On factorization algebras arising in the quantum geometric Langlands theory

Gaitsgory, Dennis

doi:10.1016/j.aim.2021.107962

Cited by 2 publications

(3 citation statements)

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“…Similarly, we define Ω DK, q := F DK Gr (F 0 ). When q avoids small torsion, we have Ω DK, q Ω DK q (see [Gai21b,Theorem 3.6.2]). By the same proof as that of Proposition 6.4.7, we have the following result.…”

Section: Grconfmentioning

confidence: 99%

“…In this section, we recall the factorization algebra given in [Gai21b, § 2.3]. It is the category of modules over this factorization algebra that is expected to be equivalent to the Whittaker category on affine flags.…”

Section: Factorization Algebra and Factorization Modulementioning

confidence: 99%

“…Note that (1.1.2) only makes sense when , otherwise we cannot define D-modules and the mixed quantum group simultaneously. In this case, can be realized as the category of factorization modules [Gai21b] over a factorization algebra . Instead of working with , we compare with this factorization module category.…”

Section: Introductionmentioning

confidence: 99%

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

Yang

2024

Compositio Math.

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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.

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Section: Grconfmentioning

confidence: 99%

Section: Factorization Algebra and Factorization Modulementioning

confidence: 99%