The spectral gluing theorem revisited

We strengthen the gluing theorem occurring on the spectral side of the geometric Langlands conjecture. While the latter embeds $IndCoh_N(LS_G)$ into a category glued out of 'Fourier coefficients' parametrized by standard parabolics, our refinement explicitly identifies the essential image of such embedding.


Introduction
Let X be a smooth connected complete curve over a ground field k, algebraically closed and of characteristic zero. Let G be a connected reductive group over k and LS G the derived stack of de Rham G-local systems on X. We choose a Borel subgroup B ⊆ G, regarded as fixed throughout.
1.0.1. The DG category IndCoh N (LS G ) of ind-coherent sheaves on LS G with nilpotent singular support (see [Gai13b] and [AG15]) is one of the two protagonists of the geometric Langlands program. In [AG15], it is shown that IndCoh N (LS G ) is bigger than the more familiar QCoh(LS G ), in the precise sense that there is a colocalization 1
Ξ Ψ It is also explained that the difference between these two DG categories is a manifestation of the noncommutativity of G, accounted for by the existence of proper parabolic subgroups. For instance, the two DG categories coincide iff G is a torus, in which case there are no proper parabolic subgroups.
1.0.2. This idea takes a more precise form in the spectral gluing theorem of [AG18], which amounts to: • a glued category Glue := laxlim P ∈Par op I(G, P ) consisting of 2 rank(G) pieces, one for each standard 2 parabolic subgroup; • a fully faithful functor γ : IndCoh N (LS G ) → Glue.
Here Par denotes the poset of standard parabolics, including G, with respect to inclusions.

1.0.3.
In analogy with the extended Whittaker category construction (see [Gai15a], [Ber19a]), the gluing components I(G, P ) should be regarded as categories of Fourier modes and γ should be regarded as a Fourier decomposition. For instance, the category of G-Fourier modes I(G, G) coincides with QCoh(LS G ).
The present work originates from the observation that such Fourier decomposition is imperfect, in that γ is not essentially surjective. The goal of this paper is to correct this imperfection by identifying exactly which collections of Fourier coefficients belong to the essential image of γ.

The case of G = GL 2
Let us explain the construction of [AG18] and our improvement in the simplest case of G of semisimple rank 1. For definiteness, we set G = GL 2 and keep this assumption in place until Section 1.3.

1.1.2.
The definition of IndCoh 0 goes as follows. For a map Y → Z of quasi-smooth stacks, set: where the map QCoh(Y) → IndCoh(Y) is the natural inclusion and IndCoh(Z ∧ Y ) → IndCoh(Y) is the ind-coherent pullback along Y → Z ∧ Y . In [Ber17b], [Ber19b] and [Ber19c], we gave explanations for the raison d'être of IndCoh 0 . By construction, IndCoh 0 (Z ∧ Y ) belongs to a natural sequence of colocalizations: As mentioned, the starting point of the present work is the observation that such inclusion is not an equivalence. More precisely, while QCoh(LS G ) is too small to match IndCoh N (LS G ), the glued DG category Glue is too big. 1.1.6. In this paper, we state and prove a strong spectral gluing theorem which explicitly identifies the essential image of the inclusion γ. For G = GL 2 , our result can be stated now: informally, rather than gluing along (1.1), we glue along the cospan(=op-correspondence) below. Theorem 1.1.7 (Strong spectral gluing for G = GL 2 ). For G = GL 2 , the DG category IndCoh N (LS G ) is naturally equivalent to the limit of the diagram .
Remark 1.1.8. Let us explicitly describe the target DG category in the two versions of the spectral gluing theorem (still in the case of GL 2 ). The target of the version in [AG18] is the DG category Glue, whose objects are triples F ∈ QCoh(LS G ), G ∈ I(G, B), η : where the arrow η is not required to be an isomorphism. The target of our version is the limit of the op-correspondence above, which is the full subcategory of Glue spanned by objects satisfying the following additional requirement: the morphism η : ( p B ) * (F) → Ψ G,B (G) in QCoh((LS G ) ∧ LS B ), obtained from η by adjunction, must be an isomorphism.

A baby version
The point of the above theorem is that γ : IndCoh N (LS G ) → Glue factors as the composition As we are about to illustrate, a similar phenomenon occurs in a much simpler (yet entertaining) situation.
The following discussion might appear unrelated to the strong spectral gluing for GL 2 ; we will explain later that there is a tight relation between the two.

Consider the cartesian square
where V A n is a finite dimensional vector scheme, Bl its blow-up at the origin and E the exceptional divisor. The baby situation we will describe amounts to expressing D(V ) as a DG category glued out D(Bl) and D(pt) Vect k .

1.2.2.
Let us look at the following table of analogies, where the baby version is on the left and the adult version on the right:

Consider now the functor
is the obvious one obtained by adjunction. One can prove directly that γ baby is fully faithful (but not an equivalence). This is the baby version of the original spectral gluing theorem.

1.2.4.
On the other hand, let us look at the pushout prestack pt E Bl and at its DG category of D-modules: There is an evident inclusion The baby version of the strong spectral gluing theorem states that γ baby factors as the composition of the equivalence

The case of general G
Let us now describe the shape of the strong spectral gluing theorem in the case G has higher semisimple rank. To formulate the statement, it will be necessary to borrow some notions from the theory of H, developed in [Ber17b] and [Ber19b]. The main points of this theory are reviewed in Section 1.3.4.

