My question is motivated by the following well-known computation. Let $M$ be an even dimensional Riemannian spin manifold and let $D$ be the spinor Dirac operator on $M$. Lichnerowicz showed that $D^2 = \nabla^* \nabla + \kappa/4$ where $\nabla$ is the spin connection on the spinor bundle and $\kappa$ is the scalar curvature of $M$. It is not hard to show that $\nabla^* \nabla$ is a positive operator, and thus if $\kappa > 0$ there is an interval containing $0$ in the real line which avoids the spectrum of $D^2$ (and therefore $D$). A corollary is the well-known fact that the Fredholm index of $D$ vanishes if $M$ has positive scalar curvature.
This example is a starting point for the entire theory of positive scalar curvature obstructions. The machinery involved gets more sophisticated, but in the end all one really needs about positive scalar curvature metrics on spin manifolds is the fact that they create a gap around 0 in the spectrum of the spinor Dirac operator.
So I am wondering if there are other geometric causes for gaps in the spectrum of a Dirac operator. Note that I do not want to restrict my attention to the spinor Dirac operator; I think the question is particularly interesting for the signature operator or the Dolbeault operator, for example. I am aware that elliptic operator theory of this sort helps produce a proof of the Weyl character formula, so conceivably the sort of answer that I'm looking for could come from representation theory.
Any ideas?
