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 Gaitsgory (Astérisque 370:1–112, 2015, Theorem 4.5.2).

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 [24]. 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 (K 2 ) Zar is the sheafification of the presheaf of abelian groups that assigns to an affine scheme S = Spec(A) the group K 2 (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 Gr G = 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 Bun G that classifies principal G-bundles on X . The main object of investigation is the category of sheaves on Bun G .
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 O x of X at x with k[[t]], we have the map where we interpret Gr G 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 Bun G , and the corresponding category Shv G (Bun G ) of twisted sheaves. Now, if we want to consider the local versus 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 Bun G 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 Gr G of the group G.
In a way, this answer is more elementary than (0.3) in that we are dealing with étale 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 Gr G , we attach to it a certain reductive group H , a gerbe G Z H on X with respect to the center Z H of H , and a character : ±1 → Z H . We refer to the triple as the metaplectic Langlands dual datum corresponding to G.
The datum of G Z 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 Dmodules), we will propose a conjecture (namely, Conjecture 8.6.2) that says that the monoidal category QCoh LocSys The geometric input for such an action is provided by the metaplectic geometric Satake functor, see Sect. 8.
Presumably, in the arithmetic context, the above notion of twisted H -local system coincides with that of homomorphism of the (arithmetic) fundamental group of X to Weissman's L-group. 0.2. "Metaplectic" versus "Quantum". In the paper [10], a program was proposed towards the quantum Langlands theory. Let us comment on the terminological difference between "metaplectic" and "quantum", and how the two theories are supposed to be related. 0.2.1. If Y is a scheme (resp., or more generally, a prestack) we can talk about E ×gerbes on it. As was mentioned above, such gerbes on various spaces associated with the group G and the geometry of the curve X are parameters for the metaplectic Langlands theory.
Let us now assume that k has characteristic 0, and let us work in the context of D-modules. Then, in addition to the notion of E × -gerbe on Y, there is another one: that of twisting (see Sect. 1.9.5 for what the word "twisting" means).
There is a forgetful map from twistings to gerbes. Roughly speaking, a gerbe G on Y defines the corresponding twisted category of sheaves (=D-modules) Shv G (Y) = D-mod G (Y), while if we lift our gerbe to a twsiting, we also have a forgetful functor D-mod G (Y) → QCoh(Y). 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-mod κ (Bun G ), which is the same as Shv G (Bun G ), 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-mod κ (Bun G ) → QCoh(Bun G ).
In the TQFT interpretation of geometric Langlands, this forgetful functor is called "the big brane". It allows us to relate the category D-mod κ (Bun G ) 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.
In the global quantum geometric theory one expects to have an equivalence of categories D-mod κ (Bun G ) D-mod −κ −1 (BunǦ). (0.6) We refer to (0.6) as the global quantum Langlands equivalence.
0.2.1. 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 Gr G (resp., GrǦ) corresponding to κ (resp., −κ −1 ). We conjecture that the metaplectic Langlands dual data (H, G Z H , ) corresponding to G andǦ are isomorphic.
Furthermore, we conjecture that the resulting actions of
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 Gr G . This is done in Sect. 2. -We formulate the classification result that describes factorization gerbes on Gr G in terms of étale 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 Gr G and those on Gr M , 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 Gr G to those on Gr M 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, attached to a given geometric metaplectic datum G. We introduce the notion of G Z Htwisted H -local system on X ; when we work with D-modules, these local systems are k-points of a (derived) algebraic stack, denoted LocSys H . This is done in Sect. 6.
-We show that a factorization gerbe on Gr G 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 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 [6] of M. Finkelberg and the second-named author and the paper [19] 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 [6,19], 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 [6,19] in a way tailored for the needs of the geometric metaplectic theory.
The paper [19] develops the theory of factorization gerbes on Gr G (in loc. cit. they are called "symmetric factorizable gerbes"). One caveat is that in the setting of [19] one works with schemes over C and sheaves in the analytic topology, while in the present paper we work over a general ground field and étale sheaves.
The main points of the theory developed in [19] 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 Gr G 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 [19] 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 [6,19], metaplectic geometric Satake is stated as an equivalence of certain abelian categories. In [6], 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 [19] 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 [19]. 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 [24] 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 Gr G . 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 Gr G . Factorization gerbes over Gr G 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 Gr G in terms of étale 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 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 Gr G 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 Gr G and those on Gr M , 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 Gr G , we can restrict it to Gr P , and the resulting gerbe will uniquely come from a gerbe on Gr M . This procedure gives a map from the space of gerbes Gr G to that on Gr M . 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 Gr G gives rise to a metaplectic Langlands dual datum, i.e., a reductive group H , a gerbe G Z H on X with respect to the center Z H of H , and a character ±1 → Z H .
We define the notion of G Z 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 Gr G , 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 ), G Z 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 F q . However, since all the constructions in this paper are canonical, the results over F q can be deduced from those over F q 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 [15,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 the coweight lattice of G and byˇ its dual, i.e., the weight lattice.
We will denote by α i ∈ (resp.,α i ∈ˇ ) 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 [16,17] for the foundations of the theory, or [15, 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 π 0 (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 c 0 , c 1 ∈ C, we let Maps C (c 0 , c 1 ) ∈ 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., over-category).
In several places in the paper we will need the notion of left (resp., right) Kan extension. Let F : C → D be a functor, and let E is an ∞-category with colimits. The notion of right Kan extension is obtained similarly: it is the right adjoint of (0.7); the formula for it is given by We let DGCat denote the ∞-category of DG categories over E, see [15,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 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 Gr G , 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 Bun G or quotients of Gr G 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 π i (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, the étale 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 groupprestack H. Such an object is the same as a map of group-prestacks

