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Steinberg's tensor product theorem and exceptional groups

Let X be a simple, simply connected algebraic group over an algebraically closed field k of characteristic p. Steinberg's tensor product theorem states that if V is a finite dimensional irreducible kXmodule then V = V₁ ^(q1) ⊗ ... ⊗ V_n ^(qn) where each V_i is a restricted kXmodule and the q_i are distinct powers of p. This theorem can be reformulated in terms of rational representations instead of modules as follows: any irreducible rational representation Φ : X → SL(V) can be factorised as Φ : X →^ψ X x X x ... x X →^μ SL(V) with Ψ : x → (x^(q_1) , ... , x^(q_j)) where x^(q_i) denotes the image of x under the standard Frobenius q_i map and μ restricts to a completely reducible restricted representation on each factor. Liebeck and Seitz proved a generalisation of this theorem where the target group SL(V) is replaced by an arbitrary simple algebraic group G over k in good characteristic. A consequence of this theorem is that every connected simple Gcompletely reducible (Gcr) subgroup X of G is contained in a uniquely determined commuting product R₁...R_n in G such that each Rᵢ is a simple restricted subgroup of the same type as X and each projection X →Rᵢ/Z(Rᵢ) is non trivial and involves a different field twist. Here saying that X is Gcr means that whenever X is contained in a parabolic subgroup P of G, it is contained in a Levi subgroup of P. The first new theorem of this thesis is a converse of this result: if G is of exceptional type in good characteristic and X is contained in such a product R₁ ... R_n, then X is Gcr, with two exceptions. If G is of classical type we prove a similar theorem with the extra assumption that the Rᵢ are Gcr. In view of these results, it is of interest to determine the products R₁ ... R_n of more than one restricted subgroup of the same type in exceptional algebraic groups; this is achieved in the rest of this thesis. The above results have a number of consequences. First, every restricted connected simple subgroup of G is Gcr with two exceptions. We also classify all nonrestricted Gcr simple subgroups of exceptional groups. We find the centralisers of all products of more than one subgroup of the same type and of all nonrestricted Gcr simple subgroups. We also prove that the conjugacy classes of such subgroups are uniquely determined by the restriction of the Lie algebra of G.
