Lie Group Actions in Complex Analysis

1 Lie theory.- 1.1 Complex spaces.- 1.2 Lie group actions.- 1.3 One-parameter transformation groups.- 1.4 Vector fields.- 1 5 Infinitesimal transformations.- 1.6 Analyticity of Lie group actions.- 1.7 Lie homomorphism.- 1.8 Global actions.- 2 Automorphism groups.- 2.1 Topology in Hol(X, Y).- 2.2 Local linearization of a compact group with a fixed point.- 2.3 The automorphism group of a compact complex space.- 2.4 Automorphisms of fiber bundles.- 2.5 Proper actions.- 2.6 The automorphism group of a bounded domain.- 2.7 The automorphism groups of the polydisk and the ball.- 2.8 A characterization of the ball.- 2.9 Bounded domains with compact quotient D/Aut(D).- 3 Compact homogeneous manifolds.- 3.1 Flag manifolds.- 3.2 Equivariant projective embeddings.- 3.3 Automorphism groups of flag manifolds.- 3.4 Parallelizable manifolds.- 3.5 Tits fibration.- 3.6 Manifolds fibered by tori.- 3.7 The role of the fundamental group.- 3.8 An estimate of the dimension of Aut(X).- 3.9 Compact homogeneousKähler manifolds.- 4 Homogeneous vector bundles.- 4.1 Coherent analytic G-sheaves.- 4.2 Holomorphic vector G-bundles.- 4.3 Theorem of R.Bott. Proof of the Borel-Weil theorem.- 4.4 Application of the Leray spectral sequence.- 4.5 Proof of the theorem of R.Bott.- 4.6 Invertible sheaves on G/P for P maximal parabolic.- 4.7 Computations in root systems.- 4.8 Cohomology of the tangent sheaf.- 5 Function theory on homogeneous manifolds.- 5.1 Representations of compact Lie groups on Fréchet spaces.- 5.2 Differentiable vectors and Fourier series in O(X).- 5.3 Reductive complex Lie groups.- 5.4 Quasi-affine homogeneous varieties.- 5.5 Holomorphically separable homogeneous manifolds.- 5.6 Stein homogeneous manifolds.- 5.7 Observable subgroups.- 5.8 Invariant plurisubharmonic functions and geodesic convexity.- Concluding remarks.- Index of notations.- Index of terminology.

Prof. Dr. Dimitri Akhiezer lehrt Mathematik an einer Moskauer Universität.