Poincaré series for non-Riemannian locally symmetric spaces

Abstract

We initiate the spectral analysis of pseudo-Riemannian locally symmetric spaces Γ\\G/H, beyond the classical cases where H is compact (automorphic forms) or Γ 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 Γ whose action on X is “sufficiently proper”, we construct L2-eigenfunctions of the Laplacian on XΓ:=Γ\\X for an infinite set of eigenvalues. These eigenfunctions are obtained as generalized Poincaré series, i.e. as projections to XΓ of sums, over the Γ-orbits, of eigenfunctions of the Laplacian on X.We prove that the Poincaré series we construct still converge, and define nonzero L2-functions, after any small deformation of Γ inside G, for a large class of groups Γ. 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 Teichmüller space of a compact Riemann surface.We actually construct joint L2-eigenfunctions for the whole commutative algebra of invariant differential operators on XΓ.