Abstract
In this article we discuss a peculiar interplay between the representation theory of the holonomy group of a Riemannian manifold, the Weitzenbformula for the Hodge-Laplace operator on forms and the Lichnerowicz formula for twisted Dirac op- erators. For quaternionic Kahler manifolds this leads to simple proofs of eigenvalue estimates for Dirac and Laplace operators. Moreover, it enables us to determine which representations can contribute to harmonic forms. As a corollary we prove the vanish- ing 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.
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