# An Example Using Chevalley Restriction

Here is a classic problem of geometric invariant theory: let $G$ be a reductive linear algebraic group such as $\mathrm{GL}_n$ and let $\mathfrak{g}$ be its Lie algebra. Determine the invariant functions $k[\mathfrak{g}]^G$, where $G$ acts on $\mathfrak{g}$ via the adjoint action. This problem is motivated by the search for quotients: What is the quotient $\mathfrak{g}/G$? Here, the action of $G$ on $\mathfrak{g}$ is given by the adjoint action. More explicitly, an element $g\in G$ acts via the differentiation of $\mathrm{Int}_g$, where $\mathrm{Int}_g$ is conjugation by $g$ on $G$.

For simplicity, we will stay in the realm of varieties over an algebraically closed field $k$ of characteristic zero.

What should $\mathfrak{g}/G$ even mean?