Let $\mathbb{G}$ be a connected reductive group over $\mathbb{C}$. Let $G:=\mathbb{G}(\mathbb{C}(\!(t)\!))$. Let $T$ be a maximal torus in $G$.

Question: What do we know about the normalizer $N_G(T)$ and the quotient $N_G(T)/T$?

If $T$ is split then $N_G(T)/T$ is (isomorphic to) "the" Weyl group $W$ of $\mathbb{G}$. So the question is what happens for non-split tori.

If it helps, recall that conjugacy classes of tori in $G$ are in bijection with conjugacy classes in $W$. If it makes a difference, we can consider $\mathbb{G}=\mathbb{GL}_n$ first.

yourquotient is contained in $W(K)$. To describe your quotient you need Galois cohomology. $\endgroup$ – Mikhail Borovoi Jun 2 '16 at 9:03