Arthur parameter - Knowlpedia

Let $F$ be a local field and $G$ a connected reductive $F$ -group. A local Arthur parameter is an admissible $L$ -homomorphism

\psi:L_F\times\operatorname{SL}_2(\mathbb C) \longrightarrow {}^L G

such that the restriction to $L_F$ is bounded modulo the center and the restriction to the displayed $\operatorname{SL}_2(\mathbb C)$ is algebraic. Parameters are considered up to $\widehat G$ -conjugacy.

Two SL2 factors at a nonarchimedean place

If the local Langlands group is written

L_F=W_F\times\operatorname{SL}_2(\mathbb C),

then an Arthur parameter has two $\operatorname{SL}_2$ -factors: the Deligne factor already present in $L_F$ and the additional Arthur factor. The latter records controlled nontemperedness.

Associated Langlands parameter

An Arthur parameter gives an ordinary local parameter by inserting the norm cocharacter into the Arthur factor:

\varphi_\psi(w) = \psi\!\left( w, \begin{pmatrix} |w|^{1/2}&0\\ 0&|w|^{-1/2} \end{pmatrix} \right),

with the evident extension over the Deligne factor and with $|w|$ depending on the stated reciprocity convention. If the Arthur $\operatorname{SL}_2$ is trivial, the associated parameter is tempered; for nonarchimedean $F$ , this is the parameter-side condition corresponding to a tempered representation.

Global form

A general global Langlands group is conjectural over number fields. In the classification of classical groups, global Arthur parameters are therefore encoded concretely as formal isobaric sums

\psi=\boxplus_i\,\pi_i[d_i],

where the $\pi_i$ are suitable self-dual cuspidal automorphic representations of general linear groups and $[d_i]$ denotes the $d_i$ -dimensional irreducible representation of $\operatorname{SL}_2(\mathbb C)$ , subject to dimension, parity, and ellipticity conditions.

Role

The parameter determines an $A$ -packet and a component group. A global character of that component group enters the Arthur multiplicity formula. This formalism captures both cuspidal and residual discrete automorphic representations.

Status

Arthur parameters and packets are theorems for major classical families, including Arthur's symplectic and orthogonal classification and Mok's quasi-split unitary classification. Their expected general form for every reductive group remains conjectural.

References

James Arthur, The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups, AMS Colloquium Publications 61, 2013. AMS.
James Arthur, “Unipotent automorphic representations: conjectures,” Astérisque 171–172 (1989), 13–71. Numdam.