Quantum parameters of the geometric Langlands theory

Fix a smooth, complete algebraic curve X over an algebraically closed field k of characteristic zero. To a reductive group G over k, we associate an algebraic stack ParG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {Par}}_G$$\end{document} of quantum parameters for the geometric Langlands theory. Then we construct a family of (quasi-)twistings parametrized by ParG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {Par}}_G$$\end{document}, whose module categories give rise to twisted D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {D}}$$\end{document}-modules on BunG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {Bun}}_G$$\end{document} as well as quasi-coherent sheaves on the DG stack LocSysG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {LocSys}}_G$$\end{document}.

1. Introduction 1.1. The geometric Langlands conjecture. The goal of the Langlands program can be broadly described as establishing a bijective correspondence between automorphic forms attached to a reductive group G and Galois representations valued in the Langlands dual group G.
1.1.1. In the (global, unramified) geometric theory, we fix a smooth, connected, projective curve X over a field k. Automorphic functions correspond to certain sheaves on the stack Bun G parametrizing G-bundles over X, and the role of Galois representations is played by G-local systems on X. If we further specialize to the case where k is algebraically closed of characteristic zero, thenǦ-local systems also form a moduli stack, denoted by LocSysǦ.
1.1.2. Unlike Bun G , the stack LocSysǦ is not smooth; furthermore, it is a DG algebraic stack in general and the correct formulation of the geometric Langlands conjecture has to take into account its DG nature.
After Arinkin and Gaitsgory [AG15], one conjectures an equivalence of DG categories: (1.1) Here, the right-hand-side is the DG category of ind-coherent sheaves on LocSysǦ whose singular support is contained in the global nilpotent cone. This DG category is an enlargement of QCoh(LocSysǦ), and the appearance of singular support is the geometric incarnation of Arthur parameters.
1.2. What do we mean by "quantum"? The quantum geometric Langlands theory seeks to simultaneously deform both sides of (1.1) in a way to make them look more symmetric. The main idea, due to Drinfeld and expounded on by Stoyanovsky [St06] and Gaitsgory [Ga16], is to consider the DG category of twisted D-modules on Bun G .
1.2.1. To explain this approach, let us assume G is simple, and let L G,det be the determinant line bundle over Bun G . To every value c ∈ k one can associate the DG category D-Mod c (Bun G ) of L c−h ∨ 2h ∨ G,det -twisted D-modules over Bun G , where h ∨ denotes the dual Coxeter number of G. Let r = 1, 2, or 3 be the maximal multiplicity of arrows in the Dynkin diagram of G. One expects an equivalence of DG categories: The equivalence L (c) G should vary continuously in c, and degenerate to (1.1) as c tends to zero. 1 For a survey on the conjecture (1.2), see [Sc14].
1.2.2. However, (1.2) is conjectured prior to the formulation of (1.1). For the correct degeneration to IndCoh Nilp (LocSysǦ) to take place, one has to renormalize the DG category D-Mod c (BunǦ).
The renormalized DG category D-Mod c ren (Bun G ) has apparently different nature depending on the rationality and positivity of c, so fitting them in a family is not a trivial matter. Yet, the author expects that they do, and the construction of this family of categories should follow a two-step procedure: (a) Construct a family of non-commutative algebras A over Bun G , whose generic fiber (at c < ∞) is a TDO on Bun G and whose special fiber (at c = ∞) is O LocSys G ; (b) Choose certain objects in module category of A and let them generate the "renormalized" module category that should appear in the quantum geometric Langlands conjecture.
In the present article, we fulfill step (a).
1.3. What's in this article? Let us acknowledge right away that for a simple group G, the space of quantum parameters is just a copy of P 1 , and when the genus of the curve X is at least 2, the stack LocSys G is classical. In this case, the P 1 -family of step (a) has already been constructed by Stoyanovsky [St06], making use of the line bundle L G,det .
1.3.1. Nonetheless, the approach taken in the present article is independent of [St06]. It is motivated by the following considerations for a general reductive group G: -In order to treat geometric Eisenstein series and constant term functors, we need to introduce additional quantum parameters to account for anomalies; these parameters give rise to TDOs that do not arise from the determinant line bundle.
-The DG nature of LocSys G requires us to consider (generalizations of) TDO's whose underlying O-modules are chain complexes; as complexes interact poorly with explicit formulas, we are forced to make a geometric construction, inspired by the theory of twistings developed in [GR14].
The result is a construction of A that completely dispenses of the line bundle L G,det and contains more information as soon as the center Z(G) is nontrivial. The key steps in this construction are summarized by the following chart: 2 quantum parameter (g κ , E) Lie- * algebra g The family of algebras A ultimately arises as the universal enveloping algebra of T (κ,E) G , when we vary the quantum parameter. From our point-of-view, the family of quasi-twistings T (κ,E) G is a more fundamental object than A.
1.4. Organization of this article. We now give a more detailed summary of the content in each section. In particular, we will explain what quasi-twistings are and how they enter naturally into the picture.
1.4.1. We start in §2 with the definition of Par G , the space of quantum parameters. It is a fiber bundle over a compactification of Sym 2 (g * ) G , with fibers being linear stacks describing the "additional parameters." The aforementioned compactification of Sym 2 (g * ) G is simply the space of G-invariant Lagrangian subspaces of g ⊕ g * , where a G-invariant symmetric bilinear form embeds as its graph. The level "at ∞" is understood as the Lagrangian subspace g ∞ := 0 ⊕ g * .
1.4.2. The main idea. Let us take a k-point in Par G , which is a Lagrangian subspace g κ ⊂ g⊕g * together with an additional parameter E. Using the theory of Lie- * algebras developed in [BD04], we construct a central extension of Lie algebroids over the scheme Bun G,∞x parametrizing G-bundles trivialized over the formal neighborhood D x of a fixed closed point x ∈ X. We refer to central extensions of Lie algebroids as classical quasi-twistings. For g κ arising from a symmetric bilinear form, the reduced universal envelope of (1.3): U red ( L (κ,E) ) := U( L (κ,E) )/(1 − 1) defines a TDO over Bun G,∞x . At (g κ , E) = (∞, 0), the algebra U red ( L (∞,0) ) becomes commutative, and identifies with the ring of functions on the ind-scheme LocSys G,∞x (X − {x}) parametrizing a point (P T , η) ∈ Bun G,∞x together with a connection ∇ over P T | X−{x} .
To obtain a central extension of Lie algebroids over Bun G , we "descend" (1.3) along the torsor Bun G,∞x → Bun G , and the algebra A (κ,E) is set to be its universal envelop. The family of algebras A is obtained by letting the point (g κ , E) in Par G vary.
1.4.3. The main challenge. There is, however, a caveat in what it means to "descend" the classical quasi-twisting (1.3). We need a procedure that simultaneously does the following: -For g κ arising from a symmetric bilinear form, it performs the strong quotient of a TDO, in the sense of [BB93]; -For g κ = g ∞ , it transforms (the ring of functions over) LocSys G,∞x (X − {x}) into the DG stack LocSys G , a procedure usually understood as symplectic reduction.
It turns out that one needs to form what we call the quotient of a classical quasi-twisting. In general (and in the way we will apply it), this notion belongs to the DG world, i.e., the quotient of a classical quasi-twisting may cease to be classical.
1.4.4. A (non-classical) quasi-twisting over a finite type scheme Y is defined as a G m -gerbe in the ∞-category of formal moduli problems under Y . They make up the geometric theory of central extensions of Lie algebroids over Y , and are studied in §3. The theory of quasitwistings is made possible by the machinery of formal groupoids and formal moduli problems, as developed in [GR16].
The quotient of quasi-twistings fits into the general paradigm of taking the quotient of an inf-scheme by a group inf-scheme. The latter procedure is rather elaborate, as it mixes prestack quotient with formal groupoid quotient. This is the content of §4.
1.4.5. Finally, we need to deal with the technical annoyance that the theory of [GR16] is built for prestacks locally (almost) of finite type, whereas Bun G,∞x is of infinite type. Hence the actual quotient process has to be performed in two steps, one classical and one geometric, along the torsors: where Bun (≤θ) G is a Harder-Narasimhan truncation of Bun G and n is sufficiently large so that Bun (≤θ) G,nx is a scheme (of finite type.) For this reason, we need to prove a number of results communicating between the classical and derived worlds in §3 and §4. It is the author's hope that an extension of [GR16] to ∞-dimensional algebraic geometry will render this trick obsolete.
1.4.6. The main results. In §5, we perform the main construction of the quasi-twisting T (κ,E) G over Bun G and check that it gives rise to the expected TDOs when g κ is the graph of a bilinear form and E = 0.
In §6, we show that the DG category of modules over T (∞,0) G recovers QCoh(LocSys G ); in doing so, we also obtain a description of the underlying quasi-coherent sheaf of the TDO at an arbitrary level. We end the article with remarks on the "meaning" of certain additional parameters at level ∞.
1.5. Quantum vs. metaplectic parameters. There is another approach of deforming the DG category D-Mod(Bun G ) 3 that undergoes the name "metaplectic geometric Langlands program" (see [GL16], for example.) We briefly explain the relation between metaplectic and quantum parameters.
For simplicity, let us focus on the points (g κ , E) of Par G where g κ arises from a symmetric bilinear form. Such quantum parameters form an open substack isomorphic to Sym 2 (g * ) G × Ext 1 (z G ⊗ O X , ω X ), and the quasi-twistings on Bun G they produce are in fact twistings.
1.5.1. Metaplectic parameters give rise to gerbes, as opposed to twistings, on Bun G . In the context of D-modules, a gerbe on a prestack Y refers to a map from Y dR to B 2 G m . Note that a gerbe on Bun G is sufficient to form the DG category of twisted D-modules, but the additional data of a twisting equip this DG category with a forgetful functor to QCoh(Bun G ). 4 1.5.2. Let Gr G denote the affine Grassmannian associated to G, regarded as a factorization prestack over the Ran space of X. Conjecturally, the spaces of quantum, respectively metaplectic, parameters have the following intrinsic meanings: they are the spaces of factorization twistings, respectively gerbes, on Gr G . The corresponding objects on Bun G then arise from descent along Gr G → Bun G .
Furthermore, there should be a fiber sequence of k-linear groupoids: relating three concepts of different origins: algebro-geometric differential-geometric topological Brylinski-Deligne extensions quantum parameters metaplectic parameters 1.6. Acknowledgement. The author is deeply indebted to his Ph.D. advisor Dennis Gaitsgory. Many ideas here arose during conversations with him-in fact, the idea of using quotient by group inf-schemes is essentially his. The author also thanks Justin Campbell for many helpful discussions.

