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.

First, we should ask:

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

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