$L$-Group and Satake Parameter

For a split reductive $G$ over a field with Galois/Weil group $\Gamma$, the $L$-group is a semidirect product ${}^LG:=\Gamma\ltimes \widehat G$, where $\Gamma$ acts on $\widehat G$ via pinned automorphisms (see pinning ).

In the letter, $\Gamma=\mathrm{Gal}(K/k)$ and ${}^LG$ is written as $\Gamma\ n_\delta\ cG$.

At an unramified prime $p$, a Hecke eigencharacter gives a Satake parameter $\alpha_p$: a semisimple conjugacy class in ${}^LG$ whose projection to $\Gamma$ is (a choice of) Frobenius .

Given a complex representation $\pi:{}^LG\to \mathrm{GL}(V)$, the associated local factor is $L_p(s,\pi,\phi)=\det(1-\pi(\alpha_p)\,p^{-s})^{-1}.$