Parameters and duality for the metaplectic geometric Langlands theory

We introduce the space of parameters for the metaplectic Langlands theory as *factorization gerbes* on the affine Grassmannian, and develop metaplectic Langlands duality in the incarnation of the metaplectic geometric Satake functor. We formulate a conjecture in the context of the global metaplectic Langlands theory, which is a metaplectic version of the"vanishing theorem"of ref. Ga5 (Theorem 4.5.2 in loc. cit.)

We can then study representations of G(F) on which the central E × acts by the tautological character. We will refer to (0.1) as a local metaplectic extension of G(F), and to the above category of representations as metaplectic representations of G(F) corresponding to the extension (0.1).
Let now F be a global field, and let A F be the corresponding ring of adèles. Let us be given a central extension equipped with a splitting over G(F) → G(A F ).
We can then study the space of E-valued functions on the quotient G(A F )/G(F), on which the central E × acts by the tautological character. We will refer to (0.2) as a global metaplectic extension of G(F), and to the above space of functions as metaplectic automorphic functions on G(F) corresponding to the extension (0.2).
There has been a renewed interest in the study of metaplectic representations and metaplectic automorphic functions, e.g., by B.Brubaker-D.Bump-S.Friedberg, P.McNamara, W.T.Gan-F.Gao.
M. Weissman has initiated a program of constructing the L-groups corresponding to metaplectic extensions, to be used in the formulation of the Langlands program in the metaplectic setting, see [We]. 0.1.2. Parameters for metaplectic extensions. In order to construct metaplectic extensions, in both the local and global settings, one starts with a datum of algebro-geometric nature. Namely, one usually takes as an input what we call a Brylinski-Deligne datum, by which we mean a central extension of sheaves of groups on the big Zariski site of F, where (K2)Zar is the sheafification of the presheaf of abelian groups that assigns to an affine scheme S = Spec(A) the group K2(A).
For a local field F, let f denote its residue field and let us choose a homomorphism Then taking the group of F-points of G and pushing out with respect to we obtain a central extension (0.1). A similar procedure applies also in the global setting.
0.1.3. The geometric theory. Let k be a ground field and let G be a reductive group over k.
In the local geometric Langlands theory one considers the loop group G((t)) along with its action on various spaces, such as the affine Grassmannian GrG = G((t))/G [[t]]. Specfically one studies the behavior of categories of sheaves 1 on such spaces with respect to this action.
In the global geometric Langlands theory one considers a smooth proper curve X, and one studies the stack BunG that classifies principal G-bundles on X. The main object of investigation is the category of sheaves on BunG.
There are multiple ways in which the local and global theories interact. For example, given a (k-rational) point x ∈ X, and identifying the local ring Ox of X at x with k[[t]], we have the map where we interpret GrG as the moduli space of principal G-bundles on X, trivialized over X − x.
0.1.4. The setting of metaplectic geometric Langlands theory. Let E denote the field of coefficients of the sheaf theory that we consider. Recall (see Sect. 1.7.4) that if Y is a space 2 and G is a E × -gerbe on Y, we can twist the category of sheaves on Y, and obtain a new category, denoted Shv G (Y).
In the local metaplectic Langlands theory, the input datum (which is an analog of a central extension (0.1)) is an E × -gerbe over the loop group G((t)) that behaves multiplicaively, i.e., one that is compatible with the group-law on G((t)).
Similarly, whenever we consider an action of G((t)) on Y, we equip Y with E × -gerbe that is compatible with the given multiplicative gerbe on G((t)). In this case we say that the category Shv G (Y) carries a twisted action of G((t)), where the parameter of the twist is our gerbe on G((t)).
In the global setting we consider a gerbe G over BunG, and the corresponding category Shv G (BunG) of twisted sheaves. Now, if we want to consider the local vs. global interaction, we need a compatibility structure on our gerbes. For example, we need that for every point x ∈ X, the pullback along (0.5) of the given gerbe on BunG be a gerbe compatible with some given multiplicative gerbe on G((t)).
So, it is natural to seek an algebro-geometric datum, akin to (0.3), that would provide such a compatible family of gerbes. 0.1.5. Geometric metaplectic datum. It turns out that such a datum (let us call it "the geometric metaplectic datum") is not difficult to describe, see Sect. 2.4.1 below. It amounts to the datum of a factorization gerbe with respect to E × on the affine Grassmannian 3 GrG of the group G.
In a way, this answer is more elementary than (0.3) in that we are dealing withétale cohomology rather than K-theory.
Moreover, in the original metaplectic setting, if the global field F is the function field corresponding to the curve X over a finite ground field k, a geometric metaplectic datum gives rise directly to an extension (0.2).
Finally, a Brylinski-Deligne datum (i.e., an extension (0.3)) and a choice of a character k × → E × gives rise to a geometric metaplectic datum.
Thus, we could venture into saying that a geometric metaplectic datum is a more economical way, sufficient for most purposes, to encode also the datum needed to set up the classical metaplectic representation/automorphic theory. 0.1.6. The metaplectic Langlands dual. Given a geometric metaplectic datum, i.e., a factorization gerbe G on GrG, we attach to it a certain reductive group H, a gerbe GZ H on X with respect to the center ZH of H, and a character : ±1 → ZH . We refer to the triple (H, GZ H , ) as the metaplectic Langlands dual datum corresponding to G.
The datum of GZ H determines the notion of twisted H-local system of X. Such twisted local systems are supposed to play a role vis-à-vis metaplectic representations/automorphic functions of G parallel to that of usualǦ-local systems vis-à-vis usual representations/automorphic functions of G.
For example, in the context of the global geometric theory (in the setting of D-modules), we will propose a conjecture (namely, Conjecture 8.6.2) that says that the monoidal category QCoh LocSys 0.2.2. For the quantum Langlands theory, our parameter will be a factorizable twisting on the affine Grassmannian, which one can also interpret as a Kac-Moody level ; we will denote it by κ.
Thus, for example, in the global quantum geometric Langlands theory, we consider the category D-modκ(BunG), which is the same as Shv G (BunG), where G is the gerbe corresponding to κ.
As was mentioned above, the additional piece of datum that the twisting "buys" us is the forgetful functor D-modκ(BunG) → QCoh(BunG). In the TQFT interpretation of geometric Langlands, this forgetful functor is called "the big brane". It allows us to relate the category D-modκ(BunG) to representations of the Kac-Moody algebra attached to G and the level κ. 0.2.3. Consider the usual Langlands dual groupǦ of G, and if κ is non-degenerate, it gives rise to a twisting, denoted −κ −1 , on the affine Grassmannian GrǦ ofǦ.
We refer to (0.6) as the global quantum Langlands equivalence.
0.2.4. How are the two theories related? The relationship between the equivalence (0.6) and the metaplectic Langlands dual is the following: Let G (resp.,Ǧ) be the gerbe on GrG (resp., GrǦ) corresponding to κ (resp., −κ −1 ). We conjecture that the metaplectic Langlands dual data (H, GZ H , ) corresponding to G andǦ are isomorphic.
Furthermore, we conjecture that the resulting actions of QCoh LocSys G Z H H on D-modκ(BunG) and D-mod −κ −1 (BunǦ), respectively (see Sect. 0.1.6 above) are intertwined by the equivalence (0.6). 0.3. What is actually done in this paper? Technically, our focus is on the geometric metaplectic theory, with the goal of constructing the metaplectic geometric Satake functor. 0.3.1. The mathematical content of this paper is the following: -We define a geometric metaplectic datum to be a factorization gerbe on the (factorization version) of affine Grassmannian GrG. This is done in Sect. 2. -We formulate the classification result that describes factorization gerbes on GrG in terms ofétale cohomology on the classifying stack BG of G. This is done in Sect. 3. This classification result is inspired by an analogous one in the topological setting, explained to us by J. Lurie.
-We study the relationship between factorization gerbes on GrG and those on GrM , where M is the Levi quotient of a parabolic P ⊂ G. This is done in Sect. 5.
The main point is that the naive map from factorization gerbes on GrG to those on GrM needs to be corrected by a gerbe that has to do with signs. It is this correction that is responsible for the fact that the usual geometric Satake does not quite produce the category Rep(Ǧ), but rather its modification where we alter the commutativity constraint by a canonical character ±1 → Z(Ǧ).
-We define the notion of metaplectic Langlands dual datum, denoted (H, GZ H , ), attached to a given geometric metaplectic datum G. We introduce the notion of GZ H -twisted H-local system on X; when we work with D-modules, these local systems are k-points of a (derived) algebraic stack, denoted . This is done in Sect. 6.
-We show that a factorization gerbe on GrG gives rise to a multiplicative gerbe over the loop group G((t)) for every point x ∈ X. Moreover, these multiplicative gerbes also admit a natural factorization structure when instead of a single point x we consider the entire Ran space. This is done in Sect. 7.
-We define metaplectic geometric Satake as a functor between factorization categories over the Ran space.
-We formulate a conjecture about the action of the monoidal category QCoh LocSys . This is done in Sect. 8. 0.3.2. A disclaimer. Although most of the items listed in Sect. 0.3.1 have not appeared in the previously existing literature, this is mainly due to the fact that these earlier sources, specifically the paper [FL] of M. Finkelberg and the second-named author and the paper [Re] of R. Reich, did not use the language of ∞-categories, while containing most of the relevant mathematics.
So, one can regard the present paper as a summary of results that are "almost known", but formulated in the language that is better adapted to the modern take on the geometric Langlands theory 4 .
We felt that there was a need for such a summary in order to facilitate further research in this area.
Correspondingly, our focus is on statements, rather than proofs. Most of the omitted proofs can be found in either [FL] or [Re], or can be obtained from other sources cited in the paper.
Below we give some details on the relation of contents of this paper and some of previously existing literature. 0.3.3. Relation to other work: geometric theory. As was just mentioned, a significant part of this paper is devoted to reformulating the results of [FL] and [Re] in a way tailored for the needs of the geometric metaplectic theory.
The paper [Re] develops the theory of factorization gerbes on GrG (in loc. cit. they are called "symmetric factorizable gerbes"). One caveat is that in the setting of [Re] one works with schemes over C and sheaves in the analytic topology, while in the present paper we work over a general ground field andétale sheaves.