The space of quantum parameters
Throughout the text, we work over an algebraically closed ground field k of characteristic zero. We write X for a smooth, connected, projective curve and G a connected, reductive group.
In this section, we define the smooth algebraic stack Par G of quantum parameters for the geometric Langlands theory. We will define a natural isomorphism Par G ∼ − → ParǦ, and explain how Par G behaves when we change G into the Levi quotient M of a parabolic of G.
2.1. The space Par G . Let g denote the Lie algebra of G.
2.1.1. Consider the symplectic form on g ⊕ g * defined by the pairing: (2.1) 2.1.2. Clearly, the space Sym 2 (g * ) G of G-invariant symmetric bilinear forms on g embeds into Gr G Lag (g ⊕ g * ), where a form κ, regarded as a linear map g → g * , is sent to its graph g κ . We will use the following notations: g ∞ denotes the k-point g * of Gr G Lag (g ⊕ g * ); g crit is the graph of the critical form crit := − 1 2 Kil, where Kil is the Killing form of g. -for every S-point g κ , the notation g κ−crit denotes the Lagrangian subbundle of (g ⊕ g * ) ⊗ O S defined by the property: Remark 2.1. Note that if κ ∈ Sym 2 (g * ) G , then g κ−crit is the graph of κ − crit, so the above notation is unambiguous; we also have g ∞−crit = g ∞ .
Remark 2.2. More generally, one may replace g κ−crit in the above construction by g κ+κ0 for any κ 0 ∈ Sym 2 (g * ) G . This construction defines an action of Sym 2 (g * ) G on Gr G Lag (g ⊕ g * ) that extends addition on Sym 2 (g * ) G .
2.1.3. We study the k-points of Par G a bit more closely. Let g = z⊕ i g i be the decomposition of g into its center z and simple factors g i .
where the summands are mutually orthogonal with respect to the symplectic form (2.1). We may also decompose L = L z ⊕ j L j , where L z is the G-fixed subspace and each L j is irreducible. Obviously, the embedding L ֒→ g ⊕ g * sends L z into z ⊕ z * as an isotropic subspace.
We claim that each embedding L j ֒→ g ⊕ g * factors through g i ⊕ g * i for a unique i. In other words, the composition L j ֒→ g ⊕ g * ։ g i ⊕ g * i must vanish for all but one i. Suppose, to the contrary, we have i = i ′ such that both are nonzero. Without loss of generality, we may assume that the projections onto the first factors L j → g i , L j → g i ′ are nonzero. Hence we have -L j ∼ = g i ∼ = g i ′ as G-representations; and -the image of L j under the projection g ⊕ g * ։ g i ⊕ g i ′ is a G-invariant subspace with nonzero projection onto both factors. The second statement implies that this image is the entire space g i ⊕ g i ′ , contradicting the equality dim(L j ) = dim(g i ) from the first statement. This prove the claim. Now, suppose j = j ′ and both embeddings L j , L j ′ ֒→ g ⊕ g * factor through the same g i ⊕ g * i . This is obviously impossible since L j ⊕ L j ′ ֒→ g ⊕ g * would factor through an isomorphism so it is not isotropic. We conclude that there is a bijection between the sets {L j } and {g i ⊕ g * i } such that each L j ֒→ g ⊕ g * factors through the corresponding item g i ⊕ g * i . Finally, since each L j is an isotropic subspace of g i ⊕ g * i , we have: Hence the equality is achieved, and each L j (resp. L z ) is a Lagrangian subspace of g i ⊕ g * i (resp. z ⊕ z * ).
Corollary 2.4. Let L be a Lagrangian, G-invariant subspace of g ⊕ g * . Then there is a (noncanonical) isomorphism L ∼ − → g of G-representations.
Note that we have an obvious morphism: sending a series of vector bundles z κ , {g κ i } over S to their direct sum z κ ⊕ i g κ i , which is a subbundle of (g ⊕ g * ) ⊗ O S .
Proof. Indeed, (2.2) is a proper morphism between smooth schemes. Lemma 2.3 shows that it is bijective on k-points, so in particular quasi-finite, and therefore finite (by properness). A finite morphism of degree 1 between smooth schemes is an isomorphism.
Furthermore, any G-invariant symmetric bilinear form on g i fixes an isomorphism Gr G is non-canonically isomorphic to the product of a Lagrangian Grassmannian together with finitely many copies of P 1 .
2.1.4. A particular consequence of Corollary 2.5 is that we have a morphism given by projection: 3) Note that z identifies with the subspace of G-invariants of g. Although z * is more naturally the space of G-coinvariants of g * , we will identify it with the invariants (g * ) G via the isomorphism (g * ) G ֒→ g * ։ z * .
More intrinsically, the morphism (2.3) is defined on S-points by: where (z ⊕ z * ) ⊗ O S is regarded as a submodule of (g ⊕ g * ) ⊗ O S .
Remark 2.6. We refer to (g κ ) G as the G-invariants of g κ . The same terminology is used in the sequel when we replace G by a different group H and g κ by an H-invariant subspace of V ⊕ V * , where V is any H-representation for which the composition (V * ) H ֒→ V * ։ (V H ) * is an isomorphism.
Since the embedding z ֒→ g canonically splits with kernel g s.s. := [g, g], there is a surjection Note that the image of g κ identifies with (g κ ) G , and the composition (g κ ) G ֒→ g κ ։ (g κ ) G is the identity. In other words, Lemma 2.8. The morphism (g κ ) G ֒→ g κ canonically splits.
We denote the complement of (g κ ) G in g κ by g κ s.s. ; it corresponds to the semisimple part of the Lie algebra g.
2.1.5. We define the stack Par G as follows: Maps(S, Par G ) is the groupoid of pairs (g κ , E), where g κ is an S-point of Gr G Lag (g ⊕ g * ), and E is an extension of O X -modules: where X := S × X, and ω X/S ∼ = O S ⊠ ω X is the relative dualizing sheaf.
In other words, Par G is a fiber bundle over Gr G Lag (g ⊕ g * ), whose fiber at a k-point g κ is the linear stack Ext((g κ ) G ⊠ O X , ω X ) of extensions over X. We think of g κ as a generalized symmetric bilinear form on g and E as an additional parameter.
Remark 2.9. The substack of Par G corresponding to the points (g κ , E) where g κ arises from a bilinear form conjecturally parametrizes factorization twistings on the affine Grassmannian Gr G , subject to a certain regularity condition (see §1.5). Hence, one may view Par G as a (partial) compactification of the stack of factorization twistings. We hope to address this conjecture in a forthcoming work.
2.2. Langlands duality of Par G . We now fix a maximal torus T ֒→ G. The Langlands dual of (G, T ) consists of a reductive groupǦ together with a maximal torusŤ ֒→Ǧ.
2.2.1. Let W := N G (T )/T denote the Weyl group of T . It acts on t ⊕ t * in the standard way. There is a symplectic isomorphism: defined using the canonical identifications t * ∼ − →ť and t ∼ − →ť * . Furthermore, (2.5) intertwines the W andW actions (again, under the identification W ∼ − →W ).
Remark 2.10. The sign in (2.5) is present not just for matching up the symplectic forms; it is a feature of the Langlands theory.
Let Gr W Lag (t ⊕ t * ) denote the smooth, projective variety parametrizing W -invariant, Lagrangian subspaces of t ⊕ t * . The isomorphism (2.5) induces an isomorphism: We denote the image of t κ under (2.6) byťκ, and view it as the graph associated to the "dual" form.

