Simply Connected Groups, the Hyperalgebra, and Verma's Conjecture

Abstract

A finite-dimensional representation of a connected affine algebraic group is determined by the behavior of the operators coming from the hyperalgebra of the group. In order to get a complete picture of the representation theory of the group in terms of that of the hyperalgebra, one questions whether every finite-dimensional representation of the hyperalgebra comes from a representation of the group (see [1] for instance). In this paper, we show the following two things: 1. A simply connected group in positive characteristic is the inductive limit of its infinitesimal neighborhoods of the identity. 2. Each finite-dimensional representation of the hyperalgebra of a simply connected group comes from a representation of the group. Each of these two properties characterizes the simply connected groups. The proof of the second fact completes an aspect of the work begun in the paper [1] on the hyperalgebra. In characteristic zero, a group is called simply connected if it cannot be covered by another group by a map with finite, non-trivial kernel. In line with this definition, Takeuchi in [3] defines a group in characteristic p to be simply connected if it has no non-trivial etale coverings. An interpretation of Theorem 1.9 of [3] leads to condition 1 above for simple connectedness in terms of infinitesimal neighborhoods. This condition in turn yields Verma's conjecture: finite-dimensional representations of the hyperalgebra of a simply connected group are rational. Let G be an affine k-group with coordinate ring A over a field k of characteristic p. Let M be the kernel of the augmentation map of A, and let