Unitary Shimura correspondences for split real groups

Abstract

We find a relationship between certain complementary series representations for nonlinear coverings of split simple groups, and spherical complementary series for (different) linear groups. The main technique is Barbasch’s method of calculating some intertwining operators purely in terms of the Weyl group.

Unitary genuine principal series of the metaplectic group

“…The latter are known from [4]; hence we obtain the non-unitarity of a large (conjecturally complete) set of parameters for Langlands quotients of genuine principal series of Mp(2n). For the pseudospherical case, this result already appears in [3]. The authors of [3] prove that a similar embedding holds for all non-trivial coverings of simple split groups, but they restrict their attention to the pseudospherical case.…”

“…For the pseudospherical case, this result already appears in [3]. The authors of [3] prove that a similar embedding holds for all non-trivial coverings of simple split groups, but they restrict their attention to the pseudospherical case. In this paper, the emphasis is exclusively on Mp(2n), but the pseudospherical restriction is removed.…”

“…In this paper, the emphasis is exclusively on Mp(2n), but the pseudospherical restriction is removed. In the specific example of the pseudospherical principal series of Mp(2n), the embedding in [3] turns out to be a bijection. We conjecture that this result holds for all genuine principal series of Mp(2n) and provide some evidence for this claim.…”

“…Here is an equivalent formulation of Theorem 1.1. As an example, we include the picture of the spherical unitary duals of SO(2, 1) 0 and SO (3,2) 0 , and of the δ 2,1 -complementary series of Mp (6). See Figure 1.…”

The omega-regular unitary dual of the metaplectic group of rank 2

“…The results presented at the beginning of the section are well known in the literature (c.f. [5], [16], [2] and [7]) and are reported here for completeness. The content of the last part is more innovative.…”