Dimensions of submanifolds of SO(n)

Question

I would like to calculate the dimension of \begin{align*} \mathcal{M}_k=\{R\in\mathsf{SO}(n,\mathbb{R})\,|\,\sigma(R)=\{-1,1\},\,m(-1)=k\}, \end{align*} where $\sigma$ is the spectrum and $m$ is the algebraic multiplicity for all $k=0,\ldots,\lfloor\frac{n}{2}\rfloor$. Clearly $\dim\mathcal{M}_0=0$ and $\dim\mathcal{M}_{\frac{n}{2}}=0$ if $n$ is even, but I do not know how to proceed from there. I can show certain other properties such that $\mathcal{M}_k$ is path connected and is separated by the trace function from $\mathcal{M}_j$, $j\neq k$. Any help would be appreciated.

Answer 1

$\newcommand{\Reals}{\mathbf{R}}\newcommand{\calm}[1][k]{\mathcal{M}_{#1}}$Hints: Since $\det R = (-1)^{k}$ for all $R$ in $\calm$, the index $k$ is even.

Each element of $\calm$ determines a splitting $\Reals^{n} = E_{-1} \oplus E_{1}$ into eigenspaces, of respective dimension $k$ and $n - k$. Conversely, each splitting $\Reals^{n} = E_{-1} \oplus E_{1}$ with $\dim E_{-1}$ even corresponds to a unique $R$ in $\calm$.

It may be helpful to read about Grassmannian manifolds if you haven't encountered the concept.