# Normalizer of non-split tori

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.

• It is a question on a reductive group $G$ over a nonclosed field $K$. You should regard the quotient $W=W(G,T)$ as an algebraic group over $K$. Then your quotient is contained in $W(K)$. To describe your quotient you need Galois cohomology. – Mikhail Borovoi Jun 2 '16 at 9:03
• For $K = \mathbf{C}(\!(t)\!)$ we have $G := \mathbf{G}(K)$ and you mean $T := \mathbf{T}(K)$ for a (unique) maximal $K$-torus $\mathbf{T}$ of $\mathbf{G}$. Then $N_G(T) = N_G(\mathbf{T}) = N_{\mathbf{G}}(\mathbf{T})(K)$ since $T$ is Zariski-dense in $\mathbf{T}$, but as Borovoi notes $N_G(T)/T$ is the kernel of $\delta: W(K) \rightarrow {\rm{H}}^1(K,\mathbf{T})$ for the finite etale $K$-group $W := N_{\mathbf{G}}(\mathbf{T})/\mathbf{T}$. What interests you: $N_G(T)/T, W(K)$, or $W$? Please clarify. Is there specific motivation which may help to provide a useful answer? – nfdc23 Jun 2 '16 at 16:36
• Put $K = \mathbb C((t))$, as suggested by @nfdc23. For $\mathbb G = \mathrm{GL}_n$, we have $T = \mathbb T(K)$ for $\mathbb T$ a product of tori of the form $\mathrm{Res}_{E/K}\mathrm{GL}_1$ with $E/K$ separable. For a single such factor, we have an obvious isomorphism $\mathrm N_G(T)/T \cong \mathrm{Aut}(E/K)$. In general, $N_G(T)/T$ also contains elements that permute distinct but isomorphic factors. – LSpice Jun 9 '16 at 23:13