Part 1 Lie algebras and flag manifolds: some structure theory; borel and parabolic subalgebras; generalized flag varieties; fibrations of generalized flag varieties. Part 2 Homogeneous vector bundles on G/P: a brief review of representation theory; homogeneous bundles on G/P; a remark on inverse images. Part 3 The Weyl group, its actions, and Hasse diagrams: the Weyl group; the affine Weyl action; the Hasse diagram of a parabolic subalgebra; relative Hasse diagrams. Part 4 The Bott-Borel-Weil Theorem: a simple proof; some examples; direct images. Part 5 Realizations of G/P: the projective realization; the cell structure of G/P; integral cohomology rings; Co-Adjoint realizations and moment maps. Part 6 The Penrose transform in principle: pulling-back cohomology; pushing-down cohomology; a spectral sequence. Part 7 The Bernstein-Gelfand-Gelfand Resolution: a prototype; translating BGG resolutions; the general case on G/B; the story for G/P; an algorithm for computation; non-standard morphisms; relative BGG resolutions. Part 8 The Penrose transform in practice: the homogeneous Penrose transform; the real thing; the Penrose transform of forms on twistor space; other bundles on twistor space; the Penrose transform for ambitwistor space; higher dimensions - conformal case; a Grassmannian example; an exceptional example; the Ward correspondence. Part 9 Constructing unitary representations: the discrete series of SU(1,1); massless field representations; the twistor point of view; the twistor transform; Hermitian symmetric spaces; towards discrete series. Part 10 Module structures on cohomology: verma modules and differential operators; invariant differential operators; the algebraic Penrose transform; K-types, local cohomology, and elementary states; homomorphisms of Verma modules.
