Automorphism Sheaves, Spectral Covers, and the Kostant and Steinberg Sections

Throughout this paper, G denotes a simple and simply connected algebraic group over C of rank r and H is a Cartan subgroup, with Lie algebras g = LieG and h = LieH. Let R be the root system of the pair (G,H), W the Weyl group, and Λ ⊆ h the coroot lattice. Fix once and for all a positive Weyl chamber, i.e. a set of simple roots ∆. The geometric invariant theory quotient of g by the adjoint action of G is identified with h/W , which by a theorem of Chevalley is isomorphic to A. Let greg ⊆ g be the Zariski open and dense subset of g consisting of regular elements, i.e. of elements X whose centralizer (in g or G) has the minimal dimension r. If X is regular, then the centralizers of X in g and in G, i.e. the stabilizer of X in G under the adjoint representation, are abelian. One can show that each fiber of the adjoint quotient morphism g → h/W contains a unique orbit of regular elements, so that the restriction of the adjoint quotient morphism to greg induces an isomorphism greg/G → h/W . In 1963, Kostant [14] proved that the morphism greg → h/W is smooth and constructed a section σ : h/W → greg of the adjoint quotient morphism, generalizing the existence of rational canonical forms of a matrix. Actually, the construction yields a family of such sections, defined as follows. Let X ∈ greg be a principal nilpotent matrix and let L ⊆ g be a linear complement to ImadX. Then the affine subvariety L + X is contained in greg and maps isomorphically to h/W , and so defines a section of the adjoint quotient morphism. One can make an analogous construction for the adjoint quotient of G, i.e. the geometric invariant theory quotient of the action of G on itself by conjugation. In this case, the adjoint quotient is equal to H/W , which is again isomorphic to A [2]. Let Greg be the Zariski open and dense subset of regular elements, where by definition g ∈ G is regular if and only if the dimension of its centralizer is r, the minimal possible dimension. If g is regular, then its centralizer is abelian. Then again each fiber of the morphism G → H/W contains a unique conjugacy class of regular elements, so that Greg/G is isomorphic to H/W . In 1965, using The first author was partially supported by NSF grant DMS-02-00810. The second author was partially supported by NSF grant DMS-01-03877.