A proof of Langlands’ conjecture on Plancherel measures; complementary series for \(p\)-adic groups.

*(English)*Zbl 0780.22005Langlands 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.

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.

Reviewer: J.G.M.Mars (Utrecht)

##### 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 |