Cohomology and the resolution of the nilpotent variety

1. Let G be a split reductive linear algebraic group over a field k of characteristic zero. Consider the variety N of the nilpotent elements of the Lie algebra g of G. It is a normal variety cf. [14] Theorem 16. It is isomorphic to the variety of the unipotent elements of G, cf. [17]. The theorem of Brieskorn-Steinberg-Tits states that the rational singularities are dense in the singular locus of N, see [1] and [18] (3.10). Here we shall prove that N has only rational singularities, cf. [t2] p. 50, i.e. we prove Theorem A. There exists a proper birational morphism z: Y~N such that Y is smooth over k, that z,((gy) = (9 s and RPz,((gr)=0 for p> = 1. This theorem admits a generalization which will be stated and proved in Section 5. In the complex analytic situation the same assertions follow by Theorem 5 of [3] exp. II. For rational singularities in that case see [2]. In [t 1] we investigated the local structure of N for the classical groups. 2. By [17] (2.2) we may assume that G is semi-simple and simply connected. Let G be split with respect o a maximal torus T and a Borel group B