1.3.1.
We start by recalling that the spectral gluing theorem of [AG18] amounts to an explicit fully faithful functor where Par is the poset of all standard parabolics subgroups of G.

1.3.3.
An object of Glue consists of: • for each P ∈ Par, an object F P ∈ IndCoh 0 ((LS G ) ∧ LS P ); • for each inclusion Q ⊆ P , an arrow η Q⊆P : ( p G,Q⊆P ) !,0 (F P ) −→ F Q 3 after choosing an Ad-invariant bilinear form on g is the obvious pullback functor; • some natural compatibilities that we do not spell out here (we will spell them out in the main body of the paper, see also [AG18, Sections 4.1-4.2]).

1.3.4.
Our improved version of the theorem identifies the essential image of γ. To define the relevant full subcategory of Glue, we first need to review the main features of the H-construction.
• The starting point is the definition of the DG category IndCoh 0 (Z ∧ Y ) attached to a map Y → Z of quasi-smooth stacks. See Section 1.1.2, as well as [AG18, Section 3.2] and [Ber17b, Section 3].
• The IndCoh 0 construction, applied to the diagonal Y → Y × Y, yields the DG category The convolution monoidal structure on IndCoh((Y × Y) ∧ Y ) restricts to a monoidal structure on H(Y).
• More generally, for a map Y ← X → Z of quasi-smooth stacks, we consider which is naturally an (H(Y), H(Z))-bimodule. We also use the notations H X→Z := H X←X→Z and H Y←X := H Y←X→X . For instance, we have: of (H(V), H(Z))-bimodule DG categories.
• In the above setting, the (H(W), H(Y))-bimodule H W←X→Y is left and right dualizable, with both duals canonically identified with H Y←X→W . It follows from this that QCoh(Y) H Y→pt is a bimodule for the left action of H(Y) and the right action of H pt←Y→pt D(Y).
• The diagonal ∆ Y induces a monoidal pushforward functor QCoh(Y) → H(Y) that admits a continuous and conservative right adjoint. This implies that H(Y) is rigid.
• For X → Y a map of quasi-smooth stacks and N a closed conical subset of Sing(Y), the monoidal DG category H(Y) acts on IndCoh(Y), IndCoh N (Y) and QCoh(Y ∧ X ).
1.3.5. Now, for Q ⊆ P , there is an H(LS P )-linear diagram Let us now push forward along p P : LS P → LS G in the H-sense, that is, we tensor up with H LS G ←LS P over H(LS P ). Thanks to the third and fourth item above, we obtain an H(LS G )-linear diagram 1.3.6. The DG category Glue was constructed by disregarding Ψ G,Q⊆P : indeed, the composition of the two arrows from left to right is exactly the functor ( p G,Q⊆P ) !,0 that appeared in Section 1.3.3.
The key proposal of the present paper is that one should instead disregard Ξ G,Q⊆P and keep Ψ G,Q⊆P . Indeed, our theorem states that an object of Glue as above belongs to the essential image of γ if and only if the following condition is satisfied: for any Q ⊆ P , the arrow η Q⊆P induces an isomorphism

1.3.7.
To state this more formally, we set and consider the DG category where the limit is taken along the cospans displayed above. The indexing 1-category Tw(Par) is the poset of twisted arrows of Par, see Section 3.2.2. The decoration "temp" stands for "tempered", see [Ber19b, Example 1.3.4] for an explanation of the terminology.

1.3.8.
We can now formulate an early version of the main result of this paper (the official version appears later as Theorem 3.3.8).
Theorem 1.3.9 (Strong spectral gluing). There is a natural H(LS G )-linear equivalence 1.3.10. In Example 4.2.8, we will show that, by restricting to the trivial G-local system, the above theorem is related to the following local (that is, independent of the curve X) analogue.
Theorem 1.3.11. Let N g ⊆ g be the nilpotent cone of the Lie algebra of G. For each pair of standard parabolics Q ⊆ P , consider the natural correspondence Remark 1.3.12. For G = GL 2 , Theorem 1.3.11 states that D(N gl 2 ) is equivalent to D(pt) × D(P 1 ) D(T * (P 1 )), the DG category obtained from the diagram pt ← P 1 → T * P 1 by pullback. This situation is very similar to the blow-up situation of Section 1.2.1 and it will be proven in the same way.

Automorphic gluing, a preview
The geometric Langlands conjecture calls for an equivalence between IndCoh N (LS G ) and the DG category D(BunǦ) of D-modules on the stack BunǦ(X) ofǦ-bundles on the same curve X. HereǦ is the Langlands dual group of G. Under geometric Langlands, the spectral gluing theorem ought to correspond to a gluing statement for D(BunǦ). We conclude the introduction with an informal 4 discussion of such a statement, which we call automorphic gluing.

1.4.1.
To reduce cuttler, let us swap G withǦ and formulate a gluing conjecture for D(Bun G ). A key ingredient is the tempered subcategory, denoted by temp D(Bun G ). The definition appears in [AG15, Section 12] and an equivalent characterization is given in [Ber19d]; see also [Ber19b, Sections 1.3-1.4]. In general, we can define the tempered subcategory of any DG category C equipped with an action of the spherical monoidal DG category Sph G . As with QCoh and IndCoh, there is a colocalization (the right adjoint is called the "temperization functor"): We denote by C • the right orthogonal of temp C inside C: this might be called the anti-tempered subcategory.

