Maximal tori in $SO(n,\mathbb{C})$

Question

What are maximal tori in $SO(n,\mathbb{C})$? (not $SO(n,\mathbb{R})$)

Can a maximal torus in $SO(n,\mathbb{C})$ be written as $T\cap SO(n,\mathbb{C})$ for some maximal torus $T$ in $SL(n,\mathbb{C})$?

Answer 1

Bear with me for a moment and consider the vector space $V\newcommand{\df}{:=}\df\newcommand{\k}{\mathbb C}\k[x_1,\ldots,x_n]_2$ of quadratic $n$-forms over $\k$ and the usual action of $G\df\newcommand{\GL}{\operatorname{GL}}\GL_n(\k)$ on $V$ given by precomposition, i.e. $g.f\df f\circ g$ for $f\in V$ and $g\in G$. We study the stabilizer of a form $f\in V$. First, we recall that $f$ is given by a symmetric matrix $a\in\k^{n\times n}$ in the sense that $f=\newcommand{\trans}[1]{#1^T}\trans xax$ for $x=(x_1,\ldots,x_n)$. We then have $g.f=f$ for $g\in G$ if and only if \begin{align*} && \trans xax = f = g.f = \trans{(gx)}a(gx) &= \trans x \trans g a g x \\ &\Longleftrightarrow& \trans g a g &= a %\\ &\Longleftrightarrow& \forall i\le j\colon \trans{g_i} a g_j &= a_{ij} \end{align*} Let us assume that $a$ is invertible and diagonalizable. Since $a$ is symmetric and $\k$ contains all square roots, there exists some $h\in G$ with \begin{align*} e &= \trans hah = \trans h \trans g a g h = \trans h \trans g h^{-T} h^{-1} g h = \trans{(h^{-1} g h)}\cdot(h^{-1} g h) \end{align*} Therefore, $G_f=hOh^{-1}$ where $O=\{ g\in G\mid \trans gg = e\}$ is the variety of orthogonal matrices.

So, instead of considering $\operatorname{SO}(n,\mathbb C)$ you can consider the intersection of $\operatorname{SL}(n,\k)$ with the stabilizer of any apropriate quadratic form.

Let $a$ be the matrix corresponding to the order-reversing permutation on $n$ symbols. It is the product of all the transpositions $(n-k+1,k)$ for $1\le k\le \lceil\tfrac{n}2\rceil$. Consider the group $$ S := \{ g\in\operatorname{SL}(n,\k) \mid g^Ta g = a \}$$

For this group, you should be able to verify that $T\cap S$ is a maximal torus of $S$, when $T$ denotes the torus of diagonal matrices in $\operatorname{SL}(n,\k)$.

For $n=2k$, the matrices in this intersection are of the form $\operatorname{diag}(t_1,\ldots,t_k,t_k^{-1},\ldots,t_1)$ and for $n=2k+1$ they are of the form $\operatorname{diag}(t_1,\ldots,t_k,1,t_k^{-1},\ldots,t_1^{-1})$. Now since $h^{-1}Sh=\operatorname{SO}(n,\k)$ for the matrix $h$ with $\trans h a h = e$, you also get that a maximal torus of $\operatorname{SO}(n,\k)$ is given by $h^{-1}(S\cap T)h$.