Bibikov P. Differential invariants and representations of semisimple algebraic groups (abstract)

Speaker: Pavel Bibikov

Title: Differential invariants and representations of semisimple algebraic groups

Abstract:
In this talk we discuss an approach to the study of orbits of actions of semisimple algebraic groups in their irreducible complex representations, which is based on differential invariants on the one hand, and on geometry of reductive homogeneous spaces on the other hand.

We will start from the well-known problem of $\mathrm {SL} _{2}$-classification of binary forms, which was studied by many famous mathematicians during XIX and XX centuries. Classical invariant theory starts from this problem. But it appears that the full solution of this problem can be obtained with the help of differential equations and differential invariants. Namely, we represent each binary form of degree $n$ as a solution of the so-called Euler equation $xu_{x}+yu_{y}=nu$, and study differential invariants for the $\mathrm {SL} _{2}$-actions on the prolongations of this equation. We prove that the dependence between basic differential invariant and its derivations uniquely defines the $\mathrm {SL} _{2}$-orbit of a given binary form. We also present some examples.

It the second part of the talk we generalize this approach for the problem of classification of $G$-orbits of a given connected semisimple algebraic group $G$ in its irreducible representation. According to the Borel-Weil-Bott theorem, every irreducible representation of connected semisimple Lie group is isomorphic to the action of this group on the module of holomorphic sections of some one-dimensional bundle over the flag variety $G/B$. Using this, we give a complete description of the structure of the field of differential invariants for this action and obtain a criterion which separates regular $G$-orbits. Also we discuss the applications of these results to the classical invariant theory.

In collaboration with Valentin Lychagin.

Slides: BibikovTrieste2018slides.pdf, BibikovTrieste2018slides-article.pdf

Event: Local and Nonlocal Geometry of PDEs and Integrability, 8-12 October 2018, SISSA, Trieste, Italy.
The conference in honor of Joseph Krasil'shchik's 70th birthday.