Study of a Z-form of the Coordinate Ring of a Reductive Group

In his famous paper [C1] Chevalley associated to any root datum of adjoint type and to any field k a certain group (now known as a Chevalley group) which in the case where k = C was the usual adjoint Lie group over C and which in the case where k is finite led to some new families of finite simple groups. Let O′ be the coordinate ring of a connected semisimple group G over C attached to a fixed (semisimple) root datum. In a sequel [C2] to [C1], Chevalley defined a Zform of O′. Later, another construction of such a Z-form was proposed by Kostant [Ko]. Kostant notes that O′ can be viewed as a “restricted” dual of the universal enveloping algebra U of Lie G; he defines a Z-form UZ of U (“the Kostant Zform”) and then defines the Z-form OZ as the set of all elements in O ′ which take integral values on UZ. Then for any commutative ring A with 1 he defines OA as A ⊗OZ; this is naturally a Hopf algebra over A. It follows that the set GA of A-algebra homomorphisms OA → A has a natural group structure. Thus the root datum gives rise to a family of groups GA, one for each A as above. Unlike Chevalley’s approach which was based on a choice of a faithful representation of G, Kostant’s approach is direct (no choices involved) and generalizes to the quantum case. In this paper we develop the theory of Chevalley groups following Kostant’s approach. We shall prove that: (I) If A is an algebraically closed field, then OA is the coordinate algebra of a connected semisimple algebraic group over A corresponding to the given root datum. (We treat the reductive case at the same time.) Note that (I) was stated without proof in [Ko]. In this paper we note that Kostant’s definition can be reformulated by replacing U by a “modified enveloping algebra”. The theory is then developed using extensively the theory of canonical bases of such modified enveloping algebras (presented in [L1]), coming from quantum groups. (See the Notes in [L1] for references to original sources concerning canonical bases.) We now present the content of this paper in more detail. Let A be a fixed commutative ring with 1 with a given invertible element v ∈ A.