Discrete Components of Some Complementary Series (II)

TL;DR

This paper investigates the behavior of complementary series representations of SO(n,1) under restriction to SO(n-1,1) and explores implications for eigenvalues of the Laplacian on hyperbolic space quotients, linking representation theory with geometric analysis.

Contribution

It demonstrates the persistence of complementary series representations under restriction and connects spectral properties of the Laplacian to representation theory conjectures.

Findings

Complementary series close to cohomological representations contain similar series upon restriction.

Boundedness of Laplacian eigenvalues in middle degree implies boundedness in all degrees.

Reduces conjectures of Clozel and Bergeron to middle degree cases.

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.