TITLE: Action of symmetric matrices under $\mathrm{O}(n)$
QUESTION [8 upvotes]: $\DeclareMathOperator\Sym{Sym}\DeclareMathOperator\O{O}\DeclareMathOperator\GL{GL}$Let $k$ be an algebraically closed field of characteristic 0 (it can even be $\mathbb{C}$ if you like), and let $n\in\mathbb{N}$.  Let $\Sym(n)$ denote the space of $n\times n$ symmetric matrices with entries in $k$.  Let $\O(n)\subseteq \GL(n)$ denote the orthogonal group, that is all $n\times n$ matrices $A$ with entries in $k$ such that $AA^t=A^tA=I_n$.
Then $\O(n)$ acts on $\Sym(n)$ by conjugation.  I would like nice representatives for the orbits of this action.
Let $B\in \Sym(n)$.  If $B$ is diagonalizable, then it is a theorem that it can be diagonalized by an orthogonal matrix, so all the 'semisimple orbits' have a diagonal matrix as a representative.
So my question is about when $B$ is not diagonalizable — what is a 'nice' form we can put it in under orthogonal conjugation?  (For an example of a non-diagonalizable symmetric matrix, see https://math.stackexchange.com/questions/1658393/why-a-complex-symmetric-matrix-is-not-diagonalizible.)
Seeking explanations or references.  Thank you!

REPLY [7 votes]: The adjoint action of $SO(n,\mathbb C)$ on $\mathrm{Sym}_0(n,\mathbb C)$ (we can replace $O(n,\mathbb C)$ by $SO(n,\mathbb C)$, and $\mathrm{Sym}(n,\mathbb C)$
by the subspace of traceless matrices, since the
action splits off the scalar multiples of the identity) is the isotropy representation of the complex symmetric space
$SL(n,\mathbb C)/SO(n,\mathbb C)$. Kostant and Rallis (1971) investigated such
representations. This was later generalized by Vinberg (1976) to the case of a $\theta$-group (the $\mathfrak g_0$-action on $\mathfrak g_1$, where we consider the decomposition of a complex Lie algebra $\mathfrak g=\mathfrak g_0 
+\mathfrak g_1+\cdots + \mathfrak g_m$ into eigenspaces of an automorphism
$\theta$ of order $m$ ($1$-parameter group of automorphisms in case $m=\infty$).
Among other things, they found that for such a representation $(G_0,V)$:

There is a Jordan decomposition: every $x\in V$ can be uniquely written
as $x_s+ x_n$, where $x_s$ is a semisimple element (this means that $ad_{x_s}$
is a semisimple transformation) , $x_n$ is a nilpotent element (this means that
$\mathrm{ad}_{x_n}$ is a nilpotent transformation),
and they commute.
A maximal Abelian subspace $c$ of $V$ consisting of semisimple elements
is called a Cartan subspace. Any two Cartan subspaces are $G_0$-conjugate.
Semisimple elements are characterized by the fact that their $G^0$-orbits are closed, or they lie in a Cartan subspace, or they are $G^0$-conjugate
to an element in a given Cartan subspace. In your example, a Cartan subspace is given by the diagonal matrices, and the semisimple elements are exactly the
diagonalizable symmetric matrices.
Nilpotent elements are characterized by the fact that $0$
lies in the closure of their orbits. Nilpotent elements of course come in
$G^0$-orbits, and there are only finitely many nilpotent orbits.
In the case of a complex symmetric space $\mathfrak g=\mathfrak k + \mathfrak p$
(under $\theta$),
the Kostant-Sekiguchi correspondence establishes a bijection between the set of nilpotent $K$-orbits on $\mathfrak p$ and the set of nilpotent $G_{\mathbb R}$-orbits on $\mathfrak g_{\mathbb R}$, where $\mathfrak g_{\mathbb R}$ is the noncompact real form of $\mathfrak g$ whose Cartan involution induces $\theta$ on $\mathfrak g$. In turn, the nilpotent $G_{\mathbb R}$-orbits on $\mathfrak g_{\mathbb R}$ are parametrized by partitions.

More explicitly, in our case
we have the symmetric space $\mathfrak{sl}(n,\mathbb C)=\mathfrak{so}(n,\mathbb C)+\mathrm{Sym}_0(n,\mathbb C)$ and the nilpotent $SO(n,\mathbb C)$-orbits in
$\mathrm{Sym}(n,\mathbb C)$ are in bijection with the nilpotent  $SL(n,\mathbb R)$-orbits in $\mathfrak{sl}(n,\mathbb R)$. The latter can be understood in terms of the Jordan canonical form.
As an explicit example of the Kostant-Sekiguchi correspondence, the nilpositive element $\begin{pmatrix}0&1\\0&0\end{pmatrix}$
of an $\mathfrak{sl}_2$-triple in $\mathfrak{sl}(2,\mathbb R)$ corresponds under the Cayley transform to the nilpositive element  $\frac12\begin{pmatrix}i&1\\1&-i\end{pmatrix}$ of a normal $\mathfrak{sl}_2$-triple in $\mathfrak{sl}(2,\mathbb C)$; that is essentially the example given by @JohannesEbert. One can work the details of the correspondence more generally, see e.g. the book
https://www.amazon.com/Nilpotent-Orbits-Semisimple-Lie-Algebra/dp/0534188346