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.
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.
In fact there is a one-one correspondence between intermediate subalgebras and K-invariant subspaces of given by
