In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural generalization of the notion of Riemannian symmetric space from real manifolds to complex manifolds.
Every Hermitian symmetric space is a homogeneous space for its isometry group and has a unique decomposition as a product of irreducible spaces and a Euclidean space. The irreducible spaces arise in pairs as a non-compact space that, as Borel showed, can be embedded as an open subspace of its compact dual space. Harish Chandra showed that each non-compact space can be realized as a bounded symmetric domain in a complex vector space. The simplest case involves the groups SU(2), SU(1,1) and their common complexification SL(2,C). In this case the non-compact space is the unit disk, a homogeneous space for SU(1,1). It is a bounded domain in the complex plane C. The one-point compactification of C, the Riemann sphere, is the dual space, a homogeneous space for SU(2) and SL(2,C).
Irreducible compact Hermitian symmetric spaces are exactly the homogeneous spaces of simple compact Lie groups by maximal closed connected subgroups which contain a maximal torus and have center isomorphic to the circle group. There is a complete classification of irreducible spaces, with four classical series, studied by Cartan, and two exceptional cases; the classification can be deduced from Borel–de Siebenthal theory, which classifies closed connected subgroups containing a maximal torus. Hermitian symmetric spaces appear in the theory of Jordan triple systems, several complex variables, complex geometry, automorphic forms and group representations, in particular permitting the construction of the holomorphic discrete series representations of semisimple Lie groups.
Hermitian symmetric spaces of compact type
Definition Let H be a connected compact semisimple Lie group, σ an automorphism of H of order 2 and Hσ the fixed point subgroup of σ. Let K be a closed subgroup of H lying between Hσ and its identity component. The compact homogeneous space H / K is called a symmetric space of compact type. The Lie algebra admits a decomposition
where , the Lie algebra of K, is the +1 eigenspace of σ and the –1 eigenspace. If contains no simple summand of , the pair (, σ) is called an orthogonal symmetric Lie algebra of compact type.
Any inner product on , invariant under the adjoint representation and σ, induces a Riemannian structure on H / K, with H acting by isometries. A canonical example is given by minus the Killing form. Under such an inner product, and are orthogonal. H / K is then a Riemannian symmetric space of compact type.See:
The symmetric space H / K is called a Hermitian symmetric space if it has an almost complex structure preserving the Riemannian metric. This is equivalent to the existence of a linear map J with J2 = −I on which preserves the inner product and commutes with the action of K.
Symmetry and center of isotropy subgroup If (,σ) is Hermitian, K has non-trivial center and the symmetry σ is inner, implemented by an element of the center of K.
In fact J lies in and exp tJ forms a one-parameter group in the center of K. This follows because if A, B, C, D lie in , then by the invariance of the inner product on
Replacing A and B by JA and JB, it follows that
Define a linear map δ on by extending J to be 0 on . The last relation shows that δ is a derivation of . Since is semisimple, δ must be an inner derivation, so that
with T in and A in . Taking X in , it follows that A = 0 and T lies in the center of and hence that K is non-semisimple. The symmetry σ is implemented by z = exp πT and the almost complex structure by exp π/2 T.
The innerness of σ implies that K contains a maximal torus of H, so has maximal rank. On the other hand, the centralizer of the subgroup generated by the torus S of elements exp tT is connected, since if x is any element in K there is a maximal torus containing x and S, which lies in the centralizer. On the other hand, it contains K since S is central in K and is contained in K since z lies in S. So K is the centralizer of S and hence connected. In particular K contains the center of H.
Irreducible decomposition The symmetric space or the pair (, σ) is said to be irreducible if the adjoint action of (or equivalently the identity component of Hσ or K) is irreducible on . This is equivalent to the maximality of as a subalgebra.See:
In fact there is a one-one correspondence between intermediate subalgebras and K-invariant subspaces of given by
Any orthogonal symmetric algebra (, σ) of Hermitian type can be decomposed as an (orthogonal) direct sum of irreducible orthogonal symmetric algebras of Hermitian type.See:
In fact can be written as a direct sum of simple algebras
each of which is left invariant by the automorphism σ and the complex structure J, since they are both inner. The eigenspace decomposition of coincides with its intersections with and . So the restriction of σ to is irreducible.
This decomposition of the orthogonal symmetric Lie algebra yields a direct product decomposition of the corresponding compact symmetric space H / K when H is simply connected. In this case the fixed point subgroup Hσ is automatically connected. For simply connected H, the symmetric space H / K is the direct product of Hi / Ki with Hi simply connected and simple. In the irreducible case, K is a maximal connected subgroup of H. Since K acts irreducibly on (regarded as a complex space for the complex structure defined by J), the center of K is a one-dimensional torus T, given by the operators exp tT. Since each H is simply connected and K connected, the quotient H/K is simply connected.
Complex structure
if H / K is irreducible with K non-semisimple, the compact group H must be simple and K of maximal rank. From Borel-de Siebenthal theory, the involution σ is inner and K is the centralizer of its center, which is isomorphic to T. In particular K is connected. It follows that H / K is simply connected and there is a parabolic subgroup P in the complexification G of H such that H / K = G / P. In particular there is a complex structure on H / K and the action of H is holomorphic. Since any Hermitian symmetric space is a product of irreducible spaces, the same is true in general.
