Abstract: Geometric invariant theory (GIT) provides a construction of quotients for projective algebraic varieties equipped with an action of a reductive algebraic group. Since its foundation by Mumford, GIT plays an important role in the construction of moduli (or parameter) spaces. More recently, its methods have been successfully applied to problems of representation theory. The first part of the lectures will present basic results and examples on GIT; the second part will concern the recent developments related to representation theory.

Introduction to geometric invariant theory by Michel Brion.

The aim of this course is to present basic results and examples on geometric invariant theory; it will also serve as a preparation for the course of Nicolas Ressayre. The following topics will be covered :

1. Reductive algebraic groups and their representations.
2. Geometric invariant theory : basic notions (finite generation of algebras of invariants; stable and semi-stable points; geometric quotients).
3. The Hilbert-Mumford criterion and optimal one-parameter subgroups.
4. The Bialynicki-Birula decomposition.

• M. Brion: Introduction to actions of algebraic groups
• S. Mukai, An introduction to invariants and moduli, Cambridge Studies in Advanced Mathematics, 81. Cambridge University Press, Cambridge, 2003; Chapters 1 to 7.
• I. Dolgachev, Lectures on invariant theory. London Mathematical Society Lecture Note Series, 296. Cambridge University Press, Cambridge, 2003.
• F. Kirwan, Cohomology of quotients in symplectic and algebraic geometry. Mathematical Notes, 31. Princeton University Press, Princeton, NJ, 1984.
• D. Mumford, J. Fogarty and F. C. Kirwan, Geometric invariant theory, Third edition, Springer, Berlin, 1994.

Geometric Invariant Theory and eigenvalue problem, by Nicolas Ressayre

Abstract: This lecture concerns two apparently unrelated questions. The first one, known as the Horn problem is "What can be said about the eigenvalues of the sum of two Hermitian matrices in terms of the eigenvalues of the summands ?".

The second one is known as the branching problem: Let $H\subset G$ be two connected complex reductive groups. Let $V_G$ be an irreducible representation of $G$. In an obvious way, $V_G$ is also an $H$-representation, but it is not irreducible. A natural question is "What are the irreducible $H$-subrepresentations of $V_G$?

We will see that the first question can be interpreted in terms of Hamiltonian action. The second one will be interpreted in terms of Geometric Invariant Theory. These two theories are related by Kempf-Ness' theorem in such a way that the first question will appear as a particular case of the second one. Then, we will show how to apply Geometric Invariant Theory to these questions.

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