G. , O. G-)-on-v-,-i.e,-v-??-?(g, and O. , V satisfies some coassociative conditions. If we want to categorify this notation, we need to replace functions by sheaves. There are two kinds of sheaf theories: quasi-coherent sheaves and D-modules. Given a finite type G, * -pullback along with the multiplication map provides QCoh(G) a comonoidal category structure, Definition A.4. For a DG category C with a weak G-action, we mean C is a comodule category of (QCoh(G), m * ). By duality, The coalgebra structure is given by the pullback along with the multiplication map m. A structure of G-representation on V is a coaction of the coalgebra ?

, But when we define actions of pro-finite group scheme, such as G(O), weak action definition is different

, The category of categories with weak G(O)-action is defined to be: G(O) ? mod weak := colim n QCoh(K n ) ? mod ?

, Given an ind-pro group scheme G. According to Definition A.3, we could define two DG-categories D * (G) and D ! (G). The following lemma in, Remark The functor G(O) m od weak ?? QCoh

A. Lemma, The * -direct image functor along with m : G × G ? G defines D * (G) a monoidal category structure m * : D * (G)

. D(g)-*-?-??-d-*-(g-×-g,

, Dually, m ! provides D ! a structure of comonoidal category

, For a DG category C with a strong G-action(or infinitesimal trivialized action), we mean C is a module category of (D * (G), m * ). Dually

. Then,

B. Lemma, 1. Given a smooth scheme X and an effective Cardier s divisor Y inside. Then, the following functor is bijective. k/Z ?? tame gerbes on X dR with a trivialization on

, What's more

B. Lemma, 2. a).(descend along with finite surjective morphism) If X ?? Y is a finite, surjective map between smooth schemes, then, the pullback defines an equivalence, Grb reg (Y ) Grb reg (X)

, A 1 -homotopy) There is a canonical equivalence: Grb reg (X) Grb reg (X × A 1 )

, And we call the object inside the regular factorization gerbe on Gr G,Ran . References [AB] S. Arkhipov and R.Bezrukavnikov, Perverse sheaves on affine flags and langlands dual group, Isr. J. Math, vol.170, p.135, 2009.

S. Arkhipov and D. , Gaitsgory Another realization of the category of modules over the small quantum group

S. Arkhipov, R. Bezrukavnikov, A. Braverman, D. Gaitsgory, and I. , Mirkovi? Modules over the small quantum group and semi-infinite flag manifold

D. Abramovich and F. Oort, Stable maps and Hurwitz schemes in mixed characteristics, Advances in algebraic geometry motivated by physics, vol.276, pp.89-100, 2001.

A. Beilinson, J.Bernstein A proof of Jantzen conjectures

A. Beilinson and J. Bernstein, A generalization of Casselman's submodule theorem, Representation theory of reductive groups, Progr. Math, vol.40, pp.35-52, 1982.

A. Beilinson and V. Drinfeld, Quantization of Hitchin's integrable system and Hecke eigensheaves, pp.1297-1301, 1991.

A. Beilinson and V. Drinfeld, Chiral algebras, vol.51, 2004.

R. Bezrukavnikov, Perverse sheaves on affine flags and nilpotent cone of the Langlands dual group

R. Bezrukavnikov, Cohomology of tilting modules over quantum groups and t-structures on derived categories of coherent sheaves, Inv. Math, vol.166, pp.327-357, 2006.

R. Bezrukavnikov, On two geometric realizations of an affine Hecke algebra

J. M. Beck, Triples, algebras and cohomology

A. Beilinson, Constructible sheaves are holonomic, Sel. Math. New Ser, vol.22, p.1797, 2016.

R. Bezrukavnikov and A. , Lachowska The small quantum group and the Springer resolution

D. Beraldo, Loop Group Actions on Categories and Whittaker Invariants

A. Braverman, M. Finkelberg, D. Gaitsgory, and I. , Mirkovi? Intersection cohomology of Drinfeld's compactifications, Selecta Mathematica, vol.8, pp.381-418, 2002.

R. Bezrukavnikov, M. Finkelberg, and V. Schechtman, Factorization algebras and quantum groups, p.1691, 1998.

A. Braverman and D. , Gaitsgory Geometric Eisenstein series, Invent. math, vol.150, p.287, 2002.

A. Beauville and Y. Laszlo, Un lemme de descente, Comptes Rendus de l'Academie des Sciences-Serie I-Mathematique, vol.320, pp.335-340, 1995.

A. Beligiannis and I. Reiten, Homological and homotopical aspects of torsion theories, Mem. Amer. Math. Soc, vol.188, issue.883, 2007.

R. Bott and H. Samelson, Applications of the theory of Morse to symmetric spaces, American Journal of Mathematics, vol.80, issue.4, pp.964-1029, 1958.

M. Barr and C. Wells, Toposes, Triples and Theories, vol.278, 1983.

J. Campbell, A resolution of singularities for Drinfeld's compactification by stable maps

P. Deligne, SGA 4 1/2-Cohomologie étale, Lecture Notes in Mathematics, vol.569, 1977.

V. Drinfeld and D. , Gaitsgory On a theorem of Braden

V. Drinfeld and D. Gaitsgory, On some finiteness questions for algebraic stacks, GAFA, vol.23, pp.149-294, 2013.

V. Drinfeld and D. , Gaitsgory Compact generation of the category of Dmodules on the stack of G-bundles on a curve, vol.3, pp.19-125, 2015.

