In this paper, we prove a conjecture of Wei Zhang on comparison of certain local spherical characters from which we draw some consequences for the Ichino-Ikeda conjecture for unitary groups.

Comparison of local spherical characters and the Ichino-Ikeda conjecture for unitary groups

https://hal.archives-ouvertes.fr/hal-01941343/document

In this article, we consider the links between parabolic induction and the local Langlands correspondence. We enunciate a conjecture about the (enhanced) Langlands parameters of supercuspidal representation of split reductives $p$-adics groups. We are able to verify this in those known cases of the local Langlands correspondence for linear groups and classical groups. Furthermore, in the case of classical groups, we can construct the "cuspidal support" of an enhanced Langlands parameter and get a decomposition of the set of enhanced Langlands parameters a la Bernstein. We check that these constructions match under the Langlands correspondence and as consequence, we obtain the compatibility of the Langlands correspondence with parabolic induction.

/pdf/centre-de-bernstein-dual-pour-les-groupes-classiques-c67icvjcxs.pdf

Centre de Bernstein dual pour les groupes classiques

We develop a general procedure to study the combinatorial structure of Arthur packets for $p$-adic quasisplit $Sp(N)$ and $O(N)$ following the works of Mœglin. This allows us to answer many delicate questions concerning the Arthur packets of these groups, for example the size of the packets.

A combinatorial solution to M{\oe}glin's parametrization of Arthur packets for p-adic quasisplit $Sp(N)$ and $O(N)$

A combinatorial solution to mœglin’s parametrization of arthur packets for -adic quasisplit and

Zelevinsky’s classification theory of discrete series of p-adic general linear groups has been well known. Mœglin and Tadic gave the same kind of theory for p-adic classical groups, which is more complicated due to the occurrence of nontrivial structure of L-packets. Nonetheless, their work is independent of the endoscopic classification theory of Arthur (also Mok in the unitary case), which concerns the structure of L-packets in these cases. So our goal in this paper is to make more explicit the connection between these two very different types of theories. To do so, we reprove the results of Mœglin and Tadic in the case of quasisplit symplectic groups and orthogonal groups by using Arthur’s theory.

On the cuspidal support of discrete series for p-adic quasisplit $\textit{Sp}(N)$ and $\textit{SO}(N)$

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On the cuspidal support of discrete series for p-adic quasisplit \(\textit{Sp}(N)\) and \(\textit{SO}(N)\)