construction: if we follow section 6 of [@MR546608] then in the crucial Proposition 6.7 Borel has chosen the $\sigma$ that appears there to be an *arbitrary* generator of the Galois group of a finite unramified extension of $k$ which splits $G$, and we are free to choose $\sigma$ to be either an arithmetic or a geometric Frobenius. We shall let $\sigma$ denote a geometric Frobenius: now an easy check (unravelling the definitions in [@MR546608] and [@cartier:corvallis]) shows that if $G={\operatorname{GL}}_1$ and $\pi$ is an unramified representation of ${\operatorname{GL}}_1(k)$ then the corresponding Galois representation $W_k\to{\operatorname{GL}}_1({\mathbb{C}})$ is the one induced by our given isomorphism $k^\times=W_k^{\operatorname{ab}}$. See Remark \[swapping geometric and arithmetic frobenius via an involution\] for some comments about what would have happened had we chosen an arithmetic Frobenius here, and normalised our class field theory isomorphisms so that they sent arithmetic Frobenii to uniformisers.

\[unramlltwist\] The local Langlands correspondence behaves in a natural way under certain unramified twists. Let $\xi:G\to{{\mathbb{G}_m}}$ be a character of $G$ defined over $k$. Then $\xi(K)$ is a compact subset of $k^\times$ and is hence contained in ${\mathcal{O}}^\times$. If $s\in{\mathbb{C}}$ then we can define $\chi:G(k)\to{\mathbb{C}}^\times$ by $\chi(g)=|\xi(g)|^s$; then $\chi(K)=1$. Hence for an irreducible $\pi$ with a $K$-fixed 