K-theory for Sp(n,1)

The classification of unitary irreducible representations of G=Sp(n,1),(n⩾2), is done by Baldoni-Silva [Duke Math. J. 48 (3) (1981) 549-583]. In particular, there are Langlands quotients that do not appear in the continuous complementary series. They are called isolated series. Let λ:Cmax*(G)→Cred*(G) be the regular representation. In this article we show that the representation λ⊕(⊕π), where π runs over the set of isolated series, induces an isomorphism in K-theory. In particular, the kernel of the map induced by λ in K-theory, is a free Z-module with a set of generators in bijective correspondence with the set of isolated series. Let K be a maximal compact subgroup of G, and let R(K) be its representation ring. We then compute the range of the full Baum-Connes map R(K)→K0(Cmax*(G)) in terms of these generators.