The main points of the theory developed in [Re] are the description of the homotopy groups of the space of factorization gerbes (but not of the space itself; the latter is done in Sect. 3 of the present paper), and the fact that a factorization gerbe on GrG gives rise to a multiplicative gerbe on (the factorization version of) the loop group (we summarize this construction in Sect. 7 of the present paper).
The proofs of the corresponding results in [Re] are obtained by reducing assertions for a reductive group G to that for its Cartan subgroup, and an explicit analysis for tori. We do not reproduce these proofs in the present paper.
In both [FL] and [Re], metaplectic geometric Satake is stated as an equivalence of certain abelian categories. In [FL], this is an equivalence of symmetric monoidal categories (corresponding to a chosen point x ∈ X), for a particular class of gerbes (namely, ones obtained from the determinant line bundle).
In [Re] more general gerbes are considered and the factorization structure on both sides of the equivalence is taken into account. Our version of metaplectic geometric Satake is a statement at the level of DG categories; it is no longer an equivalence, but rather a functor in one direction, between monoidal factorization categories. In this form, our formulation is a simple consequence of that of [Re]. 0.3.4. Relation to other work: arithmetic theory. As was already mentioned above, our notion of the metaplectic Langlands dual datum is probably equivalent to the datum constructed by M. Weissman in [We] for his definition of the L-group. 0.4. Structure of the paper. The paper begins with a section that contains some background on prestacks, ∞-categories, sheaf theories, etc. The reader who has a rudimentary familiarity with this material can safely skip it.
As has been mentioned already, our geometric metaplectic datum is a gerbe on the affine Grassmannian, denoted GrG. We will need the factorization (a.k.a. Beilinson-Drinfeld, Ran space) version of the affine Grassmannian. Its key feature is that it is not a scheme, and not even an ind-scheme. Rather, this version of the affine Grassmannian is what we call a prestack. In Sect. 1 we recall the definition of what a prestack is, and what gerbes on a prestack are.
In addition, in the same section we recall what we mean by the category of sheaves on a prestack, and by a sheaf of categories over a prestack. Both these notions are necessary for the statement of metaplectic geometric Satake. 0.4.1. In Sect. 2 we recall the definition of the Ran space of a given curve X (denoted Ran). We define the notions of a factorization prestack over Ran, a factorization gerbe over a factorization prestack and a (pre)factorization sheaf of categories over Ran.
Our main example of a factorization prestack is the affine Grassmannian GrG. Factorization gerbes over GrG are the main object of study in this paper; they provide an input data for the metaplectic geometric Langlands theory. 0.4.2. In Sect. 3 we discuss the parameterization of the set (more precisely, space) of factorization gerbes on GrG in terms ofétale cohomology of the classifying stack BG of G.
We will see that to a factorization gerbe we can associate a combinatorial invariant, denoted q, which a quadratic form on the coweight lattice Λ of G with coefficients in E × (here E is our field of coefficients), invariant with respect to the Weyl group.
The space of factorization gerbes with a fixed parameter q is of local nature with respect to our curve X. More precisely, it is acted on simply transitively by (the commutative group in spaces of) gerbes on X with respect to the group Hom(π 1,alg (G), E × ). 0.4.3. In Sect. 4 we make our analysis of factorization gerbes on GrG more explicit in the case when G is a torus. In fact, most of proofs of statements left unproved in this paper go by reduction to this case. 0.4.4. In Sect. 5 we study the interaction between factorization gerbes on GrG and those on GrM , where M is the Levi quotient of a parabolic P ⊂ G.
The two affine Grassmannians are related by the diagram where the map q has contractible fibers. Hence, given a gerbe on GrG, we can restrict it to GrP , and the resulting gerbe will uniquely come from a gerbe on GrM .
This procedure gives a map from the space of gerbes GrG to that on GrM . However, this map is not quite what we want. Namely, it differs from the "right" one by a certain gerbe that has to do with signs. 0.4.5. In Sect. 6 we explain how the datum of a factorization gerbe G on GrG gives rise to a metaplectic Langlands dual datum, i.e., a reductive group H, a gerbe GZ H on X with respect to the center ZH of H, and a character ±1 → ZH .
We define the notion of GZ H -twisted local system on X, and when we work with D-modules over a field of characteristic 0, we construct the (derived) algebraic stack LocSys G Z H H that classifies such local systems. 0.4.6. In Sect. 7 we introduce (the factorization version of) the loop group, denoted L(G). We explain a construction that, starting from a factorization gerbe G on GrG, produces a multiplicative structure on the pullback of G to L(G). This multiplicative structure is what allows us to define the convolution product on the metaplectic spherical Hecke category. 0.4.7. In Sect. 8 we state the existence of the metaplectic geometric Satake functor, which maps the factorization category, built out of (Rep(H), GZ H , ) (here Rep(H) denotes the category of representations of H), to the metaplectic spherical Hecke category.
Finally, we state our "metaplectic vanishing conjecture", Conjecture 8.6.2 about the action of QCoh LocSys 0.5.1. Algebraic geometry. In the main body of the paper we will be working over a fixed ground field k, assumed algebraically closed.
For arithmetic applications one would also be interested in the case of k being a finite field Fq. However, since all the constructions in this paper are canonical, the results over Fq can be deduced from those over Fq by Galois descent.
We will denote by X a smooth connected algebraic curve over k (we do not need X to be complete).
For the purposes of this paper, we do not need derived algebraic geometry, with the exception of Sects. 6.6 and 8.6 (which are devoted to the D-module situation).
We let Sch aff denote the category of (classical!) affine schemes over k, and by Sch aff ft its full subcategory consisting of affine schemes of finite type.
In the main body of the paper we will make an extensive use of algebro-geometric objects more general than schemes, namely, prestacks. We recall the definition of prestacks in Sect. 1.1.3, and refer the reader to [GR2, Chapter 2] for a more detailed discussion. 0.5.2. Coeffients. We will work with various sheaf theories on schemes, see Sect. 1.1.
Our sheaves will have coefficients in E-vector spaces, where E is a field of coefficients, assumed algebraically closed and of characteristic zero. 0.5.3. Groups. We will work with a fixed connected algebraic group G over k; our main interest is the case when G is reductive.
We will denote by Λ the coweight lattice of G and byΛ its dual, i.e., the weight lattice.
We will denote by αi ∈ Λ (resp.,αi ∈Λ) the simple coroots (resp., roots), where i runs over the set of vertices of the Dynkin diagram of G.
If G is reductive, we denote byǦ its Langlands dual, viewed as a reductive group over E. 0.5.4. The usage of higher category theory. Although, as we have said above, we do not need derived algebraic geometry, we do need higher category theory. However, we only really need ∞-categories for one type of manipulation: in order to define the notion of the category of sheaves on a given prestack (and a related notion of a sheaf of categories over a prestack); we will recall the corresponding definitions in Sects. 1.1.3 and 1.6), respectively. These definitions involve the procedure of taking the limit, and the language of higher categories is the adequate framework for doing so.
In their turn, sheaves of categories on prestacks appear for us as follows: the metaplectic spherical Hecke category, which is the recipient of the metaplectic geometric Satake functor (and hence is of primary interest for us), is a sheaf of categories over the Ran space. 0.5.5. Glossary of ∞-categories. We will now recall several most common pieces of notation, pertaining to ∞-categories, used in this paper. We refer the reader to [Lu1,Lu2] for the foundations of the theory, or [GR2, Chapter 1] for a concise summary.
We denote by Spc the ∞-category of spaces. We denote by * the point-space. For a space S, we denote by π0(S) its set of connected components. If S is a space we can view it as an ∞-category; its objects are also called the points of S.
For an ∞-category C and two objects c0, c1 ∈ C, we let Maps C (c0, c1) ∈ Spc denote the mapping space between them.
For an object c ∈ C we let C c/ (resp., C /c ) denote the corresponding under-category (resp., overcategory).
In several places in the paper we will need the notion of left (resp., right) Kan extension. Let The notion of right Kan extension is obtained similarly: it is the right adjoint of (0.7); the formula for it is given by lim We let DGCat denote the ∞-category of DG categories over E, see [GR2, Chapter 1, Sect. 10]. We assume all our DG categories to be cocomplete and we allow only colimit-preserving functors as 1-morphisms. 0.5.6. Prestacks. By definition, a prestack is a functor (Sch aff ) op → Spc .
There will be two types of prestacks in this paper: the "source" type and the "target" type. The source type will be various geometric objects associated to the group G and the curve X, such as the Ran space, affine Grassmannian GrG, the loop group L(G), etc. These prestacks have the feature that the corresponding functors on (Sch aff ) op take values in the full subcategory Sets ⊂ Spc .
There will be a few other source prestacks (such as BunG or quotients of GrG by groups acting on it) and they will have the feature that the corresponding functors on (Sch aff ) op take values in the full subcategory of Spc spanned by 1-groupoids (these are spaces S, for which for any choice of s : * → S, the homoropy groups πi(S, s) vanish for i > 1).
When we talk about the category of sheaves on a prestack, the prestack in question will be typically of the source type.
The target prestacks will be of the form B n (A) (see Sect. 1.3.7 below), where A is a prestack that takes a constant value A, where A is a discrete abelian group (or its sheafification in, say, theétale topology, denoted B n et (A)). Such prestacks take values in n-truncated spaces and they form a (n + 1)category. When n is small, they can be described in a hands-on way by specifying objects, 1-morphisms, 2-morphisms, etc; in this paper n will be ≤ 4, and in most cases ≤ 2.
For example, we will often use the notion of a multiplicative A-gerbe on a group-prestack H. Such an object is the same as a map of group-prestacks 0.6. Acknowledgements. The first author like to thank J. Lurie for many illuminating discussions related to factorization gerbes.
We would like to thank A. Beilinson, D. Clausen and M. Groechenig for helping us with K-theory.
We would like to thank the referee for some very helpful comments.

