Discrete components of some complementary series representations

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

Branching Laws for Some Unitary Representations of SL(4,R)

In this paper we consider the restriction of a unitary irreducible representation of type Aq(λ) of GL(4,R) to reductive subgroups H which are the fixpoint sets of an involution. We obtain a formula for the restriction to the symplectic group and to GL(2,C), and as an application we construct in the last section some representations in the cuspidal spectrum of the symplectic and the complex general linear group. In addition to working directly with the cohmologically induced module to obtain the branching law, we also introduce the useful concept of pseudo dual pairs of subgroups in a reductive Lie group

Uniqueness of Bessel models: the archimedean case

In the archimedean case, we prove uniqueness of Bessel models for general linear groups, unitary groups and orthogonal groups.Comment: 22 page

World Spinors - Construction and Some Applications

The existence of a topological double-covering for the $GL(n,R)$ and diffeomorphism groups is reviewed. These groups do not have finite-dimensional faithful representations. An explicit construction and the classification of all $\bar{SL}(n,R)$, $n=3,4$ unitary irreducible representations is presented. Infinite-component spinorial and tensorial $\bar{SL}(4,R)$ fields, "manifields", are introduced. Particle content of the ladder manifields, as given by the $\bar{SL}(3,R)$ "little" group is determined. The manifields are lifted to the corresponding world spinorial and tensorial manifields by making use of generalized infinite-component frame fields. World manifields transform w.r.t. corresponding $\bar{Diff}(4,R)$ representations, that are constructed explicitly.Comment: 19 pages, Te