Simple linear algebraic groups: representation theory and subgroup structure
This research project concerns the representation theory and the subgroup structure of simple linear algebraic groups. These two topics are very closely related, as for example the study of subgroups of classical groups is essentially the study of representations of groups. Two major open problems for researchers in this field are understanding the irreducible representations of simple linear algebraic groups and classifying their reductive subgroups up to conjugacy. Although considerable progress has been made towards both problems in the past decades, our understanding is still lacking and a complete solution seems to be unattainable for the foreseeable future. Let G be a simple linear algebraic group over an algebraically closed field. The aim of this research project is to make further progress on open problems which will improve our understanding of the representation theory and the subgroup structure of G. The main focus is on problems concerning unipotent elements of G and their overgroups. The project builds upon and continues the research work of the applicant, previously supported by Swiss National Science Foundation grants 200021_146223 (Ph.D.) and P2ELP2_181902 (postdoc). The project will make use of the results and methods from the Ph.D. thesis (2018) and published work (J. Algebra, 2017; J. Group Theory, 2018; Proc. Amer. Math. Soc., 2019; Transform. Groups, 2019) of the applicant.