Preliminaries
This section is included for the reader's convenience: we review some constructions in algebraic geometry that involve higher category theory. The reader having a basic familiarity with this material should feel free to skip it.
1.1. The sheaf-theoretic context. Although most of this paper is devoted to the study of gerbes, we need to discuss categories of sheaves on various geometric objects because they appear in the formulation of metaplectic geometric Satake. where DGCat is the ∞-category of (presentable) DG categories over a fixed field of coeffiecients E, assumed algebraically closed and of characteristic 0. We stipulate that where Vect is the DG category of vector spaces over E. We also require that the functor (1.1) satifieś etale descent.
The right-lax symmetric monoidal structure on Shv amounts to a compatible collection of functors : Shv(S1) ⊗ Shv(S2) → Shv(S1 × S2). (a) For any ground field k let E = Q , where is assumed to be invertible in k. First, for a finite extension E of Q we consider the category Shv(S, E ) equal to the ind-completion of the category of constructibleétale sheaves on S with E -coefficients, see [GL1, Sect. 2.3]. We take Shv(S) to be the colimit of these categories over E ⊂ E.
(b) When the ground field is C, then for an arbitrary algebraically closed field E of characteristic 0, we can take Shv(S) to be the ind-completion of the category of constructible sheaves 5 on S with E-coefficients.
(c) When the ground field k has characteristic 0, and E = k, we take Shv(S) to be the category of D-modules on S. Thus, if Y ∈ PreStk lft is written as

1.2.
What about non-finite type? At a certain point in this paper we will encounter the loop group G((t)), along with its various subgroups N ((t)), G [[t]]. In order to extend our sheaf theory to these objects, we proceed as follows. We define the functor Shv : (PreStk) op → DGCat to be the right Kan extension of (1.2) along the Yoneda embedding Sch aff → PreStk.
1.3. Digression: some higher algebra. To facilitate the reader's task, in this subsection we will review some notions from higher algebra that will be used in this paper. The main reference for this material is [Lu2].
1.3.1. Monoids and groups. In any ∞-category C that contains finite products (including the empty finite product, i.e., a final object), it makes sense to consider the category Monoid(C) of monoid-objects in C. This is a full subcategory in the category of simplicial objects of C (i.e., Funct(∆ op , C)) that consists of objects, satisfying the Segal condition. Similarly, one defines the category commutative monoids ComMonoid(C) in C.
For example, take C = ∞ -Cat. In this way we obtain the notion of monoidal (resp., symmetric monoidal) category.
1.3.2. If c is an object of an ∞-category C, then Maps C (c, c) has a natural structure of monoid in Spc.
For H ∈ Monoid(Spc), an action of H on c is by definition a homomorphism H → Maps C (c, c) of monoid objects in Spc.
Let Ptd(C) be the category of pointed objects in C, i.e., C * / , where * denotes the final object in C. We have the loop functor For an object S ∈ Ptd(Spc), its i-th homotopy group πi(S) is defined to be where Ω i (S) is viewed as a mere object of Spc. For C = Spc, this coincides with the notion from Sect. 1.3.2.
1.3.6. For k ≥ 0, we introduce the category E k (C) of E k -objects in C inductively, by setting the full subcategory of group-like objects, defined to be the preimage of We have a pair of mutually adjoint functors For i ≤ k we let B i denote the resulting functor 1.3.7. One shows that the forgetful functor is an equivalence.
This implies that for every k we have a canonically defined functor and these functors are compatible with the forgetful functors E k (C) → E k−1 (C). Thus, we obtain a canonically defined functor It is known (see [Lu2, Remark 5.2.6.26]) that the functor (1.3) is an equivalence.
The category identifies with that of connective spectra.
For any i ≥ 0, we have the mutually adjoint endo-functors By an action of A on an ∞-category C we shall mean an action of B(A) on C as an object of ∞ -Cat.
For example, taking A = E × ∈ ComGrp(Spc), we obtain an action of E × on any DG category. Explicitly, we identify B(E × ) with the space of E × -torsors, i.e., lines, and the action in question sends a line to the endofunctor c → ⊗ c.
1.4.1. Let Y be a prestack, and let A be a group-like En-object in the category PreStk /Y , for n ≥ 1. In other words, for a given (S {y} is a group-like En-object of Spc, in a way functorial in (S, y).
We include the case of n = ∞, when we stipulate that A is a commutative group-object of PreStk /Y . I.e., (1.4) should be a commutative group-object of Spc, i.e., a connective spectrum.
Chapter 2, Sect. 2.3]). We will be interested in spaces of the form Note that (1.5) is naturally a group-like En−i-space (resp., a commutative group object in Spc if n = ∞).

1.4.3.
In most examples, we will take A to be of the form A × Y, where A is a torsion abelian group, considered as a constant prestack. In this case In other words, it is the cohomology of the object Note also that in this case the functor , (Sch aff ) op → Spc identifies with the left Kan extension of its restriction to (Sch aff ft ) op . I.e., if an affine scheme S is written as a (filtered) limit is an isomorphism 7 .
1.4.4. For k = 1, the points of the space are by definition A-torsors on Y.
1.4.5. Our primary interest is the cases of k = 2. We will call objects of the space When A is of the form A × Y (see Sect. 1.4.3 above), we will simply write GeA(Y).
1.5. Gerbes coming from line bundles. In this subsection we will be studying gerbes for a constant commutative group-prestack, corresponding to a torsion abelian group A.
Here µn is the group of n-th roots of unity in k, where the integer n is assumed invertible in k. The above colimit is taken with respect to the maps µ n x →x n n µn, for n | n .
For future reference, denote also where colimit is taken with respect to the maps µn → µ n , for n | n .
7 The latter assertion means that B i et (A) is locally of finite type as a prestack.
1.5.2. We claim that to any line bundle L on a prestack Y and an element a ∈ A(−1) one can canonically associate an A-gerbe, denoted L a , over Y.
It suffices to perform this construction for A = µn and a coming from the identity map µn → µn. In this case, the corresponding µn-gerbe will be denoted L 1 n .
By definition, for an affine test scheme S over Y, the value of L 1 n on S is the groupoid of pairs where L is a line bundle on S.
Note that if L admits an n-th root L , then this L determines a trivialization of L 1 n .
Remark 1.5.3. We emphasize the notational difference between the µn-gerbe L 1 n , and the line bundle L ⊗ 1 n , when the latter happens to exist. Namely, a choice of L ⊗ 1 n defines a trivialization of the gerbe L 1 n .
1.5.4. Let Y be a smooth scheme, and let Z ⊂ Y be a subvariety of codimension one. Let Zi, i ∈ I denote the irreducible components of Z. For every i, let O(Zi) denote the corresponding line bundle on Y , trivialized away from Z.
We obtain a homomorphism Lemma 1.5.5. Assuming that the orders of elements in A are prime to char(k), i.e., that A has no p-torsion, where p = char(k). Then the map (1.8) is an isomorphism in Spc.
Proof. The assertion follows from the fact that theétale cohomology group H i et,Z (Y, A) identifies with Maps(I, A(−1)) for i = 2 and vanishes for i = 1, 0.
1.6. Presheaves of categories. (Pre)sheaves of categories appear in this paper as a language in which we formulate the metaplectic geometric Satake functor. The reader can skip this subsection on the first pass, and return to it when necessary.
The discussion in this section is essentially borrowed from [Ga1, Sect. 1.1].
1.6.1. Note that the diagonal morphism for affine schemes defines on every object of (Sch aff ) op a canonical structure of commutative algebra.
1.6.2. By a presheaf of DG categories C over Y ∈ PreStk we will mean a functorial assignment where Shv(S)-mod denotes the category of modules in the (symmetric) monoidal category DGCat for the (commutative) algebra object Shv(S).
A basic example of a sheaf of categories is Shv /Y , defined by setting Shv /Y (S, y) := Shv(S).
1.6.3. An example. Let Z be a prestack over Y. We define a presheaf of categories Shv 1.6.4. Part of the data of a presheaf of DG categories is a compatibility of actions for morphisms between affine schemes: must intertwine the action of Shv(S2) on C(S2, y2) with the action of Shv(S1) on C(S1, y1) via the monoidal functor f ! : Shv(S2) → Shv(S1).
1.6.5. We will say that a presheaf of DG categories is quasi-coherent if the functors (1.10) are equivalences for all f : S1 → S2.
Typically, presheaves of categories of the form of Sect. 1.6.3 are not quasi-coherent, even if the morphism Z → Y is schematic. This is because in the context of -adic sheaves, for a pair of affine schemes S1 and S2, the functor Shv(S1) ⊗ Shv(S2) → Shv(S1 × S2) is fully faithful, but not an equivalence (however, it is an equivalence in the context of D-modules).
1.6.6. Forgetting the module structure, a presheaf of DG categories C over Y defines a functor We shall say that C is a sheaf if it satisfiesétale descent, i.e., if the functor (1.11) satisfiesétale descent.
For example, presheaves of categories arising as in Sect. 1.6.3 are sheaves of categories.
1.6.7. Applying to the functor (1.11) the procedure of right Kan extension along we obtain that for every prestack Z over Y there is a well-defined DG category C(Z).
We will refer to C(Z) as the "category of sections of C over Z". By construction the DG category C(Z) is naturally an object of Shv(Z)-mod.
1.6.8. When Z is Y itself, we will refer to C(Y) as the "category of global sections of C".
For C as in Sect. 1.6.3, we have C(Y) Shv(Z).
1.7. Some twisting constructions. The material in this subsection may not have proper references in the literature, so we provide some details. The reader is advised to skip it and return to it when necessary.
1.7.1. Twisting by a torsor. Let Y be a prestack, and let H (resp., F) a group-like object in PreStk /Y (resp., an object in PreStk /Y , equipped with an action of H). In other words, these are functorial assignments and an action of H(S, y) on F(S, y).
Let T be an H-torsor on Y. In this case, we can form a T-twist of F, denoted F T , and which is ań etale sheaf. Here is the construction 8 : Consider the category Split(T) formed by (S, y) ∈ (Sch aff ) /Y equipped with a datum of a lift of the The forgetful functor Split(T) → (Sch aff ) /Y forms a basis of theétale topology, so it is sufficient to specify the restriction of F T to Split(T).
We interpret the data of the action of H on F as an object F ∈ PreStk /B(H) (see Sect. 1.3.5). The sought-for functor F T | Split(T) is given by sending (S, y, z) to * × where * ∈ B(H)(S, y) corresponds to the given map z.

