Restrictions of Unitary Representations to Lattices and AssociatedC*-Algebras

LetGbe a connected simple Lie group with finite centre, and letΓbe a lattice inG. Letπbe an irreducible unitary representation ofG, and letπ∣1 be the restriction ofπtoΓ. A result by Cowling and Steger states thatπ∣1 remains irreducible ifπis not a discrete series representation and thatπ∣1 is determined byπ. Our first result shows that even the weak equivalence class ofπ∣1 is determined byπwhenπis a complementary series representation. LetC*π(Γ) denote theC*-algebra generated by allπ(γ) forγinΓ. We show thatC*π(Γ) has a unique maximal two-sided ideal and a unique normalized trace.