Twisted Whittaker category on affine flags and the category of representations of the mixed quantum group (Q6543252)

Let \(G\) be a reductive group over an algebraically closed field \(\mathsf{k}\). Fix a pair \((B, B^-)\) of opposite Borel subgroups, and denote by \(N\) and \(N^-\) as their unipotent radicals, respectively. Denote by \(\mathcal{K} = \textsf{k}(\!(t)\!)\) the field of Laurent series and by \(\mathcal{O} = \textsf{k}[\![t]\!]\) the ring of formal power series. Denote by \(G(\mathcal{K})\) the loop group of \(G\), by \(I\) the Iwahori subgroup, and by \(\text{Fl}_G := G(\mathcal{K})/I\) the affine flags.\N\NLet \(\check{G}\) be its Langlands dual group. \textit{S. Arkhipov} and \textit{R. Bezrukavnikov} [Isr. J. Math. 170, 135--183 (2009; Zbl 1214.14011)] proved that the Whittaker category on the affine flags \(\text{Fl}_G\) is equivalent to the category of \(\check{G}\)-equivariant quasi-coherent sheaves on the Springer resolution of the nilpotent cone. This paper proves this theorem in the quantum case. The author shows that the twisted Whittaker category on \(\text{Fl}_G\) and the category of representations of the mixed quantum group are equivalent. In particular, he proves that the quantum category \(\mathsf{O}\) is equivalent to the twisted Whittaker category on \(\text{Fl}_G\) in the generic case. The strong version of the author's main theorem claims a motivic equivalence between the Whittaker category on \(\text{Fl}_G\) and a factorization module category, which holds in the de Rham setting, the Betti setting, and the \(\ell\)-adic setting.