Let G be a simple algebraic group over an algebraically closed field of characteristic p ≥ 0. In this thesis we address the following problem: classify pairs of closed subgroups H, J ≤ G such
that there is a finite numbers of (H, J)-double cosets in G.
We restrict our attention to the case where G is a classical group and both H and J are connected.
A result of Brundan says that if H and J are maximal reductive then the existence of finitely
many (H, J)-double cosets is equivalent to G factorizing as G = HJ. Such factorizations have
been classified by Liebeck, Saxl and Seitz. On the other hand if both H and J are parabolic
subgroups, by the Bruhat decomposition there are finitely many (H, J)-double cosets.
We therefore assume that H is reductive and J is not. Moreover, we only consider the case
where H is simple and J is a maximal parabolic. With this setup, the problem has been solved
for G = SL(V ) by Guralnick, Liebeck, Macpherson and Seitz. When G = SL(V ), the maximal
parabolic subgroups correspond to stabilizers of subspaces, which implies that the problem is
equivalent to determining modules for simple algebraic groups with finitely many orbits on
subspaces.
We deal with the case G = Sp(V ) or G = SO(V ), where the maximal parabolic subgroups
are the stabilizers of totally singular subspaces. We do this by classifying all self-dual faithful
irreducible modules for simple algebraic groups with finitely many orbits on totally singular
subspaces of dimension 1 or 2.
Furthermore, we determine a complete list of candidates for simple algebraic groups with finitely many orbits on k-dimensional totally singular subspaces for k > 2.
We also prove that when H is simple, k = 1 or k = 2 and V is a faithful self-dual irreducible
H-module, there are finitely many orbits on totally singular k-spaces of V if and only if there is a dense orbit.
Finally, we extend the results for k = 1 to the case H semisimple and maximal among connected
subgroups of G.