1.4.2.
For any P ∈ Par, we also need to introduce a DG category I(G, P ) aut as follows. (The superscript "aut" stands for "automorphic".) Consider the prestack Bun P −gen G of G-bundles on X equipped with a generic reduction to P , see [Bar14] for the precise definition. There are canonical maps where M is the Levi quotient of P . We set 5 I(G, P ) aut := D(Bun where the left arrow is the pullback f ! P and the right arrow is the (fully faithful) composition As in [Gai15a, Section 6], there is an action of Sph G on I(G, P ): this allows us to later take the tempered and anti-tempered subcategories of I(G, P ).

1.4.3.
Now, the automorphic analogue of the spectral gluing of [AG18] states that the DG categories I(G, P ) aut assemble into a glued DG category Glue aut := laxlim P ∈Par op I(G, P ) aut and that there is a fully faithful functor γ aut : D(Bun G ) → Glue aut . The strong automorphic gluing conjecture amounts to an explicit description of the essential image of γ aut as a limit over Tw(Par), similarly to what is done in the present paper for the spectral side. Below, we give the details for G of semisimple rank 1.

1.4.4.
From now until the end of the introduction, let G be of semisimple rank one. In this case, Glue aut has two terms: temp D(Bun G ) and Example 1.4.6. As evidence for the validity of the latter statement, consider the example of ω Bun G . This object has been proven to be anti-tempered (for any nonabelian G, not just in semisimple rank 1) in [Ber19d]. Now, the fact that the counit of the adjunction

I(G, B) aut D(Bun
Bun G is an isomorphism is a quick consequence of the contractibility of the space of rational maps from X to G/B, proven in [Gai13a] and [Bar14]. This example is very close to being a proof: indeed, for G of rank one, we expect that D(Bun G ) • is generated under colimits by ω Bun G .
1.4.7. Now note that the functor appearing on the RHS of (1.6) factors as Moreover, the inclusion temp I(G, B) aut → I(G, B) aut admits a right adjoint, see (1.5). The strong automorphic gluing in semisimple rank 1 reads as follows: is a fiber square.

1.4.9.
This conjecture boils down to proving that the functor • is an equivalence. As above, we expect that I(G, B) aut • is generated under colimits by ω Bun B−gen G . This would make the equivalence manifest.

Organization of the paper
In Section 2, we construct the action of D(Sing(Y )) ⇒ on IndCoh(Y ) and discuss some of its properties. Section 3 is devoted to the definition of the term appearing in the main Theorem 1.3.9: our glued DG category and of the gluing functor that ought to realize the equivalence. Section 4 explains how to reduce the proof of the main theorem to a simpler statement about categories of D-modules on schemes: this is made possible by the results of Section 2. Finally, Section 5 extends the combinatorics of [AG18] to prove the statements left open in Section 4.

Acknowledgements
I would like to thank Dima Arinkin and Dennis Gaitsgory for their patient explanations and their insight: I owe them a great deal. I am obliged to Ian Grojnowski and Sam Raskin for several useful conversations, as well as to the anonymous referees for their comments and corrections.

Global complete intersections
After a preliminary section on the shift of grading trick, we discuss the relationship between ind-coherent sheaves and spaces of singularities in the case of global complete intersections.

Shift of grading
Consider the ∞-category [AG15] or [Ber17a]. We will make extensive usage of the "shift of grading" automorphism of G m -rep weak , defined in [AG15] and denoted by C C ⇐ . We denote by C C ⇒ its inverse.

2.1.1.
The definition of C ⇐ goes as follows: • take the invariant category C G m , equipped with its action of Rep(G m ); • twist this action of Rep(G m ) by the symmetric monoidal autoequivalence Rep(G m ) → Rep(G m ) induced by the assignment where V i,n denotes the component in cohomological degree i and weight n of a graded DG vector space V ; • finally, set Example 2.1.2. If G m acts trivially on C, then C ⇐ C.
Example 2.1.3. Let A be a graded DG algebra. Then C = Amod is naturally an object of G m -rep weak . As above, we write A = {A i,n } where A i,n is the component in cohomological degree i and grading degree n.
We have: where A ⇐ is the graded DG algebra with components (A ⇐ ) i,n = A i+2n,n .

2.1.4.
The latter example shows that the shift/unshift automorphisms do not commute with the forgetful functor G m -rep → DGCat. On the other hand, by construction, (C ⇐ ) G m and C G m are equivalent as DG categories.

2.1.5.
Recall that G m -rep weak is symmetric monoidal (compatibly with the forgetful functor to DGCat). The shift automorphism preserves the symmetric monoidal structure. Hence, it also preserves relative tensor products within G m -rep weak .
Example 2.1.6. Let V be a finite dimensional vector space and consider the dilation G m -action on V * , which corresponds to the tautological grading on Sym V . This induces G m -actions on QCoh(V * ), on ΩV := pt × V pt and on IndCoh(pt × V pt). Under the Koszul duality equivalence elements of Sym n V are placed in bidegree (2n, n). Hence, IndCoh(ΩV ) QCoh(V * ) ⇒ . This equivalence swaps the convolution and the pointwise monoidal structures. By the theory of singular support, we deduce that QCoh(ΩV ) QCoh((V * ) ∧ 0 ) ⇒ . Example 2.1.7. In the situation of the previous example, there is also a G m -action on D(V * ). We obtain an equivalence Proposition 2.1.8. Consider again the scheme Y = ΩV and recall the action of QCoh(Y ) on IndCoh(Y ), as well as the action of QCoh(Y ) on Vect given by pullback along the inclusion pt → ΩV . There is a canonical equivalence Proof. We use the above Koszul duality equivalence, together with the fact that the shift automorphism is symmetric monoidal: where the third equivalence follows (for example) by formal smoothness of (V * ) ∧ 0 , together with proper descent for IndCoh.
Remark 2.1.9. In fact, the above equivalence is the simplest nontrivial instance of the equivalence valid for any quasi-smooth stack and any k-point y ∈ Y. This will be addressed in another paper.

