Suppose that N is an irreducible 2n-dimensional Riemannian symmetric space. We may realise N as a coset space N = G/K with Gτ ⊂ K ⊂ (Gτ)0 for some involution τ of G. Now K is (a covering of) the holonomy group of N and similarly the coset fibration G → G/K covers the holonomy bundle P → N. In this setting, J(N) is associated to G:
This focusses our attention on the zero-set of NJ which we denote by Z. In favourable circumstances, the structure of this set can be completely described. We begin by assuming that N is of compact type so that G is compact and semi-simple. We also assume that N is inner i.e. that τ is an inner involution of G or, equivalently, that rankG = rankK. The class of inner symmetric spaces include the even-dimensional spheres, the Hermitian symmetric spaces, the quaternionic Kähler symmetric spaces and indeed all symmetric G-spaces for G = SO(2n+1), Sp(n), E7, E8, F4 and G2. Moreover, all inner symmetric spaces are necessarily even-dimensional and so fit into our framework.
Let N = G/K be a simply-connected inner Riemannian symmetric space of compact type. Then Z consists of a finite number of connected components on each of which G acts transitively. Moreover, any G-flag manifold is realised as such an orbit for some N.
The proof for the above requires a detour into the geometry of flag manifolds and reveals an interesting interaction between the complex geometry of flag manifolds and the real geometry of inner symmetric spaces. For this, we begin by noting that a coset space of the form G/C(T) admits several invariant Kählerian complex structures in general. Using a complex realisation of G/C(T) as follows: having fixed a complex structure, the complexified group GC acts transitively on G/C(T) by biholomorphisms with parabolic subgroups as stabilisers. Conversely, if P ⊂ GC is a parabolic subgroup then the action of G on GC/P is transitive and G ∩ P is the centraliser of a torus in G. For the infinitesimal situation: let F = G/C(T) be a flag manifold and let o ∈ F. We have a splitting of the Lie algebra of G
with m ≅ ToF and h the Lie algebra of the stabiliser of o in G. An invariant complex structure on F induces an ad h-invariant splitting of mC into (1, 0) and (0, 1) spaces mC = m+ ⊕ m− with [m+, m+] ⊂ m+ by integrability. One can show that m+ and m− are nilpotent subalgebras of gC and in fact hC ⊕ m− is a parabolic subalgebra of gC with nilradical m−. If P is the corresponding parabolic subgroup of GC then P is the stabiliser of o and we obtain a biholomorphism between the complex coset space GC/P and the flag manifold F.
Conversely, let P ⊂ GC be a parabolic subgroup with Lie algebra p and let n be the conjugate of the nilradical of p (with respect to the real form g). Then H = G ∩ P is the centraliser of a torus and we have orthogonal decompositions (with respect to the Killing inner product)
The relationship between a flag manifold F = GC/P and an inner symmetric space comes from an examination of the central descending series of n. This is a filtration 0 = nk+1 ⊂ nk ⊂…⊂ n1 = n of n defined by ni = [n, ni−1].
which can be proved by demonstrating the existence of an element ξ ∈ h with the property that, for each i, adξ has eigenvalue √−1i on gi. This element ξ (necessarily unique since g is semi-simple) is the canonical element of p. Since ad ξ has eigenvalues in √−1Z, ad exp πξ is an involution of g which we exponentiate to obtain an inner involution τξ of G and thus an inner symmetric space G/K where K = (Gτξ)0. Clearly, K has Lie algebra given by
and so contains H whence we obtain a homogeneous fibration G/H → G/K of our flag manifold over our inner symmetric space. Moreover, this fibration is essentially unique: the only ambiguity in the prescription is that several points in the symmetric space might have the same stabiliser K (e.g. antipodal points on a sphere). However, the number of such points is finite and so we only get a finite number of such fibrations.
Let us emphasise that this construction depends on nothing but the conjugacy class of p ⊂ gC and the choice of compact real form g. Equivalently, it depends solely on the choice of invariant complex structure on F.