arXiv subjects
Publications and source records attributed to W. Ethan Duckworth.
This paper describes how to use subgroups to parameterize unipotent classes in the classical algebraic group in characteristic 2. These results can be viewed as an extension of the Bala-Carter Theorem, and give a convenient way to compare unipotent classes in a group $G$ with unipotent classes of a subgroup $X$ where $G$ is exceptional and $X$ is a Levi subgroup of classical type.
Let $G$ be a classical algebraic group, $X$ a maximal rank reductive subgroup and $P$ a parabolic subgroup. This paper classifies when $X\G/P$ is finite. Finiteness is proven using geometric arguments about the action of $X$ on subspaces of the natural module for $G$. Infiniteness is proven using a dimension criterion which involves root systems.
This paper provides new, relatively simple proofs of some important results about unipotent classes in simple linear algebraic groups. We derive the formula for the Jordan blocks of the Richardson class of a parabolic subgroup of a classical group. This result was originally due to Spaltenstein. Secondly, we derive, for good characteristic, the description of the natural partial order of unipotent classes of a classical group in terms of their Jordan blocks. This result was originally due to Gerstenhaber and Hesselink. As a consequence we obtain a proof of the Bala--Carter Theorem which holds even in certain bad characteristics (this proof requires the prior classification of unipotent classes, unlike the original proofs due to Bala, Carter and Pommerening).
Let $G$ be a linear algebraic group defined over an algebraically closed field. The double coset question addressed in this paper is the following: Given closed subgroups $X$ and $P$, is the double coset collection $X\backslash G/P$ finite or infinite? We limit ourselves to the case where $X$ is maximal rank and reductive and $P$ parabolic. This paper presents a criterion for infiniteness which involves only dimensions of centralizers of semisimple elements. This result is then applied to finish the classification of those $X$ which are spherical. Finally, excluding a case in $F_4$, we show that if $X\backslash G/P$ is finite then $X$ is spherical or the Levi factor of $P$ is spherical. This implies that it is rare for $X\backslash G/P$ to be finite. The primary method of proof is to descend to calculations at the finite group level and then to use elementary character theory.