Discrete Components of Some Complementary Series (II)

Abstract

We show that complementary series of SO(n,1) which are sufficiently close to a cohomological representation in the Fell topology, upon restriction to SO(n-1,1), contain discretely, complementary series for SO(n-1,1) which are also sufficiently close to cohomological representations. As a global application, we show that if the non-zero eigenvalues of the Laplacian for differential forms of middle degree on congruence quotients of the hyperbolic n-space remain bounded away from zero (for all even n), then nonzero eigenvalues of the Laplacian on forms of arbitrary degree remain bonded away from zero; this reduces conjectures of Clozel and Bergeron to the case of middle degree forms.