Let \(G\) be a reductive complex algebraic group, this includes all finite groups, tori, semisimple groups, classical groups. Consider a finite \(\mathbb{C}\)-space \(V\) where \(G\) acts linearly.
We know that when we have a reductive group action on affine spaces, there will be a lot of orbits, some of them are closed. If \(Gv\) is a closed orbit, we know that \(G_v\) is also reductive by Matsushima’s theorem.
This means, \(V\) can be decomposed into direct sum of \(G_v\)-modules, since \(T_vGv \subset V\) we have \(V=T_vGv \oplus N_v\), where the normal space \(N_v\) is the slice representation at \(v\).
A basic idea of slice theorem is that, the \(G\) action near the closed orbit \(Gv\) should be determined by the \(G_v\) action on the slice representation \(N_v\).