Minuscule representations, invariant polynomials, and spectral covers

Given a minuscule representation of a simple Lie algebra, we find an algebraic model for the action of a regular element and show that these models can be glued together over the adjoint quotient, viewed as the set of all regular conjugacy classes of the Lie algebra. There are partial results in the case of a quasiminuscule representation, and a conjecture in the case of a general irreducible finite-dimensional representation. The method of proof is to relate the question to a problem concerning holomorphic principal bundles over cuspidal cubic curves.