First order differential operators on a locally symmetric space

Abstract

The basic result is the classication of rst order invariant elliptic dierential operators on a quotient of a spin symmetric space by a suitable discrete group: Such operators are all twisted Dirac operators. As a consequence we obtain conditions for the spectral symmetry to be equivariant. We also show that the characteristic numbers of these spaces vanish, a result previously obtained by Hirzebruch and Slodowy from the study of elliptic genera. Here we prove that any rst order invariant elliptic operator on a space nM, where M = G=H is a spin symmetric space and G is a discrete subgroup, is a twisted Dirac operator. Furthermore, we study the spectral symmetry of such operators when is co-compact giving simple conditions for spectral symmetry and, if G is compact, when this spectral symmetry is equivariant with respect to G. A symmetric space is given as M = G=H where G and H are Lie groups such that there is an involution on the Lie algebra, g ,o f Gwith xed set the Lie algebra h of H; and now M is compact if and only if G is compact. (Since every compact group has a noncompact dual we see that every compact symmetric space has a noncompact dual, and conversely.) When G is compact we may take = f1g and if G is not compact there are always co-compact subgroups , see [ 9]. Given a self-adjoint elliptic operator on a compact Riemannian manifold, there is a complete set of eigenvalues with nite dimensional eigenspaces. The eta function is dened as