V. Drinfeld and C. Simpson, B-structures on G-bundles and local triviality, Mathematical Research Letters, vol.2, issue.6, pp.823-829, 1995.

C. De-concini and V. , Kac Representations of quantum groups at roots of 1, Operator algebras, unitary representations, enveloping algebras, and invariant theory, Birkhäuser Boston, vol.92, pp.471-506, 1989.

B. Feigin, M. Finkelberg, A. Kuznetsov, and I. Mirkovic, Semi-infinite flags. II. Local and global intersection cohomology of quasimaps' spaces, Differential topology, infinite-dimensional Lie algebras, and applications, vol.2, pp.113-148, 1999.

E. Frenkel and D. Gaitsgory, D-modules on the affine flag variety and representations of affine Kac-Moody algebras

E. Frenkel, D. Gaitsgory, and K. Vilonen, Whittaker patterns in the geometry of moduli spaces of bundles on curves, Annals of Math, vol.153, issue.3, pp.699-748, 2001.

W. Fulton and R. , Pandharipande Notes on stable maps and quantum cohomology

J. Francis, The tangent complex and Hochschild cohomology of E n -rings, Compositio Math, vol.149, pp.430-480, 2013.

J. Francis and D. , Gaitsgory Chiral Koszul duality, vol.51, 2004.

D. Gaitsgory, A conjectural extension of the Kazhdan-Lusztig equivalence

D. Gaitsgory, Twisted Whittaker model and factorization algebras, Sel. math., New ser, vol.13, p.617, 2008.

D. Gaitsgory, The semi-infinite intersection cohomology sheaf

D. Gaitsgory, The semi-infinite intersection cohomology sheaf-II: the Ran space version

D. Gaitsgory, The local and global versions of the Whittaker category

D. Gaitsgory, On factorization algebras arising in the quantum geometric Langlands theory

D. Gaitsgory, The Atiyah-Bott formula for the cohomology of the moduli space of bundles on a curve

D. Gaitsgory, On a vanishing conjecture appearing in the geometric Langlands correspondence, Ann. Math, vol.160, pp.617-682, 2004.

D. Gaitsgory, Sheaves of categories and the notion of 1-affineness

D. Gaitsgory, Ind-coherent sheaves

D. Gaitsgory, Introduction to quantum local geometric Langlands

D. Gaitsgory and S. Lysenko, Metaplectic Whittaker category and quantum groups : the "small, FLE

D. Gaitsgory and S. Lysenko, Parameters and duality for the metaplectic geometric Langlands theory

D. Gaitsgory and N. Rozenblyum, A study in derived algebraic geometry, vol.1, 2017.

D. Gaitsgory and N. Rozenblyum, Crystals and D-modules

P. Quoc, Ho Factorization algebras and categories

J. C. Jantzen, Lectures on quantum groups Graduate Studies in Mathematics, vol.6, 1996.

G. R. Kempf, Linear systems on homogeneous spaces, Annals of Mathematics, pp.557-591, 1976.

V. G. Kac, Infinite-dimensional Lie algebras, 1990.

D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras, J. Amer. Math. Soc, vol.6, pp.335-453, 1993.

M. Lanini, Semi-infinite combinatorics in representation theory

S. Lysenko, Twisted Whittaker models for metaplectic groups, GAFA, vol.27, pp.289-372
URL : https://hal.archives-ouvertes.fr/hal-01227086

S. Lysenko, Twisted geometric Langlands correspondence for a torus, IMRN, vol.18, pp.8680-8723, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01227148

J. Lurie, Higher Algebra

J. Lurie, Higher topos theory, 2009.

J. Lurie, Derived Algebraic Geometry II: Noncommutative Algebra

G. Lusztig, Introduction to quantum groups, 2010.

A. Ramanathan, Equations defining Schubert varieties and Frobenius splittings of diagonals, Publications Mathématiques de l'IHÉS, vol.65, pp.61-90, 1987.

, S.Raskin Chiral Principal Series Categories

S. Raskin-w-algebras and W. Categories,

, S.Raskin D-modules on infinite dimensional varieties, 2015.

R. C. Reich, Twisted geometric Satake equivalence via gerbes on the factorizable grassmannian

S. Riche, Tutorial on quantum groups

N. Rozenblyum, Tutorial on factorization vs braided monoidal categories

T. Saito, The characteristic cycle and the singular support of a constructible sheaf

C. Sorger, Lectures on moduli of principal G-bundles over algebraic curves, Lectures on moduli of principal G-bundles over algebraic curves

Y. Fu, Tutorial 1: Group Actions on Categories

Y. Zhao, Quantum parameters of the geometric Langlands theory

Y. Zhao, Notes o quantum parameters (GL-2)

Q. Zhou, Convex Polytopes for the Central Degeneration of the Affine Grassmannian

X. Zhu, An introduction to affine Grassmannians and the geometric Satake equivalence

, Nous notons par KL?(?) la catégorie des modules Kac-Moody?(O) -intégrables. Ici,?? désigne l'extension centrale de?((t)) par?. Et selon Kazhdan-Lusztig