Singular support for global complete intersections
We show that for Y a global complete intersection (see the definition below), the DG category IndCoh(Y ) admits an action of D(Sing(Y )) ⇒ , which encodes the notion of singular support for coherent sheaves.

2.2.1.
Recall from [AG15, Section 2.3] that, for a quasi-smooth scheme Y , the scheme of singularities Sing(Y ) is a classical scheme living over Y cl , the classical truncation of Y . It is defined as the relative spectrum of the O Y cl -algebra Sym O Y cl H 1 (T Y ). We always equip Sing(Y ) with the G m -action coming from the obvious grading of the above symmetric algebra. Such G m -action induces a strong (and in particular a weak) G m -action on D(Sing(Y )): this structure allows us to consider the shifted DG category D(Sing(Y )) ⇒ .

2.2.2.
We say that a DG scheme Y is a global complete intersection 6 if it is presented as a fiber product Y = U × V pt, with U smooth affine and V A n a vector space. A global complete intersection is obviously quasi-smooth and conversely any quasi-smooth scheme is Zariski locally of this form, see e.g. [AG15, Corollary 2.1.6].

2.2.3.
Let Y = U × V pt be a global complete intersection. Such Y , together with its presentation, is regarded as fixed throughout the remainder of Section 2. Observe that as classical schemes, where the left map in the fiber product is the dual of the differential and the right one is the zero section. In particular, we have G m -equivariant closed embeddings where the G m -action on V * and on T * U is dilation (along the fibers of T * U → U in the second case).

2.2.4.
Consider the pullback action of QCoh(U ) IndCoh(U ) on IndCoh(Y ), as well as the convolution action of IndCoh(pt × V pt) on IndCoh(Y ). A simple diagram chase shows that these two actions commute. It follows that transforms this into an action of Then the monoidal functor induced by pullback along this is the action we were looking for.
Corollary 2.2.6. For Y as above and N a closed conical subset of Sing(Y ), the following full subcategories of IndCoh(Y ) are equivalent: Proof. Tautological from [AG15, Corollary 5.4.7].
Example 2.2.7. In particular: . 6 We warn the reader that this definition of global complete intersection is not standard: for us, the presentation as a fiber product is part of the data.

The singular codifferential
In this section, we use the D(Sing(Y )) ⇒ -action on IndCoh(Y ) constructed above to study the DG be a global complete intersection as before, X a quasi-smooth scheme, and f : X → Y an arbitrary map. We consider the standard correspondence where the left map is called singular codifferential, see [AG15, Section 2.4].

The exterior tensor product yields a natural equivalence
To see this, it suffices to combine [AG18, Proposition 3.1.2] with the 1-affineness of Y dR and [Gai15b, Proposition 3.1.9]. In our case, since D(Y ) acts on IndCoh(Y ) via D(Y ) → D(Sing(Y )) ⇒ the monoidal pullback functor, we obtain that The next result shows that IndCoh 0 (Y ∧ X ) can be expressed is a similar way. Proposition 2.3.3. Let X be a quasi-smooth scheme equipped with a map f : X → Y . Under the equivalence (2.4), the subcategory , it is easy to see that the natural functor is an equivalence. Now, the Arinkin-Gaitsgory action of D(PSing(Y )) on IndCoh(Y ) • , see Example 2.2.8, yields By [AG18, Section 3.2.10], we also have Then the assertion follows by plugging in (2.3).
Remark 2.3.4. In the above proposition, we do not require that X be a global complete intersection.

Let
be a string of quasi-smooth schemes, with Y a global complete intersection as always in this section. Observe first that there is a natural correspondence where we emphasize that the right arrow

Corollary 2.3.6. Under the equivalences
of the above proposition, the pullback functor ξ !,0 is induced by the D ⇒ -module pull-push along the correspondence (2.5).
Proof. This is simply because the equivalence is functorial in X under pullbacks.

Strong spectral gluing
In this section, we construct our glued DG category The construction follows a general paradigm, which we eventually apply to the case of local systems.

Some preliminary constructions
obviously factors as Abusing notation, the first inclusion , denoted by the same symbol.
Remark 3.1.2. Let us show that the above functor is D(X)-linear. This fact will be used implicitly later, especially at the end of Lemma 3.2.9. Factoring Ψ as and noticing that the left arrow is D(X)-linear, it suffices to treat the second arrow. As the latter is right adjoint to a D(X)-linear functor, it is a priori lax D(X)-linear. To verify that such lax linearity is actually strict, it suffices to work smooth-locally on Y . Thus, we may assume that Y is a scheme. In this case, we are dealing with the functor which is evidently D(X)-linear.