At the Lie algebra level, there is a symmetric decomposition
where is a real vector space with a complex structure J, whose complex dimension is given in the table. Correspondingly, there is a graded Lie algebra decomposition
where is the decomposition into +i and −i eigenspaces of J and . The Lie algebra of P is the semidirect product . The complex Lie algebras are Abelian. Indeed, if U and V lie in , [U,V] = J[U,V] = [JU,JV] = [±iU,±iV] = –[U,V], so the Lie bracket must vanish.
The complex subspaces of are irreducible for the action of K, since J commutes with K so that each is isomorphic to with complex structure ±J. Equivalently the centre T of K acts on by the identity representation and on by its conjugate.
The realization of H/K as a generalized flag variety G/P is obtained by taking G as in the table (the complexification of H) and P to be the parabolic subgroup equal to the semidirect product of L, the complexification of K, with the complex Abelian subgroup exp . (In the language of algebraic groups, L is the Levi factor of P.)
Classification Any Hermitian symmetric space of compact type is simply connected and can be written as a direct product of irreducible hermitian symmetric spaces Hi / Ki with Hi simple, Ki connected of maximal rank with center T. The irreducible ones are therefore exactly the non-semisimple cases classified by Borel–de Siebenthal theory.
Accordingly, the irreducible compact Hermitian symmetric spaces H/K are classified as follows.
G H Kcomplex dimensionrank geometric interpretation pq min(p,q) Grassmannian of complex p-dimensional subspaces of Space of orthogonal complex structures on n Space of complex structures on compatible with the inner product n 2 Grassmannian of oriented real 2-dimensional subspaces of 16 2 Rosenfeld projective plane 27 3 Space of symmetric submanifolds of the Rosenfeld projective plane that are isomorphic to
In terms of the classification of compact Riemannian symmetric spaces, the Hermitian symmetric spaces are the four infinite series AIII, DIII, CI and BDI with p = 2 or q = 2, and two exceptional spaces, namely EIII and EVII.
Classical examples The irreducible Hermitian symmetric spaces of compact type are all simply connected. The corresponding symmetry σ of the simply connected simple compact Lie group is inner, given by conjugation by the unique element S in Z(K) / Z(H) of period 2. For the classical groups, as in the table above, these symmetries are as follows:
AIII: in S(U(p)×U(q)), where αp+q=(−1)p. DIII: S = iI in U(n) ⊂ SO(2n); this choice is equivalent to . CI: S=iI in U(n) ⊂ Sp(n) = Sp(n,C) ∩ U(2n); this choice is equivalent to Jn. BDI: in SO(p)×SO(2).
The maximal parabolic subgroup P can be described explicitly in these classical cases. For AIII
in SL(p+q,C). P(p,q) is the stabilizer of a subspace of dimension p in Cp+q.
The other groups arise as fixed points of involutions. Let J be the n × n matrix with 1's on the antidiagonal and 0's elsewhere and set
Then Sp(n,C) is the fixed point subgroup of the involution θ(g) = A (gt)−1 A−1 of SL(2n,C). SO(n,C) can be realised as the fixed points of ψ(g) = B (gt)−1 B−1 in SL(n,C) where B = J. These involutions leave invariant P(n,n) in the cases DIII and CI and P(p,2) in the case BDI. The corresponding parabolic subgroups P are obtained by taking the fixed points. The compact group H acts transitively on G / P, so that G / P = H / K.
Hermitian symmetric spaces of noncompact type
Definition As with symmetric spaces in general, each compact Hermitian symmetric space H/K has a noncompact dual H*/K obtained by replacing H with the closed real Lie subgroup H* of the complex Lie group G with Lie algebra
Borel embedding Whereas the natural map from H/K to G/P is an isomorphism, the natural map from H*/K to G/P is only an inclusion onto an open subset. This inclusion is called the Borel embedding after Armand Borel. In fact P ∩ H = K = P ∩ H*. The images of H and H* have the same dimension so are open. Since the image of H is compact, so closed, it follows that H/K = G/P.See:
Cartan decomposition The polar decomposition in the complex linear group G implies the Cartan decomposition H* = K ⋅ exp in H*.
Moreover, given a maximal Abelian subalgebra in t, A = exp is a toral subgroup such that σ(a) = a−1 on A; and any two such 's are conjugate by an element of K. A similar statement holds for . Moreover if A* = exp , then
These results are special cases of the Cartan decomposition in any Riemannian symmetric space and its dual. The geodesics emanating from the origin in the homogeneous spaces can be identified with one parameter groups with generators in or . Similar results hold for in the compact case: H= K ⋅ exp and H = KAK.
The properties of the totally geodesic subspace A can be shown directly. A is closed because the closure of A is a toral subgroup satisfying σ(a) = a−1, so its Lie algebra lies in and hence equals by maximality. A can be generated topologically by a single element exp X, so is the centralizer of X in . In the K-orbit of any element of there is an element Y such that (X,Ad k Y) is minimized at k = 1. Setting k = exp tT with T in , it follows that (X,[T,Y]) = 0 and hence [X,Y] = 0, so that Y must lie in . Thus is the union of the conjugates of . In particular some conjugate of X lies in any other choice of , which centralizes that conjugate; so by maximality the only possibilities are conjugates of .
The decompositions