Derived categories and symmetry: geometric invariant theory

Geometric invariant theory allows us to form useful quotients of varieties by the action of Lie groups, even when the group action is not free, by choosing a stability condition and removing “unstable” points. Recent work by Halpern–Leistner and by Ballard, Favero and Katzarkov, partly inspired by geometric quantisation, provides an elegant general description of the derived categories of the resulting quotients. Specifically, given a reductive group $G$ acting on a smooth projective-over-affine variety $X$, we may realise the derived category of the quotient as a subcategory of the $G$-equivariant derived category of $X$ containing objects lying in a certain “grade restriction window”.
We review this construction, show how it can give equivalences of derived categories corresponding to variation of stability condition, and in particular apply it to the Grassmannian example discussed previously. If there is time, we will show how it can also be used to produce symmetries of derived categories, and offer some ideas on how these symmetries can be described geometrically.