1.7.2.
A twist of a presheaf of categories by a gerbe. Let now C be a presheaf of DG categories over Y, and let A be a group-like E2-object in PreStk /Y .
Let us be given an action of A on C. In other words, we are given a functorial assignment for every (S, y) ∈ (Sch aff ) /Y of an action of A(S, y) on C(S, y), see Sect. 1.3.8. In particular, we obtain that the prestack of groups B(A) acts on C.
Let G be a A-gerbe on Y, i.e., a B(A)-torsor. Repeating the construction of Sect. 1.7.1, we obtain that we can form the twist C G of C by G, which is a sheaf of DG categories over Y.
Explicitly, for every (S y → Y) ∈ (Sch aff ) /Y and a trivialization of G|S we have an identification The effect of change of trivialization by a point a ∈ A(S, y) has the effect of action of a ∈ End(Id C(S,y) ). 1.7.3. Let E ×,tors denote the subgroup of elements of E × that have a finite order prime to char(k).
Let us take A to be the constant group-prestack Y×E ×,tors . In this case, the tautological embedding E ×,tors → E × gives rise to an action of A on any presheaf of DG categories.
Thus, for every G ∈ Ge E ×,tors (Y) and any presheaf of categories C over Y, we can form its twisted version C G . 1.7.4. The category of sheaves twisted by a gerbe. We apply the above construction to C := Shv /Y . Thus, for any (S, y) ∈ (Sch aff ) /Y we have the twisted version of the category Shv(S), denoted Shv G (S).
As in Sect. 1.6.7, the procedure of right Kan extension defines the category Shv G (Z) for any Z ∈ PreStk /Y . 1.8. Character sheaves and twisted equivariance. 8 Note that when T is the trivial torsor, the output of this construction is theétale sheafification of F. 1.8.1. Recall that for S ∈ Sch aff , the category Shv(S) has a natural symmetric monoidal structure.
By a graded local system on S we will mean an object of Shv(S) that is invertible in the sense of the above symmetric monoidal structure. By a local system on S we will mean a graded local system all of whose fibers are lines in cohomological degree 0.
(Graded) local systems on S form a Picard category, i.e., a symmetric monoidal category in which every object is invertible.
1.8.2. Let LS denote the group object of PreStk that assigns to S ∈ Sch aff the Picard category of 1-dimensional local systems (within Shv(S)).
Let H be another group-object of PreStk. By a character sheaf on H we will mean a map of group prestacks H → LS.
Let Y be a prestack acted on by H, which we interpret as a prestack Y, equipped with a map to B(H). Given a chracter sheaf χ on H, we can thus view Y as equipped with a map to B(LS), i.e., with an LS-torsor, denoted Tχ.
Note that the presheaf of categories Shv / Y is naturally acted on by LS. Applying a variant of the twisting construction of Sect. 1.7.1, we obtain a twist of this sheaf of categories by the above LS-torsor Tχ.
In particular, for (S, y) ∈ (Sch aff ) / Y , we obtain a well-defined category Shv Tχ (S). By applying the procedure of right Kan extension, we obtain a well-defined category Shv Tχ (Z) for any Z ∈ PreStk / Y , and in particular for Z = Y.
1.8.3. The above construction and one in Sect. 1.7.2 are interrelated. Namely, note that we have a tautological map of group prestacks In particular, there exists a tautological character sheaf χtaut over B(E ×,tors ).
Given a prestack Y and a E ×,tors -gerbe G over Y , we can form the prestack equipped with an action of Bet(E ×,tors ); this is the "total space" of G; We have 1.8.4. An example. Let n be an integer invertible in k and let a be an element of order n in (E ×,tors )(−1), see Sect. 1.5.1.
Using the element a we obtain a homomorphism Gm → Bet(E ×,tors ).
Let χa := χtaut| Gm denote the corresponding character sheaf (known as the Kummer sheaf) on Gm.
Let L be a line bundle over a prestack Y. Let L a be the corresponding E ×,tors -gerbe over Y. Then the category Shv L a (Y) can be explicitly described as follows: where L − {0} is the total space of L with zero-section removed, viewed as a Gm-torsor over Y.
1.9.1. Suppose for a moment that our ground field k is C, and our sheaf theory is that of constructible sheaves with E-coefficients. When working with schemes of finite type, instead of considering the group E ×,tors and gerbes locally trivial in theétale topology, one can consider E × -gerbes locally trivial in the analytic topology.
For a prestack Y, we denote the corresponding 2-groupoid of E × -gerbes by Ge E × (Y).
Given a prestack Y and G ∈ Ge E × (Y), we have a well-defined functor If L is a line bundle on a prestack Y and a is an element of E × , we let L a denote the corresponding E × -gerbe on Y.
The assertion of Lemma 1.5.5 holds mutatis mutandis. The rest of the theory is unchanged.
1.9.2. For a finite type scheme S we have a canonical map At the level of π0, the image of this map consists of torsion elements in H 2 an (S, E × ).
Note, however, that the map (1.12) is not fully faithful: at the level of π1 it corresponds to the map whose image consists of torsion elements. In other words, automorphisms of a given E × -gerbe is the Picard category of all E × -torsors (i.e., 1-dimensional local systems with coefficients in E), and for a E ×,tors -gerbe we allow those local systems that become trivial when raised to some power.
Note that in this case we can identify E ×,tors (−1) with E ×,tors itself; this is because the fundamental group of Gm is identified with Z via the exponential map.
1.9.3. Let now k be an arbitrary field of characteristic 0, and let our sheaf theory be that of Dmodules, so that E = k. Recall that for a scheme S of finite type, the category Shv(S) = D-mod(S) is by definition where S dR is the de Rham prestack of S.
In this case, the counterpart of the notion of E × -gerbe from Sect. 1.9.1 is the notion of O × -gerbe on S dR .
For a prestack Y we denote the corresponding 2-groupoid by Given a prestack Y and G ∈ Ge O × (Y dR ), we have a well-defined functor If L is a line bundle on a prestack Y and a is an element of k/Z, the construction of [GR1, Example 6.4.6] defines an object L a ∈ Ge O × (Y dR ). The assertion of Lemma 1.5.5 holds mutatis mutandis. The rest of the theory is unchanged.
1.9.4. For a finite type scheme S we have a canonical map It has the same properties as the map (1.12).
Note that in this case, k ×,tors (−1) identifies with Q/Z, which we regard as a subgroup in k/Z.
1.9.5. In addition to O × -gerbes on Y dR for a scheme Y, one can consider the notion of twisting on Y in the sense of [GR1, Sect. 6]. By definition, this is a O × -gerbe on Y dR , equipped with a trivialization of its pullback to Y. We denote the space of twistings on Y by Tw(Y).
Let L be again a line bundle on Y, and let κ be an element of k. To this data the construction of [GR1, Sect. 6] attaches an object L κ ∈ Tw(Y). The image of L κ under the tautological projection is L a , where a is the image of κ under k → k/Z. 1.9.6. In what follows we will stay in the context ofétale sheaves and gerbes, leaving it to the reader to make appropriate modifications for the other sheaf-theoretic contexts.

