definitions. {#sec:review and notation}
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In this section we give an overview of various standard facts concerning $L$-groups, the Satake isomorphism, the archimedean local Langlands correspondence, and basic Hodge–Tate theory, often with a specific emphasis on certain arithmetic aspects that are not considered relevant in many of the standard references. In summary: our $L$-groups will be over ${\overline{{\mathbb{Q}}}}$, we will keep track of the two different ${\mathbb{Q}}$-structures in the Satake isomorphism, and our local Langlands correspondence will concern representations of $G({\mathbb{R}})$ or $G(\overline{{\mathbb{R}}})$, where $\overline{{\mathbb{R}}}$ is an algebraic closure of the reals which we do not identify with ${\mathbb{C}}$ (note that on the other hand, all our representations will be on ${\mathbb{C}}$-vector spaces). This section is relatively elementary but contains all of the crucial local definitions.

The $L$-group.
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We briefly review the notion of an $L$-group. We want to view the $L$-group of a connected reductive group as a group over ${\overline{{\mathbb{Q}}}}$, rather than the more traditional ${\mathbb{C}}$, as we shall later on be considering representations into the ${\overline{{\mathbb{Q}}}_p}$-points of the $L$-group. We review the standard definitions from the point of view that we shall be taking.

We take the approach to dual groups explained in section 1 of 