The derived algebra of a stabilizer, families of coadjoint orbits, and sheets (Q2927933)

Let \(L\) be a finite dimensional real or complex Lie algebra and let \(\mu\) be an element of the dual space, \(L^*\). The adjoint group of \(L\) acts on \(L^*\) by the coadjoint action. Let \(L_k^*\) be the union of all orbits of codimension k under this action. Further, let \(L_{\mu} = \{x\in L \mid \text{ad}_x^* (\mu)=0\}\), \(\mu\in L^*\). Then any \(\alpha \in L^*\) which is tangent to \(L_k^*\) at a point \(\mu \in L^*\) vanishes on the derived algebra of \(L_{\mu}\). It follows that each element of the tangent space vanishes on the derived algebra of \(L_{\mu}\) and the dimension of this derived algebra is less than or equal to the codimension of \(L_k^*\) where \(k\) is the dimension of \(L_{\mu}\). Conditions equivalent to equality in the last statement are shown.