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) satifies étale descent. Thus, to an affine scheme of finite type S we assign a DG-category Shv(S), and to a morphism f : S 1 → S 2 a colimit-preserving pullback functor The right-lax symmetric monoidal structure on Shv amounts to a compatible collection of functors The examples of sheaf theories that we are interested in are: (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 constructible étale sheaves on S with E -coefficients, see [12,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 we have by definition

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. Thus, if an affine scheme S is written as a (filtered) limit we have by definition The functor (1.2) inherits a right-lax symmetric monoidal structure.

Let PreStk denote the category of all (accessible 6 ) functors
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 [17].

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 π i (S) is defined to be where i (S) is viewed as a mere object of Spc.
1.3.5. For C as above, an object c ∈ C and H ∈ Grp(C), one defines the notion of action of H on C. By definition, such a data consists of an object c ∈ C /B(H ) together with an identification c.
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 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 Thus, we obtain a canonically defined functor identifies with that of connective spectra.
For any i ≥ 0, we have the mutually adjoint endo-functors is an object of Grp(Spc). 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.

Gerbes.
1.4.1. Let Y be a prestack, and let A be a group-like E n -object in the category PreStk /Y , for n ≥ 1. In other words, for a given (S is a group-like E n -object of Spc, in a way functorial in (S, y).
We include the case of n = ∞, when we stipulate that A is a commutative groupobject of PreStk /Y . I.e., (1.4) should be a commutative group-object of Spc, i.e., a connective spectrum. For . We will be interested in spaces of the form Note that (1.5) is naturally a group-like E n−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 Note that Here In other words, it is the cohomology of the object Note also that in this case the functor 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 A-gerbes on Y.
When A is of the form A × Y (see Sect. 1.4.3 above), we will simply write Ge A (Y). 7 The latter assertion means that B i et (A) is locally of finite type as a prestack.

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 μ n 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 For future reference, denote also where colimit is taken with respect to the maps μ n → μ n , for n | n .

