I also suggest reading Chapter 3 (Invariants of maximal unipotent subgroups) of:
Frank D. Grosshans, $\textit{Algebraic homogeneous spaces and invariant theory}$, Lecture Notes in Mathematics, vol. 1673, Springer-Verlag, Berlin, 1997.
It gives an explicit description of $\mathbb{C}[\mathfrak{g}]^P$ under the adjoint action and of $\mathbb{C}[\mathfrak{g}]^N$ under left translation in the case when $\mathfrak{g}=\mathfrak{gl}_n$.