Kostantʹs Conjecture and the Geometry of Cartan Subalgebras in the Rank Two and Exceptional Cases Original Research Article

This paper deals with certain aspects of a conjecture made by B. Kostant in 1983 relating the Coxeter number to the occurrence of the simple finite groupsL(2, q) in simple complex Lie groups. In particular, we examine how the conjecture gives rise to certain presentations of the Lie algebra as a sum of Cartan subalgebras for the rank two and exceptional cases. The presentations studied are those with invariant properties with respect to Kostant and Kac elements. The techniques also connect certain presentations of theq+1 dimensional representations ofL(2, q) to certain properties of Jacobi sums.