$L$-packet - Knowlpedia

Definition

Assume the basic local Langlands correspondence for a connected reductive group $G$ over a local field. For a relevant parameter $\varphi$ , its $L$ -packet is the fiber

\Pi_\varphi(G)= \{\pi\in\operatorname{Irr}(G(F)):\varphi_\pi=\varphi\}.

Thus an $L$ -packet is not an arbitrary family of similar representations: it is the finite set sharing one local $L$ -parameter.

Packet size

Packets for tori and $\operatorname{GL}_n$ are singletons. For groups with nontrivial endoscopy, several inequivalent representations can share a parameter. Their internal distinctions are encoded by a finite component group.

Stable character

For a tempered packet of a $p$ -adic group, an appropriately weighted sum of the Harish–Chandra characters of its members is expected to be a stable distribution. Other linear combinations are related by endoscopic transfer. This character-theoretic structure is part of what makes the partition into packets mathematically meaningful.

Inner forms

The notation $\Pi_\varphi(G)$ fixes one group $G(F)$ . A compound packet ranges over suitable pure or rigid inner forms of a quasi-split group. The refined correspondence uses the compound packet because its component-group parametrization naturally records which inner form contains each member.

References

Tasho Kaletha, “Representations of reductive groups over local fields,” §§2.1–2.2, 2022. arXiv.
James Arthur, The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups, American Mathematical Society, 2013. AMS.