Arthur parameters and cuspidal automorphic modules of classical groups. (English) Zbl 1443.11084

Jiang, Dihua; Zhang, Lei

Ann. Math. (2) 191, No. 3, 739-827 (2020).

Let us denote by \(F\) a number field and by \(\mathbb{A}\) its ring of adeles. Let \(G\) stand for a classical group over \(F\). By the endoscopic theory, intially established by Arthur, irreducible automorphic representations of \(G(\mathbb{A})\) occurring in the discrete spectrum can be parametrized by global Arthur parameters, up to global Arthur packets. The trace formula method establishes character relation between irreducible cuspidal automorphic representations of classical groups and their global Arthur parameters. A crucial question, posed by Arthur and Schmidt, is how to construct a concrete model for any irreducible cuspidal automorphic representation in terms of its global Arthur parameter.
In the paper under the review, authors provide a general construction of concrete models for cuspidal automorphic representations of general classical groups. They introduce a theory of concrete modules, under certain conjectures, for cuspidal automorphic representations with generic global Arthur parameters. A key ingredient is an extension of the method of automorphic descent [D. Ginzburg et al., The descent map from automorphic representations of \(\text{GL}(n)\) to classical groups. Hackensack, NJ: World Scientific (2011; Zbl 1233.11056)] to general classical groups and general cuspidal automorphic representations with generic global Arthur parameters. To do so, authors establish the global non-vanishing of the twisted automorphic descents constructed from given data and prove irreducibility of modules constructed using the automorphic descent. The later is obtained using the local Gan-Gross-Prasad conjecture as input. Along the way, authors prove one direction of the global Gan-Gross-Prasad conjecture in full generality and the other direction with the global assumption.
The approach used in the paper provides a uniform treatment of the global Gan-Gross-Prasad conjecture for unitary and orthogonal groups, and using the Fourier-Jacobi periods one can obtain it for symplectic and metaplectic groups. It is naturally related to the theory of twisted automorphic descents and the Rankin-Selberg method, and does not use the cuspidal multiplicity one assumption, assumed in [W. T. Gan et al., Astérisque 346, 1–109 (2012; Zbl 1280.22019)].

Reviewer: Ivan Matić (Osijek)

MSC:

11F70	Representation-theoretic methods; automorphic representations over local and global fields
22E50	Representations of Lie and linear algebraic groups over local fields

Keywords:

cuspidal automorphic modules; Arthur parameters; classical groups; Bessel-Fourier coefficients; twisted automorphic descent; global Gan-Gross-Prasad conjecture

Citations:

Zbl 1233.11056; Zbl 1280.22019