Let F be a non-archimedean local field of characteristic zero. Let E be a quadratic extension of F. Then G(F) is a natural subgroup of G(E). For the complex representations, assuming the enhanced local Langlands correspondence for a quasi-split group G, Dipendra Prasad gave a conjectural identity for the multiplicity \({\rm Hom}_{G(F)}(\pi,\chi_G)\) for a smooth irreducible representation \( \pi \) of G(E), in terms of the enhanced parameter of \(\pi\), where \(\chi_G\) is a quadratic character of G depending on the quadratic extension E/F. We are trying to verify the Prasad conjecture for G=PGL(2) in the modular setting under the local Langlands correspondence set up by Vigneras. It turns out that it fails. Then we propose another solution using non-nilpotent Weil-Deligne representations. This is joint work with Peiyi Cui and Thomas Lanard.
The Prasad conjecture for PGL(2) in the modular setting
15.03.2022 13:15 - 14:45
Organiser:
H. Grobner, A. Minguez-Espallargas, A. Mellit
Location: