My main field of research is to say something about actions of these groups. For example, GL(n,k) acts in a natural way on the vector space k^n, but also on the set of all subspaces (Graßmannian), the set of quadratic forms and so on.
A very special but important class of actions are called sperical varieties. These generalize symmetric spaces which have been investigated for a long time. Much of my research was devoted on generalizing previous results on symmetric spaces to spherical varieties (and then sometimes to arbitrary actions). This is by no means straightforward.
One of my favorite theorems is: Let G be a connected reductive group acting on a smooth affine variety X. Then the center of the ring of G-invariant differential operators on X is a polynomial ring. For details and a proof see the paper.
Other papers are available electronically, as well.