A technical note.
Recall the standard functor Υ defined in [Gai13b] and [GR17]. There are two possible realizations of IndCoh 0 (Y ∧ X ): the one of [AG18] uses the embedding Ξ X : QCoh(X) → IndCoh(X), while the one of [Ber17b] uses the embedding Υ X : QCoh(X) → IndCoh(X). For a moment, let us denote the former by IndCoh (Ξ) 0 (Y ∧ X ) and the latter by IndCoh . These two DG categories are obviously equivalent (indeed, for X is quasi-smooth, the functors Ξ X and Υ X differ only by a shifted line bundle), however their functoriality under pullbacks is slightly different. To be consistent with [AG18], we use the Ξ-realization. For this reason, the functoriality of IndCoh 0 developed in [Ber17b] must be reinterpreted accordingly, that is, by conjugating with the natural equivalences σ : IndCoh with both triangles commutative.

3.1.5.
Assume now that X f − → Y is a map of quasi-smooth stacks over a third quasi-smooth stack Z. Denote by f : Z ∧ X → Z ∧ Y the induced arrow. We will tensor up the above diagram with H Z←Y over H(Y ), see Section 1.3.4. Thanks to the two natural equivalences 7 we obtain the diagram (3.1) with both triangles again commutative.
3.1.6. We need a notation for the DG category appearing on the bottom left of the above diagram and for the vertical arrow. We set: . When regarding IndCoh 0 (Z ∧ X ) Y −temp as a full subcategory of IndCoh 0 (Z ∧ X ), we refer to it as the Y -tempered subcategory. The next lemma gives an explicit characterization of IndCoh 0 (Z ∧ X ) Y −temp . Lemma 3.1.7. The inclusion IndCoh 0 (Z ∧ X ) Y −temp → IndCoh 0 (Z ∧ X ) restricts to the equivalence Proof. We proceed as in [AG18, Corollary 3.2.5]. A routine base-change computation shows that the adjunction induced by H(Y ) H Z←Y goes over, under the equivalence , to the monadic adjunction 7 these are instances of the main theorem of [Ber19b], see also the fourth item of the list in Section 1.3.4 As a consequence of the computation, the functor ( f ) IndCoh * preserves the IndCoh 0 -subcategories. It suffices to check that the latter adjunction restricts to an adjunction This is clear for the right adjoint. As for the left adjoint, it suffices to prove that the monad ( f ) ! ( f ) IndCoh * preserves QCoh(Y ∧ X ). The monad in question admits a nonnegative filtration with associated graded being the functor of tensoring with Sym(T QCoh Y /Z ), in the sense of the action of QCoh(Y ) on IndCoh(Y ∧ X ) by pullback. The assertion follows.

3.1.8.
We denote by arrow appearing in (3.1). Any map f : X → Y of quasi-smooth stacks over Z yields a cospan

3.1.9.
If Y happens to be a scheme, then IndCoh 0 (Z ∧ X ) Y −temp can be rewritten more simply as

This follows from the natural H(Y )-linear equivalence
which is valid when Y is a scheme, together with the (H(Y ), D(Y ))-bimodule structure of QCoh(Y ). Thus, in this case, the pullback ( f ) !,Y −temp is obtained by the D-module pullback D(Y ) → D(X) upon tensoring up.

Constructing the glued DG category
We retain the notation from the previous section.

3.2.1.
Regarding the quasi-smooth stack Z as fixed throughout, let us consider the full subcategory QSmooth /Z ⊆ Stk /Z spanned by quasi-smooth stacks mapping schematically to Z. We need to study the functoriality of the assignment from arrows in QSmooth /Z to cospans in H(Z)-mod.

3.2.2.
To this end, recall the ∞-category Tw(I) of twisted arrows associated to an ∞-category I, see [Lur11, Section 4.2]. Our convention is that is covariant in the first argument and contravariant in the second argument. The answer to the functoriality question posed in Section 3.2.1 is that there is a functor (to be constructed below) in Tw(QSmooth /Z ) to the cospan appearing in (3.2).

3.2.3.
To construct IndCoh Tw 0 , we need an auxiliary notion: the simplicial space of descending grids associated to an ∞-category E. For n ≥ 0, let Pictorially, Grid ↓ n (E) is the space of commutative diagrams (henceforth called descending n-grids) Grid ↓ n (E) is naturally a simplicial space, which we will denote by Grid ↓ • (E).

3.2.4.
Similarly, we have the notion of ascending grid, obtained from the above by reversing (only) the vertical arrows. Precisely, we consider We use this to define our functor IndCoh Tw 0 : Tw(QSmooth /Z ) op → H(Z)-mod: we will provide a simplicial assignment 8 of a descending n-grid grid ↓ n (X i ) in H(Z)-mod to any [n] ∈ ∆ and any string X 0 → X 1 → · · · → X n of quasi-smooth stacks mapping schematically to Z.

3.2.6.
To construct grid ↓ n (X i ), we proceed in two steps. We first construct an ascending version grid ↑ n (X i ), which is easier to handle. Secondly, we prove that each of the squares forming grid ↑ n (X i ) is vertically right adjointable: this means that the vertical arrows all admit continuous (and automatically H(Z)-linear, since H(Z) is rigid) right adjoints and that the resulting lax-commutative squares are commutative. This yields a well-defined descending grid grid ↓ n (X i ), and it will be clear that the assignment determines a map of simplicial spaces as desired.

3.2.7.
Let us now define grid ↑ n (X i ). • The DG category in the entry (i, j) is • For i ≤ j ≤ k, the horizontal arrow corresponding to (j, k) ← (i, k) is the pullback Thanks to Lemma 3.1.7, such pullback does indeed preserve the X k -tempered subcategories.
• For i ≤ j ≤ k, the vertical arrow corresponding to (i, k) ← (i, j) is the obvious inclusion In view of the functoriality of IndCoh 0 under pullbacks, this datum underlies an ascending n-grid. In fact, for each i ≤ j, the (!, 0)-pullback along Z ∧ X i → Z ∧ X j gives the diagonal arrow (j, j) → (i, i) pointing north-east, and it is clear that these diagonal arrows, together with the vertical ascending inclusions, determine the rest of the grid uniquely.

Consider the arrow N
which appears as a general square of grid ↑ n (X i ). In the following lemma, we prove the promised property of such a square.
Lemma 3.2.9. The above square is vertically right adjointable, meaning that the vertical arrows admit continuous right adjoints and that the lax commutative square obtained by changing the vertical arrows with these right adjoints is commutative.
Proof. The assertion about the continuity of the right adjoints, to be denoted simply by Ψ , is clear. Indeed, by tensoring up with H Z←X over H(X ), and the latter functor evidently admits a continuous right adjoint, given by the composition It remains to prove that the lax commutative square is commutative. As a first simplication, we can assume that the map X = Z is an isomorphism: indeed the above square is obtained from by tensoring up with H Z←X over H(X ). By pulling back to an atlas of X , we can assume that all the stacks in question are schemes (recall that all the maps to Z = X are schematic). Then, thanks to the observation of Section 3.1.9, the latter diagram can be rewritten as This diagram is manifestly commutative, as the vertical and horizontal arrows are "decoupled".

The gluing functor and the statement of the main theorem
In this section, we use the above construction to define our gluing functor and state our main theorem. Here we are using the notation "P − temp" instead of the more cumbersome "LS P − temp".

In the previous section, we have constructed a functor IndCoh
Remark 3.3.2. The 1-category Tw := Tw(Par) is easy to grasp: it is a poset consisting of 3 rank(G) elements. For G of semisimple rank one, Tw is simply a correspondence diagram For G of semisimple rank two, the Hasse diagram of Tw is where P and Q denote the two maximal parabolic subgroups. Like any category of twisted arrows, Tw has morphisms of two kinds: the ones that keep the first argument fixed (to be called morphisms of the first type) and the ones that keep the second argument fixed (to be called morphisms of the second type).

The limit DG category
is the glued DG category that will feature in the strong gluing theorem. To complete the statement of that theorem, it remains to exhibit an H(LS G )-linear functor from IndCoh N (LS G ) to lim(IndCoh Tw,Par 0 ). Proof. Unraveling the definitions, it suffices to verify that, for Q ⊆ P , the obviously commutative square

As a short digression, let us discuss the relation between lim(IndCoh
pullback pullback is vertically right adjointable, see Lemma 3.2.9. The proof goes as in that lemma: pass to an atlas of LS G and use the observation of Section 3.1.9 to reduce to a decoupled diagram.

3.3.7.
By pre-composing with the inclusion IndCoh N (LS G ) → IndCoh(LS G ), we finally get our gluing functor By construction, γ strong is assembled out of the H(LS G )-linear functors for any arrow [Q ⊆ P ] in Tw. Our main theorem reads: Thus, for G of semisimple rank 1, the theorem states that the commutative diagram is a fiber square. This is the diagram we have been considering in Theorem 1.1.7.

A D-module reformulation
To prove Theorem 3.3.8, we follow the strategy of [AG18]. In the first step, we use the D(Sing(Y )) ⇒ -action on IndCoh(Y ), present for a complete intersection scheme Y , to reduce the statement to a gluing statement for DG categories of D-modules on various schemes of singularities. The second step, perfomed in Section 5, uses the combinatorics of [AG18] where by abuse of notation s P denotes the singular codifferential of p P : LS P → LS G . In words, N Q⊆P is the stack of pairs (σ Q , A), where σ Q is a Q-local system and A a horizontal section of (u P ) σ Q .

4.1.2.
The functoriality of shifted cotangent bundles implies that the assignment [Q ⊆ P ] N Q⊆P extends to a functor Tw := Tw(Par) → Stk, given by compatible diagrams By pulling back along the natural maps we obtain a D(Sing(LS G ))-linear functor Remark 4.1.3. Since the maps appearing in (4.1) are proper, the RHS above can also be written as a colimit by passing to the left adjoints of the transition functors. Furthermore, since the maps µ Q⊆P are all proper, µ ! admits a D(Sing(LS G ))-linear left adjoint.

4.1.4.
For a technical reason (the fact that N dR is not 1-affine) that will appear evident later, we need to consider a slightly different functor: namely, we need to repeat the above construction after having pulled back to an atlas Y LS G . In order to apply the theory of Section 2, we will choose an atlas that is a global complete intersection. To do so explicitly, let us fix a point x ∈ X once and for all. The choice of x ∈ X gives rise to a map LS G → BG and, since X is assumed to be connected, to a canonical atlas Even more explicitly, we can write Y = Y × g pt, where Y is a certain smooth subscheme of LS RS G × g/G g, see [AG15, Section 10.6] for more details.

