Algebraic Groups I. Unipotent radicals and reductivity

In class, we have proved the important fact that over any field k, a non-solvable connected reductive group containing a 1-dimensional split maximal k-torus is k-isomorphic to SL2 or PGL2. That proof relied on knowing that maximal tori remain maximal after a ground field extension to k, and so relies on Grothendieck’s theorem. But for algebraically closed fields there is no content to Grothendieck’s theorem, so for k = k this rank-1 classification is simpler to prove. The aim of this handout is to use the rank-1 classification (usually just over algebraically closed fields) to prove some important results on the behavior of unipotent radicals and the property of reductivity with respect to two ubiquitous operations on smooth connected affine groups over an arbitrary field k: the formation of quotient k-groups (modulo normal k-subgroup schemes) and the formation of centralizers of k-tori (which we have seen are always smooth and connected). Recall that it was proved in class by elementary means that reductivity is inherited by smooth connected normal k-subgroups. More specifically, we proved that if N ⊆ G is a smooth connected normal k-subgroup then Ru(Nk) ⊆ Ru(Gk) (so reductivity of G implies that of N). In fact, the inclusion Ru(Nk) ⊆ Nk∩Ru(Gk) of subgroup schemes of Gk (using scheme-theoretic intersection) is always an equality, but the proof rests on some non-trivial structural properties of reductive groups which have not yet been proved. (A proof is given in Proposition A.4.8 of “Pseudo-reductive groups”, working over k there.) The main input is the non-obvious fact that the scheme-theoretic center of a connected reductive group is a subgroup scheme of a torus (see Corollary 2.2 below), and so has no nontrivial subgroup schemes which can arise as subgroup schemes of smooth unipotent groups (HW5, Exercise 1). Notation. In what follows, G always denotes a smooth connected affine group over an arbitrary field k, unless we indicate otherwise. Also, following tradition, we often denote characters and cocharacters of tori in additive notation, for instance writing −λ rather than λ−1 for the composition of a homomorphism λ : Gm → T with inversion and likewise writing 0 to denote the trivial character of T . The reason for doing this is that it is convenient to work with the Q-vector space X(T )Q and to view the collections of characters and cocharacters as Z-lattices.