|
In this paper, we give a
criterion for the irreducibility of certain induced representations, including, but not
limited to, degenerate principal series. More precisely, suppose G is the F-rational
points of a split, connected, reductive group over F, with F = ℝ or p-adic. Fix a
minimal parabolic subgroup Pmin = AU ⊂ G, with A a split torus and U unipotent.
Suppose M is the Levi factor of a parabolic subgroup P ⊃ Pmin, and ρ an
irreducible representation of M. Further, we assume that ρ has Langlands data
(A,λ) in the subrepresentation setting of the Langlands classification (so that
ρ↪IndPmin∩MM(λ ⊗ 1)). The criterion gives the irreducibility of IndPG(ρ ⊗ 1) if a
collection of induced representations, induced up to Levi factors of standard
parabolics, are all irreducible. This lowers the rank of the problem; in many cases, to
one.
|
