Lie group actions in complex analysis. (English) Zbl 0845.22001

Akhiezer, Dmitri N.

Aspects of Mathematics. E27. Braunschweig: Vieweg. vii, 201 p. (1995).

From the introduction: “The main topic of the book is the study of the interaction between two major subjects of modern mathematics, namely the theory of Lie groups with its specific methods and ways of thinking on the one hand and complex analysis with all its analytic, algebraic and geometric aspects on the other.” The book is organized as follows: Chapter 1. Lie theory. The local and global Lie group actions on complex spaces are defined. A local action of a Lie group \(G\) on a complex space \(X\) is shown to be real analytic and to give rise to the Lie homomorphism which is a map from the Lie algebra of \(G\) into the Lie algebra of vector fields on \(X\). In accordance with the second fundamental theorem of Sophus Lie the local action can be recovered from this homomorphism. The proof of this theorem is given and some sufficient conditions for a local action to extend to a global one are established. Chapter 2. Automorphism groups. It is shown that there are two important classes of complex spaces \(X\) for which the automorphism group \(\operatorname{Aut}(X)\) has Lie group structure: the first class consists of all (not necessarily reduced) compact spaces, the second one is the class of all bounded domains in \(\mathbb{C}^n\). Chapter 3. Compact homogeneous manifolds. The geometric properties of two kinds of compact homogeneous complex manifolds, namely flag manifolds and parallelizable ones, are studied. The role of these two classes is explained by the normalizer theorem. This theorem states that if \(H\) is a closed complex subgroup of a connected complex Lie group \(G\) and if \(X = G/H\) is compact, then \(X\) admits a fibration, called the Tits fibration, whose base is a flag manifold and whose fiber is a parallelizable manifold. Chapter 4. Homogeneous vector bundles. The proof and some applications are given of a theorem of R. Bott which determines the induced representations \(I^q \varphi\) in the case when \(H\) is a parabolic subgroup of a semisimple group \(G\) and \(\varphi : H \to GL (V)\) is an irreducible representation. As an application the representations induced by the characters of maximal parabolic subgroups are considered. Chapter 5. Function theory of homogeneous manifolds. Holomorphic functions in \(K\)-invariant domains \(\Omega \subset G/H\) are studied. Here \(K\) is a connected compact group and \(G = K_C\) a reductive linear algebraic group obtained by complexification and \(H \subset G\) is a closed complex Lie subgroup. As starting point a theorem of Harish-Chandra is chosen which extends the classical Fourier expansion to the representation theory of compact Lie groups on Fréchet vector spaces \({\mathcal O} (\Omega)\). As a consequence a description of the class of so-called observable subgroups is obtained by using methods of geometric invariant theory.

Reviewer: A.A.Bogush (Minsk)

MSC:

22-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to topological groups
32-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to several complex variables and analytic spaces
14-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry
53-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry

Keywords:

homogeneous vector bundles; holomorphic functions; Lie groups; complex analysis; Lie group actions; complex spaces; Lie homomorphism; Lie algebra; automorphism group; compact homogeneous complex manifolds; flag manifolds; normalizer theorem; Tits fibration; induced representations; characters; linear algebraic group; complexification; Fourier expansion; Fréchet vector spaces; geometric invariant theory