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What is known about local structure of actions of semi-simple groups? More precisely, suppose I have a semi-simple group $G$ acting on a variety $V$, and $x\in V$. Assume that the stabilizer of $x$ is a parabolic subgroup $P\subset G$. Can I always find a slice $x\in V'\subset V$ such that the natural map $G\times_P V'\to V$ is a local isomorphism near $x$?

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The Luna slice theorem says something like this, but in the étale topology. A local isomorphism in the Zariski topology is too much to ask for. You can find a good exposition of the Luna slice theorem in Appendix A of Geometric Invariant Theory by Mumford-Fogarty-Kirwan. – Michael Thaddeus Aug 29 '11 at 16:32
Dear Michael, for Luna's slice theorem to hold the variety has to be affine, and the stabilizer must be reductive. Neither hypothesis is satisfied in this case, except when $P = G$. – Angelo Aug 29 '11 at 17:03
The counterexample I had written below was complete nonsense, I removed it. – Angelo Aug 31 '11 at 7:04
up vote 2 down vote accepted

[Edit]: my previous counterexample was irredeemably wrong; hopefully this one works.

Suppose that there exists an étale $G$-equivariant map $V' \times^{P}G \to V$; then there exists an invariant neighborhood $U$ of $x$ in $V$ (the image of this map) such that the connected component of the identity in the stabilizer of any point of $U$ is conjugate to a subgroup of $P$.

Let $G = \mathrm{GL}_3$ acting in the natural way on the space $V$ consisting of pairs $(p, L)$, where $p$ is a point of $\mathbb P^2$ and $L$ is a line in $\mathbb P^2$. There are two orbits, one consisting of pair $(p, L)$ such that $p \in L$, call it $I$, and its complement $V \smallsetminus I$. The stabilizer of a point $x$ of $I$ is a Borel subgroup $P$ of $G$. However, the stabilizer $H$ of any point of $V \smallsetminus I$ is connected, and is not contained in any conjugate of $P$. In fact, it $H$ was conjugate to a subgroup of $P$, it would stabilize a point of $I$; however, it is easy to see that the only fixed point of $V$ under the action of $H$ is $x$ itself, and $x \notin I$. So in this case there cannot be an étale slice.

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Honestly speaking, I don't understand this example. How does $\mathrm{PGL}_n$ acts on the total space of $\mathcal O(1)$? It seems to me that $\mathrm{SL}_n$ acts but the center acts non-trivially. – Roman Fedorov Aug 30 '11 at 17:01
Dear Roman, you are absolutely right, what I wrote above is trash. – Angelo Aug 31 '11 at 7:03

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