Dimension of maximal split subtorus and fixed point subspace of Lie algebra

Question

Let $F = \mathbb{C}((t))$. Let $G$ be a complex semisimple algebraic group. Then conjugacy classes of maximal tori in $G(F)$ are in bijection with conjugacy classes in $W$, the Weyl group of $G$ with respect to a fixed maximal torus $T$ of $G$.

Let $T_w$ be a maximal torus of type $[w]$, $A_w$ a maximal $F$-split subtorus. Is it then true that $$ \dim A_w = \dim \mathfrak{h}^w? $$ Here, $\mathfrak{h}= \operatorname{Lie}{T}$ and the superscript denotes fixed point set.

Answer 1

Yes, $X_*(A_w) = X_*(T_w)^{\operatorname{Gal}(\overline{\mathbb C((t))}/\mathbb C((t)))} = X_*(T)^w$ and $\mathfrak t^w = (X_*(T) \otimes_{\mathbb Z} \mathbb C)^w = X_*(T)^w \otimes_{\mathbb Z} \mathbb C$.