Linear Algebraic Groups without the Normalizer Theorem

One can develop the basic structure theory of linear algebraic groups (the root system, Bruhat decomposition, etc.) in a way that bypasses several major steps of the standard development, including the self-normalizing property of Borel subgroups. An awkwardness of the theory of linear algebraic groups is that one must develop a lot of material about general linear algebraic groups before one can really get started. Our goal here is to show how to develop the root system, etc., using only the completeness of the flag variety and some facts about solvable groups. In particular, one can skip over the usual analysis of Cartan subgroups, the fact that G is the union of its Borel subgroups, the connectedness of torus centralizers, and the normalizer theorem (i.e., a Borel subgroup is self-normalizing). The main idea is a new approach to the structure of rank 1 groups; the key step is lemma 5. All algebraic geometry is over a fixed algebraically closed field. G always denotes a connected linear algebraic group with Lie algebra g, T a maximal torus, and B a Borel subgroup containing it. We assume the structure theory for connected solvable groups, and the completeness of the flag variety G/B and some of its consequences. Namely: that all Borel subgroups (resp. maximal tori) are conjugate; that G is nilpotent if one of its Borel subgroups is nilpotent; that CG(T )0 lies in every Borel subgroup containing T ; and that NG(B) contains B of finite index and (therefore) is self-normalizing. We also assume known that the centralizer of a torus has the expected dimension, namely, that of the subspace of g where the torus acts trivially. For these results we refer to Borel [1], Humphreys [2] and Springer [3]. In section 1 we develop a few properties of solvable groups, and in section 2 we treat the structure of rank 1 groups. The root system, etc., can then be developed in essentially the standard way, so after Date: August 27, 2007. 2000 Mathematics Subject Classification. 20GXX, 14GXX. Partly supported by NSF grants DMS-024512 and DMS-0600112.