Shahidi’s strong tempered packet conjecture

Establish Shahidi’s strong tempered packet conjecture for a quasi-split connected reductive group G over a local field: for any tempered L-parameter φ and any fixed Whittaker datum w for G, prove that the L-packet Πφ contains exactly one w-generic member πw and that its pairing with the component group Sφ satisfies ·,πwφ = 1.

Background

Appendix D formulates Shahidi’s strong tempered packet conjecture in Conjecture D.1.1 for general quasi-split connected reductive groups. The conjecture asserts uniqueness of a w-generic member in the tempered L-packet and compatibility of the internal parameterization with the trivial character of the component group.

The authors prove a relative result (Theorem D.1.2) that deduces the conjecture for G from its validity for endoscopic factorizations, and then conclude the conjecture for classical groups (Corollary D.1.3). The conjecture thus remains a general open problem beyond the cases treated in the paper.