About the strength of representation-theoretic obstructions for orbit closure problems

2012-09-03T16:18:42Z

Let $G$ be a reductive, affine, algebraic group over $\newcommand{\C}{\mathbb C}\C$ . Let $X$ be a $G$ -variety. For $x\in X$ , we write $$G_x:=\{ g\in G\mid g.x=x\}$$ for its stabilizer and for any subgroup $H\subseteq G$ , we write $$X^H:=\{x\in X\mid H.x=x\}$$ for the $H$ -invariants of $X$ . We say that $x\in X$ is characterized by its stabilizer if $X^{G_x}=\{x\}$ . Let $\{V_\lambda\mid \lambda\in\Lambda\}$ be the irreducible $G$ -modules.

Given two points $x,y\in X$ , then $x\in\overline{G.y}$ implies $\overline{G.x}\subseteq\overline{G.y}$ . Hence, $\C[\overline{G.y}]\twoheadrightarrow\C[\overline{G.x}]$ and thus,

$$\DeclareMathOperator{\mult}{mult}\forall \lambda\in\Lambda:\quad \mult\nolimits_\lambda(\C[\overline{G.x}])\le\mult\nolimits_\lambda(\C[\overline{G.y}])$$

Finding $\lambda\in\Lambda$ violating the above is therefore an "obstruction" for the inclusion of orbit closures.

My question now is the following: If $x$ and $y$ are characterized by their respective stabilizers, does the converse hold? I.e., does the above inequality imply that $x\in\overline{G.y}$ ? I have been trying to come up with a counterexample, but without success so far.

Intuition: If $G$ acts on a variety $Y$ and $y\in Y$ is characterized by its stabilizer, then you can very easily find counterexamples if you give up the condition that both points are characterized by their respective stabilizers: Consider $X:=Y\times\{z_1,z_2\}$ with $G$ acting trivially on $Z=\{z_1,z_2\}$ . Now, the points $x_i:=(y,z_i)$ satisfy $x_1\notin\overline{G.x_2}$ and $\C[\overline{G.x_1}]\cong\C[\overline{G.x_2}]$ . In the cases of interest to me, however, both points are characterized by their stabilizer and the question arises whether there are counterexamples under this additional condition.

Answer by Alexander Premet for About the strength of representation-theoretic obstructions for orbit closure problems

2012-09-04T07:37:43Z

Take the nilpotent cone $\mathcal N$ in $g={\rm Lie}(G)$ and the $G$-orbit of a regular semisimple element, $h$ say. The categorical quotient $g\rightarrow g//G\cong\mathbb{A}^l$, $l={\rm rk}(G),$ is equidimensional and each of its fibres is an trreducible complete intersection and contains a unique open $G$-orbit. Since the orbit $Gh$ is closed it coincides with one of the fibres and the algebra of regular functions $\mathbb{C}[Gh]$ is just a filtered deformation of the graded algebra $\mathbb{C}[\mathcal{N}]=\mathbb{C}[g]/(f_1,\ldots,f_l)$ whose defining ideal is generated by $f_1-\lambda_1,\ldots, f_l-\lambda_l$ for some $\lambda_i\in\mathbb{C}$ (here $f_1,\ldots, f_l$is a set of free homogeneous generators for $\mathbb{C}[g]^G$). Since $G$ is reductive, we are in characteristic $0$ and the action of $G$ on $\mathbb{C}[Gh]$ is rational, we have that $\mathbb{C}[\mathcal{N}]\cong \mathbb{C}[Gh]$ as $G$-modules. So all multiplicities will be the same in both cases. However, $Gh$ is not contained in $\mathcal N$ (and vice versa).

However this example does not answer the question as the stabilisers $G_x$ of regular elements $x\in g$ are not self-normalising (I have completely overlooked the extra condition on $x$ in the first reading, which implies that $N_G(G_x)=G_x$).

About the strength of representation-theoretic obstructions for orbit closure problems - MathOverflow

About the strength of representation-theoretic obstructions for orbit closure problems

Answer by Alexander Premet for About the strength of representation-theoretic obstructions for orbit closure problems