We define a morphism
by sending an S-point g κ to (g κ ) T , the T -invariants of g κ . An argument similar to the one in §2.1.4 shows that we have a well-defined map Gr G Lag (g ⊕ g * ) → Gr Lag (t ⊕ t * ); it is clear that the image lies in the W -fixed locus.
Proof. Indeed, a decomposition of g = z ⊕ i g i into simple factors induces a decomposition t = z ⊕ i t i , where each t i is the maximal torus of the factor g i . Note that t i is irreducible as a W -representation. An analogue of Corollary 2.5 asserts an isomorphism Gr W , making the following diagram commute: . Note that the bottom arrow is an isomorphism since the choice of a G-invariant, symmetric bilinear form on g i (hence a W -invariant form on t i ) identifies both Gr G Lag (g i ⊕g * i ) and Gr W Lag (t i ⊕ t * i ) with P 1 . Remark 2.12. Using T , we may also rewrite (2.3) as the two-step procedure of first taking T -invariants and then taking W -invariants: This isomorphism again follows from the description of fibers of g κ in Lemma 2.3.
2.2.3. We will consider a slight variant of the isomorphism (2.7) which takes into account the critical shift: There is an isomorphism between Gr G Lag (g ⊕ g * ) and the corresponding space forǦ, making the following diagram commute: We denote the image of g κ in GrǦ Lag (ǧ ⊕ǧ * ) byǧκ.
whereĚ is the extension of (ǧκ)Ǧ ⊠ O X induced from E via the identification of O S×X -modules: where the middle isomorphism comes from the identification of (g κ−crit ) T and (ǧκ −crit )Ť under (2.6). We refer to (2.9) as the Langlands duality for the parameter space Par G .
Example 2.13. Suppose G is simple, and we fix a k-valued parameter (g κ , 0) of Par G corresponding to some bilinear form κ on g. Then κ = λ·Kil G for some λ ∈ k. Write λ = (c−h ∨ )/2h ∨ for some c ∈ k, where h ∨ denotes the dual Coxeter number of G. Assume c = 0. Then under the isomorphism (2.9): we claim thatǧκ also arises from a bilinear formκ, defined by the formulae: where r = 1, 2 or 3 denotes the maximal multiplicity of arrows in the Dynkin diagram of G. Indeed, one sees this from the fact that (1/2h ∨ ) · Kil G is the minimal bilinear form and r is the ratio of the square lengths of long and short roots of G.
2.3. Parabolics and anomaly. We now explain how to incorporate, via an additional parameter, the anomaly term that appears in the study of constant term functors (see [Ga15,§3.3]). This discussion requires further fixing: -a Borel subgroup B containing T ; -a theta characteristic on the curve X, i.e., a line bundle θ together with an isomorphism A standard parabolic is a parabolic subgroup of G containing B.
is canonically split; this is because the composition Z 0 (G) ֒→ G ։ G/[G, G] is an isogeny, giving rise to the projection z M → z. It follows that we have a canonical map: from the W M -invariants of t ⊕ t * to its W -invariants. In particular, given any Lagrangian, compatible with (2.10).
2.3.2. There is a reduction morphism given by the composition where the isomorphisms are supplied by (2.8) for G, respectively M . In other words, the image of g κ under (2.12) is an S-point m κ such that (m κ−crit ) T and (g κ−crit ) T are canonically isomorphic as subbundles of (t ⊕ t * ) ⊗ O S . LetŽ 0 (M ) denote the Langlands dual torus of Z 0 (M ). We use ωρ M X to denote theŽ 0 (M )bundle on X induced from θ under 2ρ M (regarded as a cocharacter ofŽ 0 (M )). Then the Atiyah bundle of ωρ M X fits into an exact sequence: → T X → 0 Its monoidal dual gives rise to an extension of O X -modules for every S (recall the notation X := S × X): For each S-point m κ of Gr M Lag (m ⊕ m * ), let E + G→M denote the extension of (m κ ) M induced from (2.13) along the canonical map G→M is the anomaly term at level m κ .

The reduction morphism for quantum parameters is defined by
where m κ is the image of g κ under (2.12), and E G→M is the Baer sum of the following two extensions of (m κ ) M : -an extension induced from E (which is an extension of (g κ ) G ) via the map: where the map in the middle comes from (2.11) for t κ := (m κ−crit ) T ∼ = (g κ−crit ) T ; -the anomaly term E + G→M at level m κ .
Remark 2.14. The image of (g ∞ , E) under (2.14) is simply the unadjusted one (m ∞ , E). In particular, we see that (2.14) is incompatible with Langlands duality for quantum parameters, i.e., if we letM be the group dual to M , the following diagram does not commute: Par G (2.9) G G (2.14)
Remark 2.15. For P = B and M = T , the character 2ρ is the sum of positive roots, and splittings of (2.13) form a t * ⊗ ω X -torsor Conn(ωρ X ) commonly known as the Miura opers.
2.4. Structures on g κ . We now note some structures on an S-point g κ of Gr G Lag (g ⊕ g * ) that will be used later. These structures are functorial in S.

2.4.1.
There is an O S -bilinear Lie bracket: defined by the formula (on the ambient bundle (g ⊕ g * ) ⊗ O S ): One checks immediately that the image lies in g κ and the required identities hold. Note that (2.15) factors through the embedding g κ s.s. ֒→ g κ . 2.4.2. There is an O S -bilinear symmetric pairing: defined by the formula: Remark 2.16. The pairing (2.16) gives rise to a canonical central extension of the loop algebra g κ ((t)): 0 → O S → g κ → g κ ((t)) → 0 whose cocycle is given by the residue pairing Res(−, d−). This is the prototype of a generalized Kac-Moody extension. We will return to it in §5 (although in the D-module setting).
2.4.3. Fixing an S-point (g κ , E) of Par G , there is an extension of O X -modules: In other words, g (κ,E) is the direct sum of E and g κ s.s. ⊠ O X , corresponding to the decomposition g κ ∼ − → g κ ⊕ g κ s.s. .

Quasi-twistings
In this section, we make sense of a central extension of Lie algebroids in the DG setting; such objects are called quasi-twistings. A dynamic theory of Lie algebroids in such generality has been built by Gaitsgory and Rozenblyum [GR16], and our results in §3 and §4 are no more than a modest extension of their theory.
We work over a fixed base scheme S locally of finite type over k.
3.1. The classical notion. Let Y ∈ Sch /S be a scheme over S.
The morphism σ is called the anchor map of L. The category of Lie algebroids over Y is denoted by LieAlgd /S (Y ). A Picard algebroid is a central extension of the tangent Lie algebroid T Y /S by O Y ; they are equivalent to twisted differential operators (TDO) over Y (see [BB93]).
of Lie algebroids. We say that T cl is based at the Lie algebroid L. Classical quasi-twistings with a fixed base L form a k-linear category, denoted by QTw cl /S (Y /L). The following is obvious: Lemma 3.1. A classical quasi-twisting T cl is a Picard algebroid if and only if the anchor map of L is an isomorphism.
where U( L) is the universal enveloping algebra of L, and 1 denotes the image of the unit in O Y . A module over T cl is a U(T cl )-module, or equivalently, a module over the Lie algebroid L on which 1 acts by the identity.
3.2. Some ∞-dimensional geometry. When Y is not locally of finite type over S, the above notion of Lie algebroids is not very amenable to study. We will occasionally encounter some ∞-type schemes, for which we need the notion of a Lie algebroid "on Tate module".
3.2.1. Let R be a (discrete) ring over k. The notion of Tate R-modules is developed in [Dr06]. We briefly recall the definitions. An elementary Tate R-module is a topological R-module isomorphic to P ⊕ Q * , where P and Q are discrete, projective R-modules. 5 A Tate R-module is topological R-module isomorphic to a direct summand of some elementary Tate R-module.
There are two important types of submodules of a Tate R-module M : -a lattice is an open submodule L + with the property that L + /U is finitely generated for all open submodule U ֒→ L + . -a co-lattice is a submodule L − such that for some lattice L + , both L + ∩ L − and M/(L + + L − ) are finitely generated.
Example 3.2. Clearly, every profinite R-module is an elementary Tate R-module. The Laurent series ring R((t)) is also an elementary Tate module (but not profinite). 5 The topology on Q * is generated by opens of the form U ⊥ where U is a finite generated R-submodule of Q.

