Orbits and generic stabilizers for actions of algebraic groups

This project is in the area of the representation theory of algebraic groups, a mainstream part of modern algebra. The simple algebraic groups over algebraically closed fields were classified by Chevalley, and correspond to the simple Lie algebras. Each has an irreducible root system with connected Dynkin diagram, and can be studied using a variety of methods from group theory, commutative algebra, topology and geometry. A representation of an algebraic group is a homomorphism from the group to a group of linear transformations of a vector space. Thus representations give concrete realisations of algebraic groups as groups of transformations, or matrices. One can then study the geometry of these vector spaces to learn about the algebraic groups. This project seeks to do this by studying the orbits of algebraic groups on the vector spaces arising from their representations. Specifically, if G is a simple algebraic group, and G -> GL(V) is a representation of G as a group of linear transformations of a vector space V, we seek answers to the following questions: (1) for which pairs (G,V) does G have only a finite number of orbits on V, or on P(k,V), the Grassmanian variety of k-dimensional subspaces of V? (2) does G always have a 'generic stabilizer' in its action on V (that is, can we find a dense open set U of vectors such that all the stabilizers in G of vectors in U are conjugate - this would be the generic stabilizer? (3) the same questions, but for 'totally singular' vectors and k-subspaces, when G preserves a bilinear form on V. These questions lead to many interesting sub-problems in algebraic group theory of a geometric and representation theoretic nature. Also a positive solution to (2) and (3) would give some of the first general instances of the existence of generic stabilizers in arbitrary characteristics - these are known to exist for all G-actions on varieties in characteristic zero, but are much rarer in positive characteristic.