Unitary principal series of split orthogonal groups

Alessandra Pantano, Annegret Paul and Susana Salamanca Riba

Vol. 271 (2014), No. 2, 479–510

DOI: 10.2140/pjm.2014.271.479

Abstract

We prove the nonunitarity of a large set of parameters for Langlands quotients of minimal principal series of the orthogonal group $SO\left(n+1,n\right)$, by showing that the set of unitary principal series parameters of $SO\left(n+1,n\right)$ embeds into a (known) union of spherical unitary parameters for certain split orthogonal groups. In an earlier paper, we proved the nonunitarity of the genuine principal series of the metaplectic group $Mp\left(2n\right)$ attached to the same set of parameters. We conjecture that the set of parameters is complete in both cases and prove the conjecture for small rank groups and in the case of unipotent parameters.

Keywords

orthogonal groups, intertwining operators, petite K-types, complementary series, theta correspondence, unipotent representations, spherical unitary dual, Weyl group representations

Mathematical Subject Classification 2010

Primary: 22E45

Milestones

Received: 7 July 2013

Revised: 24 June 2014

Accepted: 28 June 2014

Published: 20 September 2014