Abstract: We provide a uniform construction of “mixed versions” or “graded lifts” in the sense of Beilinson-Ginzburg–Soergel which works for arbitrary Artin stacks. Our new theory associates to each Artin stack of finite type Y over F_q a symmetric monoidal DG-category of graded sheaves on Y along with the six-functor formalism, a perverse t-structure, and a weight (or co-t-)structure in the sense of Bondarko and Pauksztello, compatible with the six-functor formalism, perverse t-structures, and Frobenius weights on the category of (mixed) l-adic sheaves.

As an application, we show that the category of graded character sheaves on GL_n is equivalent to a category of coherent sheaves on the Hilbert schemes Hilb_n of n-points on A^2, yielding a proof of a conjecture of Gorsky–Negut–Rasmussen which relates HOMFLY-PT link homology and the spaces of global sections of certain coherent sheaves on Hilb_n. As an important computational input, we also establish a conjecture of Gorsky–Hogancamp–Wedrich on the formality of the Hochschild homology of Soergel bimodules. This is a joint work with Quoc P. Ho.