Galois representations, we shall fix a prime number $p$ and an embedding ${\overline{{\mathbb{Q}}}}\to{\overline{{\mathbb{Q}}}_p}$. This will enable us to talk about the groups ${{{}^{L}G}}({\mathbb{C}})$ and ${{{}^{L}G}}({\overline{{\mathbb{Q}}}_p})$.

Satake parameters. {#satakeparams}
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In this section, $k$ is a non-archimedean local field with integers ${\mathcal{O}}$, and we again fix a separable closure $k^{{\operatorname{sep}}}$ of $k$ and set $\Gamma_k={\operatorname{Gal}}(k^{{\operatorname{sep}}}/k)$. We normalise the reciprocity map $k^\times\to\Gamma_k^{\operatorname{ab}}$ of local class field theory so that it takes a uniformiser to a geometric Frobenius. We follow Tate’s definitions and conventions for Weil groups—in brief, a Weil group $W_k=W_{k^{{\operatorname{sep}}}/k}$ for $k$ comes equipped with maps $W_k\to\Gamma_k$ and $k^\times\to W_k^{\operatorname{ab}}$ such that the induced map $k^\times\to\Gamma_k^{\operatorname{ab}}$ is the reciprocity homomorphism of class field theory, normalised as above.

Let $G/k$ be connected reductive group which is furthermore unramified (that is, quasi-split, and split over an unramified extension of $k$). Then $G(k)$ has hyperspecial maximal compact subgroups (namely ${\mathcal G}({\mathcal{O}})\subseteq G(k)$, where ${\mathcal G}$ is any reductive group over ${\mathcal{O}}$ with generic fibre $G$); fix one, and call it $K$. Nothing we do will depend on this choice, but we will occasionally need to justify this. Let $B=B_k$ be a Borel in $G$ defined over $k$, let $T=T_k$ be a maximal torus of 