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.