Factorization gerbes on the affine Grassmannian
In this section we introduce our main object of study: factorization gerbes on the affine Grassmannian, which we stipulate to be the parameters for the metaplectic Langlands theory.
2.1. The Ran space. The Ran space of a curve X is an algebro-geometric device (first suggested in [BD1]) that allows us to talk about factorization structures relative to our curve.
2.1.1. Let X be a fixed smooth algebraic curve. We let Ran ∈ PreStk be the Ran space of X. By definition, for an affine test scheme S, the space Maps(S, Ran) is discrete (i.e., is a set), and equals the set of finite non-empty subsets of the (set) Maps(S, X). where I runs through the category opposite to that of non-empty finite sets and surjective maps 9 . For a surjection φ : I1 → I2, the corresponding map X I 2 → X I 1 is the corresponding diagonal morphism, denoted ∆ φ . An S-point of Ran J , given by Ij ⊂ Maps(S, X), j ∈ J belongs to Ran J disj if for every j1 = j2 and i1 ∈ Ij 1 , i2 ∈ Ij 2 , the corresponding two maps S ⇒ X have non-intersecting images.
2.2. Factorization patterns over the Ran space. Let Z be a prestack over Ran. At the level of k-points, a factorization structure on Z is the following system of isomorphisms: For a k-point x of Ran corresponding a finite set x1, ..., xn of k-points of X, the fiber Zx of Z over the above point is supposed to be identified with where {xi} are the corresponding singleton points of Ran.
We will now spell this idea, and some related notions, more precisely.
2.2.1. By a factorization structure on Z we shall mean an assignment for any finite set J of an isomorphism where the morphism Ran J → Ran is given by (2.1).
We require the isomorphisms (2.2) to be compatible with surjections of finite sets in the sense that for I φ J the diagram 9 We note that this category is not filtered, and hence Ran is not an ind-scheme. (2.3) where Ij := φ −1 (j), is required to commute. Furthermore, if Z takes values in ∞-groupoids (rather than sets), we require a homotopy-coherent system of compatibilities for higher order compositions, see [Ras1, Sect. 6].
2.2.2. Let C be a presheaf of DG categories over Ran. By a pre-factorization structure on C we shall mean a functorial assignment for any finite set J and an S-point of Ran We require the functors (2.4) to be compatible with surjections J1 J2 via the commutative diagrams analogous to (2.3). A precise formulation of these compatibilities is given in [Ras1, Sect. 6].
We will say that prefactorization structure on C is a factorization structure if the functors (2.4) are equivalences.
2.2.3. For example, let Z be a factorization prestack over Ran. Then the presheaf of categories Shv(Z) / Ran , given by has a natural prefactorization structure.
Typically, this prefactorization structure is not a factorization structure, for the same reason as one given in Sect. 1.6.5.
2.2.4. Let Z be a factorization prestack over Ran, and let A be a torsion abelian group. Let G be an A-gerbe on Z. By a factorization structure on G we shall mean a system of identifications where the underlying spaces are identified via (2.2).
The identifications (2.5) are required to be compatible with surjections J1 J2 via the commutative diagrams (2.3). Note that since gerbes form a 2-groupoid, we only need to specify the datum of (2.5) up to |J| = 3, and check the relations up to |J| = 4. Factorization gerbes over Z naturally form a space (in fact, a 2-groupoid), equipped with a structure of commutative group in Spc (i.e., connective spectrum), to be denoted FactGeA(Z).
Remark 2.2.5. Note that the diagrams (2.2) include those corresponding to automorphisms of finite sets. I.e., the datum of factorization gerbe includes equivariance with respect to the action of the symmetric group. For this reason what we call "factorization gerbe" in [Re] was called "symmetric factorizable gerbe".
2.2.6. Let Z be a factorization prestack over Ran, and let G be a factorization E ×,tors -gerbe over it. Then the presheaf of categories Shv G (Z) / Ran defined by is a sheaf of categories, and has a natural prefactorization structure.
2.2.7. By a similar token, we can consider factorization line bundles over factorization prestacks, and also Z-or Z/2Z-graded line bundles 10 .
If L is a (usual, i.e., not graded) factorization line bundle and a ∈ A(−1), we obtain a factorization gerbe L a .
2.3. The Ran version of the affine Grassmannian. In this subsection we introduce the Ran version of the affine Grassmannian, which plays a crucial role in the geometric Langlands theory.
2.3.1. For an algebraic group G, we define the Ran version of the affine Grassmannian of G, denoted GrG, to be the following prestack.
For an affine test scheme S, the groupoid (in fact, set) Maps(S, GrG) consists of triples where I is an S-point of Ran, PG is a G-bundle on S × X, and α is a trivialization of PG over the open subset UI ⊂ S × X equal to the complement of the union of the graphs of the maps S → X corresponding to the elements of I ⊂ Maps(S, X).
2.3.2. The basic feature of the prestack GrG is that it admits a natural factorization structure over Ran, obtained by gluing bundles.
Hence, for a torsion abelian group A, it makes sense to talk about factorization A-gerbes over GrG. We denote the the resulting space (i.e., in fact, a connective 2-truncated spectrum) by FactGeA(GrG). This example is important because there is a canonical factorization line bundle on GrG, denoted detG; we will encounter it in Sect. 5.2.1.

Assume for a moment that X is proper.
Let BunG denote the moduli stack of G-bundles on X. Note that we have a tautological projection (2.6) GrG → BunG .
Recall now that [GL2, Theorem 3.2.13] says 11 that the map (2.6) is a universal homological equivalence. This implies that any gerbe on GrG uniquely descends to a gerbe on BunG.
In particular, this is the case for factorization gerbes.
2.4. The space of geometric metaplectic data.
2.4.1. We stipulate that the space is the space of parameters for the metaplectic Langlands theory. We also refer to it as geometric metaplectic datum.
This includes both the global case (when X is complete), and the local case when we take X to be a Zariski neighborhood of some point x.

2.4.2.
Given an E ×,tors -factorization gerbe G on GrG, we can thus talk about the prefactorization sheaf of categories, denoted Shv G (GrG) / Ran , whose value on S, I ⊂ Maps(S, X) is Shv G (S × Ran GrG).

Parameterization of factorization gerbes
From now on we let A be a torsion abelian group whose elements have orders prime to char(k). The main example is A = E ×,tors .
The goal of this section is to describe the set of isomorphism classes (and, more ambitiously, the space) of A-factorization gerbes on GrG in terms of more concise algebro-geometric objects.
3.1. Parameterization viaétale cohomology. In this subsection we will create a space, provided by the theory ofétale cohomology, that maps to the space FactGe E ×,tors (GrG), thereby giving a parameterization of geometric metaplectic data.
3.1.1. Let Bet(G) := pt /G be the stack of G-torsors. I.e., this is the sheafification in theétale topology of the prestack B(G) that attaches to an affine test scheme S the groupoid * / Maps(S, G).

Consider the space of maps
Maps(Bet(G) × X, B 4 et (A(1))), which is the same as Maps(B(G) × X, B 4 et (A(1))). We claim that there is a naturally defined map For an affine test scheme S and an S-point (I, PG, α) of GrG, we need to construct a A-gerbe GI on S.
Moreover, for φ : I J, such that the point hits Ran J disj , we need to be given an identification We claim that such a datum indeed gives rise to a A-gerbe GI on S, equipped with identifications (3.3).

First off, since
H i et (S × X, A(1)) and H i−1 et (UI , A(1)) for i = 3 and i = 4 vanishétale-locally on S, we obtain that the prestack that sends S to the space of maps (3.4), equipped with a trivialization of (3.5), identifies with B 2 et of the prestack that sends S to the space of maps (3.6) S × X → B 2 et (A(1)), equipped with a trivialization of (3.7) UI → B 2 et (A(1)).
3.1.6. Thus, given a map (3.6), equipped with a trivialization of (3.7), we need to construct a locally constant map S → A whose dependence on (3.6) and the trivialization of (3.7) respects the structure of commutative group on A(1).
Let ΓI denote the complement of UI (the scheme structure on ΓI is irrelevant). We need to construct the trace map where for a scheme Y , we denote by AY the constantétale sheaf on Y with value A.
3.1.7. In its turn, the map (3.9) is obtained by the (π * , π ! )-adjunction from the isomorphism where the latter comes from the identification where pX : X → pt is the projection.

3.2.
Analysis of homotopy groups of the space of factorization gerbes.
3.2.1. We have the following assertion that results from [Re, Theorem II.7.3] and the computation of the homotopy groups of the left-hand side of (3.1) (the latter is given below): Proposition 3.2.2. The map (3.1) is an isomorphism.
Remark 3.2.3. As was explained to us by J. Lurie, the assertion of Proposition 3.2.2 is nearly tautological if one works over the field of complex numbers and in the context of sheaves in the analytic topology.
3.2.4. From Proposition 3.2.2 we will obtain the following more explicit parameterization of the 2groupoid FactGeA(GrG).
Namely, πi(FactGeA(GrG)) = H 4−i et (B(G) × X; pt ×X, A(1)). Let us analyze what these cohomology groups look like. For the duration of this subsection we will assume that A is divisible, unless G is a torus.

We have:
Remark 3.2.7. We note that the natural map is injective, but in general it is not surjective.

For a given
FactGe q A (GrG) denote the fiber of the map In particular, we can consider the commutative group in Spc In Corollary 4.4.7 we will construct a canonical isomorphism: (3.10) FactGe 0 A (GrG) Maps(X, B 2 et (Hom(π 1,alg (G), A))).

Parametrization of factorization line bundles.
This subsection is included for the sake of completeness, in order to make contact with the theory of metaplectic extensions developed in [We].
Recall from Sect. 2.3.3 that given a factorization line bundle L on GrG and an element a ∈ A(−1) we can produce a factorization gerbe L a . In this subsection we will describe a geometric data that gives rise to factorization line bundles 12 on GrG.
3.3.1. Let K2 denote the prestack over X that associates to an affine scheme S = Spec(A) mapping to X the abelian group K2(A). Let (K2)Zar be the sheafification of K2 in the Zariski topology.
On the one hand, we consider the space CExt(G, (K2)Zar) (in fact, an ordinary groupoid) of Brylinski-Deligne data, which are by definition central extensions The operation of Baer sum makes CExt(G, (K2)Zar) into a commutative group in spaces, i.e., into a Picard category.
On the other hand, consider the Picard category Given such a map, for an affine scheme S and an S-point (I, PG, α) of GrG, we need to construct a line bundle LI on S. By [DrSi, Theorem 2], after passing to anétale cover of S, the G-bundle PG becomes Zariski locally trivial. Hence, we can assume that (I, PG, α) is a map equipped with a trivialization of the composition where UI is as in Sect. 2.3.1.

Composing (3.13) with (3.12) we obtain a map
(3.15) S × X → B 2 Zar (K2), equipped with a trivialization of the composition To this data we need to associate a line bundle LI on S.

Consider the exact triangle of categories
is the full subcategory spanned by objects set-theoretically supported on ΓI .
The long exact cohomology sequence gives rise to a map  3.4. Relationship between the two parameterizations. This subsection is also included for the sake of completeness; its contents will not be used in the sequel.
We will give a cohomological construction a map As was explained to us by D. Clausen, the map (3.22) cannot be lifted to a map of commutative group objects in PreStk 3.4.2. Let us consider B 2 et (µ ⊗2 n ) as a sheaf of commutative group objects in Spc on the big Zariski site. Let us consider its Postnikov truncation τ ≤1 (B 2 et (µ ⊗2 n )), again as a sheaf on the big Zariski site. We have the following fiber sequence As was explained to us by A. Beilinson, the map (3.22) does lift to a map of presheaves The map (3.24) gives rise to a map

Note now that we have a fiber sequence
, from which it follows that the induced map is an isomorphism.
Combining with (3.25) we obtain a map

The case of tori
In this section we let G = T be a torus. We will perform an explicit analysis of factorization gerbes on the affine Grassmannian GrT , and introduce two related objects (multiplicative and commutative factorization gerbes) that would play an important role in the sequel.

