TITLE: Elliptic Operators on Vector Bundles
QUESTION [5 upvotes]: I know the kernel of an elliptic operator on a compact manifold has finite dimension. Is the kernel of an elliptic operator on sections of a vector bundle a finite dimensional space?

REPLY [4 votes]: In general, no. As an example, the spin Dirac operator on even-dimensional complex hyperbolic space has nontrivial $L^2$-sections in its kernel. Because the space is symmetric, in particular homogeneous, one can conclude that the $L^2$-kernel must be infinite-dimensional. The proof uses some results of Harish-Chandra on representations of semi-simple noncompact Lie groups. For more details, see this article or arXiv:math/9905089.