Restriction for general linear groups: The local non-tempered Gan–Gross–Prasad conjecture (non-Archimedean case)

We prove a local Gan–Gross–Prasad conjecture on predicting the branching law for the non-tempered representations of general linear groups in the case of non-Archimedean fields. We also generalize to Bessel and Fourier–Jacobi models and study a possible generalization to Ext-branching laws.

A necessary condition for discrete branching laws for Klein four symmetric pairs

We show a necessary condition for Klein four symmetric pairs [Formula: see text] satisfying the condition (D.D.); that is, there exists at least one infinite-dimensional simple [Formula: see text]-module that is discretely decomposable as a [Formula: see text]-module. This work is a continuation of [A criterion for discrete branching laws for Klein four symmetric pairs and its application to [Formula: see text], Int. J. Math. 31(6) (2020) 2050049]. Moreover, we define associated Klein four symmetric pairs, and we may use these tools to compute that a class of Klein four symmetric pairs do not satisfy the condition (D.D.); for example, [Formula: see text].

Branching laws for classical groups: the non-tempered case

This paper generalizes the Gan–Gross–Prasad (GGP) conjectures that were earlier formulated for tempered or more generally generic L-packets to Arthur packets, especially for the non-generic L-packets arising from Arthur parameters. The paper introduces the key notion of a relevant pair of Arthur parameters that governs the branching laws for ${{\rm GL}}_n$ and all classical groups over both local fields and global fields. It plays a role for all the branching problems studied in Gan et al. [Symplectic local root numbers, central critical L-values and restriction problems in the representation theory of classical groups. Sur les conjectures de Gross et Prasad. I, Astérisque 346 (2012), 1–109] including Bessel models and Fourier–Jacobi models.

A criterion for discrete branching laws for Klein four symmetric pairs and its application to E6(−14)

Let [Formula: see text] be a noncompact connected simple Lie group, and [Formula: see text] a Klein four-symmetric pair. In this paper, we show a necessary condition for the discrete decomposability of unitarizable simple [Formula: see text]-modules for Klein for symmetric pairs. Precisely, if certain conditions hold for [Formula: see text], there does not exist a unitarizable simple [Formula: see text]-module that is discretely decomposable as a [Formula: see text]-module. As an application, for [Formula: see text], we obtain a complete classification of Klein four symmetric pairs [Formula: see text], with [Formula: see text] noncompact, such that there exists at least one nontrivial unitarizable simple [Formula: see text]-module that is discretely decomposable as a [Formula: see text]-module and is also discretely decomposable as a [Formula: see text]-module for some nonidentity element [Formula: see text].