Parametrization of the Image of Normalized Intertwining Operators

In studying residual automorphic representations, we need to parametrize the image of normalized local intertwining operators. This has been done by Moeglin in the case of the residual spectrum attached to the trivial character of the torus for split classical groups. In this paper, we extend her result to non-trivial characters of the torus. To do this, we use Roche’s Hecke algebra isomorphisms and Barbasch-Moy’s graded algebra isomorphisms to reduce to the case of the trivial character. Along the way, we need to show that Roche’s Hecke algebra isomorphisms are compatible with induction in stages, construct a generalized Iwahori-Matsumoto involution, and show that the images of intertwining operators behave well with respect to the Hecke algebra and graded algebra isomorphisms. We note that this also gives a parameterization of the square-integrable and tempered representations supported on the Borel subgroup.