Syllabus
1) Universal enveloping algebra of a Lie algebra and the theorem of Poincaré--Birkhoff--Witt. Filtration, associated gradation, and the Noether feature of universal enveloping algebras. 2) Verma modules: Recall of representation theory of simple Lie algebras - Cartan subalgebra, roots, co-roots, positive and simple roots, fundamental weights, Weyl group and Bruhat ordering.
Weights) of representations of semi-simple Lie groups, semi-lattice of non-negative weights. Verma modules - definition, weight property, irreducibility characterization. Description of irreducible and finite-dimensional simple Lie algebra modules. Citation of Bernstein--Gelfand--Gelfand theorem on a connection of homomorphisms of Verma modules and Bruhat ordering. 3) Theorem of (Bott--)Borel--Weil (solutions of Laplace equation on complex flag manifolds): smooth locally trivial fibrations - vector, principal and associated fibrations. Holomorphic manifolds and fibrations. Flag manifolds - Borel and compact presentation of flag manifolds: spheres, projective spaces, Grassmannians, especially Gr_2(4, C). Some results of the structure and representation theory of semi-simple Lie groups. Holomorphic sections for Borel presentations. Formulation of the Borel--Weil theorem and its proof for the complex projective line.
Eventually, the unitary dual of SL(2,R).
Annotation
Harmonic analysis generalizes the classical Fourier analysis of partial differential equations in R^n for other groups than the abelian R^n.
Second part of lecture.