Discrete components of some complementary series representations

Abstract

We show that the restriction of the complementary series representations of SO(n, 1) to SO(m, 1) (m < n) contains complementary series representations of SO(m, 1) discretely, provided that the continuous parameter is sufficiently close to the first point of reducibility and the representation of M- the compact part of the Levi- is a sufficiently small fundamental representation. We prove, as a consequence, that the cohomological representation of degree i of the group SO(n, 1) contains discretely, for i ≤ m/2, the cohomological representation of degree i of the subgroup SO(m, 1) if i ≤ m/2. As a global application, we show that if G/Q is a semisimple algebraic group such that G(R) = SO(n, 1) up to compact factors, and if we assume that for all n, the tempered cohomological representations are not limits of complementary series in the automorphic dual of SO(n, 1), then for all n, non-tempered cohomological representations are isolated in the automorphic dual of G. This reduces conjectures of Bergeron to the case of tempered cohomological representations.