A proof of Langlands’ conjecture on Plancherel measures; complementary series for \(p\)-adic groups. (English) Zbl 0780.22005

Shahidi, Freydoon

Ann. Math. (2) 132, No. 2, 273-330 (1990).

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.

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

Keywords:

local intertwining operators; irreducible unitary representation; reductive group; existence of \(\gamma\)-factors; generic representation; representations; maximal parabolic subgroup; irreducible unitary supercuspidal generic representation