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 = μ n and a coming from the identity map μ n → μ n . In this case, the corresponding μ n -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 μ n -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.
We obtain a homomorphism Proof The assertion follows from the fact that the étale 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 [7, 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. Hence, the right-lax symmetric monoidal structure on Shv naturally gives rise to a functor In particular, for every S ∈ Sch aff , the category Shv(S) has a natural symmetric monoidal structure, and for every f : 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 1.6.4. Part of the data of a presheaf of DG categories is a compatibility of actions for morphisms between affine schemes: For f : S 1 → S 2 , y 2 : S 2 → Y and y 1 = y 2 • f , the corresponding functor must intertwine the action of Shv(S 2 ) on C(S 2 , y 2 ) with the action of Shv(S 1 ) on C(S 1 , y 1 ) via the monoidal functor f ! : Shv(S 2 ) → Shv(S 1 ).
In particular, the functor (1.9) gives rise to a functor of Shv(S 1 )-module categories where ⊗ is the operation of tensor product of DG categories (see, e.g., [15, Chapter 1, Sect. 10.4]).
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 : 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 S 1 and S 2 , the functor 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 (1.11) We shall say that C is a sheaf if it satisfies étale descent, i.e., if the functor (1.11) satisfies étale 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 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 an étale sheaf. Here is the construction 8 : Consider the category Split(T) formed by (S, y) The forgetful functor Split(T) → (Sch aff ) /Y forms a basis of the étale 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 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 E 2 -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 for any Z ∈ PreStk /Y .
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 T χ .
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 T χ .
In particular, for (S, y) ∈ (Sch aff ) / Y , we obtain a well-defined category Shv T χ (S). By applying the procedure of right Kan extension, we obtain a well-defined category Shv T χ (Z ) for any Z ∈ PreStk / Y , and in particular for Z = Y.
We define the category of (H, χ)-twisted equivariant sheaves on Y as 1.8.4. An example. Let n be an integer invertible in k and let a be an element of order Using the element a we obtain a homomorphism Let χ a := χ taut | G m denote the corresponding character sheaf (known as the Kummer sheaf) on G m . 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 G m -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 the étale 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 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 (1.12) At the level of π 0 , 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 π 1 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 G m 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 D-modules, 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 [14, 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 (1.13) 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 [14,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 [14,Sect. 6] attaches an object L κ ∈ Tw(Y). The image of L κ under the tautological projection In what follows we will stay in the context of étale 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 [2]) 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 ).
For a finite set J we have a map given by the union of the corresponding finite subsets.
2.1.2. The Ran space admits the following explicit description as a colimit (as an object of PreStk): where I runs through the category opposite to that of non-empty finite sets and surjective maps. 9 For a surjection φ : I 1 → I 2 , the corresponding map X I 2 → X I 1 is the corresponding diagonal morphism, denoted φ .
belongs to Ran J disj if for every j 1 = j 2 and i 1 ∈ I j 1 , i 2 ∈ I j 2 , the corresponding two maps S ⇒ X have non-intersecting images.

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 x 1 , ..., x n of k-points of X , the fiber Z x of Z over the above point is supposed to be identified with where {x i } 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 where I j := 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 [22,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 J disj , given by where I = j∈J I j .
We require the functors (2.4) to be compatible with surjections J 1 J 2 via the commutative diagrams analogous to (2.3). A precise formulation of these compatibilities is given in [22,Sect. 6].
We will say that prefactorization structure on C is a factorization structure if the functors (2.4) are equivalences. 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.

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 J 1 J 2 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 FactGe A (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 [19] was called "symmetric factorizable gerbe".
2.2.6. Let Z be a factorization prestack over Ran, and let G be a factorization E ×,torsgerbe 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 Zor 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 Gr G , to be the following prestack.
For an affine test scheme S, the groupoid (in fact, set) Maps(S, Gr G ) consists of triples where I is an S-point of Ran, P G is a G-bundle on S × X , and α is a trivialization of P G over the open subset U I ⊂ 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 Gr G 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 Gr G . We denote the the resulting space (i.e., in fact, a connective 2truncated spectrum) by This example is important because there is a canonical factorization line bundle on Gr G , denoted det G ; we will encounter it in Sect. 5.2.1.

Assume for a moment that X is proper.
Let Bun G denote the moduli stack of G-bundles on X . Note that we have a tautological projection (2.6) Recall now that [13, Theorem 3.2.13] says 11 that the map (2.6) is a universal homological equivalence. This implies that any gerbe on Gr G uniquely descends to a gerbe on Bun G .
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 Gr G , we can thus talk about the prefactorization sheaf of categories, denoted Shv G (Gr G ) / Ran , whose value on S, I ⊂ Maps(S, X ) is

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 Gr G in terms of more concise algebro-geometric objects.
3.1 Parameterization via étale cohomology. In this subsection we will create a space, provided by the theory of étale cohomology, that maps to the space FactGe E ×,tors (Gr G ), thereby giving a parameterization of geometric metaplectic data. For an affine test scheme S and an S-point (I, P G , α) of Gr G , we need to construct a A-gerbe G I on S.
Moreover, for φ : I J , such that the point hits Ran J disj , we need to be given an identification and a trivialization of the resulting map where U I is as in Sect. 2.3.1.
We claim that such a datum indeed gives rise to a A-gerbe G I on S, equipped with identifications (3.3). 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 I denote the complement of U I (the scheme structure on I is irrelevant). We need to construct the trace map

First off, since
where for a scheme Y , we denote by A Y the constant étale 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 p X : X → pt is the projection.

