We compute the Euler-Poincaré characteristic of the homogeneous compact manifolds that can be described as minimal orbits for the action of a real form in a complex flag manifold.

Let (V, (·, ·)) be a pseudo-Euclidean vector space and S an irreducible Cℓ(V )-module. An extended translation algebra is a graded Lie algebra m = m−2 + m−1 = V + S with bracket given by ([s, t], v) = b(v · s, t) for some nondegenerate so(V )-invariant reflexive bilinear form b on S. An extended Poincaré structure on a manifold M is a regular distribution D of depth 2 whose Levi form Lx : Dx ∧ Dx → TxM/Dx at any point x ∈ M is identifiable with the bracket [·, ·] : S ∧ S → V of a fixed extended translation algebra m. The classification of the standard maximally homogeneous manifolds with an extended Poincaré structure is given, in terms of Tanaka prolongations of extended translation algebras and of appropriate gradations of real simple Lie algebras.2000 Mathematics Subject Classification. 53C30, 58A30, 53C27, 53C80.

Let V be a complex vector space with a non-degenerate symmetric bilinear form and S an irreducible module over the Clifford algebra Cℓ(V ) determined by this form. A supertranslation algebra is a Z-graded Lie superalgebra m = m−2 ⊕m−1, where m−2 = V and m−1 = S⊕· · ·⊕S is the direct sum of an arbitrary number N ≥ 1 of copies of S, whose bracket. We consider the maximal transitive prolongations in the sense of Tanaka of supertranslation algebras. We prove that they are finitedimensional for dim V ≥ 3 and classify them in terms of super-Poincaré algebras and appropriate Z-gradings of simple Lie superalgebras.the Clifford algebra determined by this form, with its natural Z 2 -grading. We adopt the conventions used in [24], in particular, the product in Cℓ(V ) satisfies vu + uv = −2(v, u)1 for any v, u ∈ V and the natural inclusion of the orthogonal Lie algebra in the Clifford algebra is given by the mapwhere v ∧ u is the anti-symmetric endomorphism (v, ·)u − (u, ·)v. Note that some authors prefer to use different conventions, cf. Deligne's lectures [12].

We prove a subelliptic estimate for systems of complex vector fields under some assumptions that generalize the essential pseudoconcavity for CR manifolds, that was first introduced by two of the Authors, and the Hörmander's bracket condition for real vector fields. Applications are given to prove the hypoellipticity of first order systems and second order partial differential operators. Finally we describe a class of compact homogeneous CR manifolds for which the distribution of (0, 1) vector fields satisfies a subelliptic estimate.

We define and investigate a class of compact homogeneous CR manifolds, that we call -reductive. They are orbits of minimal dimension of a compact Lie group K (0) in algebraic affine homogeneous spaces of its complexification K. For these manifolds we obtain canonical equivariant fibrations onto complex flag manifolds, generalizing the Hopf fibration . These fibrations are not, in general, CR submersions, but satisfy the weaker condition of being CR-deployments; to obtain CR submersions we need to strengthen their CR structure by lifting the complex stucture of the base