Let $G$ be a connected reductive group over $\mathbb{C}$ of (reductive) rank $\ell$. Let $P$ be a parabolic of $G$ and let $P=LN$ denote the Levi decomposition. Let $\mathfrak{g}, \mathfrak{p}, \mathfrak{l}$, and $\mathfrak{n}$ denote the corresponding Lie algebras. Recall that $N$ is normal in $P$; therefore, $P$ and $L$ act on $\mathfrak{n}$. 


There is a nice description of $\mathbb{C}[\mathfrak{g}]^G$: it is a polynomial algebra on $\ell$ homogenous generators whose degrees are canonically defined. 


As Steven points out, $\mathbb{C}[\mathfrak{n}]^L$ is trivial, so this means that $\mathbb{C}[\mathfrak{n}]^P$ is also trivial, since we have an inclusion $\mathbb{C}[\mathfrak{n}]^P\subset \mathbb{C}[\mathfrak{n}]^L$. 


Question: Is there a description of $\mathbb{C}[\mathfrak{g}]^P$? 


For instance, We have an obvious injective map 
$\mathbb{C}[\mathfrak{g}]^G \rightarrow \mathbb{C}[\mathfrak{g}]^P$. 
What is known about this map? 



Invariant theory for parabolics
The map $\mathbb C[\mathfrak g]^G\to\mathbb C[\mathfrak g]^P$ is an isomorphism for trivial reasons: In any *quasi-affine* $G$-variety, $P$ and $G$ have the same fixed points. Just look at the orbit map of a $P$-fixed point which factors through the complete variety $G/P$. Applied to representations, this means $V^G=V^P$ for any rational $G$-module. This holds for any field.