Analysis of homotopy groups of the space of factorization gerbes.
3.2.1. We have the following assertion that results from [19, 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 2-groupoid FactGe A (Gr G ). Namely, 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.
3.2.5. Let π 1,alg (G) denote the algebraic fundamental group of G. Explicitly, π 1,alg (G) can be described as follows: Choose a short exact sequence where T 2 is a torus and [ G 1 , G 1 ] is simply connected. Set . Let 1 and 2 be the coweight lattices of T 1 and T 2 , respectively. Then π 1,alg (G) 1 / 2 .

We have:
where Quad( , Z) W is the abelian group of W -invariant integer-valued quadratic forms on .

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

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

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 [24].
Recall from Sect. 2.3.3 that given a factorization line bundle L on Gr G 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 Gr G .
3.3.1. Let K 2 denote the prestack over X that associates to an affine scheme S = Spec(A) mapping to X the abelian group K 2 (A). Let (K 2 ) Zar be the sheafification of K 2 in the Zariski topology.
On the one hand, we consider the space CExt(G, (K 2 ) Zar ) (in fact, an ordinary groupoid) of Brylinski-Deligne data, which are by definition central extensions The operation of Baer sum makes CExt(G, (K 2 ) Zar ) into a commutative group in spaces, i.e., into a Picard category.
On the other hand, consider the Picard category equipped with a trivialization of the composition Given such a map, for an affine scheme S and an S-point (I, P G , α) of Gr G , we need to construct a line bundle L I on S. By [5,Theorem 2], after passing to an étale cover of S, the G-bundle P G becomes Zariski locally trivial. Hence, we can assume that (I, P G , α) is a map equipped with a trivialization of the composition where U I is as in Sect. 2.3.1.

Composing (3.13) with (3.12) we obtain a map
equipped with a trivialization of the composition To this data we need to associate a line bundle L I on S.

As in Sect. 3.1.5, it suffices to construct an invertible function on S, starting from the data of a map
S × X → B Zar (K 2 ), (3.17) equipped with a trivialization of the composition The desired map comes from the residue map constructed as follows (Zariski sheafification is automatic since I = S × X − U I is finite over S).

Consider the exact triangle of categories
where is the full subcategory spanned by objects set-theoretically supported on I .
The long exact cohomology sequence gives rise to a map Now, the direct image functor has the property that it sends Perf(S × X ) I to Perf(S). Thus, we obtain a map

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 where the left-hand side is the space of Brylinski-Deligne data, and the right-hand side is the space parameterizing factorization gerbes on the affine Grassmannian.

Let n be an integer invertible in k. Then the construction of [21] defines a map
that depends functorially on S ∈ Sch aff . As was explained to us by D. Clausen, the map (3.22) cannot be lifted to a map of commutative group objects in PreStk  )), 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 (3.24) The map (3.24) gives rise to a map (3.25)

Note now that we have a fiber sequence
from which it follows that the induced map is an isomorphism.

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 Gr T , 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 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 → ; in what follows we will denote by λ i ∈ is the value of λ I on i ∈ I .
A morphism (J, λ J ) → (I, λ I ) is a surjection φ : I J such that Consider the prestack Gr T,comb := colim The prestack Gr T,comb endowed with its natural forgetful map to Ran, also has a natural factorization structure.
There is a canonical map Gr T,comb → Gr T , (4.2) 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.
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 I j .
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 This description implies that the sheaf of categories Shv G (Gr T ) / Ran is quasicoherent (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. Given two elements λ 1 , λ 2 ∈ , consider I = {1, 2} and the map Consider the corresponding gerbe G λ 1 ,λ 2 := G λ I over X 2 . By (4.4) it is identified with G λ 1 G λ 2 over X 2 − (X ). By Lemma 1.5.5, there exists a well-defined element a ∈ A(−1) such that We let

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 λ 1 , λ 2 , λ 3 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

Finally, let us recover the quadratic form
For a given λ ∈ , consider the gerbes G λ,λ and G λ G λ on X 2 . They are both equipped with a structure of S 2 -equivariance, and they are identified as such over X 2 − . In addition, the induced equivariance structure on both is the tautological one.
We note that the datum of a gerbe on X 2 , equipped with a structure of S 2equivariance, 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,
The relation is checked in a way similar to Sect. 4.2.2.

