# Invariant functions on the dual Lie algebra

Let $G$ be a real Lie group and $\mathfrak{g}$ the corresponding Lie algebra. Let $\mathfrak{g}^*$ be the dual of the Lie algebra. Then we have the coadjoint action of $G$ on $\mathfrak{g}^*$.

Consider a $G$-invariant (w.r.t. the coadjoint action) embedded submanifold $U \subset \mathfrak{g}^*$, such that the maximal-dimensional orbits in $U$ are embedded submanifolds. Let $$k := \dim U, \qquad \ell := \max_{x \in U} \dim (G.x),$$ and fix a point $z \in U$ such that $\dim (G.z) = \ell$.

Can we find $G$-invariant polynomials $f_1, \dots, f_{k-\ell} \colon U \to \mathbb{R}$ such that the function $F :=(f_1, \dots, f_{k-\ell}) \colon U \to \mathbb{R}^{k-\ell}$ is a submersion at $z$?

Edit: By polynomials on $U$, I mean polynomials on $\mathfrak{g}^*$ restricted to $U$. And by $G$-invariant, that the restricted polynomials are $G$-invariant on $U$ and not necessarily on the whole of $\mathfrak{g}^*$.

• Having a $G$-invariant polynomial on $\mathfrak{g}^*$ and then restricting it to $U$, shouldn't we still get a $G$-invariant function? So my polynomials don't need to be $G$-invariant on all of $\mathfrak{g}^*$, only when restricted to $U$. Although I think it doesn't hurt, if they are $G$-invariant on all of $\mathfrak{g}^*$. Jun 8, 2016 at 12:14