Minuscule Representations, Invariant Polynomials, and Spectral Covers

Let G be a simple and simply connected complex linear algebraic group, with Lie algebra g. Let ρ : G → AutV be an irreducible finite-dimensional representation of G, and let ρ∗ : g → EndV be the induced representation of g. The goal of this paper is to study ρ∗, and in particular to give normal forms for the action of ρ∗(X) for regular elements X of g. Of course, when X is semisimple, the action of ρ∗(X) on V can be diagonalized, and its eigenvalues are given by evaluating the weights of ρ on an element in the Cartan subalgebra of g conjugate to X. If on the other hand X is a principal nilpotent element of g, then X can be completed to an sl2-triple (X,h0,X−), where h0 is regular and semisimple. In this case, if h is the Cartan subalgebra containing h0, then the eigenvalues for ρ∗(h0) together with their multiplicities, which are given by evaluating the weights of ρ with respect to h on h0, completely determine V as an sl2-module and hence determine the action of ρ∗(X) on V . A minor modification of these ideas will then describe the action of ρ∗(X) for every regular element X. In this paper, we attempt to give a different algebraic model for the action of ρ∗(X), where X is regular, and to glue these different models together over the set of all regular elements. While we are only successful in case ρ is minuscule, and partially successful in case ρ is quasiminuscule, we believe that the techniques of this paper can be extended to give information about an arbitrary ρ. Related methods also handle the case of ρ(g), where g is a regular element of G. To explain our results in more detail, we begin by recalling some results of Kostant on the adjoint quotient of g. Let h be the Lie algebra of a Cartan subgroup H of G and let W be the Weyl group of the pair (G,H). Kostant has shown [8] that the GIT quotient of g by the adjoint action of G is isomorphic to h/W . Moreover, he has constructed an explicit cross-section Σ of the natural morphism g → h/W , such that the image of Σ is contained in the dense open subset greg of regular elements of g. The first author was partially supported by NSF grant DMS-99-70437. The second author was partially supported by NSF grant DMS-97-04507.