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.
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.
, invariant under the adjoint representation and σ, induces a Riemannian structure on H / K, with H acting by isometries.
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.
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.
(In the language of algebraic groups, L is the Levi factor of P.) 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.
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.
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).
, 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
[13] The decompositions can be proved directly by applying the slice theorem for compact transformation groups to the action of K on H / K.[14] In fact the space H / K can be identified with a closed submanifold of H, and the Cartan decomposition follows by showing that M is the union of the kAk−1 for k in K. Since this union is the continuous image of K × A, it is compact and connected.
Any positive noncompact root then has the form β = α1 + c2 α2 + ⋅⋅⋅ + cn αn with non-negative coefficients ci.
is irreducible for K so is spanned by vectors obtained by successively applying the lowering operators E–α for simple compact roots α.
[15] (As Sugiura later showed, having fixed T, the set of strongly orthogonal roots is uniquely determined up to applying an element in the Weyl group of K.[16]) Maximality can be checked by showing that if for all i, then cα = 0 for all positive noncompact roots α different from the ψj's.
SL(2,C), the complexification of SU(2), also acts by Möbius transformations and the stabiliser of 0 is the subgroup B of lower triangular matrices.
The middle term is the orbit of 0 under the upper unitriangular matrices Now for each root ψi there is a homomorphism of πi of SU(2) into H which is compatible with the symmetries.
It extends uniquely to a homomorphism of SL(2,C) into G. The images of the Lie algebras for different ψi's commute since they are strongly orthogonal.
It extends to a homomorphism of SL(2,C)r into G. The kernel of π is contained in the center (±1)r of SU(2)r which is fixed pointwise by the symmetry.
By the Cartan decompositions in SU(2) and SU(1,1), the polysphere is the orbit of TrA in H / K and the polydisk is the orbit of TrA*, where Tr = π(Tr) ⊆ K. On the other hand, H = KAK and H* = K A* K. Hence every element in the compact Hermitian symmetric space H / K is in the K-orbit of a point in the polysphere; and every element in the image under the Borel embedding of the noncompact Hermitian symmetric space H* / K is in the K-orbit of a point in the polydisk.
In the classical cases (I–IV), the noncompact group can be realized by 2 × 2 block matrices[21] acting by generalized Möbius transformations The polydisk theorem takes the following concrete form in the classical cases:[22] The noncompact group H* acts on the complex Hermitian symmetric space H/K = G/P with only finitely many orbits.
The compact Hermitian symmetric spaces are projective varieties, and admit a strictly larger Lie group G of biholomorphisms with respect to which they are homogeneous: in fact, they are generalized flag manifolds, i.e., G is semisimple and the stabilizer of a point is a parabolic subgroup P of G. Among (complex) generalized flag manifolds G/P, they are characterized as those for which the nilradical of the Lie algebra of P is abelian.
Although the classical Hermitian symmetric spaces can be constructed by ad hoc methods, Jordan triple systems, or equivalently Jordan pairs, provide a uniform algebraic means of describing all the basic properties connected with a Hermitian symmetric space of compact type and its non-compact dual.
The theory is easiest to describe when the irreducible compact Hermitian symmetric space is of tube type.
It must admit an action of SU(2) which only acts via the trivial and adjoint representation, both types occurring.
It is a three-graded complex Lie algebra, with the Weyl group element of SU(2) providing the involution.
Any two are related by an automorphism of E, so that the integer m is an invariant called the rank of E. Moreover, if A is the complexification of E, it has a unitary structure group.
The usual Cayley transform and its inverse, mapping the unit disk in C to the upper half plane, establishes analogous maps between D and T. The polydisk corresponds to the real and complex Jordan subalgebras generated by a fixed Jordan frame.
The classification of Hermitian symmetric spaces of tube type reduces to that of simple Euclidean Jordan algebras.
Koecher (1969) constructed the irreducible Hermitian symmetric space of non-tube type from a simple Euclidean Jordan algebra together with a period 2 automorphism.