  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.
