One of the classical questions in invariant theory is the classification of binary forms, i.e., the description of polynomial invariants of the ${\rm SL}_2(\mathbb{C})$-action on ${\rm Sym}^d \mathbb{C}^2$. In low degrees, we know all the invariants and relations among them, i.e., we have equations for the GIT-quotient.
A more modern view on this would possibly be to study the quotient stack and its relation to the GIT-quotient; the modern theory also tells us that we might want to look at the derived category of the stack -- or for now the equivariant derived category of projective space $\mathbb{P}({\rm Sym}^d\mathbb{C}^2)$ with the induced ${\rm SL}_2$-action.
I would like to know if the structure of the equivariant derived category has been discussed in the low degree cases where we know invariants and GIT-quotients completely. Probably this is well-known to the experts, but I haven't found something like this after searching the literature. It would be great, if someone could answer some parts of the following questionnaire:
What do we know about the structure of the equivariant derived category for the binary forms action? (decompositions, maybe generating objects, exceptional sequences, derived equivalences to other interesting group actions, ...)
How does the equivariant derived category relate to the structure of the GIT-quotient? Is there a general dictionary how properties of the equivariant derived category relate to classical statements concerning the fundamental invariants for binary forms? How much is forgotten in the passage to the equivariant derived category, or can we completely recover information about the actual invariants?
What (if anything) can the equivariant derived category tell us about the non-semistable points and the group orbits that are discarded in the GIT approach?
Maybe there is no connection. If so, why, and what other things could we possibly learn from studying the equivariant derived category of the binary forms action?
