Deligne-Lusztig duality on the stack of local systems

In the setting of the geometric Langlands conjecture, we argue that the phenomenon of divergence at infinity on Bun_G (that is, the difference between $!$-extensions and $*$-extensions) is controlled, Langlands-dually, by the locus of semisimple $\check{G}$-local systems. To see this, we first rephrase the question in terms of Deligne-Lusztig duality and then study the Deligne-Lusztig functor DL_G^\spec acting on the spectral Langlands DG category IndCoh_N(LS_G). We prove that DL_G^\spec is the projection IndCoh_N(LS_G) \to QCoh(LS_G), followed by the action of a coherent D-module St_G which we call the {Steinberg} D-module. We argue that St_G might be regarded as the dualizing sheaf of the locus of semisimple $G$-local systems. We also show that DL_G^\spec, while far from being conservative, is fully faithful on the subcategory of compact objects.


Introduction and main results
The subjects of the present paper are: • the phenomenon of divergence at infinity on the stack Bun G ; • the locus of semisimpleǦ-local systems; • the Deligne-Lusztig functors on the two sides of the geometric Langlands correspondence.
In the introduction we explain how these items are related and state our main results: Theorems A, C, D, E, F, as well as the conditional proof of Conjecture B.
1.1. Divergence at infinity on the stack of G-bundles.
1.1.1. Denote by Bun G := Bun G (X) the stack of G-bundles on a smooth complete curve X defined over . Here and always in this paper, denotes an algebraically closed field of characteristic zero and G a connected reductive group over . Note that Bun G is never quasi-compact (unless G is the trivial group): by bounding the degree of instability of G-bundles, one obtains an exhausting sequence of quasi-compact open substacks of Bun G .
The failure of quasi-compactness leads to the phenomenon of divergence at infinity on Bun G , to be explained below. The goal of this paper is to describe this phenomenon from the Langlands dual point of view.
1.1.2. We denote by D(Y) the DG category of D-modules on an algebraic stack Y, see e.g. [21]. In particular, we are interested in D(Bun G ) and in its variants discussed below.
Given U ⊆ Bun G a quasi-compact open substack, denote by j U the inclusion functor. Let D(Bun G ) * -gen be the full subcategory of D(Bun G ) generated under colimits by objects of the form (j U ) * ,dR (F U ), for all quasi-compact opens U ⊆ Bun G and all F U ∈ D(U ). Similarly, let D(Bun G ) ! -gen be the full subcategory of D(Bun G ) generated under colimits by objects of the form (j U ) ! (F U ), for all quasi-compact opens U ⊆ Bun G and all F U ∈ D(U ) for which (j U ) ! (F U ) is defined.
MSC 2010: 14D24, 14F05, 18F99, 22E57. [12] that D(Bun G ) ! -gen ≃ D(Bun G ): that is, any object can be written as a colimit of !-extensions from quasi-compact opens. The phenomenon of divergence at infinity on Bun G is the fact that the inclusion D(Bun G ) * -gen ⊆ D(Bun G ) is strict, as soon as G is not abelian. This statement is an 1.2.4. The locus LS sš G of semisimpleǦ-local systems is only constructible in LSǦ, hence the formal completion of LSǦ at LS sš G does not make sense. Nevertheless, in Section 1.4 we will define a full subcategory QCoh(LSǦ) ss ⊆ QCoh(LSǦ) which plays the role of the category of quasi-coherent sheaves on LSǦ settheoretically supported on LS sš G . With such definition, we propose:

It is proven in
Conjecture B. Under Langlands duality, D(Bun G ) * -gen is equivalent to QCoh(LSǦ) ss .
We will "prove" this conjecture in Section 1.5 by first reformulating it as Conjecture B ′ , and then by showing that the latter follows from combining the geometric Langlands conjecture with a natural conjecture about Drinfeld's compactification of the diagonal of Bun G .
1.3. Cuspidal objects, ⋆-extensions, and tempered objects. This section, which can be skipped by the reader, explains how Conjecture B is related to the more standard versions of the geometric Langlands conjecture. where D(Bun G ) cusp is the DG category of cuspidal D-modules on Bun G . 1.3.2. Let us comment on the inclusions on the automorphic side. The third inclusion is the only tautological one. The first inclusion follows from [11,Proposition 1.4.6]. The second inclusion appears to be nontrivial: it says that any * -extension is tempered. While the present paper was undergoing the publication process, a proof of this fact has become available, see [7]: it relies on some of the methods of [5], but it required a different point of view on the notion of temperedness. Remark 1.3.3. Note that in this case an obvious fact on the spectral side, namely the inclusion QCoh(LSǦ) ss ⊆ QCoh(LSǦ), informed us about something that is not evident on the automorphic side (namely, the fact that any * -extension is tempered). For an instance of the inverse direction, the reader might look ahead at Theorem E and the remark following it. the induction functors. In spite of the notation, the map i P : LS M → LS P is not at all an embedding. Yet, by the contraction principle, the functor (i P ) * ,dR : D(LS M ) ֒→ D(LS P ) is fully faithful. We will recall the contraction principle in Section 5.1.