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 Gr T , and then when the lattice = Hom(G m , T ) is replaced by a general finitely generated abelian group.  In the above corollary, the notation Grp(−) (resp., ComGrp(−)) means groupobjects (resp., commutative group-objects) in a given ∞-category.

(a) We have a canonical isomorphism
We are going to use Corollary 4.3.3 to describe the spaces

FactGe mult
A (Gr T ) and FactGe com A (Gr T ) more explicitly.

Note that the Kummer map
which is a map of commutative group-prestacks, gives rise to a map and hence We also note that the looping map is an isomorphism. Combining with Corollary 4.3.3(b), we obtain a map Maps(X, B 2 et (Hom( , A))) → FactGe com A (Gr T ). (4.9) We claim: is the preimage of the subset of Quad( , A), consisting of those quadratic forms, whose associated bilinear form is zero.
Proof Follows from Corollary 4.3.3, combined with the following lemma: is an isomorphism. The map is injective and has as its image the set of quadratic forms whose associated bilinear form vanishes. is an isomorphism.

More general abelian groups.
In this section we generalize the discussion of Sect. 4.3 to the case when instead of a lattice (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 be a finitely generated abelian group. We define the commutative groupprestack over Ran Gr ⊗G m as follows. Write as 1 / 2 , where 1 ⊃ 2 are lattices. Let T 1 and T 2 be the corresponding tori. We define Gr ⊗G m as a quotient of Gr T 1 by Gr T 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 as a quotient.
The group-prestack Gr ⊗G m has a natural factorization structure over Ran.

4.4.2.
Let now G be a connective reductive group. Let = 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 ⊗G m over Ran.
The following results from Proposition 4.3.7: Then the datum of descent of the gerbe G 1 to a factorization gerbe G on Gr ⊗G m exists only if the restriction of q 1 to 2 is trivial, and in the latter case is equivalent to the trivialization of G 2 := G 1 | Gr T 2 as a factorization gerbe on Gr T 2 . Moreover: (a) The gerbe G admits a multiplicative structure if and only if b 1 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 q 1 is trivial. In the latter case, the commutative multiplicative structure is unique up to a unique isomorphism.
From here we obtain:
(b) There is a diagram of fiber sequences where Hom( , A(−1)) 2 -tors is identified with the kernel of the map 4.4.6. Let now be the algebraic fundamental group π 1,alg (G) of a reductive group G, and recall the map (4.10) Consider the composite map

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 In order to define the sought-for splitting, by functoriality, it suffices to consider the case of = Z/2Z, A = ±1 and we need to produce a multiplicative factorization gerbe on Gr Z/2Z⊗G m that gives rise to the tautological map Z/2Z → ±1. 4.5.2. We will first construct the corresponding multiplicative factorization gerbe on Gr G m , i.e., for = 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 = 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 G m ,Ran .
We require that the restriction of L to X ⊂ Ran be such that its further restriction to the connected component of corresponding to 1 ∈ Z, is odd. An example of such an L is the determinant line bundle, corresponding to the tautological action of G m 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 L 1 and L 2 be two different choices for L. We note that their ratio L := L 1 ⊗ 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( , A(−1)) 2 -tors , we will denote by G the resulting multiplicative factorization gerbe on Gr ⊗G m .

For a given object G ∈ FactGe mult
A (Gr ⊗G m ) let us denote by the map that measures the obstruction of G to belong to FactGe com A (Gr ⊗G m ).
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 Gr G and those on Gr M , 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 N P denote the unipotent radical of P.

