Unipotent elements in algebraic groups and finite groups of Lie type

University Of Oregon Eugene, Eugene OR

Investigators

Abstract

The project concerns several closely related problems in the theory of unipotent elements in algebraic groups, with applications to the structure theory of algebraic groups and finite groups of Lie type. A new approach to the analysis of centralizers of unipotent elements will be carried out, based on commuting pairs of reductive groups as developed in a recent paper of Seitz. New information resulting from this analysis will be applied to the study of a certain abelian subgroup canonically associated with a unipotent element, to obtain new information on saturation, to establish saturation results for certain sets of unipotent elements, and to study a new partial order on unipotent classes. These results will be used to study problems on the subgroup structure of algebraic groups and their finite analogs, in particular the modular version of the problem of determining conjugacy classes of finite simple subgroups of exceptional groups over the complex numbers. The theory of groups is sometimes described as the mathematics of symmetry. It is a fundamental subject within mathematics and has applications in a variety of scientific fields. Certain types of groups, those arising from Lie theory, occur time and again in widely different contexts and so it is important to establish general results about these groups. This project is aimed at obtaining such information for algebraic groups and associated finite groups of Lie type. These groups contain special elements, called unipotent elements, which play a particularly important role in the general theory. In spite of their fundamental importance, certain mysteries regarding these elements remain to be understood and some basic results are in an unsatisfactory state. The project is aimed at developing a new approach to this important theory, clarifying existing results and establishing new results. Seitz is in excellent position to address these problems and has recently obtained new results in the area.