Automorphism sheaves, spectral covers, and the Kostant and Steinberg sections

Kostant constructed a section from the adjoint quotient morphism of a simple Lie algebra to the open set of regular elements, and Steinberg constructed such a section for the adjoint quotient of a simply connected and simple algebraic group. In this paper, we show that all sections of the adjoint quotient are conjugate via a morphism from the adjoint quotient to the group. In particular, the sections constructed via the parabolic construction are conjugate either to the Kostant section or to the Steinberg section. The method of proof consists in studying automorphism sheaves of certain principle bundles over faamilies of cuspidal or nodal plane cubic curves.