Unipotent subgroups of reductive algebraic groups

• Main Author: Proud, Richard
• Published: University of Warwick 1997
• Subjects: 510; QA Mathematics
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.263492

Let G be a connected reductive algebraic group defined over an algebraically closed field of good characteristic p>0. Suppose uEG has order p. In [T2] it is shown that u lies in a closed reductive subgroup of G of type Al. This is the best possible group theoretic analogue of the Jacobson-Morozov theorem for simple Lie algebras. Testerman's key result is a type of `exponentiation process'. For our given element u, this process constructs a 1-dimensional connected abelian unipotent subgroup of G, hence isomorphic to Ga, containing u. This in turn yields the required Al overgroup of u. Now let 1#uEG be an arbitrary unipotent element. Such an element has order pt, for some tEN. In this thesis we extend the above result, and show that u lies in a t-dimensional closed connected abelian unipotent subgroup of G, provided p> 29 when G' contains a simple component of type E8, and that p is good for the remaining components. The structure of the resulting unipotent overgroup is also explicitly given. This is the best possible result, in terms of `minimal dimension', which we could hope for. In Chapter 1 we discuss the theory of Witt vectors, associated with a commutative ring with identity. They are closely related to the study of connected abelian unipotent algebraic groups. The unipotent overgroups are constructed using a variation of the usual exponen- tiation process. The necessary material on formal power series rings is given in 1.3. The Artin-Hasse exponentials of 1.4 play a crucial role in this construction. The connection between Witt groups and Artin-Hasse exponentials is discussed in 1.5. In Chapter 2 we apply the techniques of Chapter 1 to the various simple algebraic groups. For each type, a particular isogeny class is chosen and the required overgroup is constructed for the regular (and subregular) classes. In 2.9 we pass to the adjoint case. In Chapter 3 we extend the results of Chapter 2 to include all unipotent classes in all reductive algebraic groups (under certain restrictions). In 3.1 the Cayley Transform for the classical groups is combined with the ideas of Chapter 1 to give an explicit construction of the unipotent overgroups for every unipotent class. In 3.2 we discuss semiregular unipotent elements. Finally, in 3.3, we prove the main theorem of this thesis.