Abstract

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 P_min = AU ⊂ G, with A a split torus and U unipotent. Suppose M is the Levi factor of a parabolic subgroup P ⊃ P_min, 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↪Ind_Pmin∩M^M(λ× 1)). The criterion gives the irreducibility of Ind_P^G(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.

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