Vanishing theorems for quaternionic Kahler manifolds

In this article we discuss a peculiar interplay between the representation theory of the holonomy group of a Riemannian manifold, the Weitzenbock formula for the Hodge-Laplace operator on forms and the Lichnerowicz formula for twisted Dirac operators. For quaternionic Kahler manifolds this leads to simple proofs of eigenvalue estimates for Dirac and Laplace operators. We determine which representations may contribute to harmonic forms and prove the vanishing of certain odd Betti numbers on compact quaternionic Kahler manifolds of negative scalar curvature. We simplify the proofs of several related results in the positive case.