Hermitian symmetric space

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.[1]

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.[2]

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.

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.[2]

Accordingly, the irreducible compact Hermitian symmetric spaces H/K are classified as follows.

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.

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:[10]

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.

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

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 PH = K = PH*. 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.[11]

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

Two roots α and β are said to be strongly orthogonal if ±α ±β are not roots or zero, written α ≐ β. The highest positive root ψ1 is noncompact. Take ψ2 to be the highest noncompact positive root strongly orthogonal to ψ1 (for the lexicographic order). Then continue in this way taking ψi + 1 to be the highest noncompact positive root strongly orthogonal to ψ1, ..., ψi until the process terminates. The corresponding vectors

for all i, then cα = 0 for all positive noncompact roots α different from the ψj's. This follows by showing inductively that if cα ≠ 0, then α is strongly orthogonal to ψ1, ψ2, ... a contradiction. Indeed, the above relation shows ψi + α cannot be a root; and that if ψi – α is a root, then it would necessarily have the form β – ψi. If ψi – α were negative, then α would be a higher positive root than ψi, strongly orthogonal to the ψj with j < i, which is not possible; similarly if β – ψi were positive.

In the case of H = SU(2) the symmetry σ is given by conjugation by the diagonal matrix with entries ±i so that

where B+ and TC denote the subgroups of upper triangular and diagonal matrices in SL(2,C). 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. Thus there is a homomorphism π of the direct product SU(2)r into H compatible with the symmetries. 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. So the image of the center under π lies in K. Thus there is an embedding of the polysphere (SU(2)/T)r into H / K = G / P and the polysphere contains the polydisk (SU(1,1)/T)r. The polysphere and polydisk are the direct product of r copies of the Riemann sphere and the unit disk. 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.[17]

which is injective, so (2) follows. For the special case H = SU(2), H* = SU(1,1) and G = SL(2,C) the remaining assertions are consequences of the identification with the Riemann sphere, C and unit disk. They can be applied to the groups defined for each root ψi. By the polysphere and polydisk theorem H*/K, X/P and H/K are the union of the K-translates of the polydisk, Cr and the polysphere. So H* lies in X, the closure of H*/K is compact in X/P, which is in turn dense in H/K.

Note that (2) and (3) are also consequences of the fact that the image of X in G/P is that of the big cell B+B in the Gauss decomposition of G.[18]

A bounded domain Ω in a complex vector space is said to be a bounded symmetric domain if for every x in Ω, there is an involutive biholomorphism σx of Ω for which x is an isolated fixed point. The Harish-Chandra embedding exhibits every Hermitian symmetric space of noncompact type H* / K as a bounded symmetric domain. The biholomorphism group of H* / K is equal to its isometry group H*.

Conversely every bounded symmetric domain arises in this way. Indeed, given a bounded symmetric domain Ω, the Bergman kernel defines a metric on Ω, the Bergman metric, for which every biholomorphism is an isometry. This realizes Ω as a Hermitian symmetric space of noncompact type.[20]

The irreducible bounded symmetric domains are called Cartan domains and are classified as follows.

In the classical cases (I–IV), the noncompact group can be realized by 2 × 2 block matrices[21]

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 orbit structure is described in detail in Wolf (1972). In particular the closure of the bounded domain H*/K has a unique closed orbit, which is the Shilov boundary of the domain. In general the orbits are unions of Hermitian symmetric spaces of lower dimension. The complex function theory of the domains, in particular the analogue of the Cauchy integral formulas, are described for the Cartan domains in Hua (1979). The closure of the bounded domain is the Baily–Borel compactification of H*/K.[23]

The boundary structure can be described using Cayley transforms. For each copy of SU(2) defined by one of the noncompact roots ψi, there is a Cayley transform ci which as a Möbius transformation maps the unit disk onto the upper half plane. Given a subset I of indices of the strongly orthogonal family ψ1, ..., ψr, the partial Cayley transform cI is defined as the product of the ci's with i in I in the product of the groups πi. Let G(I) be the centralizer of this product in G and H*(I) = H* ∩ G(I). Since σ leaves H*(I) invariant, there is a corresponding Hermitian symmetric space MI H*(I)/H*(I)∩KH*/K = M . The boundary component for the subset I is the union of the K-translates of cI MI. When I is the set of all indices, MI is a single point and the boundary component is the Shilov boundary. Moreover, MI is in the closure of MJ if and only if IJ.[24]

Every Hermitian symmetric space is a Kähler manifold. They can be defined equivalently as Riemannian symmetric spaces with a parallel complex structure with respect to which the Riemannian metric is Hermitian. The complex structure is automatically preserved by the isometry group H of the metric, and so any Hermitian symmetric space M is a homogeneous complex manifold. Some examples are complex vector spaces and complex projective spaces, with their usual Hermitian metrics and Fubini–Study metrics, and the complex unit balls with suitable metrics so that they become complete and Riemannian symmetric. 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. Thus they are contained within the family of symmetric R-spaces which conversely comprises Hermitian symmetric spaces and their real forms. The non-compact Hermitian symmetric spaces can be realized as bounded domains in complex vector spaces.

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. This theory is described in detail in Koecher (1969) and Loos (1977) and summarized in Satake (1981). The development is in the reverse order from that using the structure theory of compact Lie groups. It starting point is the Hermitian symmetric space of noncompact type realized as a bounded symmetric domain. It can be described in terms of a Jordan pair or hermitian Jordan triple system. This Jordan algebra structure can be used to reconstruct the dual Hermitian symmetric space of compact type, including in particular all the associated Lie algebras and Lie groups.

In general a Hermitian symmetric space gives rise to a 3-graded Lie algebra with a period 2 conjugate linear automorphism switching the parts of degree ±1 and preserving the degree 0 part. This gives rise to the structure of a Jordan pair or hermitian Jordan triple system, to which Loos (1977) extended the theory of Jordan algebras. All irreducible Hermitian symmetric spaces can be constructed uniformly within this framework. Koecher (1969) constructed the irreducible Hermitian symmetric space of non-tube type from a simple Euclidean Jordan algebra together with a period 2 automorphism. The −1 eigenspace of the automorphism has the structure of a Jordan pair, which can be deduced from that of the larger Jordan algebra. In the non-tube type case corresponding to a Siegel domain of type II, there is no distinguished subgroup of real or complex Möbius transformations. For irreducible Hermitian symmetric spaces, tube type is characterized by the real dimension of the Shilov boundary S being equal to the complex dimension of D.