Given a map of (discrete) rings
where U ranges over open submodules of M . Tate R-modules are local objects for the flat topology ([Dr06, Theorem 3.3].) In particular, we may define a Tate O Y -module F over a scheme Y (or more generally, an algebraic stack) as a compatible system of Tate O Z -modules F Z for every affine scheme Z mapping to Y .
where each Y i is a scheme of finite type, and the connecting morphisms Y j → Y i are smooth surjections. We call a placid scheme Y pro-smooth, if we can furthermore choose each Y i to be smooth.
If Y is a pro-smooth placid scheme, then the tangent sheaf T Y /S is naturally a Tate O Ymodule. Indeed, locally on Y there is an isomorphism: has the structure of a Lie algebroid with σ as its anchor map.
Example 3.3. The tangent sheaf T Y /S has the structure of a Lie algebroid on Tate module.
A classical quasi-twisting on Tate modules T cl over Y is a central extension (3.1) of Lie algebroids on Tate modules where all the morphisms are continuous.
Remark 3.4. The above notion is very naïve, as it does not indicate how the Lie bracket interacts with the topology on L. However, it suffices for our purpose since in the construction of T (κ,E) G in §5, the first quotient step will reduce the classical quasi-twisting on Tate modules T (κ,E) G into a discrete, classical quasi-twisting over Bun (≤θ) G,nx . Remark 3.5. We will frequently refer to a classical quasi-twisting on Tate modules simply as a classical quasi-twisting, as the Tate structures should be clear from the context. 3.3. Some infinitesimal geometry. All materials here are taken from [GR16]. We use the language of ∞-categories as developed in [Lu09], [Lu12]; a DG category means a module object over the monoidal ∞-category Vect of complexes of k-vector spaces. We use DGCat cont to denote the ∞-category of all DG categories with continuous functors between them. It admits a symmetric monoidal structure given by the Lurie tensor product.
− → Y such that the following conditions are satisfied: -for every n ≥ 2, the map is an isomorphism.
A groupoid object R • is a formal groupoid if all morphisms in R • are nil-isomorphisms, i.e., inducing isomorphism on the reduced prestacks. We denote the ∞-category of formal groupoids (relative to S) by FrmGpd /S ; this is a full subcategory of that of simplicial objects R • in PStk laft-def /S . We have the functor whose fiber at Y ∈ PStk laft-def /S is by definition the ∞-category of formal groupoids acting on Y, and is denoted by Example 3.6 (Infinitesimal groupoid). Completion along the main diagonals In particular, FrmMod /S is a full subcategory of the functor category Fun(∆ 1 , PStk laft-def /S ). We have a functor whose fiber at Y ∈ PStk laft-def /S is by definition the ∞-category of formal moduli problems under Y, and is denoted by FrmMod /S (Y). The functor (3.3) is also a Cartesian fibration of ∞-categories, whose Cartesian arrows are commutative diagrams on the left whose induced square on the right is Cartesian: Remark 3.7. Analogously, one may consider the Cartesian fibration FrmMod /S → PStk laft-def /S sending Y ♯ → Y to Y, whose fiber is called the ∞-category of formal moduli problems over Y.
Such objects are also studied in [GR16, §IV.1], but we will not need them for this paper.
Straightening (3.3) gives rise to a pullback functor for every morphism f : Y → Z in PStk laft-def /S : We denote the functor inverse to Ω by B : FrmGpd /S → FrmMod /S . Their restrictions to the fiber at Y ∈ PStk laft-def /S are denoted by Ω Y and B Y .

TheČech nerve construction defines a functor Ω :
In particular, given any group object H ∈ PStk laft-def /S , there is a canonical short exact sequence of group prestacks: Here is a simple Corollary of Theorem 3.8: 3.3.4. However, we point out that the quotient of R • in PStk laft-def /S may not agree with that in PStk /S , which is one of the main technical complications for us.
Example 3.11. Let S = pt for simplicity. TheČech nerve of the object pt where the colimit is taken in PStk /S , is an isomorphism. Recall that a morphism X → Y of prestacks is called formally smooth if for every affine DG scheme T over Y, and a nilpotent embedding T ֒→ T ′ , the map Let T * X/Y x denote the cotangent complex at a T -point x : T → X. It is proved in loc.cit. that if X → Y admits (relative) deformation theory, then formal smoothness is equivalent to where F ∈ QCoh(T ) ♥ and T is any affine DG scheme with a morphism x : T → X.
Proof of Lemma 3.12. The authors of [GR16] give the following explicit description of B Y (R • ): let U be an affine DG scheme, then Maps(U, B Y (R • )) identifies with the space of: -a formal moduli problem U over U ; -a morphism from theČech nerve of U → U to R • , such that the following diagram is Cartesian for each of the vertical arrows: On the other hand, Maps(U, colim ∆ op R • ) classifies the above data satisfying the condition that U → U admits a section. Now, since U → U is a nil-isomorphism, we obtain a section over U red . A lift of this section to U exists if the morphism U → U is formally smooth. Now, let T be affine DG scheme equipped with a mapũ : T → U . The Cartesian diagrams: Hence the formal smoothness of R 1 over Y implies that of U over U .
In particular, let h be a (classical) Lie algebra over O S , such that exp(h) acts on some There is a symmetric monoidal functor: . More generally, we may regard IndCoh(−) as a functor IndCoh : PStk laft /S → DGCat cont , 6 We use the notation QCoh(Y ) to denote the DG category of complexes of O Y -modules. In contrast, the abelian category of O Y -modules is denoted by QCoh(Y ) ♥ , understood as the heart of a natural t-structure on QCoh(Y ). The same principle applies to variants of this notation.
where a morphism f : X → Y of laft prestacks gives rise to the functor of !-pullback: . It furthermore has a left adjoint f IndCoh * and the pair (f IndCoh * , f ! ) is monadic. One deduces from this a descent property: Then the canonical functor: is an equivalence.
Proof. This is Proposition 3.3.3 in loc.cit..

The DG category of modules over an object
Note that IndCoh(Y ♭ ) is tensored over QCoh(S). By the above discussion, there is a conservative functor oblv : IndCoh(Y ♭ ) → IndCoh(Y) given by !-pullback along Y → Y ♭ . Furthermore, Proposition 3.13 provides an equivalence of categories: The following result is [GR16, IV.4, Theorem 9.1.5]: Theorem 3.14. If Y ∈ Sch ft /S , 7 then we have a fully faithful functor: Composing (3.8) with B Y , we obtain a fully faithful functor (3.9) whose essential image consists of those formal moduli problems . Furthermore, given a smooth morphism π : Y ′ → Y in Sch ft /S , the following diagram commutes: LieAlgd is the pullback of Lie algebroids (as described in [BB93]), and π ! FrmMod is the functor described in §3.3.2.
Remark 3.15. In what follows, we will frequently use the fact that π ! LieAlgd (L) has underlying O Y ′ -module given by π * L × π * T Y /S T Y ′ /S . 7 The notation Sch ft /S (resp. Sch lft /S ) means classical scheme (locally) of finite type over S.
Notation 3.16. We shall refer to the image Y ♭ of a Lie algebroid L under (3.9) as the formal moduli problem associated to L, and denote it by Y ♭ := L F .
Note that when Y is smooth, IndCoh(Y ♭ ) identifies with the DG category of complexes of (quasi-coherent) L-modules.
3.5. Quasi-twistings. Let Y ∈ PStk laft-def /S . We use G m to denote the formal completion of G m at identity. It is a group formal scheme.
3.5.1. A quasi-twisting T over Y consists of the following data: - Remark 3.17. For an abelian group prestack A over S, the notion of an A-gerbe here is taken in the naïve sense: the prestack B 2 A classifies A-gerbes (on an affine S-scheme) that are globally nonempty, and an A-gerbe on a prestack Y is an object of where T ranges through affine S-schemes mapping to Y. We will later show that usingétale locally trivial G m -gerbes in the definition of a quasi-twisting produces the same class of objects.
Remark 3.18. Alternatively, one can think of a quasi-twisting T as consisting of two formal moduli problems Y ♭ → Y ♭ under Y, equipped with the structure of a G m -gerbe.
The ∞-groupoid of quasi-twistings T based at Y ♭ can be defined as a fiber of ∞-groupoids: More generally, we use QTw A /S (Y/Y ♭ ) to denote an analogously defined category, with the abelian group prestack A acting as the structure group instead of G m .
3.5.2. We now show that quasi-twistings can be defined using different structure groups. The same results about twistings are obtained in [GR14].
Lemma 3.19. The functor of inducing an A-gerbe from an A { 1} -gerbe gives rise to an equivalence of categories QTw Therefore, a section of the A dR/S -gerbe Y ♭ A dR /S amounts to filling in the dotted arrow Y dR/S 8 Abuse of notation: we should really be thinking about S × Gm as a group formal scheme over S.
making the lower-right triangle commute. However, the structure of a quasi-twisting on It follows from Lemma 3.19 that the following functors are equivalences: Corollary 3.20. The tautological functor We use the G a -incarnation of quasi-twistings, as well as their counterparts defined bý etale locally trivial gerbes. For an affine S-scheme T , there holds et G a (classifyingétale locally trivial G a -gerbes on an affine Sscheme) are equivalent. It follows that the corresponding notions of quasi-twistings are also equivalent.
3.6. Modules over a quasi-twisting. We continue to assume Y ∈ PStk laft-def /S and T is a quasi-twisting over Y. Our goal now is to define T-Mod as a DG category tensored over QCoh(S).
3.6.1. We first proceed more generally and define ind-coherent sheaves "twisted" by a G mgerbe.
Let Z ∈ PStk laft-def /S , and Z be a G m -gerbe over Z. Consider the canonical action of B G m on Vect, which induces an action of B G m . More formally, Vect can be regarded as a co-module object in DGCat cont over the co-algebra (IndCoh(B G m ), m ! ), where m is the multiplication map on B G m . The co-action for notions pertaining to group actions on DG categories.) Note that IndCoh( Z) admits a B G m -action, so the product IndCoh( Z) ⊗ Vect is again acted on by B G m . The corresponding co-simplicial system {IndCoh( Z × B G ×n m )} [n]∈∆ has the following first few terms: We define the DG category IndCoh(Z) Z of Z-twisted ind-coherent sheaves on Z by the totalization of the above co-simplicial system. One sees immediately that IndCoh(Z) Z is tensored over QCoh(S).
Since the functors associated to each face map [n] → [m] all admit left adjoints, we obtain: where we use the left adjoints to form the colimit.
Remark 3.21. The above colimit is taken in DGCat cont , and the forgetful functor from DGCat cont to plain ∞-categories does not commute with colimits.
Remark 3.22. Note that any (global) trivialization of the gerbe Z → Z gives rise to an equivalence Remark 3.23. In [GL16, §1.7], a definition of a twisted presheaf of DG categories is given. We relate their definition to ours. For the presheaf over Z: and a G m -gerbe Z, the twisted sheaf of DG categories (IndCoh /Z ) Z is defined by -specifying its values on the category Split( Z) of affine DG schemes S → Z equipped with a lift to Z, using the canonical Maps(S, B G m )-action on IndCoh(S); and then -applying h-descent 9 along the basis Split( Z) → DGSch aff /Z to obtain a sheaf (in the h-topology) over DGSch aff /Z , denoted by (IndCoh /Z ) Z . Thus we may calculate the global section Γ(Z, (IndCoh /Z ) Z ) by the covering Z → Z. The resulting co-simplicial system identifies with (3.12). Hence the definition of Z-twisted indcoherent sheaves in [GL16, §1.7] (adjusted to the h-topology) agrees with ours.
3.6.2. Let T be a quasi-twisting over Y, represented by the G m -gerbe Y ♭ → Y ♭ . We denote by Y the G m -gerbe over Y pulled back along Y → Y ♭ ; it is equipped with a canonical trivialization.
We define the DG category of T-modules by: There is a canonical functor: since Y ♭ is trivialized over Y, and Remark 3.22 identifies the corresponding twisted category with IndCoh(Y).
Proposition 3.24. The functor oblv T admits a left adjoint ind T , and the pair (ind T , oblv T ) is monadic.
Proof. The functor oblv T is by definition the totalization of the !-pullback functors: Each (π (n) ) ! admits a left adjoint π (n) * ,IndCoh . Furthermore, the diagram induced from an arbitrary face map: which a priori commutes up to a natural transformation, actually commutes. Hence oblv T admits a left adjoint ind T := Tot(π (n) * ,IndCoh ). We now prove: oblv T is conservative; this is because all other arrows in the following commutative diagram: The authors of [GL16] work with theétale topology instead. are conservative, hence so is oblv T .
oblv T preserves colimits; this is obvious as we work in DGCat cont . It follows that that the pair (ind T , oblv T ) is monadic, by the Barr-Beck-Lurie theorem.
We may regard U(T) := oblv T • ind T as an algebra object in End(IndCoh(Y)), and the DG category T-Mod identifies with that of U(T)-module objects in IndCoh(Y). We call U(T) the universal envelope of T.

