Invariant differential operators. Volume 1: Noncompact semisimple Lie algebras and groups
The book under review presents a point of view on the theory of invariant differential operators that uses the perspective of Lie groups, Lie algebras and their representations. The present volume, which is the first one out of two, is divided into nine chapters. The first chapter is a fairly informal introduction which describes the general notion of symmetry, discusses the notion of invariant differential operators and clarifies the aims and the structure of the text. The second chapter is intended as a concise introduction to the general structure theory of Lie algebras. This chapter starts with basic definitions, covers classical results on nilpotent and solvable Lie algebras and then proceeds to the discussion of weights, roots, root systems and Weyl groups. All this leads naturally to the classification of semi-simple finite dimensional Lie algebras and the construction and realization of Kac-Moody Lie algebras, which are illustrated by many explicit examples, in particular, of all exceptional simple finite dimensional Lie algebras. The chapter finishes with a discussion of some basics on representations of Lie algebras, in particular on highest weight modules, Verma modules and characters. The third chapter is devoted to real semi-simple Lie algebras. It contains a classification of noncompact semi-simple Lie algebras, a discussion of parabolic subalgebras and description of real forms. This chapter contains many explicit examples where various classical real algebras are described in detail on a case-by-case basis. The fourth chapter of the book discusses invariant differential operators. It starts with generalities on Lie groups which include a description of classical groups, Cartan subgroups and Cartan and Iwasawa decomposition. This is followed by basic facts from representation theory, covering induced, elementary and unitary representations and the theory of singular vectors. The latter is illustrated by examples in all Dynkin types. Chapter~5 considers the particular example of the anti-de Sitter group which is treated in detail. The chapter contains an explicit descriptions of the Lie algebra, finite-dimensional realizations, structure theory, Lie group, elementary representations, invariant differential operators, singular vectors and characters. The chapter after that discusses, in detail, the example of the conformal group and algebra in four-dimensional space-time. It provides a detailed description of many structural components and elementary representations and gives explicit expressions for invariant differential operators. The seventh chapter contains some general discussions around Kazhdan-Lusztig polynomials and their applications, in particular, to the study of submodules of Verma modules which are not generated by highest weight vectors, and their applications to conditionally invariant equations. Chapter~7 addresses questions related to invariant differential operators for noncompact Lie algebras that are parabolically related to conformal Lie algebras. Finally, the last chapter discusses multilinear invariant differential operators from certain versions of generalized Verma modules. The book mostly focuses on listing facts rather than on explaining them or providing details of proofs. The second volume is expected to touch upon various generalizations of Lie algebras, including quantum groups and Lie superalgebras together with their infinite dimensional counterparts.
scientific article; zbMATH DE number 6608460