On Reductive Subgroups of Algebraic Groups and a Question of Külshammer
Degree GrantorUniversity of Canterbury
Degree NameDoctor of Philosophy
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.