i < j ≤ m

or

j > i ≥ m + 1},

S(θ) = SO0 (2m, 2l − 2m)/SO(2m) × SO(2l − 2m). DIII) Kac diagram: 1 •H

•

HH ©• ©©

•· · ·

··· •

•

− → p = (1, 0, . . . , 0, 1),

© ©© •H HH1 •

R0+ = {²i − ²j | i < j}, S(θ) = SO∗ (2l)/U(l). E6 ) System of positive roots: R+ = {²i − ²j , ²i + ²j + ²k + ² , 2² | 1 ≤ i < j < k ≤ 6}, Simple roots: αi = ²i − ²i+1 , i = 1, . . . , 5, α6 = ²4 + ²5 + ²6 + ², → Highest root: µ = 2² = α + 2α + 3α + 2α + α + 2α , i.e. − n = (1, 1, 2, 3, 2, 1, 2), 1

2

3

4

5

6

˜ Extended Dynkin diagram Π:

1 • α1

2 • α2

3 •

α3

2 • α6 1 • α0

18

2 • α4

1 • α5

EII) Kac diagram: 1 •

•

•

•

•

• • − → p = (0, 0, 1, 0, 0, 0, 0), R0+ = {²1 − ²2 , ²i − ²j | 3 ≤ i < j ≤ 6} ∪ {²1 + ²2 + ²k + ² | 3 ≤ k ≤ 6} ∪ {²i + ²j + ²k + ² | 3 ≤ i < j < k ≤ 6} ∪ {2²}, S(θ) = E6(2) /SU(2)SU(6). EIII) Kac diagram: 1 •

•

•

1 •

•

• • − → p = (0, 1, 0, 0, 0, 1, 0), R0+ = {²1 −²6 , ²i −²j , ²i +²j +²6 +² , 2² | 2 ≤ i < j ≤ 5}∪{²1 +²j +²k +² | 2 ≤ j < k ≤ 5}, S(θ) = E6(−14) /SO(2)Spin(10). E7 ) System of positive roots: R+ = {²i − ²j | 1 ≤ i < j ≤ 7} ∪ {−²i + ²8 | i < 8} ∪{²i + ²j + ²k + ²8 | 1 ≤ i < j < k ≤ 7}, Simple roots: αi = ²i − ²i+1 , i = 1, . . . , 6, α7 = ²5 + ²6 + ²7 + ²8 , → Highest root: µ = −² + ² , − n = (1, 1, 2, 3, 4, 3, 2, 2), 7

8


1 • α1

2 • α2

3 • α3

4 •

α4

2 • α7 EV) Kac diagram:

19

3 • α5

2 • α6

1 • α0

•

•

•

•

•

•

•

•

•

•

1• − → p = (0, 0, 0, 0, 0, 0, 0, 1), R0+ = {²i − ²j | 1 ≤ i < j ≤ 7} ∪ {−²i + ²8 | i < 8}, S(θ) = E7(7) /SU(8). EVI) Kac diagram: 1 •

•

•

• •

− → p = (0, 0, 1, 0, 0, 0, 0, 0), R0+ = {²1 − ²2 , ²i − ²j | 3 ≤ i < j ≤ 7} ∪ {−²k + ²8 , ²1 + ²2 + ²k + ²8 | 3 ≤ k ≤ 7} ∪{²i + ²j + ²k + ²8 | 3 ≤ i < j < k ≤ 7}, S(θ) = E7(−5) /SU(2)Spin(12). EVII) Kac diagram: 1 •

•

•

•

•

1 •

•

• − → p = (1, 1, 0, 0, 0, 0, 0, 0), R0+ = {²1 − ²7 , ²i − ²j | 2 ≤ i < j ≤ 7} ∪ {²i + ²j + ²k + ²8 | 2 ≤ i < j < k ≤ 7}, S(θ) = E7(−25) /SO(2)E6 . E8 ) R+ = {²i − ²j | 1 ≤ i < j ≤ 9} ∪ {²i + ²j + ²k | 1 ≤ i < j < k ≤ 9}, Simple roots: αi = ²i − ²i+1 , i = 1, . . . , 7, α8 = ²6 + ²7 + ²8 , → Highest root: µ = ² − ² , − n = (1, 2, 3, 4, 5, 6, 4, 2, 3), 1

9


1 • α0

2 • α1

3 • α2

4 • α3

5 • α4

6 •

α5

3 • α8 EVIII) Kac diagram: 20

4 • α6

2 • α7

•

•

•

•

•

•

•

1 •

• − → p = (0, 0, 0, 0, 0, 0, 0, 1, 0), R0+ = {²i − ²j , ²i + ²j + ²8 | 1 ≤ i < j ≤ 7} ∪ {²i − ²9 , ²i + ²8 + ²9 | i ≤ 7}, S(θ) = E8(8) /Spin(16). EIX) Kac diagram: 1 •

•

•

•

•

•

•

•

• − → p = (0, 1, 0, 0, 0, 0, 0, 0, 0), R0+ = {²1 − ²9 , ²i − ²j | 2 ≤ i < j ≤ 8} ∪ {²i + ²j + ²k | 2 ≤ i < j < k ≤ 8} ∪{²1 + ²j + ²9 | 2 ≤ j ≤ 8}, S(θ) = E8(−24) /SU(2)E7 . F4 ) System of positive roots: R+ = {²i | 1 ≤ i ≤ 4} ∪ {²i ± ²j | 2 ≤ i < j ≤ 4} ∪ {²1 ± ²j | 2 ≤ j ≤ 4} ∪ {(²1 + s2 ²2 + s3 ²3 + s4 e4 )/2 | si = ±1}, Simple roots: α1 = (²1 − ²2 − ²3 − ²4 )/2, α2 = ²4 , α3 = ²3 − ²4 , α4 = ²2 − ²3 , → Highest root: µ = ² + ² , − n = (1, 2, 4, 3, 2),, 1

2

˜ Extended Dynkin diagram Π: 4 • < α2

2 • α1

3 • α3

2 • α4

1 • α0

F I) Kac diagram: •

•

S(θ) = SL(3, R)/SO(3). ˜ (ρ) = B ñ : D) gC = Dn+1 , n ≥ 2. Extended Dynkin diagram Π 2 • < β0

2 • β1

2 •· · · β2 27

2 2 ··· • • βn−2 βn−1

2 • > βn

• >

DI.0) Kac diagram: 1 •

··· •

•

• >

S(θ) = SO0 (2n + 1, 1)/SO(2n + 1). DI.m) Kac diagram (1 ≤ m ≤ n − 1): •