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.

1. Is there a description of $\mathbb{C}[\mathfrak{n}]^P$?

2. We have an obvious injective map $\mathbb{C}[\mathfrak{n}]^P \rightarrow \mathbb{C}[\mathfrak{n}]^L. $ What is known about this map?

3. As $L$ is reductive, we know that $\mathbb{C}[\mathfrak{n}]^L$ is a finitely generated algebra. Is it actually trivial? (If that is the case, it answers the above two questions automatically.)

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

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

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?