Factorization Grassmannian for a torus.
In this section we will show that the affine Grassmannian of a torus can be approximated by a prestack assembled from (=written as a colomit of) powers of X.
4.1.1. Recall that Λ denotes the coweight lattice of G = T . Consider the index category whose objects are pairs (I, λ I ), where I is a finite non-empty set and λ I is a map I → Λ; in what follows we will denote by λi ∈ Λ is the value of λ I on i ∈ I. λi.
Consider the prestack Gr T,comb := colim (I,λ I ) The prestack Gr T,comb endowed with its natural forgetful map to Ran, also has a natural factorization structure.
There is a canonical map (4.2) Gr T,comb → GrT , compatible with the factorization structures.
Namely, for each (I, λ I ) the corresponding T -bundle on X I × X is where ∆i is the divisor on X I × X corresponding to the i-th coordinate being equal to the last one. [Ga2,Sect. 8.1] one shows that the map (4.2) induces an isomorphism of the sheafifications in the topology generated by finite surjective maps. In particular, for any S → Ran, the map

As in
is an isomorphism, and hence, so is the map FactGeA(GrT ) → FactGeA(Gr T,comb ).
Furthermore, for a given G ∈ FactGe E ×,tors (GrT ), the corresponding map of sheaves of categories is also an isomorphism. 4.1.3. The datum of a factorization gerbe on Gr T,comb can be explicitly described as follows: For a finite set I and a map we specify a gerbe G λ I on X I .
For a surjection of finite sets I φ J such that (4.1) holds, we specify an identification The identifications (4.3) must be compatible with compositions of maps of finite sets in the natural sense.
Let now I φ J be a surjection of finite sets, and let be the corresponding open subset. For j ∈ J, let λ I j be the restriction of λ I to Ij.
We impose the structure of factorization that consists of isomorphisms The isomorphisms (4.4) must be compatible with compositions of maps of finite sets in the natural sense.
In addition, the isomorphisms (4.4) and (4.3) must be compatible in the natural sense. 4.1.4. For a factorization gerbe G on Gr T,comb , the value of the category Shv G (Gr T,comb ) / Ran on X I corresponding to a given λ I identifes with Shv G λ I (X I ).
This description implies that the sheaf of categories Shv G (GrT ) / Ran is quasi-coherent (see Sect. 1.6.5 for what this means), and that its prefactorization structure is actually a factorization structure.
Note that the corresponding facts would be false for a group G that is not a torus. We note that a factorization Z/2Z-graded line bundle is evenly (i.e., trivially) graded if and only if the corresponding θ-datum is even, i.e., if the corresponding symmetric bilinear Z-valued form on Λ comes from a Z-valued quadratic form.
We also note that [BD1, Proposition 3.10.7] says that restriction along Gr T,comb → GrT defines an equivalence between the Picard categories of factorization (Z/2Z-graded) line bundles.

4.2.
Making the parameterization explicit for tori. In this subsection we will show explicitly how a multiplicative A-gerbe on GrT gives rise to an A-valued quadratic form q : Λ → A(−1).

It is easy to see that the resulting map
is symmetric. The fact that it is bilinear form can be seen as follows. For a triple of elements λ1, λ2, λ3 consider the corresponding gerbes They are identified away from the main diagonal ∆1,2,3, and hence this identification extends to all of X 3 , since ∆1,2,3 has codimension 2. Restricting to ∆1,2, we obtain an identification as gerbes over X 2 . Comparing with the identification we obtain the desired b(λ1, λ3) + b(λ2, λ3) = b(λ1 + λ2, λ3).

Finally, let us recover the quadratic form
For a given λ ∈ Λ, consider the gerbes G λ,λ and G λ G λ on X 2 . They are both equipped with a structure of S2-equivariance, and they are identified as such over X 2 − ∆. In addition, the induced equivariance structure on both G λ,λ |∆ and (G λ G λ )∆ is the tautological one.
We note that the datum of a gerbe on X 2 , equipped with a structure of S2-equivariance, whose restriction to ∆ is the tautological equivariance structure is equivalent to the datum of a gerbe on X (2) , where the latter is the symmetric square of X. Hence, we obtain a well-define gerbe G λ (2) over X (2) , trivialized away from the diagonal, so that compatibly with the trivializations on X 2 − ∆.
By Lemma 1.5.5, G is canonically of the form O(∆ ) a for some a ∈ A(−1), and where ∆ denotes the diagonal in X (2) . Set a =: q(λ). By construction, compatibly with the trivializations on X 2 − ∆.
4.3. The notion of multiplicative/commutative factorization gerbe. In order to be able to state the metaplectic version of geometric Satake, we will need to discuss the notion of multiplicative/commutative factorization gerbe, first on GrT , and then when the lattice Λ = Hom(Gm, T ) is replaced by a general finitely generated abelian group. 4.3.1. Note that since T is commutative, GrT is naturally a (commutative) group-prestack over Ran. Hence, along with FactGeA(GrT ), we can consider the corresponding spaces (in fact, commutative groups in spaces) (4.5) FactGe mult A (GrT ) and FactGe com A (GrT ) that correspond to gerbes that respect the group (resp., commutative group structure) on GrT over Ran.
We are going to use Corollary 4.3.3 to describe the spaces FactGe mult A (GrT ) and FactGe com A (GrT ) more explicitly.

Note that the Kummer map
A × Gm → Bet(A(1)), which is a map of commutative group-prestacks, gives rise to a map

4.4.
More general abelian groups. In this section we generalize the discussion of Sect. 4.3 to the case when instead of a lattice Λ (thought of as a lattice of cocharacters of a torus) we take a general finitely generated abelian group.
We need this in order to state the metaplectic version of geometric Satake.
4.4.1. Let Γ be a finitely generated abelian group. We define the commutative group-prestack over Ran Gr Γ⊗Gm as follows. Write Γ as Λ1/Λ2, where Λ1 ⊃ Λ2 are lattices. Let T1 and T2 be the corresponding tori. We define Gr Γ⊗Gm as a quotient of GrT 1 by GrT 2 , viewed as commutative group-prestacks over Ran.
It is easy to see that this definition (as well as other constructions we are going to perform) is canonically independent of the presentation of Γ as a quotient.
The group-prestack Gr Γ⊗Gm has a natural factorization structure over Ran. 4.4.2. Let now G be a connective reductive group. Let Γ = π 1,alg (G). The description in Sect. 3.2.5 implies that there is a canonically defined map that correspond to gerbes that respect that group (resp., commutative group structure) on Gr Γ⊗Gm over Ran.
The following results from Proposition 4.3.7: Corollary 4.4.4. Let Γ be written as a quotient of two lattices as in Sect. 4.4.1. Let G1 be a factorization A-gerbe on GrT 1 , and let b1 and q1 be the associated bilinear and quadratic forms on Λ1, respectively.
Then the datum of descent of the gerbe G1 to a factorization gerbe G on Gr Γ⊗Gm exists only if the restriction of q1 to Λ2 is trivial, and in the latter case is equivalent to the trivialization of G2 := G1|Gr T 2 as a factorization gerbe on GrT 2 . Moreover: (a) The gerbe G admits a multiplicative structure if and only if b1 is trivial. In the latter case, the multiplicative structure is unique up to a unique isomorphism.
(b) The gerbe G admits a commutative multiplicative structure if and only if q1 is trivial. In the latter case, the commutative multiplicative structure is unique up to a unique isomorphism.
From Corollary 4.4.5 and the calculation of homotopy groups in Sect. 3.2.8 we obtain: Corollary 4.4.7. The map (4.12) is fully faithful and is an isomorphism into FactGe 0 A (GrG), thereby inducing an isomorphism Maps(X, B 2 et (Hom(π 1,alg (G), A))) FactGe 0 A (GrG). 4.5. Splitting multiplicative gerbes. In this subsection we will assume that char(k) = 2. We will need to perform one more manipulation: it turns out that the fiber sequence since the group µ2 is canonically ±1 = Z/2Z.
In order to define the sought-for splitting, by functoriality, it suffices to consider the case of Γ = Z/2Z, A = ±1 and we need to produce a multiplicative factorization gerbe on Gr Z/2Z⊗Gm that gives rise to the tautological map Z/2Z → ±1. 4.5.2. We will first construct the corresponding multiplicative factorization gerbe on Gr Gm , i.e., for Γ = Z. It will be clear from the construction that its pullback under the isogeny is canonically trivial. This will give rise to the sought-for gerbe for Γ = Z/2Z by Corollary 4.4.4(a). 4.5.3. In order to perform the construction we will choose a datum of a Z/2Z-graded factorization line bundle L on Gr Gm,Ran .
We require that the restriction of L to X ⊂ Ran be such that its further restriction to the connected component of Gr Gm,X := X × Ran Gr Gm , corresponding to 1 ∈ Z, is odd. An example of such an L is the determinant line bundle, corresponding to the tautological action of Gm on a 1-dimensional vector space.
We now consider the line bundle L ⊗2 , and the ±1-gerbe (L ⊗2 ) 1 2 (see Remark 1.5.3 for our notational convention). By unwinding the construction of the quadratic form in Sect. 4.2.3, it is easy to see that this factorization gerbe has the required property. 4.5.4. We now claim that the gerbe (L ⊗2 ) 1 2 is canonically independent of the choice of L. Indeed, let L1 and L2 be two different choices for L. We note that their ratio L := L1 ⊗ L ⊗−1 2 is a usual factorization line bundle (i.e., it is Z/2Z-graded, but the grading is even). So, the gerbe ( L ⊗2 ) 1 2 is canonically trivialized by means of the line bundle ( L ⊗2 ) ⊗ 1 2 = L.
Remark 4.5.5. We note that, by construction, the gerbe (L ⊗2 ) 1 2 admits a canonical trivialization. But this factorization is not compactible with the factorization structure. 4.5.6. In what follows, for a given element ∈ Hom(Γ, A(−1))2 -tors, we will denote by G the resulting multiplicative factorization gerbe on Gr Γ⊗Gm . 4.5.7. For a given object G ∈ FactGe mult A (Gr Γ⊗Gm ) let us denote by the map Γ → A(−1)2 -tors that measures the obstruction of G to belong to FactGe com A (Gr Γ⊗Gm ). We obtain that, canonically attached to G, there exists an object where G is as in Sect. 4.5.6.

Jacquet functors for factorization gerbes
In this section we take G to be reductive. We will study the interaction between factorization gerbes on GrG and those on GrM , where M is the Levi quotient of a parabolic of G.
5.1. The naive Jacquet functor. Let P be a parabolic subgroup of G, and we let P M be its Levi quotient. Let NP denote the unipotent radical of P .

