Component group of an $L$-parameter

Definition

Let $\varphi:L_F\to{}^LG$ be a local $L$ -parameter and set

S_\varphi= Z_{\widehat G}\bigl(\operatorname{im}\varphi\bigr).

The raw component group of $\varphi$ is the finite group

A_\varphi=\pi_0(S_\varphi).

Here $Z_{\widehat G}(\operatorname{im}\varphi)$ is the centralizer of the parameter image in the dual group. It measures the disconnectedness of the dual-group symmetries that commute with the parameter and is the starting point for the internal parametrization of the $L$ -packet.

Quotient convention

For a fixed quasi-split group, the packet-indexing group is often

\mathcal S_\varphi= \pi_0\!\left( S_\varphi/Z(\widehat G)^{W_F} \right),

or a closely related quotient. Removing the fixed center prevents central symmetries already accounted for by the group from artificially enlarging the packet.

Refined cover

For packets across inner forms, one uses the preimage $S_\varphi^+$ of $S_\varphi$ in an appropriate cover of $\widehat G$ , and representations of $\pi_0(S_\varphi^+)$ with a prescribed central character. This larger group retains the cohomological data that identifies the inner form.

Consequently, the notations $A_\varphi$ , $\mathcal S_\varphi$ , and $S_\varphi^+$ are not interchangeable. A theorem must specify which component-group convention it uses.

References

Tasho Kaletha, “Representations of reductive groups over local fields,” §2.2, 2022. arXiv.
James Arthur, The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups, Chapter 1, American Mathematical Society, 2013. AMS.

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Component group of an $L$ -parameter

Definition