During the past two decades Roger Penrose's twistor theory has been a continuing source of mathematical inspiration. It is an ambitious theory which aims to reformulate the foundations of physics using conformal and complex geometry. At the heart of twistor theory lies a geometrical transform now known as the Penrose transform. This book is an exposition of this transform in a fairly general setting. This setting is provided by complex homogeneous spaces and mathematical input is taken from the representation theory of Lie groups. The book consists mostly of original research not available elsewhere and is intended for physicists and mathematicians with interests in geometry and symmetry. Whilst the material is presented in full generality, there are many examples throughout and special attention is directed towards the twistor theory of Minkowski space.

Table of Contents:
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.