Local Langlands parameter

Definition

Let $F$ be a local field, let $W_F$ denote its Weil group, and let $G$ be a connected reductive $F$ -group. The local Langlands group used here is

L_F= \begin{cases} W_F, & F\text{ archimedean},\\ W_F\times\operatorname{SL}_2(\mathbb C), & F\text{ nonarchimedean}. \end{cases}

A local Langlands parameter, or local $L$ -parameter, is an admissible continuous homomorphism

\varphi:L_F\longrightarrow{}^LG

to the $L$ -group whose composite with ${}^LG\to W_F$ is the natural projection, whose restriction to $\operatorname{SL}_2(\mathbb C)$ is algebraic when that factor is present, and whose Weil-group elements have semisimple dual-group part. Parameters are considered up to conjugation by $\widehat G$ .

Relevance

For a non-quasi-split inner form, not every parameter for the common $L$ -group corresponds to a representation of that particular form. A parameter is relevant to $G$ when every parabolic subgroup of ${}^LG$ containing its image corresponds to a parabolic subgroup defined for $G$ . Basic local Langlands uses relevant parameters.

Tempered and discrete parameters

A parameter is tempered when the image of $W_F$ in $\widehat G$ is bounded after the standard projection, and discrete when its image is not contained in any proper Levi subgroup of ${}^LG$ . These conditions parallel tempered and essentially square-integrable representations.

A parameter trivial on the Deligne $\operatorname{SL}_2$ has zero monodromy. An unramified parameter is additionally trivial on the inertia subgroup and is determined by a Satake parameter.

Weil–Deligne form for

\mathrm{GL}_n

For $G=\operatorname{GL}_n$ , a parameter is equivalently an $n$ -dimensional Frobenius-semisimple Weil–Deligne representation.

References

Armand Borel, “Automorphic $L$ -functions,” in Automorphic Forms, Representations and $L$ -Functions, Proceedings of Symposia in Pure Mathematics 33, part 2, 1979, §§8–10.
Tasho Kaletha, “Representations of reductive groups over local fields,” §2.1, 2022. arXiv.