Consider the diagram of the Grassmannians
We claim that pullback along q defines an equivalence, Thus, if G G is a factorization A-gerbe on GrG, and G M is the corresponding the factorization A-gerbe on GrM , the corresponding quadratic forms q : Λ → A(−1) coincide.
5.1.4. We now take A := E ×,tors . Given a factorization E ×,tors -gerbe G G over GrG, consider its pullback to GrP , denoted G P . We let G M denote the canonically defined factorization gerbe on GrM , whose pullback to GrP gives G P .
By construction, for any S → Ran, we have a well-defined pullback functor Furthermore, since the morphism q is ind-schematic, we have a well-defined push-forward functor Thus, the composite q * • p ! defines a map between prefactorization sheaves of categories We will refer to (5.2) as the naive Jacquet functor.

5.2.
The critical twist. The functor (5.2) is not quite what we need for the purposes of geometric Satake. Namely, we will need to correct this functor by a cohomological shift that depends on the connected component of GrM (this is needed in order to arrange that the corresponding functor on the spherical categories maps perverse sheaves to perverse sheaves). However, this cohomological shift will destroy the compatibility of the Jacquet functor with factorization, due to sign rules. In order to compensate for this, we will apply an additional twist of our categories by the square root of the determinant line bundle.
The nature of this additional twist will be explained in the present subsection.
For the rest of this subsection we will assume that char(k) = 2.
5.2.1. Let detG denote the determinant line bundle on GrG, corresponding to the adjoint representation. It is constructed as follows. For an affine test scheme S and an S-point I ⊂ Maps(S, X) of Ran, consider the corresponding G-bundle PG on S × X, equipped with an isomorphism α : PG P 0 G over UI ⊂ S × X. Consider the corresponding vector bundles associated with the adjoint representation is a well-defined line bundle 13 on S.
This construction is compatible with pullbacks under S → S, thereby giving rise to the sought-for line bundle detG on GrG.
It is easy to see that detG is equipped with a factorization structure over Ran. M |Gr P . However, this identification will be compatible with the factorization structures only up to a sign.

Taking
In fact, we claim that the ratio of the line bundles detG |Gr P and detM |Gr P admits a square root, to be denoted det Letting L be the pullback of ω ⊗ 1 2 X , we thus need to construct an isomorphism . However, this follows from the (relative to S) local Serre duality on S × X: 5.3. The corrected Jacquet functor. We will now use the square root gerbe det 5.3.2. Given a factorization E ×,tors -gerbe G G on GrG and the corresponding factorization gerbe G M on GrM (see Sect. 5.1.4), we will now define the corrected Jacquet functor as a map between prefactorization sheaves of categories: Namely, for an affine test scheme S and an S-point of Ran, the corresponding functor is the composition of the following four operations: (i) The pullback functor given by the isomorphism of gerbes (5.4); (iii) The pushforward functor q * : Shv

The metaplectic Langlands dual datum
In section we take G to be reductive. Given a factorization gerbe G on GrG, we will define the metaplectic Langlands dual datum attached to G, and the corresponding notion of twisted local system on X.
6.1. The metaplectic Langlands dual root datum. The first component of the metaplectic Langlands dual datum is purely combinatorial and consists of a certain root datum that only depends on the root datum of G and q. This is essentially the same as the root datum defined by G. Lusztig as a recipient of the quantum Frobenius.
6.1.1. Given a factorization A-gerbe G G on GrG, let be the associated quadratic and bilinear forms, respectively. Let Λ ⊂ Λ be the kernel of b.
6.1.2. We let ∆ be equal to ∆ as an abstract set. For each element α ∈ ∆, we let the corresponding element α ∈ ∆ be equal to ord(q(α)) · α ∈ Λ, and the corresponding elementα ∈∆ be The fact that q ∈ Quad(Λ, A(−1)) lies in the image of the map (see Remark 3.2.7) implies that α andα defined in this way indeed belong to Λ ⊂ Λ andΛ ⊂Λ ⊗ Z Q, respectively. 6.1.3. Since q was W -invariant, the action of W on Λ preserves Λ . Moreover, for each α ∈ ∆, the action of the corresponding reflection sα ∈ W on Λ equals that of s α . This implies that restriction defines an isomorphism from W to the group W of automorphisms of Λ generated by the elements s α .
Hence, (6.1) is a finite root system with Weyl group W , isomorphic to the original Weyl group W .
It follows from the constriction that if αi are the simple coroots of ∆, then the corresponding elements α i ∈ Λ form a set of simple roots of ∆ .
6.1.4. We let G denote the reductive group (over k) corresponding to (6.1).
6.2. The "π1-gerbe". Let G G be as above. In this subsection we will show that in addition to the reductive group G , the datum of G G defines a certain multiplicative factorization gerbe on the affine Grassmannian corresponding to the abelian group π 1,alg (G ).
6.2.1. Let G T be the factorization gerbe on GrT , corresponding to G G via Sect. 5.1.4. Consider the corresponding torus T .
Let G T be the factorization gerbe on Gr T ,Ran equal to the pullback of G T under T → T . By Proposition 4.3.7(b), the gerbe G T carries a canonical multiplicative structure.
We claim that there exists a canonically defined multiplicative factorization A-gerbe G π 1,alg (G )⊗Gm on Gr π 1,alg (G )⊗Gm , whose pullback under (6.2) identifies with G T . 6.2.2. By Corollary 4.4.4, we need to show that for every simple coroot αi, the pullback of G T to Gr Gm under By the transitivity of the construction in Sect. 5.1.4, we can replace G by its Levi subgroup Mi of semi-simple rank 1, corresponding to αi. Furthermore, using the map SL2 → Mi, we can thus assume that G = SL2. 6.2.3. Note that by Sect. 3.2.8, any factorizable A-gerbe on GrSL 2 is canonically of the form (detSL 2 ) a for some element a ∈ A(−1).
Let us first calculate the resulting A-gerbe on Gr Gm , where we think of Gm as the Cartan subgroup of SL2.
For an integer k let det Gm,k denote the determinant line bundle on Gr Gm associated with the action of Gm on the one-dimensional vector space given by the k-th power of the tautological character. This a Z-graded factorization line bundle, and we note that the grading is even if k is even.
By Sect. 5.2.4, the factorization gerbe on Gr Gm , corresponding to (detSL 2 ) a is given by (det Gm,2 ) 2a . The associated quadratic form q : Z → A takes value 4a on the generator 1 ∈ Z. Let n := ord(4a).
We need to show that the pullback of (det Gm,2 ) 2a under the isogeny is canonically trivial as a factorization gerbe on Gr Gm,Ran .
6.2.4. Note that the pullback of det Gm,2 under the above isogeny is the factorization line bundle det Gm,2n . We need to provide a canonical trivialization of the factorization gerbe For that it is sufficient to show that the factorization line bundle det Gm,2n on Gr Gm admits a canonical 2n-th root. Thus, the line bundle (det Gm,1 ) ⊗2n ⊗ Fact(ω ⊗ 2n−1 2 X ) gives the desired 2n-th root.
6.2.6. Example. Suppose that G G is trivial, in which case T = T and G = G. In this case G π 1,alg (G )⊗Gm is also trivial.
we obtain a new sheaf of (symmetric monoidal) categories over X, denoted Let Rep(H) G Z (X) denote the (symmetric monoidal) category of its global sections (see Sect. 1.6.8). The category Rep(H) G Z (X) carries a naturally defined t-structure. 6.3.5. We now introduce the notion of twisted local system for the metaplectic Langlands dual datum, understood as a triple (6.4). Namely, this is by definition a symmetric monoidal t-exact functor In Sect. 8.5 we will formulate a precise relationship between twisted local systems in the above sense and objects appearing in the global metaplectic geometric theory.
Remark 6.3.6. Presumably, twisted local systems as defined above are the same as Galois representations into the metaplectic L-group, as defined in [We].
6.4. Digression: (pre)factorization categories arising from symmetric monoidal categories. In this subsection we will explain a procedure that produces prefactorization categories from symmetric monoidal categories. The source of the metaplectic geometric Satake functor will be a factorization category obtained in this way.
6.4.1. Let C be a symmetric monoidal DG category.
We define the sheaf of categories Fact(C) on Ran as follows. For an affine test scheme S and an S-point of Ran given by I ⊂ Maps(S, X), let Tw(I) be the category whose objects are pairs (6.5) I J K (here J and K are sets (automatically, finite and non-empty)), and whose morphisms are commutative diagrams (6.6) (Note that the arrows between the K's go in the opposite direction.) Consider the functor (6.7) Tw(I) → DGCat that sends an object (6.5) to Shv(S × and a morphism (6.6) to where the first arrow is direct image along and the second arrow is the functor C ⊗J → C ⊗J given by the symmetric monoidal structure on C.
Finally, we let the value of Fact(C) on (S, I) be the object of DGCat equal to the colimit of the functor (6.7) over Tw(I).
By construction, Fact(C) admits a prefactorization structure as a sheaf of symmetric monoidal DG categories.
Remark 6.4.2. One can show that Fact(C), regarded as a presheaf on Ran is in fact quasi-coherent, and that its prefactorization structure is actually a factorization structure. 6.4.3. Let Fact(C)(Ran) denote the category of global sections of Fact(C) over Ran.
As in [Ga5,Sect. 4.2], the (symmetric) monoidal structure on Fact(C) and the operation of union of finite sets makes Fact(C)(Ran) into a non-unital (symmetric) monoidal category.
6.4.4. Let A be a group acting by automorphisms of the symmetric monoidal structure of C.
Let GA be an A-gerbe over X. We can twist the construction of Fact(C) and consider the sheaf on Ran of symmetric monoidal DG categories Fact(C) G A . 6.4.5. Let now be a 2-torsion element of A. Using the gerbe G from Sect. 4.5.1, we can further twist Fact(C) G A to obtain a (pre)factorization sheaf of monoidal DG categories, denoted Fact(C) G A .
Note, however, that by Remark 4.5.5, we have a canonical identification as monoidal categories.
6.5. The (pre)factorization category of representations. 6.5.2. Example of tori. Let G T be a multiplicative factorization gerbe on GrT . In this case, is naturally a sheaf of monoidal DG categories on Ran, equipped with a (pre)factorization structure.
Note also that by Proposition 4.3.7(a), we have T = T , and so H Ť . It is straightforward to show explicitly (see [Re,Proposition IV.5.2]) that we have a canonical isomorphism (6.9) Fact(Rep(Ť )) G Z Shv G T (GrT ) / Ran as sheaves of (pre)factorization monoidal categories.
6.5.3. Unwinding the construction, we obtain that the category of sections of Fact(Rep(H)) G Z on X (with respect to the canonical map X → Ran(X)) is the category introduced in Sect. 6.3.4. 6.5.4. Let now σ be a twisted local system on X as defined in Sect. 6.3.5. The functoriality of the construction in Sect. 6.4 defines a (symmetric) monoidal functor Assume now that X is complete. Composing with the functor of direct image Shv(Ran) → Vect, we thus obtain a functor (6.10) Evσ : Fact(Rep(H)) G Z (Ran) → Vect .
We will use the functor (6.10) for the definition of the notion of twisted Hecke eigensheaf with respect to σ.
6.6. The (derived) stack of twisted local systems. In this subsection we will assume that char(k) = 0, and that our sheaf-theory is that of D-modules (in particular, the field of coefficients E equals k).
Assume that X is complete. Starting from the pair (H, GZ ) appearing in the triple (H, GZ , ) of the metaplectic dual datum, we will construct the derived stack LocSys G Z H of GZ -twisted local systems on X. Its k-points will be the twisted local systems as defined in Sect. 6.3.5. 6.6.1. We will follow the strategy of [ Proposition 6.6.3. The functor (6.11) is a localization, i.e., it admits a fully faithful right adjoint.

