Local Lifts into Period Domains: Holonomies: Philosophies of Conjugacy. Part 2.

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Let F = GC/P be a flag manifold. Then there is a unique inner symmetric space G-space N associated to F together with a finite number of homogeneous fibrations F → N.

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.

Every flag manifold fibres over an inner symmetric space. Conversely, every inner symmetric space is the target of the canonical fibrations of at least one flag manifold. Let us now see how this story relates to the geometry of J(N).

So let p : F → N be a canonical fibration. By construction, the fibres of p are complex submanifolds of F and this allows us to define a fibre map ip : F → J(N) as follows: at f ∈ F we have an orthogonal splitting of TfF into horizontal and vertical subspaces both of which are invariant under the complex structure of F. Then dp restricts to give an isomorphism of the horizontal part with Tp(f)N and therefore induces an almost Hermitian structure on Tp(f)N : this is ip(f) ∈ Jp(f)N. Such a construction is possible whenever we have a Riemannian submersion of a Hermitian manifold with complex submanifolds as fibres.

ip : F → J(N) is a G-equivariant holomorphic embedding. This implies that ip (F) is an almost complex submanifold of J(N) on which J is integrable.

If j ∈ Z ⊂ J(N) then G · j is a flag manifold canonically fibred over N. In fact, G · j = ip(F ) for some canonical fibration p : F → N of a flag manifold F .

For this, the main observation is the following: at π(j), we have the symmetric decomposition g = k ⊕ q

with q ≅ Tπ(j)N. If q is the (0,1)-space for j then [q, q] ⊕ q

is the nilradical of a parabolic subalgebra p, where G · j is equivariantly biholomorphic to the corresponding flag manifold GC/P. Each canonical fibration of a flag manifold gives rise to a G-orbit in Z for some inner symmetric G-space N and that all such orbits arise in this way. But, for fixed G, there are only a finite number of biholomorphism types of flag manifold (they are in bijective correspondence with the conjugacy classes of parabolic subalgebras of gC) and each flag manifold admits but a finite number of canonical fibrations. Thus Z is composed of a finite number of G-orbits all of which are closed. In this way, we obtain a geometric interpretation of the purely algebraic construction of the canonical fibrations: they are just the restrictions of the projection π : J(N) → N to the various realisations of F as an orbit in Z.

For each non-compact real form GR of a complex semisimple group Lie group GC, there is a unique Riemannian symmetric space GR/K of non-compact type. The corresponding involution is called the Cartan involution of GR. Consider now the orbits of such a GR on the various flag manifolds F = GC/P. Those orbits which are open subsets of F are called flag domains: an orbit is a flag domain precisely when the stabilisers contain a compact Cartan subgroup of GR. It turns out that the presence of this compact Cartan subgroup is precisely what we need to define a canonical element of gR and thus an involution of gR just as in the compact case. However the involution is not necessarily a Cartan involution (i.e. the associated symmetric space need not be Riemmanian). In case that the involution is a Cartan involution, the flag domain is a canonical flag domain which is then exponentiated such that the involution gets to a Riemannian symmetric space of non-compact type and a canonical fibration of canonical flag domain over it.

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