Title:

Unipotent elements in algebraic groups

This thesis is concerned with three distinct, but closely related, research topics focusing on the unipotent elements of a connected reductive algebraic group G, over an algebraically closed field k, and nilpotent elements in the Lie algebra g = LieG. The first topic is a determination of canonical forms for unipotent classes and nilpotent orbits of G. Using an original approach, we begin by obtaining a new canonical form for nilpotent matrices, up to similarity, which is symmetric with respect to the nonmain diagonal (i.e. it is fixed by the map f : (xi;j) > (xn+1j;n+1i)), with entries in {0,1}. We then show how to modify this form slightly in order to satisfy a nondegenerate symmetric or skewsymmetric bilinear form, assuming that the orbit does not vanish in the presence of such a form. Replacing G by any simple classical algebraic group, we thus obtain a unified approach to computing representatives for nilpotent orbits for all classical groups G. By applying Springer morphisms, this also yields representatives for the corresponding unipotent classes in G. As a corollary, we obtain a complete set of generic canonical representatives for the unipotent classes of the finite general unitary groups GUn(Fq) for all prime powers q. Our second topic is concerned with unipotent pieces, defined by G. Lusztig in [Unipotent elements in small characteristic, Transform. Groups 10 (2005), 449487]. We give a casefree proof of the conjectures of Lusztig from that paper. This presents a uniform picture of the unipotent elements of G, which can be viewed as an extension of the DynkinKostant theory, but is valid without restriction on p. We also obtain analogous results for the adjoint action of G on its Lie algebra g and the coadjoint action of G on g*. We also obtain several general results about the Hesselink stratification and Fqrational structures on Gmodules. Our third topic is concerned with generalised GelfandGraev representations of finite groups of Lie type. Let u be a unipotent element in such a group GF and let Γu be the associated generalised GelfandGraev representation of GF . Under the assumption that G has a connected centre, we show that the dimension of the endomorphism algebra of Γu is a polynomial in q (the order of the associated finite field), with degree given by dimCG(u). When the centre of G is disconnected, it is impossible, in general, to parametrise the (isomorphism classes of) generalised GelfandGraev representations independently of q, unless one adopts a convention of considering separately various congruence classes of q. Subject to such a convention, we extend our result. We also present computational data related to the main theoretical results. In particular, tables of our canonical forms are given in the appendices, as well as tables of dimension polynomials for endomorphism algebras of generalised GelfandGraev representations, together with the relevant GAP source code.
