Poincar\'e series for non-Riemannian locally symmetric spaces

TL;DR

This paper extends the study of the Laplacian's discrete spectrum to non-Riemannian, reductive, locally symmetric spaces, constructing eigenfunctions via Poincaré series and demonstrating their stability under small group deformations.

Contribution

It introduces a method to construct L^2-eigenfunctions on non-Riemannian symmetric spaces using generalized Poincaré series, showing their convergence and stability under deformations.

Findings

Constructed eigenfunctions as Poincaré series for non-Riemannian spaces.

Proved convergence and non-vanishing of these series after small deformations.

Established stability of the discrete spectrum under group deformations.

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 Poincar\'e 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 Poincar\'e 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 Teichm\"uller 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}.