4.1.5.
Denote by Y Q and N the schemes obtained from LS Q and N by pulling back along the map Y LS G . We have

We set
where, by abuse of notation, we have denoted by s P : Y P × Y Sing(Y ) → Sing(Y P ) the singular codifferential of the map Y P → Y . Tautologically, N Q⊆P is the base-change along Y → LS G of the moduli stack N Q⊆P . The same procedure as above yields the D(Sing(Y ))-linear functor Remark 4.1.7. This theorem implies that the functor µ ! of (4.2) is an equivalence. For that, it suffices to prove that the natural maps µ ! µ ! → id and id → µ ! µ ! are isomorphisms. This can be checked after base-changing along Y LS G . Thanks to the fact that µ is pseudo-proper (see 4.2.1 below), this boils down to the statement of Theorem 4.1.6. 4.1.8. The above theorem, to be proven in in the remainder of this paper, is the main ingredient in the proof of Theorem 3.3.8. Assuming its validity for the time being, let us show how to deduce Theorem 3.3.8 from it.
As a preliminary observation, let us note that, since all the maps involved in the definition of µ ! Y are G m -equivariant, it makes sense to consider the functor (µ ! Y ) ⇒ , which is also an equivalence.

4.1.9.
We wish to show that the functor of Theorem 3.3.8 is an equivalence. To apply the theory developed in Section 2 and to connect with Theorem 4.1.6, we need to eliminate the stackyness of LS G and LS P . To this end, recall that the above functor is H(LS G )-linear (in particular, QCoh(LS G )-linear) and so it suffices to prove it is an equivalence after pulling back along the atlas Y LS G .

We have
and, by Section 3.1.9, Thus, it is enough to prove that the resulting functor is an equivalence.

4.1.11.
Since Y is a global complete intersection, Section 2.2 yields an action of D(Sing(Y )) ⇒ on IndCoh(Y ). Then and, by Proposition 2.3.3, These equivalences and the compatibilities of Section 2.3 imply that the functor γ strong Y is obtained from (µ ! Y ) ⇒ by tensoring up with IndCoh(Y ) over D(Sing(Y )) ⇒ . It follows that γ strong Y is an equivalence.

Two key tools
In this section, we state and prove two results that will be needed for the proof of Theorem 4.1.6. The first one is the fact that, under some properness conditions, equivalences of DG categories of D-modules can be checked on fibers. The second one, which is related to the first, is excision for D-modules.

4.2.1.
Let f : X → Y be a map of prestacks with Y a scheme. We say that f is pseudo-proper if where each f a : X a → Y is a scheme proper over Y . As pointed out in Remark 4.1.3, whenever f is pseudo-proper, the pullback functor f ! : D(Y ) → D(X) admits a left adjoint (to be denoted f ! ).

4.2.2.
Recall from [GR14] that DG categories of D-modules are defined only for Sch lft k , the 1-category of those (classical) schemes that are locally of finite type over the ground field k. Since we are going to consider field extensions, let us be more explicit with the notation: we denote by D /k : (Sch lft k ) op → DGCat k what has been denoted simply by D throughout. We are going to also consider the functor D /k : (Sch lft k ) op → DGCat k , with k a field extension of k. We have:

4.2.3.
For a k -point y of Y , we denote by (i y ) ! the functor where the right arrow is the !-pullback along the finite type map Spec(k ) → Spec(k ) × Spec(k) Y induced by y. Alternatively, the same functor (i y ) ! : D /k (Y ) → Vect k can be rewritten as The following fact, which can be proven by Noetherian induction, was used in [AG18, Lemma 6.1.7]; it will be crucial also in the proof of Proposition 4.2.5 below.
Proof. The "only if" direction is obvious. To prove the converse, let us assume that each functor ( f | y ) ! is an equivalence. We need to prove that the two natural transformations are equivalences. The latter has been dealt with in [AG18, Lemma 6.1.7] using Lemma 4.2.4, so let us focus on the former. It suffices to show that, for any M = {M a } ∈ D(X) and any b ∈ A, the resulting map is an isomorphism in D(X b ). We will prove that the arrow is an isomorphism in Vect k for any k -point i x : Spec(k ) → X b . This is enough in view of the above lemma.
Consider the following diagram (with cartesian outer rectangle): where ι b : X b → X is the structure map and y := f b (x) ∈ Y (k ). Denote by i y the composition of the two lower horizontal arrows. By extending scalars from k to k and renaming k with k, we can assume that x is a k-point. Since the outer rectangle is cartesian and f is pseudo-proper, f ! satisfies base-change against (i y ) ! . We obtain: Note that the point x ∈ X b (k) yields a section i x of f | y . Since ( f | y ) ! is an equivalence by assumption, it follows that ( f | y ) ! ( i x ) ! . Thus, the RHS of (4.5) is isomorphic to ( i x ) ! η ! (M), which is in turn is manifestly isomorphic to (i x ) ! (M b ).

4.2.6.
Let us put the above result in context. Set  Vect is an equivalence for all k and all (σ , A) ∈ N(k ), then (4.6) is an equivalence.
Example 4.2.8. In the case σ is the trivial G-local system, we obtain the statement of Theorem 1.3.11.

