DISCRETE COMPONENTS OF SOME COMPLEMENTARY SERIES

Abstract. We show that Complementary Series representations of SO(n, 1), which are sufficiently close to a cohomological repre-sentation contain discretely, Complementary Series of SO(m, 1) also sufficiently close to cohomological representations, provided that the degree of the cohomological representation does not ex-ceed m/2. We prove, as a consequence, that the cohomological represen-tation of degree i of the group SO(n, 1) contains discretely, 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 fac-tors, and if we assume that for all n, the tempered cohomologi-cal representations are not limits of complementary series in the automorphic dual of SO(n, 1), then for all n, non-tempered coho-mological representations are isolated in the automorphic dual of G. This reduces conjectures of Bergeron to the case of tempered cohomological representations