I'm trying to understand Luna's étale slice theorem by computing some examples. The theorem is usually phrased as an existence result. I wondered if there was a natural way to figure out the slice at a given point.
The particular example I had in mind is this: let $V$ denote the set of $2 \times 2$ trace-free matrices over $\mathbb{C}$, and $G=\operatorname{GL}_2$ acts on $V^n$ ($n$ copies) by simultaneous conjugation. Let $v = (A,0,0,0,\dotsc, 0)$ where $A$ is diagonal with nonzero diagonal entries $(a,-a)$. Then $Gv$ is closed in $V$, and the stabiliser of $v$ is the set of nonsingular diagonal matrices $T \subset G$.
The étale slice theorem says that there exists a $T$-subvariety $S \subset V$ containing $v$ such that
- The map $\psi: G \times_T S \rightarrow V$ is étale, and its image is a saturated open subset $U$ of $V$;
- The induced map $\psi_G: (G \times_T S)//G \rightarrow U//G$ is etale.
(2) has the consequence that $\widehat{\mathbb{C}[V^n]^G_v} \cong \widehat{\mathbb{C}[S]_v^T}$. The question is: is there a way to find an explicit description of the subvariety $S$?
