On the tempered \(L\)-functions conjecture (Q2851617)

In [Ann. Math. (2) 132, No. 2, 273--330 (1990; Zbl 0780.22005)], \textit{F. Shahidi} defined \(L\)-functions for tempered generic representations of \(p\)-adic groups and formulated the tempered \(L\)-function conjecture. Let \(G\) be a quasi-split \(p\)-adic group and \(P=MU\) a standard parabolic subgroup. Let \(\tau\) be an irreducible tempered generic representation of \(M\). The \(L\)-group \(^L\!M\) of \(M\) acts by the adjoint action on the Lie algebra of \(^L\!U\). For each irreducible component \(r_i\) of this action, Shahidi defined an \(L\)-function \(L(s, \tau, r_i)\). Shahidi's tempered \(L\)-function conjecture is the statement that the \(L\)-functions \(L(s, \tau, r_i)\) are holomorphic for \(\mathrm{Re}(s) >0\).NEWLINENEWLINEIn this paper, the authors give a general proof of Shahidi's tempered \(L\)-function conjecture, which has previously been known in all but one case. As a corollary, they prove the standard module conjecture, which states that the Langlands quotient of a standard module is generic if and only if the standard module is irreducible and the inducing data are generic.