Poincaré series for non-Riemannian locally symmetric spaces

Abstract

The discrete spectrum of the Laplacian has been extensively studied on reductive symmetric spaces and on Riemannian locally symmetric spaces. Here we examine it for the first time in the general setting of non-Riemannian, reductive, locally symmetric spaces. For any non-Riemannian, reductive symmetric space X on which the discrete spectrum of the Laplacian is nonempty, and for any discrete group of isometries Gamma whose action on X is sufficiently proper, we construct L^2-eigenfunctions of the Laplacian on X_{Gamma}:=Gamma\X for an infinite set of eigenvalues. These eigenfunctions are obtained as generalized Poincaré series, i.e. as projections to X_{Gamma} of sums, over the Gamma-orbits, of eigenfunctions of the Laplacian on X. We prove that the Poincaré series we construct still converge, and define nonzero L^2-functions, after any small deformation of Gamma, for a large class of groups Gamma. In other words, the infinite set of eigenvalues we construct is stable under small deformations. This contrasts with the classical setting where the nonzero discrete spectrum varies on the Teichmüller space of a compact Riemann surface. We actually construct joint L^2-eigenfunctions for the whole commutative algebra of invariant differential operators on X_{Gamma}.