Unipotent automorphic representations: Conjectures. (English) Zbl 0728.22014

Arthur, James

Orbites unipotentes et représentations. II: Groupes p-adiques et réels, Astérisque 171-172, 13-71 (1989).

[For the entire collection see Zbl 0694.00012.]
Let G be a connected reductive algebraic group over a field F of characteristic 0, \(G({\mathbb{A}}_ F)\) be the group of adelic points of G, which contains G(F) as a discrete subgroup. The basic analytic object is the right regular representation R of \(G({\mathbb{A}}_ F)\) and its decomposition into the irreducible components. Let \(\Pi\) (G) be the set of irreducible representations \(\pi \in \Pi_{unit}(G({\mathbb{A}}_ F))\) which occur in the decomposition of R. The purpose of this paper is to describe a conjectural extension of the theory which would account for all representations in \(\Pi\) (G). The theory of endoscopy works best for tempered representations but breaks down for nontempered distributions. This situation is discussed in §4. Let \(\Pi_{temp}(G({\mathbb{A}}_ F))\) be the subset of representations in \(\Pi_{unit}(G({\mathbb{A}}_ F))\) of the form \(\pi =\otimes_{v}\pi_ v\), \(\pi_ v\in \Pi_{temp}(G(F_ v))\), where \(\Pi_{temp}(G(F_ v))\) is the set of the local tempered distributions. The author formulates a conjectural description for the given representation \(\pi\) in the complement of \(\Pi_{temp}(G({\mathbb{A}}_ F))\) in \(\Pi_{unit}(G({\mathbb{A}}_ F))\) and considers the local analogue of this problem. The problem demands to introduce in §4, §6, §8 the parameters \(\Psi\) attached with an L-group of G. Existence of nontempered automorphic forms does mean that local and global problems and related. The relation of the parameters to the cohomology of Shimura varieties is discussed in §9.

Reviewer: A.Venkov (St.Petersburg)

MSC:

22E55	Representations of Lie and linear algebraic groups over global fields and adèle rings
11F70	Representation-theoretic methods; automorphic representations over local and global fields

Keywords:

connected reductive algebraic group; adelic points; right regular representation; irreducible representations; endoscopy; tempered representations; local tempered distributions; automorphic forms; cohomology of Shimura varieties

Citations:

Zbl 0694.00012