The course is devoted to geometric analysis on homogeneous spaces – smooth manifolds equipped with a transitive action of a Lie group. We start with fundamental notions of Lie group theory -Lie algebra, exponential map, Haar measure, universal enveloping algebra, adjoint representation and elaborate these notions on well known Lie groups of small dimension.
Classical algebraic homogeneous spaces – flag manifolds and grassmanians and related technique of Weyl group and root system– are considered on the example of general linear group. Here all the constructions arise from Gauss elemination method for solution of systems of linear equations.
Another side of the story is related to real reductive groups. Motivated by classical Fourier transform,
it generalizes harmonic analisys on two- dimensional sphere and Lobachevsky plane. By means of these carefully studied examples we demonstrate ideas of general procedure.
Depending on the abilities of the listeners we will highlight the emergence of cluster parametrizations of positively defined unipotent matrices, and pass to Poisson geometry of homogeneous spaces. Here are: coadjoint orbits and double Bruhat cells with Sklyanin bracket. We plan to finish with quantum mechanical integrable systems realized via matrix coefficients of representations of Lie group.
- מורה: סרגיי חורושקין