3.7.
Comparison with the classical notion. Suppose Y ∈ Sch ft /S is classical. Let L be a classical Lie algebroid over Y and Y ♭ ∈ FrmMod /S (Y ) be the formal moduli problem associated to L, under the embedding (3.9). : and the outer terms lie in the essential image of QCoh(Y ) ♥ . Hence the previous discussion shows that we have a functor: (3.14) Proposition 3.25. The functor (3.14) is an equivalence of categories.
Proof. We explicitly construct the functor inverse to (3.14). Namely, given a central extension L of L, we need to equip its corresponding formal moduli problem Y ♭ with the structure of a G m -gerbe over Y ♭ . As before, the action map Y ♭ × B G m → Y ♭ arises from the morphism of classical Lie algebroids over Y :

The morphism induced by action and projection
Y ♭ is an isomorphism since the same holds for the corresponding map of classical Lie algebroids: It remains to show that Y ♭ → Y ♭ admits a section over any affine DG scheme T mapping to Y ♭ . We shall deduce the existence of this section from the following claim: Claim 3.26. The morphism Y ♭ → Y ♭ is formally smooth.
Indeed, let T be any affine DG scheme with a morphism y : T → Y ♭ . By the criterion of formal smoothness (3.5), we ought to show Maps(T * Y ♭ /Y ♭ y , F) ∈ Vect ≤0 for all F ∈ QCoh(T ) ♥ . The Cartesian square: together with the isomorphism above gives: One deduces from this the required degree estimate.
Using the claim, we will construct a section of Y ♭ → Y ♭ over T → Y ♭ as follows. First consider the fiber product T × Y ♭ Y, which is equipped with a nil-isomorphism to T . We obtain a solid commutative diagram: Formal smoothness now implies the existence of the dotted arrow.
In particular, the ∞-category QTw /S (Y /Y ♭ ) is an ordinary category.
Remark 3.27. By letting L = T Y /S be the tangent Lie algebroid, we obtain from Proposition 3.25 the fact that Picard algebroids identify with twistings on classical schemes locally of finite type. The same result is established in [GR14, §6.5] using a computation involving de Rham cohomology.

How to take quotient of a Lie algebroid?
This section is devoted to the study of quotients of Lie algebroids, in both classical and DG settings. The set-up involves an H-torsor Y → Z and a Lie algebroid L over Y . With additional data on L, there exists a quotient Lie algebroid over Z. The quotient procedure we shall describe take as input a map η : k ⊗ O Y → L, where k is an arbitrary Lie algebra. It generalizes two existing notions-weak and strong quotients-both considered by Beilinson and Bernstein [BB93].
For technical reasons involving ∞-type schemes, we shall construct two quotient functors: inj , which is a classical procedure that works in the case where η is injective; -Q (H,H ♭ ) , which is its geometric counterpart for Y locally of finite type, and we check that they agree in overlapping cases. A geometric procedure that works in full generality should exist as soon as the theory in [GR16] is extended to ∞-type situations. Remark 4.1. This datum is superficially similar to that of a Harish-Chandra pair, but they serve very different purposes.   -the H-equivariance structure on L is compatible with its Lie bracket; -the anchor map σ of L intertwines the H-equivariance structures on L and T Y /S ; -the following diagram is commutative: -η is compatible with the Lie bracket on L in the following sense: given ξ ∈ k ⊗ O Y and l ∈ L, there holds: (4.1), and ξ h · l denotes the action of ξ h on l coming from the equivariance structure. We will frequently write a (k, H)-Lie algebroid as (L, η), in order to emphasize the dependence on η. The category of (k, H)-Lie algebroids on Y is denoted by LieAlgd 4.2.1. Suppose Z ∈ Sch /S and Y is an H-torsor over Z. Since H is affine, the projection π : Y → Z is an affine, faithfully flat cover (in particular, fpqc). We will define a quotient functor: Q  Consider the embedding: The Lie bracket on L will induce one on The latter identity is guaranteed by (4.5).
We omit checking that this procedure gives rise to a well-defined functor Q   Remark 4.4. The special case where the classical action pair is given by (h, H) with (4.1) being the identity map, has been studied in [BB93] under the name strong quotient. Note that when H acts freely on Y , the map η is automatically injective.
inj /S (Y ) in general. Proposition 4.6. There is a natural bijection: inj (L) → M is equivalent to an H-equivariant map φ : L/k⊗O Y → π * M preserving the Lie bracket on H-invariant sections. We claim that such datum is equivalent to a morphism φ : L → π ! LieAlgd M of (k, H)-Lie algebroids. Indeed, given φ, the map φ is uniquely determined by the properties that the following diagrams commute:  Assuming that Z := Y /H is represented by an algebraic stack. Then the quotient Lie algebroids again form a central extension: Therefore, we may regard Q (k,H) inj as a functor from QTw Remark 4.8. When Y is placid and k is a topological Lie algebra over O S , we can adapt the above definitions to make sense of a Tate (k, H)-Lie algebroid L (c.f. §3.2.4). In particular, η will be a map out of the completed tensor product k ⊗O Y → L.
We do not discuss how to keep track of the topology in the (analogously defined) quotient Q  Proof. We explicitly construct the inverse functor. Given a geometric action pair (H, H ♭ ) for which T H/H ♭ ∈ Υ H (QCoh(H) ♥ ), we can functorially associate a classical Lie algebroid L over H. The following Cartesian diagrams: consists of the following data: -Y, Y ♭ ∈ PStk laft-def /S together with a nil-isomorphism Y → Y ♭ ; -an H-action on Y, and an H ♭ -action on Y ♭ , such that the morphism Y → Y ♭ intertwines them.
Note that there is a functor where PStk H laft-def /S denotes the ∞-category of objects in PStk laft-def /S equipped with an Haction. The fiber of (4.8) at Y is denoted by FrmMod is the ∞-category of formal moduli problems Y ♭ equipped with an H ♭ -action that extends the H-action on Y. H) and (H, H ♭ ) are as in §4.3.2, and let Y ∈ Sch lft /S be acted on by H. We will now construct a functor:

Suppose (k,
which enhances the association of formal moduli problems to Lie algebroids, in the sense that the following diagram commutes: To proceed, let us be given (L, η) ∈ LieAlgd (k,H) /S (Y ). We need to construct an H ♭ -action on the formal moduli problem Y ♭ corresponding to L, expressed by some groupoid together with a map of simplicial prestacks: (4.10) between Lie algebroids over Y × S H (which would rise to act ♭ , in a way compatible with the morphism act) -check that the following diagram: H is commutative (which would affirm the commutativity of (4.10) up to 2-simplices, but the higher commutativity constraints are satisfied automatically since the corresponding ∞-categories are classical.)

