On conjugacy of stabilizers of reductive group actions

TL;DR

This paper demonstrates that a key result in reductive group actions, previously established by Wallach, is actually a special case of more general classical theorems, simplifying its proof.

Contribution

It shows that Wallach's principal orbit type theorems and the Kempf--Ness Theorem are special cases of classical Richardson and Luna theorems, providing a shorter proof.

Findings

Wallach's main result is a special case of classical theorems.

A short argument suffices to derive Wallach's result from classical theorems.

The paper clarifies the relationship between modern and classical results in reductive group actions.

Abstract

It is shown that the main result of N. R. Wallach, Principal orbit type theorems for reductive algebraic group actions and the Kempf--Ness Theorem, arXiv:1811.07195v1 (17 Nov 2018) is a special case of a more general statement, which can be deduced, using a short argument, from the classical Richardson and Luna theorems.

Full text

1. In the recent preprint [W], the following main result is obtained using the Kemf–Ness theorem to reduce it to the principal orbit type theorem for compact Lie groups:

“Let $G$ be a reductive, affine algebraic group and let $(\rho,V)$ be a regular representation of $G$. Let $X$ be an irreducible ${\mathbb{C}}^{\times}\texttimes G$ invariant Zariski closed subset such that $G$ has a closed orbit that has maximal dimension among all orbits (this is equivalent to: generic orbits are closed). Then there exists an open subset, $W$, of $X$ in the metric topology which is dense with complement of measure [math] such that if $x,y\in W$ then $({\mathbb{C}}^{\times}\texttimes G)_{x}$ is conjugate to $({\mathbb{C}}^{\times}\texttimes G)_{y}$. Furthermore, if $Gx$ is a closed orbit of maximal dimension and if $x$ is a smooth point of $X$ then there exists $y\in W$ such that $({\mathbb{C}}^{\times}\texttimes G)_{x}$ contains a conjugate of $({\mathbb{C}}^{\times}\texttimes G)_{y}$.”

Below is shown that the more general statements can be deduced, using a short argument, from the classical Richardson and Luna theorems.

2. We fix an algebraically closed ground field $k$ of characteristic [math] and use freely the standard notation of [B], [PV].

Let $G$ be a reductive algebraic group such that $G=CR$, where $C$ is a diagonalizable algebraic subgroup of the center of $G$ and $R$ is a reductive algebraic subgroup of $G$. We denote by ${\mathcal{X}}(C)$ the character group of $C$ and, given an algebraic $C$-module $M$ and a character $\alpha\in{\mathcal{X}}(C)$, by $M_{\alpha}$ the weight space of $M$ of the weight $\alpha$. Since $C$ is diagonalizable, $M$ is the direct sum of the $M_{\alpha}$’s; see [B, III.8.17].

Let $X$ be irreducible affine algebraic variety endowed with a regular (morphic) action of $G$.

1. (a)

*There exists a dense open *(in the Zariski topology ) subset $U$ of $X$ such that if $x,y\in U$, then $G_{x}$ is conjugate to $G_{y}$. 2. (b)

If the $R$-orbit $R(z)$ of a point $z\in X$ is closed, then there exists a point $y\in U$ such that $G_{z}$ contains a conjugate of $G_{y}$.

(b) Let $\overline{G(z)}$ be the closure of the $G$-orbit of $z$ in $X$. Then $B:=\overline{G(z)}\setminus G(z)$ is a closed $G$-stable subset of $X$. If $B=\varnothing$, then the existence of $y$ immediately follows from the Luna slice theorem, see [L, Rem. $4^{\circ}$ on p. 98] (cf. [PV, Thm. 6.3]). Now consider the case $B\neq\varnothing$. Since $B\cap R(z)=\varnothing$, the same argument as in the above proof of (a) shows the existence of a $G$-semi-invariant $f\mskip-3.0mu\in\mskip-3.0muk[X]^{R}$ such that $f|_{B}=0$, $f|_{R(z)}=1$. The latter equality implies that $f$ vanishes nowhere on $G(z)$. Therefore, $X_{f}$ is a $G$-stable dense open subset of $X$ containing $G(z)$, and $G(z)$ is closed in $X_{f}$. Now, since $X_{f}$ is affine, the existence of $y$ follows from the Luna slice theorem, as above. This proves (b). ∎

3. Remark. In [W, Sect. 6] is given an example of a linear action of a semisimple group, which shows that the existence of a point with trivial stabilizer does not imply triviality of stabilizers of points in general position. It should be noted that this phenomenon is not new, similar examples has long been known (perhaps the earliest one belongs to Richardson, see [L, Rem. $4^{\circ}$ on p. 98]).