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A proof of Langlands’ conjecture on Plancherel measures; complementary series for $$p$$-adic groups. (English) Zbl 0780.22005
Langlands conjectured that the local intertwining operators could be multiplied by certain factors such that they satisfy $$M(\sigma,w_ 1w_ 2) = M(w_ 2\sigma,w_ 1)M(\sigma,w_ 2)$$ and are unitary ($$\sigma$$ is an irreducible unitary representation of $$M$$, $$P = MN$$ a parabolic subgroup of the reductive group $$G$$).
The factors in question are expressed in terms of the values at 0 and 1 of (also conjectural) local $$L$$- and $$\varepsilon$$-factors. Shahidi proved this in case $$G$$ is quasi-split and $$\sigma$$ generic. It is a consequence of the main theorem of the paper, which proves the existence of $$\gamma$$-factors attached to a generic representation $$\sigma$$ of $$M$$ and certain representations of the $$L$$-group of $$M$$. These $$\gamma$$- factors are uniquely determined by a series of properties, among which there is that they are the right ones when the ground field is archimedean or $$\sigma$$ has an Iwahori fixed vector. The $$\gamma$$-factors are used to define $$L$$- and $$\varepsilon$$-factors, first for maximal $$P$$ and tempered $$\sigma$$ and then generally.
The main theorem is also used to determine the complementary series and special representations of $$G$$ which come from a maximal parabolic subgroup $$P$$ and an irreducible unitary supercuspidal generic representation of $$M$$. In the proof of the main theorem global methods are used.

MSC:
 22E50 Representations of Lie and linear algebraic groups over local fields 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11S37 Langlands-Weil conjectures, nonabelian class field theory 11R39 Langlands-Weil conjectures, nonabelian class field theory 22E35 Analysis on $$p$$-adic Lie groups 22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
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