Note that as an
The required map α is the sum of the following components: -the map pr * Y L → act ! LieAlgd (L) induced from the H-equivariance structure on L and the composition and the composition The following Lemma shows that the functor (4.9) is well-defined.
Lemma 4.10. The map α is a morphism of Lie algebroids, and the diagram (4.12) commutes.
Proof. It is obvious that α is compatible with the anchor maps. To show that α preserves the Lie bracket, we check it for sections of pr * Y L ⊕ pr * H (k ⊗ O H ) of the following types: -l 1 , l 2 ∈ pr −1 Y L; this follows from the assumptions that the equivariance structure θ : pr * Y L → act * L is compatible with the Lie bracket, and σ is a map of H-equivariant sheaves; -ξ 1 , ξ 2 ∈ k; this is clear; -l ∈ pr −1 Y L and ξ ∈ k; this is a slightly more involved calculation, which we now perform. Write θ(l) = i f i ⊗ l i , where f i ∈ O Y ×H and l i ∈ act −1 L. We need to show the vanishing of the following element in act * L × (4.14) where σ ′ denotes the composition (4.13). Note that the T Y × S H/S -component of (4.14) vanishes tautologically, so we just need to show the vanishing of its act * L-component. The latter is given (using (4.5)) by where in the second summand, ξ h acts on then (4.15) is the (negative of the) induced action of ξ h on the section i f i ⊗ l i = θ(l) in act * L. Note that pr * Y L can also be endowed with an H-equivariance structure: Hence the element ξ h · θ(l) identifies with θ(ξ h · l). On the other hand, l ∈ pr −1 L so ξ h · l = 0, from which we deduce the required vanishing of (4.15).
Checking the commutativity of (4.12) is not difficult, and we leave it to the reader.
4.3.6. We now characterize the image of the functor (4.9).
Proposition 4.11. The functor (4.9) is an equivalence onto the full subcategory: Proof. Indeed, such a formal moduli problems Y ♭ arises from some Lie algebroid L via the functor (3.9). Given the additional data of an (H, H ♭ )-action, we consider the following commutative diagrams: From these diagrams, we obtain two maps between tangent complexes: which gives rise to a morphism θ : pr * Y L → act * L; and 4.3.7. We give an alternative description of the map α that will be used in the proof of Proposition 4.15. Consider the commutative diagram: which is the "quotient" by H of the right diagram in (4.16). It produces the following map between tangent complexes: We claim that (4.19) identifies with the restriction of (4.17) to Y × 10 Suppose C is an ∞-category with finite products. Let H → K be a map of group objects in C. Suppose any object in C with an H-action admits a quotient. Then given an object Y ∈ C with a K-action, there exists a Hecke groupoid Y H × K/H acting on Y /H whose quotient, if exists, agrees with Y /K.
Regarding Y as a fixed prestack acted on by H, we denote the resulting quotient functor by where in the second expression, (Z, Z ♭ ) is equipped with the trivial (H, H ♭ )-action. Specializing to Z = Y/H, we see that the object Q (H,H ♭ ) (Y ♭ ) ∈ FrmMod /S (Y/H) is characterized by the universal property: Proposition 4.14. There is a natural isomorphism: Proof. Both sides are the quotient of (Y, Y ♭ ) by (H, H ♭ ) in the ∞-category FrmMod /S .

Suppose we have a quasi-twisting
is also an (H, H ♭ )-formal moduli problem, and the morphism Y ♭ → Y ♭ preserves this structure. We call quasi-twistings with these additional data (H, H ♭ )-quasi-twistings (based at Y ♭ ) and denote the category of them by QTw Therefore, we may view Q (H,H ♭ ) as a functor QTw Proposition 4.15. The following diagram is commutative:

Comparison of Q
where the two lower squares, as well as the dotted quadrilateral, are Cartesian. Thus, we obtain the following commutative diagram of objects in QCoh(Y ), where commutativity of the red (resp. blue) squares is derived from the red (resp. blue) arrows in the above diagram 11 : T Y /Y ♭ Furthermore, the two horizontal red triangles are exact. Note that the composition (4.19) identifies with η, so the upper horizontal triangle identifies 4.6. Example: inert quasi-twistings. We now specialize to Lie algebroids arising from abelian Lie algebras. They give rise to what we call "inert quasi-twistings." In the geometric Langlands theory, they arise naturally as degeneration of (non-inert) quasi-twistings as the quantum parameter κ tends to ∞ (see §6).
4.6.1. Recall that over any Y ∈ PStk laft-def /S , there is a functor triv : that associates to an ind-coherent sheaf F the abelian Lie algebra on F. More precisely, triv is the right inverse to the forgetful functor; because the latter is conservative and preserves limits, triv also preserves limits. We also have a pair of adjunction: where diag Y preserves fiber products. 12 It follows that the composition diag Y • triv preserves fiber products. We call Y ♭ := diag Y • triv(F) the inert formal moduli problem on F.
Remark 4.17. Let Y be a scheme (not necessarily locally of finite type) over S. The classical analogue of the above construction associates to an O Y -module F the Lie algebroid on F with zero Lie bracket and anchor map. If Y ∈ Sch lft /S , then the image of F under (3.9) agrees with diag Y • triv(Υ Y (F)). 4.6.2. For the remainder of this section, we suppose Y ∈ Sch lft /S is smooth. Then the identification Υ Y : QCoh(Y ) ∼ − → IndCoh(Y ) allows us to view the universal enveloping algebra 13 of an object Y ♭ ∈ FrmMod /S (Y ) as an algebra in QCoh(Y ). If Y ♭ = diag Y • triv(Υ Y (F)), then it is given by Sym OY (F).
Let V(F) := Spec Y Sym OY (F); it is a stack over Y fibered in linear DG schemes. We have an equivalence of DG categories: The map O Y → F gives rise to a morphism of DG schemes: We let V( F) λ=1 be the fiber of (4.23) at {1} ֒→ A 1 . Note that the analogously defined fiber V( F) λ=0 identifies with V(F). There is a canonical equivalence of DG categories: (4.24) Remark 4.18. From our point of view, the DG category QCoh(LocSys G ) is realized by modules over some quasi-twisting on Bun G . The DG stack LocSys G only appears a posteriori through (4.24). Thus, one can say that the origin of QCoh(LocSys G ) is non-geometric.
4.6.4. We now discuss how quotient interacts with inert quasi-twistings. Denote by pt the S-scheme S itself. Suppose (k, H) is a classical action pair with zero map k → h. Then we have where the formation of the semidirect product is formed by the H-action on pt / exp(k). Note that the normal subpair (pt, pt / exp(k)) of (H, H ♭ ) has quotient (H, H), since 4.6.5. We now assume that k is also abelian. Suppose the smooth scheme Y admits an Haction, and Y ♭ is the inert formal moduli problem on some H-equivariant sheaf F ∈ QCoh(Y ) ♥ .  Proof. By Proposition 4.14, we have Note that descent of O Y -modules corresponds to quotient by H on the inert formal moduli problem. Hence we only need to identify Q (pt,pt / exp(k)) (Y ♭ ) as the inert formal moduli problem on Q.
Consider theČech nerve of F → Q in QCoh(Y ), which identifies with the groupoid F ⊕ (k ⊗ O Y ) ⊕• . Since the composition diag Y • triv preserves fiber products, we see that identifies with theČech nerve of the map Y ♭ → diag Y • triv(Q). The result follows since this is also theČech nerve of Y ♭ → Q (pt,pt / exp(k)) (Y ♭ ).
Remark 4.20. When Y is any scheme over S (not necessarily locally of finite type) but η is injective, we also have an identification of Q Geometrically, the datum of η gives rise to a map φ : V(F) → Y × S k * , and V(Q) identifies with its fiber at {0} ֒→ k * . Hence we have isomorphisms of DG stacks: (4.25) 4.6.6. Suppose we have an exact sequence of H-equivariant O Y -modules: Let Y ♭ ∈ QTw /Y ♭ (Y /S) be the corresponding inert quasi-twisting. Assume that η lifts to an H- Then Proposition 4.19 shows that the quotient quasi-twisting arises from a triangle in QCoh(Y /H): where Q desc is the descent of Q := Cofib( η) to Y /H. In particular, we have isomorphisms of DG stacks: where φ λ=1 is the composition Remark 4.21. In light of (4.25) and (4.26), we would like to think of Q (H,H ♭ ) on inert quasitwistings as an analogue of symplectic reduction where φ and φ λ=1 play the role of the moment map (but of course, with no symplectic structures involved a priori.) The universal quasi-twisting

