Wednesday, September 19, 2012

1209.4075 (Fanny Kassel et al.)

Discrete spectrum for non-Riemannian locally symmetric spaces. I.
Construction and stability

Fanny Kassel, Toshiyuki Kobayashi
We study the discrete spectrum of the Laplacian on certain pseudo-Riemannian manifolds which are quotients X_{Gamma} = Gamma\X of reductive symmetric spaces X by discrete groups of isometries Gamma acting properly discontinuously. Assuming that X admits a maximal compact subsymmetric space of full rank, we construct L^2-eigenfunctions on X_{Gamma} for an infinite set of eigenvalues. In contrast to the classical setting where the nonzero discrete spectrum varies on the Teichm\"uller space of a compact Riemann surface, we prove that this infinite set of eigenvalues is stable under any small deformation of Gamma, for a large class of groups Gamma. We actually construct joint L^2-eigenfunctions for the whole commutative algebra of invariant differential operators on X_{Gamma}.
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