## F. Kassel and T. Kobayashi.
Poincaré series for non-Riemannian locally symmetric spaces.
Advances in Mathematics **287** (2016), 123-236. Published online: 20-NOV-2015.
DOI:
10.1016/j.aim.2015.08.029. arXiv:
1209.4075..

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 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 *L*^{2}-eigenfunctions of the Laplacian on *X*_{Γ}:=Γ\*X* for an infinite set of eigenvalues.
These eigenfunctions are obtained as generalized Poincaré series, as projections to *X*_{Γ} 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 *L*^{2}-functions, after any small deformation of Γ, for a large class of groups Γ.
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*_{Γ}.

[ DOI |
arXiv |
IHES-preprint |
preprint version(pdf) ]

© Toshiyuki Kobayashi