Factorization gerbes on loop groups
In this section we will perform a crucial geometric construction that will explain why our definition of geometric metaplectic datum was "the right thing to do": We will show that a factorization gerbe on GrG give rise to a (factorization) gerbe on (the factorization version of) the loop group of G.
7.1. Digression: factorization loop and arc spaces. 7.1.1. For an affine test scheme S and an S-point of Ran, given by a finite set I ⊂ Maps(S, X), letDI be the corresponding relative formal disc: By definition,DI is the formal scheme equal to the completion of S × X along the union of the graphs of the maps S → X corresponding to the elements of I.
Note that for a finite set J and a point {Ij, j ∈ J} ∈ Ran J disj , we have 7.1.2. Since S was assumed affine,DI is an ind-object in the category Sch aff . Let DI be the affine scheme corresponding to the formal schemeDI , i.e., the image ofDI under the functor colim : Ind(Sch aff ) → Sch aff .
In other words, ifD where Zα = Spec(Aα) and the colimit is taken in PreStk, then DI = Spec(A), where DI be the open subscheme of DI , obtained by removing the closed subscheme equal to the union of the graphs of the maps S → X corresponding to the elements of I. 7.1.3. Let Z be a prestack. We define the prestacks L + (Z) (resp., L(Z)) over Ran as follows.
For an affine test scheme S and an S-point of Ran, given by a finite set I ⊂ Maps(S, X), its lift to an S-point of L + (Z) (resp., L(Z)) is the datum of a map DI → Z (resp., The isomorphisms (7.1) imply that L + (Z) and L(Z) are naturally factorzation prestacks over Ran. 7.1.4. Assume for a moment that Z is an affine scheme. Note that in this case the definition of L + (Z), the datum of a map DI → Z is equivalent to that of a map of prestacksDI → Z.
Assume now that Z is a smooth scheme of finite type (but not necessarily affine). Then one shows that for every S → Ran, the fiber product is a projective limit (under smooth maps) of smooth affine schemes over S. induces a bijection on the corresponding pairs (PG, α). In the above formula, the notation UI is as in Sect. 2.3.1. stabilizer of the unit section being L + (G). Furthermore, the natural map (7.2) L(G)/L + (G) → GrG, is an isomorphism, where the quotient is understood in the sense of stacks in theétale topology.
The isomorphism (7.2) implies that for every S → Ran, the fiber product is an ind-scheme over S.
7.2.3. Recall that given a group-prestack H over a base Z, we can talk about a gerbe over H being multiplicative, i.e., compatible with the group-structure.
In particular, we can consider the spaces The following result is established in [Re, Theorem III.2.10]: Proposition 7.2.5. The map (7.3) is an isomorphism.
We will sketch the proof of this proposition in Sect. 7.5. It consists of explicitly constructing the inverse map. 7.2.6. Let us restate Proposition 7.2.5 in words. It says that, given a factorization gerbe on GrG, its pullback under the projection L(G) → GrG, carries a uniquely defined multiplicative structure that is compatible with that of factorization and the trivialization of the further restriction of our gerbe to L + (G). inverse to (7.3), consists of constructing a (canonical) structure of equivariance with respect to L + (G) on a given factorization gerbe G on GrG. We will explain this construction in the present subsection.
7.3.1. For a non-negative integer n, let Gr n G → Ran n be the n-fold convolution diagram. I.e., for an S-point of Ran n {Ij, 1 ≤ j ≤ n} ∈ Ran n , Ij ⊂ Hom(S, X), its lift to an S-point of Gr n G consists of a string of G-bundles (7.5) P 1 G , P 2 G , ..., P n G on S × X, together with identifications P 0 G |U I 1 α 1 P 1 G |U I 1 , P 1 G |U I 2 α 2 P 2 G |U I 2 , ..., P n−1 G |U In αn P n G |U In , where P 0 G denotes the trivial G-bundle. We have a naturally defined map (7.6) Gr n G → GrG ×
7.4.3. By unwinding the constructions, one shows that (7.9) equals one coming from the bilinear form attached to G and our chosen element λ ∈ Λ.
7.5. Construction of the inverse map in Proposition 7.2.5.
7.5.1. For a non-negative integer n, consider the prestack (7.10) Z n := L + (G)\( Gr n × Ran n Ran), where Ran → Ran n is the diagonal map.
It is easy to see that as n varies, the prestacks (7.10) form a simplicial object in PreStk / Ran ; denote it by Z • . Consider its geometric realization |Z • |, viewed as a prestack over Ran, equipped with a factorization structure.
By the construction in Sect. 7.3, a factorization A-gerbe on GrG gives rise to a 2-gerbe on |Z • | with respect to A, i.e., a map |Z • | → B 3 et (A), equipped with a trivialization of its restriction to Bet(L + (G)) = Z 0 → |Z • |.
Moreover, the above 2-gerbe is naturally equipped with the factorization structure. 7.5.2. Note now that we have the (simplicial) identification between (7.10) and theČech nerve of the map Bet(L + (G)) → Bet(L(G)). Thus, we obtain a 2-gerbe on Bet(L(G)), equipped with a trivialization of its restriction to Bet(L + (G)), and equipped with a factorization structure.
The latter datum is equivalent to that of a multiplicative gerbe on L(G), equipped with a (multiplicative) trivialization of its restriction to L + (G).

Metaplectic geometric Satake
In section we take G to be reductive. We will define the metaplectic geometric Satake functor and formulate the "metaplectic vanishing conjecture" about the global Hecke action.
We continue to assume that char(k) = 2.
8.1. The metaplectic spherical Hecke category. In this subsection we introduce the metaplectic spherical Hecke category, which is the recipient of the metaplectic geometric Satake functor. 8.1.1. Let G G be a factorization E ×,tors -gerbe on GrG. We define the sheaf of categories (Sph G G ) / Ran as follows. For an affine test scheme S and an S-point of Ran, we define the corresponding category by In the above formula, L + (G)|S denotes the value on S of the factorization group-scheme L + (G). The superscript L + (G)|S indicates the equivariant category with respect to that group-scheme 15 . Note that the latter makes sense due to the equivariance structure on the gerbe G G ⊗ det 1 2 G |S with respect to L + (G)|S that was constructed in Sect. 7.3. By Proposition 7.2.5, we obtain that the operation of convolution product defines on (Sph G G ) / Ran a structure of sheaf of monoidal categories over Ran.
By construction, (Sph G G ) / Ran carries a natural prefactorization structure, see Sect. 2.2.3. 15 For a prestack Y, a group-object H in PreStk /Y and Z ∈ PreStk /Y , equipped with an action of H, the corresponding equivariant category of sheaves is defined by Shv(Z) H := Shv(H\Z).
Remark 8.1.2. One can show that (Sph G G ) / Ran , regarded as a sheaf of categories on Ran is in fact quasi-coherent (see Sect. 1.6.5 for what this means), and that the above prefactorization structure is actually a factorization structure. 8.1.3. Let P be a parabolic subgroup of G with Levi quotient M . Let us denote by G M the factorization gerbe on GrM corresponding to G G .
The functor (5.5) naturally upgrades to a functor between sheaves of categories By construction, (8.2) respects the factorization structure, i.e., it is a functor between factorization sheaves of categories. The functoriality with respect to the finite sets I, as well as compatibility with factorization is built into the construction. 8.3. Example: metaplectic geometric Satake for tori. In this subsection we let G = T be a torus. The geometric Satake functor for T is the composite of (8.10) and (8.9).
8.4.1. A key feature of the assignment G G G π 1,alg ⊗Gm of Sect. 6.2.1 is compatibility with parabolics in the following sense.
Note that for a parabolic P of G with Levi quotient M , the corresponding reductive group M identifies with the Levi subgroup of G , attached to the same subset of the Dynkin diagram.
Let G M be the factorization gerbe on GrM that corresponds to G G under the map of Sect. 5.1.4. Then the multiplicative gerbe G π 1,alg (M )⊗Gm on Gr π 1,alg (M )⊗Gm attached to G M by Sect. 6.2.1 identifies with the pullback with respect to (8.12) of the multiplicative gerbe G π 1,alg (G )⊗Gm on Gr π 1,alg (G )⊗Gm attached to G G .