Discrete components of some complementary series representations
It is shown 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 under certain conditions. As a consequence, it is proved that the cohomological representation of degree \(i\) of the group \(SO(n,1)\) contains discretely, for \(i\leq m/2\), the cohomological representation of degree \(i\) of the subgroup \(SO(m,1)\) for \(i\leq m/2\). As a global application, it is shown that if \(G/\mathbb{Q}\) is a semisimple algebraic group such that \(G(\mathbb{R})=SO(n,1)\) up to compact factors and if 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 \(G\).
scientific article; zbMATH DE number 5734654