Double cosets of algebraic groups by maximal rank subgroups

Let G be a simple algebraic group with X and P closed subgroups. The double coset question asks whether XG/P is finite. Many interesting problems in the structure theory and representation theory of groups can be viewed as examples of the double coset question. This paper gives a classification of when XG/ P is finite, subject to certain conditions upon X and P. When G is a classical group most of the results are proven using the geometry of the classical groups. In general one can relate the double coset question for algebraic groups to whether certain finite groups have a bounded number of double cosets. At the finite group level elementary character theory and Deligne-Lusztig character theory are used. A theme of this approach is to apply knowledge about algebraic groups to finite groups, and vice versa.