Dobrev V.K. Invariant Differential Operators. Vol. 1 Noncompact Semisimple Lie Algebras and Groups

Boston: de Gruyter, 2016. — 423 p. — (De Gruyter Studies in Mathematical Physics 35) — ISBN 978-3-11-043542-9.Invariant differential operators play a very important role in the description of physical symmetries – recall, e.g., the examples of Dirac, Maxwell, Klein–Gordon, d’Almbert, and Schrödinger equations. Invariant differential operators played and continue to play important role in applications to conformal field theory. Invariant superdifferential operators were crucial in the derivation of the classification of positive energy unitary irreducible representations of extended conformal supersymmetry first in four dimensions, then in various dimensions. Last, but not least, among our motivations are the mathematical developments in the last 50 years and counting.
Obviously, it is important for the applications in physics to study these operators systematically. A few years ago we have given a canonical procedure for the construction of invariant differential operators. Lately, we have given an explicit description of the building blocks, namely, the parabolic subgroups and subalgebras from which the necessary representations are induced.
Altogether, over the years we have amassed considerable material which was suitable to be exposed systematically in book form. To achieve portable formats, we decided to split the book in two volumes. In the present first volume, our aim is to introduce and explain our canonical procedure for the construction of invariant differential operators and to explain how they are used on many series of examples. Our objects are noncompact semisimple Lie algebras, and we study in detail a family of those that we call “conformal Lie algebras” since they have properties similar to the classical conformal algebras of Minkowski space-time. Furthermore, we extend our considerations to simple Lie algebras that are called “parabolically related” to the initial family.Lie Algebras and Groups
Real Semisimple Lie Algebras
Invariant Differential Operators
Case of the Anti-de Sitter Group
Conformal Case in 4D
Kazhdan–Lusztig Polynomials, Subsingular Vectors, and Conditionally Invariant Equations
Invariant Differential Operators for Noncompact Lie Algebras Parabolically Related to Conformal Lie Algebras
Multilinear Invariant Differential Operators from New Generalized Verma Modules