Construction of T (κ,E) G
In this section, we construct a quasi-twisting T (κ,E) G over S × Bun G (relative to S), which depends functorially on the parameter (g κ , E) : S → Par G . We proceed by first constructing a Lie- * algebra g (κ,E) D over S × X, then twist its pullback to S × Bun G,∞x ×X by the tautological G-bundleP G . Via taking sections over • D x and using the residue theorem, we produce a classical quasi-twistingT ). As we shall see, this step requires both quotient functors constructed in §4 and their compatibility.
We then verify that for a simple group G and g κ arising from the bilinear form κ = λ · Kil, the quasi-twisting T (κ,0) G identifies with the twisting given by λ-power of the determinant line bundle L G,det over Bun G . 5.1. Recollection on Lie- * algebras. Fix a base scheme S ∈ Sch /k . Let X → S be a smooth curve relative to S with connected fibers. 14 In particular, the diagonal morphism ∆ : X → X× S X is a closed immersion. Denote by D X/S -Mod r the category of O X -modules equipped with a right action of the relative differential operators D X/S .
whereσ 12 is the transposition morphism over X × S X given by: all sections a, b, and c where σ 123 (x, y, z) = (y, z, x). Denote by Lie * (X/S) the category of Lie- * algebras on X relative to S. Clearly, for any morphism S ′ → S with X ′ := X × S S ′ , we have a functor Lie * (X/S) → Lie * (X ′ /S ′ ) acting as pulling back a D X/S -module, and equipping it with the induced Lie- * algebra structure. 5.1.2. Lie- * algebras areétale local objects. More precisely, letÉt /X be the smallétale site of X. Given B ∈ Lie * (U/S) where U ∈Ét /X and a morphismŨ → U, we may associate an object B Ũ ∈ Lie * (Ũ/S). This procedure defines a functor in groupoids: Theétale local nature of Lie- * algebras refers to the fact that (5.1) satisfies descent.
5.1.3. Let G be a presheaf of group schemes onÉt /X , and B ∈ Lie * (X/S). A G-action on L consists of the following data: -for each U ∈Ét /X , an action of G U as endomorphisms of B U ∈ Lie * (U/S); furthermore, this action is required to be functorial in U. Suppose P is anétale G-torsor over X, and B ∈ Lie * (X/S) admits a G-action. Then we can form the P-twisted Lie- * algebra B P ∈ Lie * (X/S) using the descent property of (5.1). The functor Γ dR of zeroth de Rham cohomology gives rise to functors:

Suppose we have a section
Coh → QCoh Tate (S). 15 We use ⊠ to denote tensoring over O S .
where D X/S -Mod r Coh denotes the category of D X/S -coherent modules (see [BD04,). Furthermore, given a Lie- * algebra B, the object Γ dR ( • D x , B) acquires the structure of a Lie algebra in QCoh Tate (S), whose (continuous) Lie bracket is given by the composition: 5.2. The Kac-Moody Lie- * algebra. Suppose now that S ∈ Sch /k is equipped with a morphism S → Par G , represented by (g κ , E) (see §2). We will construct a central extension of Lie- * algebras over X := S × X: together with G-actions on g 5.2.1. The Lie- * algebra g κ D has underlying D X/S -module g κ ⊠ D X/S . Its Lie- * algebra structure is defined using the Lie bracket (2.15) on g κ : where 1 D is the canonical symmetric section of ∆ ! (D X/S ). Note that the Lie- * bracket [−, −] factors through the embedding g κ s.s. ⊠ D X/S ֒→ g κ D . 16 We construct a G-action on g κ D as follows: for every U ∈Ét /X , there is an adjoint-coadjoint action of the group scheme Maps(U, G) on g κ ⊗ O U : where ξ ⊕ ϕ denotes a section of g κ ⊗ O U , regarded as a subbundle of (g ⊗ O U ) ⊕ (g * ⊗ O U ). The action (5.3) extends to an action of Maps(U, G) on g κ ⊗ O U D U/S by Lie- * algebra endomorphisms.
5.2.2. The underlying D X/S -modules of (5.2) are defined by first inducing a sequence of D X/Smodules from (2.17): and then taking the push-out along the action map In particular, the extension g (κ,E) D → g κ D splits over g κ s.s. ⊠D X/S , and we have a decomposition 16 See §2.1.4 for the notation g κ s.s. .

The
Lie- * algebra structure on g (κ,E) D is defined by the composition: where the middle map is defined using the bilinear form (2.16) and the Lie bracket (2.15) on g κ : ⊗ 1 D ; the notation 1 ′ ω denotes the canonical anti-symmetric section of ∆ ! (ω X/S ).
5.2.4. We now construct the G-action on g (κ,E) D . Let U ∈Ét /X and g U be a point of Maps(U, G).
The corresponding endomorphism g U : g is defined by the sum of the following maps (using the decomposition (5.5)): -identity on E D ; -adjoint-coadjoint action on g κ s.s. ⊠ D U/S by formula (5.3); -the composition: where the map res(g U ) is defined by the formula: It is clear from the construction that g where the notation g κ (O x ) (resp. g κ (K x )) denotes the Tate O S -module g κ ⊗O x (resp. localization at the uniformizer of O x .) The Lie bracket on g (κ,E) is given by the composition: Lemma 5.4. The central extension (5.7) canonically splits over g κ (O x ).
Proof. The result follows from applying Γ dR (D x , −) to the sequence (5.2) and observing that Γ dR (D x , ω X/S ) vanishes.
Let L x G (resp. L + x G) denote the loop (resp. arc) group of G at x. There is an action of L x G on g (κ,E) defined analogously to §5.2.4, with the composition (5.6) replaced by: where the map res(g) (g is a point of L x G) is defined by the formula: Res(f · ϕ(g −1 dg)).
Since the Lie algebra of L x G identifies with g(K x ), this L x G-action induces a g(K x )-action on g (κ,E) by O S -linear endomorphisms.
Lemma 5.5. The Lie bracket on g (κ,E) agrees with the composition: Proof. This is a straightforward computation.

