Exercise 11, § 4

We retain the notations of Exerc. 9 and 10.

a) The homogeneous space $G/H$ is symmetric (§1, Exerc. 8) if and only if Z is of order 2 or of dimension 1. The symmetric space $G/H$ is then irreducible (loc. cit.).

b) Assume that dim Z = 1; denote by X the complex-analytic manifold $G/H$. Show that we are then in one of the following situations:

(i) G is of type $A_l$ and D(H) of type $A_{p-1}\times A_{l-p}; X$ is isomorphic to the grassmannian $\mathbf{G}_p(\mathbf{C}^{l+1})$.

(ii) G is of type $B_l$ and D(H) of type $B_{l-1}; X$ is isomorphic to the submanifold of $\mathbf{P}_{2l}(\mathbf{C})$ defined by the vanishing of a separating quadratic form (smooth projective quadric).

(ii$'$) $G$ is of type $D_l$ and D(H) of type $D_{l-1}; X$ is isomorphic to a smooth projective quadric in $\mathbf{P}_{2l-1}(\mathbf{C})$.

(iii) G is of type $C_l$ and D(H) of type $A_{l-1}; X$ is isomorphic to the submanifold of $\mathbf{G}_l(\mathbf{C}^{2l})$ consisting of the maximal isotropic subspaces for the usual alternating bilinear form on $\mathbf{C}^{2l}$.

(iv) G is of type $D_l$ and D(H) of type $A_{l-1}; X$ is isomorphic to the submanifold of $\mathbf{G}_l(\mathbf{C}^{2l})$ consisting of the maximal isotropic subspaces for the usual symmetric bilinear form on $\mathbf{C}^{2l}$.

(v) G is of type $E_6$ and D(H) of type $D_5$.

(vi) G is of type $E_7$ and D(H) of type $E_6$. c) Assume that Card(Z) = 2; give the list of possible situations. If G is of type $A_l, B_l, C_l$ or $D_l$, show that the real manifold $G/H$ is isomorphic to a grassmannian manifold $G_p(K^q)$, with $K =\mathbf{R},\mathbf{C}$ or $\mathbf{H}$.