# Polynomials invariant with respect to a nilpotent Lie algebra

Let $\mathfrak{u}$ be a nilpotent Lie algebra and let $\mathbb{C}[\mathfrak{u}]$ be the space of polynomials with the natural coadjoint action of $\mathfrak{u}$.

Can one describe $\mathbb{C}[\mathfrak{u}]^{\mathfrak{u}}$?

We are interested mainly in the case where $\mathfrak{u} = \mathfrak{g}_1 \oplus \mathfrak{g}_2 \oplus \cdots \oplus \mathfrak{g}_k$ is graded nilradical of a parabolic subalgebra of a simple complex Lie algebra $\mathfrak{g}$. Our guess is that $\mathbb{C}[\mathfrak{u}]^{\mathfrak{u}} = \mathbb{C}[\mathfrak{g}_1]$.

• Perhaps this was looked into by Dixmier and his students in earlier decades (?), but it doesn't seem to be discussed directly in his book on enveloping algebras. At any rate, your guess looks reasonable. Is there representation-theoretic motivation? – Jim Humphreys Jun 3 '15 at 21:38
• @JimHumphreys: Thank you. Our motivation comes from computations of singular vectors in parabolic Verma modules. This result would imply that a "leading" term of a singular vector is from $U(\mathfrak{g}_{-1})$. – Vít Tuček Jun 4 '15 at 9:23