Consider the diagram of the Grassmannians
We claim that pullback along q defines an equivalence, 5.1.4. We now take A := E ×,tors . Given a factorization E ×,tors -gerbe G G over Gr G , consider its pullback to Gr P , denoted G P . We let G M denote the canonically defined factorization gerbe on Gr M , whose pullback to Gr P 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 pushforward 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 Gr M (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 det G denote the determinant line bundle on Gr G , 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 P G on S × X , equipped with an isomorphism Consider the corresponding vector bundles associated with the adjoint representation Then det. rel. g P G , g P 0 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 det G on Gr G .
It is easy to see that det G is equipped with a factorization structure over Ran.

5.2.2.
Taking A = ±1, we will consider the factorization gerbe det 1 2 G over Gr G . From now on we will choose a square root, denoted ω ⊗ 1 2 X of the canonical line bundle ω X on X (see again Remark 1.5.3 for our notational conventions).
Let P be again a parabolic of G. Consider the factorization gerbes det 1 2 G | Gr P and det 1 2 M | Gr P over Gr P . We claim that the choice of ω However, this identification will be compatible with the factorization structures only up to a sign.
In fact, we claim that the ratio of the line bundles det G | Gr P and det M | Gr P admits a square root, to be denoted det Let us construct the isomorphism Let us identify the vector space g/p with the dual of n(P) (say, using the Killing form). For an S-point (I, P P , P G | U I P 0 G | U I ) of Gr P , denote E := n(P) P P and E 0 := n(P) P 0 P .
Then the ratio of det G | S and det M | S identifies with the line bundle Note, however, that for any line bundle L on S × X , we have 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 Gr G and the corresponding factorization gerbe G M on Gr M (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

We claim that the functor (5.5) is a functor between factorization categories.
This follows from the fact that the natural grading on the line bundle det ⊗ 1 2 P is such that it equals d G,M on the corresponding connected component of Gr P .

The metaplectic Langlands dual datum
In section we take G to be reductive. Given a factorization gerbe G on Gr G , 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.

Given a factorization
be the associated quadratic and bilinear forms, respectively. Let ⊂ be the kernel of b.
Following [18], we will now define a new root datum ( ⊂ ,ˇ ⊂ˇ ). (6.1) 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 ∈ , and the corresponding elementα ∈ˇ be The fact that q ∈ Quad( , A(−1)) lies in the image of the map implies that α andα defined in this way indeed belong to ⊂ andˇ ⊂ˇ ⊗ Z Q, respectively. 6.1.3. Since q was W -invariant, the action of W on preserves . Moreover, for each α ∈ , the action of the corresponding reflection s α ∈ W on equals that of s α . This implies that restriction defines an isomorphism from W to the group W of automorphisms of 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 α i are the simple coroots of , then the corresponding elements α i ∈ 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 "π 1 -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 ). Consider the algebraic fundamental group π 1,alg (G ) of G , and the projection → π 1,alg (G ). Consider the corresponding map We claim that there exists a canonically defined multiplicative factorization A-gerbe G π 1,alg (G )⊗G m on Gr π 1,alg (G )⊗G m , whose pullback under (6.2) identifies with G T . For an integer k let det G m ,k denote the determinant line bundle on Gr G m associated with the action of G m 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 G m , corresponding to (det SL 2 ) a is given by (det G m ,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 G m ,2 ) 2a under the isogeny is canonically trivial as a factorization gerbe on Gr G m ,Ran .
6.2.4. Note that the pullback of det G m ,2 under the above isogeny is the factorization line bundle det G m ,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 G m ,2n on Gr G m admits a canonical 2n-th root.
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 )⊗G m is also trivial.
6.3. The metaplectic Langlands dual datum as a triple. Until the end of this section we will assume that char(k) = 2. We take A to be E ×,tors .
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 [24]. 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 I J K (6.5) (here J and K are sets (automatically, finite and non-empty)), and whose morphisms are commutative diagrams that sends an object (6.5) to 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 [11,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 G A 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.
6.5.1. We apply the construction in Sect. 6.4 to and from (6.4). We obtain the sheaves of monoidal categories over Ran Fact(Rep(H )) G Z and Fact(Rep(H )) G Z , and the corresponding monoidal categories 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 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, G Z ) appearing in the triple (H, G Z , ) of the metaplectic dual datum, we will construct the derived stack LocSys G Z H of G Z -twisted local systems on X . Its k-points will be the twisted local systems as defined in Sect. 6.3.5. One shows that LocSys G Z H defined in this way is representable by a quasi-smooth derived algebraic stack (see [1,Sect. 8.1] for what this means). 6.6.2. As in [11,Sect. 4.3], we have a canonically defined (symmetric) monoidal functor (6.11) The following is proved in the same way as [11,Proposition 4.3.4] 14 : 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 Gr G give rise to a (factorization) gerbe on (the factorization version of) the loop group of G.

