This thesis is a study of several spaces of continuous linear functionals on various function spaces with a natural norm inherited from a larger Banach space. The completeness of these normed linear spaces is studied in detail and several necessary and sufficient conditions are obtained in this regard. Since spaces of continuous linear functionals are inherently related to spaces of measures, their measure-theoretic counterparts are also studied. By using these counterparts, several necessary and sufficient conditions are obtained on the separability of these spaces of continuous linear functionals. / Ph. D.

Discusses certain one-parameter subgroups of the low-rank unitary groups called root subgroups. Unitary groups also have representations of Lie type which means they consist of transformations that act as automorphisms of an underlying Lie algebra, in this case the special linear algebra. Exploring this definition of the unitary groups, we find a correlation, via exponentiation, to the basis elements of Lie algebra.

Doctor of Philosophy / Department of Mathematics / Zongzhu Lin / Let G be a connected and simply connected semisimple algebraic group over an algebraically closed field of positive prime characteristic. Let L([lambda]) and [upside-down triangle]([lambda]) be the simple and induced finite dimensional rational G-modules with p-singular dominant highest weight [lambda]. In this thesis, the concept of Weyl filtration dimension of a finite dimensional rational G-module is studied for some highest weight modules with p-singular highest weights inside the p2-alcove when G is of type B[subscript]2. In chapter 4, intertwining morphisms, a diagonal G-module morphism and tilting modules are used to compute the Weyl filtration dimension of L([lambda]) with [lambda] p-singular and inside the p[superscript]2-alcove. It is shown that the Weyl filtration dimension of L([lambda]) coincides with the Weyl filtration dimension of [upside-down triangle]([lambda]) for almost all (all but one of the 6 facet types) p-singular weights inside the p[superscript]2-alcove. In chapter 5 we study some submodule structures of Weyl (and there translations), Vogan, and tilting modules with both p-regular and p-singular highest weights. Most results are for the p[superscript]2 -alcove only although some concepts used are alcove independent.

This Thesis is motivated by two problems, each concerning representations (homomorphisms) of groups into a connected reductive algebraic group G over an algebraically closed field k. The first problem is due to B. Külshammer and is to do with representations of finite groups in G: Let Γ be a finite group and suppose k has characteristic p. Let Γp be a Sylow p-subgroup of Γ and let ρ : Γp → G be a representation. Are there only finitely many conjugacy classes of representations ρ' : Γ → G whose restriction to Γp is conjugate to ρ? The second problem follows the work of M. Liebeck and G. Seitz: describe the representations of connected reductive algebraic H in G. These two problems have been settled as long as the characteristic p is large enough but not much is known in the case where the characteristic p is a so called bad prime for G, which will be the setting for our work. At the intersection of these two problems lies another problem which we call the algebraic version of Külshammer's question where we no longer suppose Γ is finite. This new variation of Külshammer's question is interesting in its own right, and a counterexample may provide insight into Külshammer's original question. Our approach is to convert these problems into problems in the nonabelian 1-cohomology. Let K be a reductive algebraic group, P a parabolic subgroup of G with Levi subgroup L < P, V the unipotent radical of P. Let ρ₀ : K → L be a representation. Then the representations ρ : K → P that equal ρ₀ under the canonical projection P → L are in bijective correspondence with elements of the space of 1-cocycles Z¹(K,V ) where K acts on V by xv = ρ₀(x)vρ₀(x)⁻¹. We can then interpret P- and G-conjugacy classes of representations in terms of the 1-cohomology H¹(K,V ). We state and prove the conditions under which a collection of representations from K to P is a finite union of conjugacy classes in terms of the 1-cohomology in Theorem 4.22. Unlike other approaches, we work directly with the nonabelian 1-cohomology. Even so, we find that the 1-cocycles in Z¹(K,V ) often take values in an abelian subgroup of V (Lemmas 5.10 and 5.11). This is interesting, for the question "is the restriction map of 1-cohomologies H¹(H,V) → H¹(U,V) induced by the inclusion of U in K injective?" is closely linked to the question of Külshammer, and has positive answer if V is abelian and H = SL₂k) (Example 3.2). We show that for G = B4 there is a family of pairwise non-conjugate embeddings of SL₂in G, a direction provided by Stewart who proved the result for G = F4. This is important as an example like this is first needed if one hopes to find a counterexample to the algebraic version of Külshammer's question.

The goal of this paper is to establish the dependency of the topology of a simple Lie group, specifically any of the special linear groups, on its underlying group structure. The intimate relationship between a Lie group's topology and its algebraic structure dictates some necessary topological properties, such as second countability. However, the extent to which a Lie group's topology is an "algebraic phenomenon" is, to date, still not known.

Let G be a semisimple linear algebraic group over an algebraically closed field of prime characteristic. In this thesis we outline the theory of such groups and their cohomology. We then concentrate on algebraic groups in rank 1 and 2, and prove some new results in their bounding cohomology.

We provide a new relative trace formula approach to the theorem of Waldspurger on toric periods for GL₂, with possible applications to the global Gross-Prasad conjecture for orthogonal groups.

Thesis (M.S.)--Brigham Young University. Dept of Mathematics, 2006. / Includes bibliographical references (p. 30).

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Please refer to the thesis for the abstract. / Thesis (Ph.D.)-University of Natal, Durban, 1998.