4.2.9.
When checking that (4.7) is an equivalence for a given k -point of N, we can replace N with its extension to k and then rename k by k. Thus, to prove Theorem 4.1.6, it suffices to check that the arrow (4.7) is an equivalence for any k-point (σ , A). Hereafter, the pair (σ , A) will always denote a k-point of N. is homologically contractible, that is, that the pullback functor Vect k −→ D(Spr σ ,A Glued,unip ) is fully faithful. To prove our main theorem, we need to show more: we need to show that the same functor is an equivalence. We will do so in Section 5 by revisiting the proof of the fully faithfulness given in [AG18]. Essentially, the only improvement to be made is the usage of the full force of the following excision result.
(The authors of [AG18] only use half of the statement.) Vect is an equivalence.
Proof. The properness of f implies that α admits a left adjoint, α L , which sends (F, V ) to (G, V ), where G is the colimit of To prove that α is an equivalence, we apply the Barr-Beck-Lurie theorem: since α is clearly (continuous and) conservative, it remains to show that α L is fully faithful. Denote by j : the open embedding and by C the DG category D(Y ) × D(Y 0 ) Vect. The notations j , U and C bear the obvious parallel meaning. We observe that C sits in the exact sequence of DG categories with functors defined as follows: Consider also the same exact sequence for C . By the assumptions, α : C → C extends to a functor out of exact sequences (that is, the four squares commute), with the two outer terms being equivalences. To prove that α L is fully faithful, it suffices to check that the two natural arrows are isomorphisms. Both claims are clear by inspection.
Remark 4.2.14. The DG category D(Y ) × D(Y 0 ) Vect may be regarded as the DG category of D-modules on the one-point compactification of Y − Y 0 .

Contractibility of glued Springer fibers
In this final section, we prove Theorem 4.1.6 and consequently Theorem 3.3.8. We follow [AG18, Sections 7-8] very closely.

Springer fibers
Recall Spr σ ,A Glued,unip , the prestack of glued Springer fibers, defined in Section 4.2.9 and in (4.9). By Corollary 4.2.7, it is enough to prove the following result, which is our goal until the end of the paper. Example 5.1.2. If A = 0, it is easy to verify that the prestack Spr σ ,A Glued,unip is isomorphic to pt = Spec(k). The argument appears in [AG18, Remark 7.1.10]. Hence, it suffices to treat the case A 0, which we assume from now on. Note that A 0 implies that Spr σ ,A Q⊆G,unip ∅ for any Q ∈ Par.
Example 5.1.3. Let σ be the trivial local system, so that A is just a nilpotent element of g. If A is regular, then it is elementary to verify that (at the level of reduced schemes) the latter statement corresponding to the well-known fact that a regular nilpotent element is contained in exactly one Borel subgroup. Hence, Spr σ ,A
Example 5.1.4. Let G = GL 3 and σ the trivial local system as before, but this time we take A to be a subregular nilpotent element. Denoting by P 1 and P 2 the two maximal parabolic subgroups, we have (again at the level of reduced schemes): so that Spr σ ,A Glued,unip is the colimit of the diagram pt P 1 → P 1 ∨ P 1 ← P 1 pt.
Note that P 1 ∨ P 1 is the scheme pushout of the correspondence P 1 ← pt → P 1 , which is different 10 from the prestack pushout of the same diagram. However, reasoning as in Proposition 4.2.5, the natural map from the latter to the former induces an equivalence on D-modules. In particular, we have so that D(Spr σ ,A Glued,unip ) is equivalent to the DG category of D-modules on the prestack colimit of the diagram pt P 1 ← pt → P 1 pt.
The latter colimit is evidently isomorphic to pt.

5.1.5.
For any P ∈ Par, recall the classical scheme Spr σ ,A P that parametrizes P -reductions σ P of σ with the property that A belongs to H 0 (X dR , p σ P ).
We have functors

Weyl combinatorics
It remains to prove Theorem 5.1.10, a task that will keep us busy until the end of the paper. We need some Weyl combinatorics: we refer to [AG18, Sections 8.1-8.3] for details and more on the notation.

5.2.1.
Let W denote the Weyl group and G and I the set of nodes of the Dynkin diagram. Let P 0 be the standard parabolic associated to A by the Jacobson-Morozov theorem: this makes sense because A 0 by assumption. Denoting by J 0 ⊂ I the subset corresponding to P 0 , define W := {w ∈ W : w −1 (J 0 ) ⊆ R + }.

5.2.3.
Since W contains a maximal element w 0 , the following proposition is enough to prove Theorem 5.1.10.

Proposition 5.2.4. The pullback functor
Vect −→ D(Spr σ ,A,≤w Glued ) is an equivalence for any w ∈ W . Proof. We proceed by induction on the length of w. The base case of w = 1 is settled by [AG18,Section 8.4.1], where it is proven that Spr σ ,A,≤1 Glued is isomorphic to pt. Next, let w 1 be fixed, and assume that the assertion holds for any w < w. Observe first that the pullback functor Vect −→ D(Spr σ ,A,<w Glued ) is an equivalence. Indeed, we tautologically have  Proof. We proceed in 5 steps.
Step 2. induced by those left adjoints.
Step 3. Recall the partition I = I 0 w I + w I − w , with I 0 w := I ∩ w −1 (R J 0 ) and I ± w := I ∩ w −1 (R ± − R J 0 ). Let Par w := {P ∈ Par : J P ⊆ I 0 w I − w }. The inclusion φ : Par w → Par admits a right adjoint ψ that sends P P , where P is the standard parabolic with J P = J P 0 − I + w . According to [AG18,Lemma 8.4.3], the tautological functor is an equivalence provided that, for any P ∈ Par , the natural arrow is an equivalence of DG categories. We will prove this statement in the next step.
Step 4. We need to show that, for w ∈ W − {1, w 0 }, the pullback functor  (2)], which states that the scheme of P -flags in position w with a given P 0 -flag is isomorphic to the scheme of P -flags in position w with the given P 0 -flag.