Here is a classic problem of geometric invariant theory: let be a reductive linear algebraic group such as and let be its Lie algebra. Determine the invariant functions , where acts on via the adjoint action. This problem is motivated by the search for quotients: What is the quotient ? Here, the action of on is given by the adjoint action. More explicitly, an element acts via the differentiation of , where is conjugation by on .
For simplicity, we will stay in the realm of varieties over an algebraically closed field of characteristic zero.
First, we should ask: