Let $k$ a field and $X$ a $k$ algebraic variety. Let $G$ a $k$ reductive group acting on $X$. I denote with $X_d$ the set of points in $X$ which have stabilizer of dimension $d$.
It is a fact that $X_d$ is a constructible subset of $X$. Also, $X_n$ is dense in $X$ where $n$ is the minimum dimension possible for a stabilizer of such an action.
Are there some known results which relate the dimensions of the different strata $X_d$? I'd be especially intrerested in knowing whether it is true that $ dim X_d \geq X_{d+i} +i$ where $i$ is the least positive integer such that $X_{d+i}$ is not empty ,or to see a counterexample to that.Let's say that $X$ is irreducible and even smooth if this would be relevant.
If you allow affine varieties, there are counter-examples to the inequality $\dim(X_{d+1}) \geq \dim(X_{d}) + 1$. Let $Gl_{2}(\mathbb{C})$ act on $\mathbb{C}^{2}$ in the standard way.
The stabiliser of $\{0\}$ is $Gl_{2}(\mathbb{C})$ of dimension $4$.
The stabiliser of a point $x \in \mathbb{C}^{2} \setminus \{0\}$ is two dimensional.
In particular $X_{2} = \mathbb{C}^{2} \setminus \{0\}$ and $X_{3} = \emptyset$ contradicting the inequality.
Even if we modify to set $X_{k}$ to be the points whose stabiliser has dimension at least $k$, then we still have $\dim(X_{2})=\dim(X_{3})$ contradicting the inequality.
(This is a little unsatisfactory, I think the question maybe becomes more interesting if you restrict to projective varieties.)