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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 $V\setminus\{0\}$?

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These questions point in the direction of a very broad subject: Geometric Invariant Theory (GIT), as pioneered in Mumford's book. Language used there includes "semisstable point". Few cases are easy, but in the special case of the adjoint representation of $G$ it's a basic fact that having a closed orbit is equivalent to being a semisimple element of the Lie algebra. Orbit spaces in your setting have been much studied by people like Vinberg, Popov, Kraft, usually working over an algebraically closed field of characteristic 0. – Jim Humphreys Oct 22 '13 at 12:52
up vote 6 down vote accepted

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.

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Faisal: Is not the terminology "semi-stable /stable vector" as opposed to semi-simple vector? – P Vanchinathan Oct 22 '13 at 5:34
I've seen both sets used. Stable/semistable moreso in the GIT context. – Faisal Oct 23 '13 at 19:29
My understanding of a semisimple element is that it is diagonalizable. So is it true that in nice cases the notions of semisimple elements and stable points coincide? I would like a reference if this is true. – batconjurer Nov 11 '13 at 3:32

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