The classical quasi-twistingT
(κ,E) G over Bun G,∞x . Let Bun G,∞x denote the stack classifying pairs (P G , α) where P G is a G-bundle on X and α : P G Dx x G-action on Bun G,∞x by changing α realizes Bun G,∞x as a L + x Gbundle over Bun G , locally trivial in theétale topology. In particular, Bun G,∞x is placid; see §3.2. Letx :S ֒→X (resp. x : S ֒→ X) denote the section given by x ∈ X. LetP G be the tautological G-bundle overX equipped with the trivialization α over Dx. Since g (κ,E) D and g κ D are equipped with G-actions, we can form theP G -twist of (5.8): Remark 5.6.
-Since g κ D is the DX /S -module induced from g κ ⊠ O BunG,∞x ×X and the G-action comes from one on g κ ⊠ O Bun G,∞x ×X , we see that (g κ D )P G is the DX /S -module induced from g κ PG . -the datum of α gives an isomorphism between (5.8) and (5.9) when restricted to Dx.
We apply the functors Γ dR (Σ, −) and Γ dR ( • Dx, −) to (5.9). Using the two observations above, we obtain a morphism between two triangles in QCoh Tate (S): where g κ (K x ) is (as before) an object of QCoh Tate (S).
Since Γ dR ( • Dx, ωX /S ) is canonically isomorphic to OS, we arrive at an exact sequence of Tate OS-modules: Notation 5.7. In what follows, we will show that (5.11) has the structure of a classical quasitwisting (on Tate modules) overS (relative to S; see §3.2.4), denoted byT (κ,E) G . 5.3.3. We (temporarily) use the notation g (κ,E) D,X to denote the Kac-Moody Lie- * algebra over X, constructed using the recipe in §5.2 for the relative curve X → S.
The isomorphism g D,X ⊠ O Bun G,∞x gives rise to an isomorphism in QCoh Tate (S): Observe that the G(K x )-action on Bun G,∞x gives rise to a g(K x )-action 17 on O BunG,∞x by derivations. Hence, the Lie (algebroid) bracket on Γ dR ( can be defined using the O S -linear Lie bracket on g (κ,E) (see §5.2.5): where µ denotes the image of µ ∈ g (κ,E) along g (κ,E) → g κ (K x ) → g(K x ) ⊠O S , which acts on OS ) with the structure of a Lie algebroid. The following lemma, whose proof is deferred to §5.8, extends this Lie algebroid structure to its quotient L (κ,E) : Lemma 5.8. The morphism γ realizes Γ(Σ, g κ PG ) as an ideal of Γ dR ( .
In an analogous way, we turn g κ (K x ) ⊠O BunG,∞x into an object of LieAlgd(S/S), and the map Γ dR ( ) is a morphism of such. Lemma 5.8 shows that γ also realizes Γ(Σ, g κ PG ) as an ideal of g κ (K x ) ⊠O Bun G,∞x . Hence the cokernels (5.11) is a central extension of Lie algebroids. 5.3.4. Proof of Lemma 5.8. We first give an alternative description of the Lie bracket on ). Indeed, from the identification in (5.12) and the g(K x )-action on g (κ,E) (see §5.2.5), we obtain an action of g(K x ) ⊠OS on Γ dR ( , (5.14) 17 Unlike the Tate O S -module g κ (Kx), the notation g(Kx) is reserved for the Tate vector space g⊗Kx (similar for the notation g(Ox).) where pr denotes the composition of the first two maps in (5.13). Therefore, it suffices to show that the Tate OS-submodule: is invariant under the aforementioned g(K x ) ⊠OS-action. Note that by construction, this action arises from the S × L x G-equivariance structure on Γ dR ( ). The following claim is immediate: Claim 5.9. There is also an S × L x G-equivariance structure on Γ dR (Σ, ( g (κ,E) D )P G ), defined at every T -point (s, P G,Σ , α, g) of S × Bun G,∞x ×L x G (for T ∈ Sch aff /k ) by: -first identifying the fiber of Γ dR (Σ, ( g So we have reduced the problem to showing that (5.15) preserves the S × L x G-equivariance structure. In other words, the following diagram in QCoh Tate (T ) needs to commute: ). (5.16) Here, the two horizontal compositions express the procedure of -first restricting a flat section of ( g (κ,E) D ) PG,Σ to • Dx ֒→ T × Σ; -then using the trivialization α (respectively, g · α) to identify it with a section of g (κ,E) D . However, the following diagram is tautologically commutative: so we obtain the commutativity of (5.16).
(Lemma 5.8) 5.4. Descent to Bun G . We continue to fix the S-point (g κ , E) of Par G . The goal of this section is to "descend" the classical quasi-twistingT (κ,E) G to Bun G . Recall the action of H := S × L + x G onS = S × Bun G,∞x , whose quotient is given byS/H ∼ − → S × Bun G . Let k := g κ (O x ). Then (k, H) forms a classical action pair (see §4.1.1). 18 We are slightly abusing the notation ( g 5.4.1. We now equip (5.11) with the structure of a (k, H)-action. Indeed, applying the functor Γ(Dx, −) to (5.9) and using Γ dR (Dx, ωX /S ) = 0, we obtain a commutative diagram: where the splitting η exists for obvious reasons. Since Γ(Dx, g κ ⊠ O Bun G,∞x ×X ) is canonically isomorphic to k ⊗OS, we obtain the (k, H)-action datum on L (κ,E) via the composition: which we again denote by η.
Remark 5.10. Ideally, we would like to define T (κ,E) as the quotient Q (k,H) (T (κ,E) ). However, we run into problems becauseS is not locally of finite type (so we cannot use Q (H,H ♭ ) (4.20)), and η is not injective (so we cannot use Q (k,H) inj (4.6)). In what follows, we circumvent this technical problem using a combination of the two functors. 5.4.2. For each integer n ≥ 0, let Bun G,nx denote the stack classifying pairs (P G , α n ) where P G is a G-bundle on X and α n : Remark 5.11. In particular, L nx G is a group scheme of finite type.
Set H n := S × L nx G, and we have an exact sequence of group schemes over S: Define k n := K ⊗ m n x , and k n := k/k n ∼ = K ⊗ O (n) x . Then the above sequence extends to an exact sequence of action pairs (see §4.1.2): 1 → (k n , H n ) → (H, k) → (H n , k n ) → 1.
(5.18) 5.4.3. We briefly review the Harder-Narasimhan truncation of Bun G . For this, we need to fix a Borel B ֒→ G, whose quotient torus is denoted by T . There are canonical maps Bun B p y y r r r r Let Λ G denote the coweight lattice of G, and Λ + G , Λ pos G ⊂ Λ G denote the submonoid of dominant coweights, respectively the submonoid generated by positive simple coroots. Denote by Λ +,Q G and Λ pos,Q G the corresponding rational cones. There is a partial ordering on Λ Q G , given by: Given λ ∈ Λ Q G , define Bun λ B as the pre-image of λ under the composition: 0) ), and similarly for Q κ n,desc .
6.1.2. The Atiyah bundle construction gives rise to a triangle ω Xn/Sn → At(P (n) over X n . Its pullback along the projection g κ Note that there is a canonical isomorphism Q κ n,desc Proposition 6.1. The triangle (6.1) identifies with the push-out of along the trace map R Γ(X, ω Xn/Sn (−nx))[1] → O Sn .
6.1.3. We now begin the proof of Proposition 6.1. Since both triangles in question are descent of triangles overS, we ought to establish an H n -equivariant isomorphism between the triangle: and the push-out of the analogous triangle: under the trace map R Γ(X, ω X/ S (−nx))[1] → O S .
6.1.4. We describe more explicitly the DX /S -modules underlying the extension sequence of Lie- * algebras (5.9): 0 → ωX /S → ( g (κ,0) D )P G → (g κ D )P G → 0, in the case where the E = 0. Namely, consider the DX /S -modules induced from the sequence (6.2) (where we useX instead of X (n) in the Atiyah bundle construction): be the push-out along act : (ωX /S ) D → ωX /S of the DX /S -module E κ (P G ) D .
Therefore, twisting the diagram (6.6) byP G , we obtain a push-out diagram: This proves the Lemma. 6.1.5. By construction of Q (κ,0) n and Q κ n , the required isomorphism shall follow from a general claim. We first explain the set-up (which is quite involved): let S be a scheme, and X := X × S with section x given by the closed point x ∈ X. Suppose we have an exact sequence of O Xmodules: 0 → ω X/S → E → F → 0.
Let E D denote the induced D-module of E and E push D its push-out along act : (ω X/S ) D → ω X/S . Then we may form a map between exact sequences: as well as a section γ from the residue theorem. On the other hand, let E push D (m (n) ) denote the O S -submodule of Γ dR (D x , E push D ) annihilated by the restriction to D (n) x ; we use the notation F(m (n) ) for a similar meaning. We have a triangle: Remark 6.3. For S :=S, E := E κ (P G ), and F := g κ PG , we see from the construction of (6.4) that it identifies with the triangle (6.8). which is also valid when F is replaced by any O X -module. It suffices to produce a morphism of triangles from (6.9) to (6.8), whose first and third terms are the trace map, respectively the above isomorphism. Consider the diagram defining E push D : 0 6.2.2. Note that a lift of P G to an S-point of LocSys G supplies the dotted arrow in the following commutative diagram: This arrow gives rise to a splitting of (6.10) because T S×X/S×X dR is isomorphic to T S×X/S . In other words, we have a morphism of stacks over Bun G : LocSys G → LocSys ′ G . (6.11) Proposition 6.6. The morphism (6.11) is an isomorphism.
6.2.3. Recall that any eventually co-connective affine DG scheme S fits into a sequence: where S 0 is a classical affine DG scheme, and each morphism S i → S i+1 is a square-zero extension ([GR16, §III.1, Proposition 5.5.3]). Furthermore, for any prestack Y admitting deformation theory and a point y : S → Y, a lift of y along a square-zero extension S → S ′ is governed by maps out of the cotangent complex T * Y y (see [GR16, §III, 0.1.5]). Hence Proposition 6.6 reduces to the following two lemmas.
Lemma 6.7. The morphism (6.11) is an isomorphism when restricted to classical test (affine) schemes.
We conclude the proof of Proposition 6.6.
Remark 6.11. An alternative argument (one that avoids using the results of §6.1) runs as follows: by a local computation, one identifies the universal envelope of the classical quasitwisting (5.11) with the (topological) ring of functions over LocSys G,∞x (Σ), the stack classifying (P G , α) ∈ Bun G,∞x together with a connection over P G Σ . One then shows that the closed subscheme V( Q (∞,0) ) λ=1 identifies with LocSys G,∞x , and (4.26) gives rise to isomorphisms: 6.3.1. We comment on the role of integral additional parameters at ∞, i.e., the ones arising from Z(G)-bundles. More precisely, let E := At(P Z(G) ) * for some Z(G)-bundle P Z(G) . Then E is an extension of z * G ⊗ O X by ω X , so (g ∞ , E) is a well defined k-point of Par G .
Proposition 6.12. Let E = At(P Z(G) ) * for a Z(G)-bundle P Z(G) . Then there is a canonical isomorphism of DG stacks: where the second map is the central shift − ⊗ P Z(G) .
Proof. Note that the D BunG,∞x ×X/ BunG,∞x -module (5.9) at parameter (g ∞ , E) is induced from the following sequence: 0 → ω BunG,∞x ×X/ BunG,∞x → At(P Z(G) ⊗ P G ) * → g * PG → 0 via the functor (−) D and pushing out (see §6.1). An argument similar to above shows that T (∞,E) G is the inert quasi-twisting associated to the triangle in QCoh(Bun G ): where we have a canonical isomorphism Q Remark 6.14. Specializing the hypothetical equivalence (5.22) to the parameter (g crit , 0), we obtain the usual, naïve statement of the geometric Langlands correspondence: Specializing to (g crit , E) where E = At(P Z(Ǧ) ) * , we obtain from (6.13) a hypothetical equivalence: where M is the pullback to Bun G of the line bundle on Bun G/[G,G] corresponding to P Z(Ǧ) . This equivalence can be viewed as an expected compatibility of the geometric Langlands duality with central shift. Let us reiterate that when G is not a torus, none of these equivalences are true without a renormalization process.