a group scheme over ${\overline{{\mathbb{Q}}}}$ with identity component ${{\widehat{G}}}$ and component group $\Gamma_k$. For $K$ a field containing ${\overline{{\mathbb{Q}}}}$ we have ${{{}^{L}G}}(K)={{\widehat{G}}}(K)\rtimes\Gamma_k$. Often in the literature people use ${{{}^{L}G}}$ to be the group that we call ${{{}^{L}G}}({\mathbb{C}})$.

Note that there is a fair amount of “ambiguity” in this definition. The group ${{\widehat{G}}}$ is “only defined up to inner automorphisms”, as is the lifting of $\mu_G$. So, even if we fix our choice of $k^{{\operatorname{sep}}}$, points in ${{{}^{L}G}}(K)$ are “only defined up to conjugation by ${{\widehat{G}}}(K)$”.

If $K$ is an extension of ${\overline{{\mathbb{Q}}}}$ and $\rho$ is a group homomorphism $\Gamma_k\to{{{}^{L}G}}(K)$, then we say that $\rho$ is *admissible* if the map $\Gamma_k\to\Gamma_k$ induced by $\rho$ and the surjection ${{{}^{L}G}}(K)\to\Gamma_k$ is the identity. We say two admissible $\rho$s are *equivalent* if they differ by conjugation by an element of ${{\widehat{G}}}(K)$.

We remark that readers for whom even the choice of $k^{{\operatorname{sep}}}$ is distasteful (I’m looking at you, Brian) can avoid making this choice all together by interpreting ${{\widehat{G}}}(K)$ as a scheme of groups over ${\operatorname{Spec}}(k)$ and then interpreting equivalence classes of admissible $\rho$s as above as elements of $H^1({\operatorname{Spec}}(k),{{\widehat{G}}}(K))$.

We fix once and for all an embedding ${\overline{{\mathbb{Q}}}}\to{\mathbb{C}}$. Later on, when talking about 