Overgroups of distinguished unipotent elements in reductive groups

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The proposal is in a core area of algebraic group theory and at the interdisciplinary cross roads of algebra, representation theory, and geometric invariant theory. It is a contribution to the study of the subgroup structure of reductive algebraic groups. Specifically, the aim of this proposal is to generalize the main results from recent joint work with Bate and Martin on overgroups of regular unipotent elements and work of Korhonen to reductive overgroups of distinguished unipotent elements of reductive groups (both possibly non-connected).By virtue of the central importance of Bala-Carter Theory, distinguished unipotent elements of reductive groups G and distinguished nilpotent elements of their Lie algebras Lie(G) play a pivotal role in the description and study of unipotent classes in G and nilpotent G-orbits in Lie(G), respectively. In turn, nilpotent orbits in positive characteristic are important in representation theory of finite groups of Lie type and reduced enveloping algebras. Therefore, an understanding of the properties of reductive overgroups H of such elements in G is of particular interest. We aim to show that under suitable natural circumstances in this setting such subgroups H are very special in that they are G-irreducible in the sense defined by J-P. Serre, that is, they are not contained in any proper parabolic subgroup of G. We intend to demonstrate that much of the machinery used to derive the principal results in our earlier paper, where the special case of overgroups H of regular unipotent elements is handled, can be employed in this more general setting to address the same question about G-irreducibility. Indeed, many of the key properties of reductive overgroups of regular unipotent elements of G are paralleled for their counterpart overgroups of distinguished unipotent elements. Central are the concepts of G-complete reducibility and optimal parabolic subgroups.However, we need to stress that this extension is far from mere routine, as the powerful pivotal theorems for regular unipotent elements from work of Steinberg and Spaltenstein are not available in this setting.