Let $G$ be a reductive group acting on an affine variety $X$. For simplicity, one may assume $G=SL\_n$ or $G=U\_n$ and assume the field is $\mathbb C$. Given this one can show $\mathbb C[X]^G$ is finitely generated algebra.



> 
> **Question:**
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> (1) Each of the orbit closures $\overline{Gx}$ is a finite union of $G$-orbits.
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> (2) Each of the orbit closures $\overline{Gx}$ contains a unique closed orbit which has minimal dimension among these $G$-orbits.
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I guess the two questions might be related to a result saying that
$$
\pi(x\_1)=\pi(x\_2) \ \ \ \ \Longleftrightarrow \ \ \ \ \overline{G x\_1} \cap \overline{G x\_2} \neq \varnothing
$$
where $\pi: X\to X //G\equiv Spec \mathbb C[X]^G$ is the GIT quotient.


EDIT: For question (2), suppose $Gx$ is not closed. If $y\in \overline{Gx}$ but $y\not \in Gx$, then what relation can we expect between $\overline{Gx}$ and $\overline{Gy}$? For example, do we have $\dim\overline{Gy} < \dim\overline{Gx}$, or $\dim G\_y \ge 1$?



The closure $\overline{Gx}$ for an affine variety on which an reductive algebraic group acts
The answer to (2) is affirmative: an affine $G$-variety with a dense open orbit (like $\overline{Gx}$) contains a unique closed orbit. The reason for that is that any two closed are orbits are separated by a $G$-invariant. The closed orbit is clearly the unique orbit of minimal dimension since it is contained in the closure of every other orbit.


Question (1) is more delicate. As Victor Protsak's examples show there are, in general, infinitely many orbits in an affine orbit closure. There are some positive results, though:


1. V.L. Popov has shown (Quasihomogeneous affine algebraic varieties of the group SL(2). Izv. Akad. Nauk SSSR Ser. Mat 37 (1973), 792–832) that $\overline{Gx}$ has finitely many orbits when $G=SL(2)$.


There are also criteria in terms of the isotropy subgroup $H=G\_x$:


2. $\overline{Gx}$ is a finite union of $G$-orbits if $G/H$ is spherical. In that case any embedding (affine or not) of $G/H$ has finitely many orbits. That is even equivalent to $G/H$ being spherical.
3. $\overline{Gx}$ is a finite union of $G$-orbits if $H$ is a reductive subgroup which is of finite index in its normalizer (e.g. $H=T$, the maximal torus). In that case $G/H=\overline{G/H}$ is already closed in $X$ (by Luna: Adherence d'orbites).
4. The paper Arzhantsev, I. V.; Timashev, D. A. Affine embeddings with a finite number of orbits. Transform. Groups 6 (2001), no. 2, 101–110 contains more results in this direction. In particular, the authors show that examples 2 and 3 exhaust (almost) all instances when $H$ is reductive.


