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Twisted Whittaker category on affine flags and the category of representations of the mixed quantum group
- Part of
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- Journal:
- Compositio Mathematica / Volume 160 / Issue 6 / June 2024
- Published online by Cambridge University Press:
- 13 May 2024, pp. 1349-1417
- Print publication:
- June 2024
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Let
$G$ be a reductive group, and let 
$\check {G}$ be its Langlands dual group. Arkhipov and Bezrukavnikov proved that the Whittaker category on the affine flags 
${\operatorname {Fl}}_G$ 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. We show that the twisted Whittaker category on 
${\operatorname {Fl}}_G$ and the category of representations of the mixed quantum group are equivalent. In particular, we prove that the quantum category 
$\mathsf {O}$ is equivalent to the twisted Whittaker category on 
${\operatorname {Fl}}_G$ in the generic case. The strong version of our main theorem claims a motivic equivalence between the Whittaker category on 
${\operatorname {Fl}}_G$ and a factorization module category, which holds in the de Rham setting, the Betti setting, and the 
$\ell$-adic setting.