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 ), letD I be the corresponding relative formal disc: By definition,D I 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 where Z α = Spec(A α ) and the colimit is taken in PreStk, then D I = Spec(A), where Let • D I be the open subscheme of D I , 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 D I → 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 D I → Z is equivalent to that of a map of prestacksD I → 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. In other words, the Beauville-Laszlo says that restriction along induces a bijection on the corresponding pairs (P G , α). In the above formula, the notation U I is as in Sect. 2.3.1.

7.2.2.
This interpretation of Gr G shows that the group-prestack L(G) acts naturally on Gr G , with the stabilizer of the unit section being L + (G). Furthermore, the natural map is an isomorphism, where the quotient is understood in the sense of stacks in the étale topology.
The isomorphism (7.2) implies that for every S → Ran, the fiber product is an ind-scheme over S.  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 Gr G , its pullback under the projection L(G) → Gr G , 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 Gr G . We will explain this construction in the present subsection.

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 7.3.4. For n = n 1 + n 2 , consider the prestack over Ran × Ran equal to Ran × Gr n 2 In the setting of Sects. 7.3.4-7.3.5 take n 1 = n 2 = 1, and consider the twisted product G G, which is a well-defined gerbe on Gr 2 G due to the identification of (7.7) and the chosen structure of equivariance with respect to L + (G) on G.
We require that G G should admit an identification with the pullback of G under the map which extends the already existing identification over given by the factorization structure on G via the diagram (7.8) for n 1 = n 2 = 1:

Another view on the bilinear form.
The L + (G)-equivariant structure on G gives rise to the following interpretation of the bilinear form attached to G, when G is a torus T . We obtain that the A-gerbe restriction G| Gr G,x is equivariant with respect to G(Ô x ).

7.4.2.
For G = T , since T is commutative, the action of T (Ô x ) on Gr T,x is trivial. Hence, for every λ ∈ , the action of T (Ô x ) on the corresponding point of Gr T,x defines a multiplicative A-torsor on T (Ô x ).
Since the elements of A have orders prime to char(k), the above multiplicative A-torsor is pulled back from T , and by Kummer theory, the latter is given by a homomorphism → A(−1).
Thus, we have constructed a map → Hom( , A(−1)). Thus, we obtain a 2-gerbe on B et (L(G)), equipped with a trivialization of its restriction to B et (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.
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 Gr G . 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 (8.1) In the above formula, L + (G)| S denotes the value on S of the factorization groupscheme 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. where I runs over the category of finite non-empty sets and surjective morphisms. Both sides in (8.4) are equipped with t-structures; moreover one shows that Fact(Rep(H )) G Z (X I ) identifies with the derived category of its t-structure, 17 i.e., the canonical map of [ where we denote by G T the restriction of G T along (8.6). In this case, it follows from Sect. 7.4 that the forgetful functor factors through the essential image of (8.7), thereby giving rise to a functor of (6.9). 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 ⊗G m 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.
We have a canonical surjection π 1,alg (M ) → π 1,alg (G ), (8.11) and the corresponding map of factorization Grassmannians Gr π 1,alg (M )⊗G m → Gr π 1,alg (G )⊗G m . (8.12) Let G M be the factorization gerbe on Gr M that corresponds to G G under the map of Sect. 5.1.4. Then the multiplicative gerbe G π 1,alg (M )⊗G m on Gr π 1,alg (M )⊗G m 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 )⊗G m on Gr π 1,alg (G )⊗G m attached to G G . The category of Hecke eigensheaves with respect to σ is canonically equivalent to where the Bun T -equivariance makes sense due to the above trivialization of G| Bun T . This category is canonically equivalent to the category of representations of Heis σ , on which G m acts by the standard character.
Since Heis σ is of Heinsenberg type, the above category is non-canonically equivalent to Vect. 8.6.8. At the moment, we do not have a conjecture as to how to explicitly describe the category of Hecke eigensheaves in the tempered subcategory of Shv with respect to a given σ for a general reductive G.