Arthur packet - Knowlpedia

Let $F$ be a local field. For a local Arthur parameter $\psi$ of a reductive group $G$ over $F$ , its Arthur packet or $A$ -packet $\Pi_\psi(G)$ is a finite collection, more precisely often a finite multiset, of irreducible admissible representations of $G(F)$ , where admissibility has its nonarchimedean or archimedean meaning according to $F$ . The packet is equipped with a pairing against a component group associated to $\psi$ .

Relation to L-packets

The parameter $\psi$ has an associated ordinary Langlands parameter $\varphi_\psi$ , but $\Pi_\psi$ need not equal the $L$ -packet $\Pi_{\varphi_\psi}$ . An $A$ -packet can contain nontempered unitary representations and can intersect several $L$ -packets. When the Arthur $\operatorname{SL}_2$ -factor is trivial, the expected packet is the tempered $L$ -packet.

Internal structure

Write $\mathcal S_\psi$ for an appropriate finite quotient of the component group of the centralizer of $\psi$ in $\widehat G$ . After normalization by a Whittaker datum, members of the packet carry characters or representations of $\mathcal S_\psi$ . The exact enhancement depends on the group and on inner-form data.

Global packet

A global parameter has localizations $\psi_v$ . Its global packet is built from restricted tensor products

\pi=\bigotimes_v'\pi_v, \qquad \pi_v\in\Pi_{\psi_v},

subject to almost-all unramified normalization. Not every such tensor product occurs in the discrete automorphic spectrum. The Arthur multiplicity formula imposes a global component-group character condition.

Status

For quasi-split symplectic and special orthogonal groups, Arthur constructed the relevant packets and classification; extensions cover inner forms and unitary groups in specified settings. For a general connected reductive group, $A$ -packets remain conjectural and several candidate constructions can require comparison.

References

James Arthur, The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups, AMS, 2013. AMS.
Chung Pang Mok, “Endoscopic classification of representations of quasi-split unitary groups,” Memoirs of the AMS 235 (2015). AMS.