Let $G$ be a connected, simply-connected complex semisimple linear algebraic group, and let $V$ be a finite-dimensional complex $G$-module. Is there a nice description of those $v\in V$ for which the $G$-orbit of $v$ is Zariski-closed in $V$? Another question: Which non-zero $v\in V$ have the property that the $G$-orbit is closed in $V\setminus\{0\}$?



When are orbits of semisimple group representations closed?
There's some terminology here that might be helpful for a literature search: $v$ is said to be **semisimple** if $Gv$ is closed in $V$ and said to be **nilpotent** if $v\neq0$ and $Gv$ is *not* closed in $V\setminus\{0\}$. The set of all nilpotent $v\in V$ is called **nullcone** of $V$. If $V$ is the adjoint representation, then "semisimple" and "nilpotent" have their usual meaning.


The nullcone of $V$ is of course simply the set $\{v \in V \colon 0 \in \overline{Gv}\}$, or equivalently, it's the zero set of $\mathbb C[V]\_+^G$, the nonconstant homogeneous $G$-invariant regular functions on $V$. A perhaps more useful characterization is given to us by the Hilbert–Mumford criterion: a nonzero $v\in V$ is in the nullcone of $V$ if and only if there is a 1-parameter subgroup $\lambda \colon \mathbb C^\times \to G$ such that $\lim\_{t\to 0} \lambda(t)v = 0$. There's a lot of research that's been done in this direction: you can start by looking up "nullcone" on MathSciNet.


On the other hand, I'm not aware of any nice, complete characterizations of the semisimple elements in an arbitrary $V$. But there is a nice sufficient condition due to Dadoc and Kac, *Polar representations*, J. Algebra **92** (1985), 504–524, which goes as follows. Fix a maximal torus $T\subset G$, let $V = \bigoplus\_\lambda V\_\lambda$ be the corresponding weight-space decomposition of $V$, and let $\Phi$ denote the set of roots. Now, given $v\in V$, write $v = \sum\_{i=1}^k v\_{\lambda\_i}$ with $v\_{\lambda\_i} \in V\_{\lambda\_i}\setminus\{0\}$. If


1. $0$ is in the convex hull of $\lambda\_1,\ldots, \lambda\_k$, and
2. $\lambda\_i - \lambda\_j \not\in\Phi$ for all $i\neq j$,


then $v$ is semisimple. (The first condition guarantees that $Tv$ is closed, by the Hilbert–Mumford criterion; the second condition is there to allow us to conclude that $Gv$ is then closed.) For example, every $v\in V\_0$ is semisimple.


