Poincare series for non-Riemannian locally symmetric spaces

We initiate the spectral analysis of pseudo-Riemannian locally symmetric spaces Gamma\G/H, beyond the classical cases where H is compact (automorphic forms) or Gamma is trivial (analysis on symmetric spaces). For any non-Riemannian reductive symmetric space X G/H 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 Poincare 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 Poincare series we construct still converge, and define nonzero L-2-functions, after any small deformation of Gamma inside G, for a large class of groups Gamma. Thus 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 Teichnffiller 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. (C) 2015 Elsevier Inc. All rights reserved.