The project will study reductive linear algebraic groups over a, possibly imperfect, ground field whose characteristic is a prime number p; this setting presents many important and interesting challenges. The author recently exploited a result in geometric invariant theory to give definitive existence and conjugacy results for a suitable class of rank 1 simple subgroups, called optimal, containing any given element of order p. If this element is rational over a ground field, the techniques yield an optimal rank one subgroup defined over that ground field. The project seeks to extend these results; an optimal rank 1 subgroup is G-completely reducible in the sense of J-P. Serre, and one hopes to describe any G-completely reducible rank 1 subgroup using the optimal ones. Moreover, the project will consider counterparts of these results for simple subgroups of rank greater than 1 -- e.g. one hopes to characterize the optimal ones. In a slightly different direction, the project seeks to extend recent results of the author on nilpotent orbits -- in the Lie algebra -- over ground fields. For an imperfect ground field, the author has showed for a rather general class of reductive groups that the orbits of the group of rational points on nilpotent elements -- i.e. the arithmetic nilpotent orbits -- have favorable properties; as consequences, for instance, one finds over a local ground field that there are finitely many arithmetic nilpotent orbits, and one finds that nilpotent orbital integrals converge; the convergence of orbital integrals was obtained in characteristic 0 by Deligne and Ranga Rao. The project will explore related issues; for instance, it seeks to exploit recent results of the author and E. Sommers to obtain information about nilpotent orbits for groups over the field of rational functions on an algebraic curve.
The structure and representations of linear algebraic groups - and especially the reductive ones -- are important to diverse parts of mathematics. The groups of their rational points over finite fields account for most of the finite simple groups; they are the natural transformation groups of many algebro-geometric questions; representations of the groups of their rational points over local and global fields carry deep number-theoretic information. The project will focus on important aspects of these groups. The local and global fields just mentioned are imperfect when their characteristic is positive; the author's existing results -- and those sought by the project -- on reductive groups over imperfect fields are essential to the study of linear groups in this "number theoretic" setting. Reasons for tackling the proposed problems abound: results on arithmetic nilpotent orbits for a group over a local field of positive characteristic should provide tools needed for the study of the relevant "harmonic analysis"; results on reductive subgroups defined over the ground field contribute to one's understanding and provide useful inductive tools.