1.4.2.
As a preliminary step, we define the full subcategory D(LS G ) ss ⊆ D(LS G ) of D-modules supported on LS ss G : an object F ∈ D(LS G ) belongs to D(LS G ) ss iff (p P ) !,dR (F) ∈ (i P ) * ,dR D(LS M ) , for any P .
Note that such definition mimics the definition of semisimple G-local systems: σ is semisimple iff, whenever it is reducible to P , it is also reducible to M . Next, define QCoh(LS G ) ss to be the cocompletion of the essential image of the action functor Here are some facts that support this definition: • If F ∈ D(LS G ) ss , then it is immediately checked that its (!, dR)-fiber at σ ∈ LS G ( ) is zero whenever σ is not semisimple. Similarly, if F ∈ QCoh(LS G ) ss , then its * -fiber at σ ∈ LS G ( ) is zero whenever σ is not semisimple. • In Section 1.4.7, we will define the Steinberg D-module St G and its underlying quasi-coherent sheaf St G ∈ QCoh(LS G ). The former plays the role of the dualizing sheaf on the non-existent stack LS ss G , the latter plays the role of the structure sheaf of the non-existent formal completion (LS G ) ∧ LS ss G . For instance, the geometric fibers St G | σ are zero for σ non-semisimple and 1-dimensional (but sitting in varying cohomological degree) if σ is semisimple.
• As a consequence of Theorem D ′ , any skyscraper (in either the D-module or the quasi-coherent sense) at a semisimple local system belongs to D(LS G ) ss or QCoh(LS G ) ss .
1.4.4. The next two theorems provide alternative characterizations of QCoh(LS G ) ss and D(LS G ) ss . Given a cocomplete monoidal symmetric DG category (C, ⊗), recall the notion of principal monoidal ideal generated by c ∈ C: this is the full subcategory of C consisting of objects in the essential image of c ⊗ − : C → C. Note that a principal monoidal ideal might not be closed under colimits.
Theorem C. The full subcategory QCoh(LS G ) ss ⊆ QCoh(LS G ) is a principal monoidal ideal, generated by the Steinberg object St G (defined below). Remark 1.4.5. In Section 6, we will spell out the definition of Div G , prove Theorem C ′ and then deduce Theorem C from it. The statement of Theorem C ′ , which we called "divisibility by the Steinberg object" was inspired by [25,Theorem 1.1]. The latter theorem shows that, in the modular representation theory of finite groups of Lie type, the ideal of projective representations is principal and generated by the Steinberg module.
Remark 1.4.6. Since QCoh(LS G ) ss is by its very definition closed under colimits, Theorem C implies that the same holds true for the principal monoidal ideal generated by St G . This is a consequence of Theorem E, see Lemma 6.2.2 for the proof.
On the other hand, in the D-module setting of Theorem C ′ , the existence of Div G immediately implies that the principal monoidal ideal generated by St G is closed under colimits.
1.4.7. Let us now define the Steinberg objects St G ∈ D(LS G ) ss and St G ∈ QCoh(LS G ) ss . As mentioned, they are precise analogues of a well-known object of classical representation theory (see, e.g., [27], [10], [25], [22]). By definition, St G is the coherent D-module defined as where Par ′ be the poset of proper standard (relative to a chosen Borel B, fixed throughout) parabolics of G.
Then we set St G to be the quasi-coherent sheaf underlying the Steinberg D-module St G ∈ D(LS G ).
1.4.8. From the above formula, it is not even clear that St G belongs to D(LS G ) ss , let alone a generator.
To prove that St G ∈ D(LS G ) ss (and consequently that St G belongs to QCoh(LS G ) ss ), we will show that the Steinberg construction enjoys the following "functional equation", which relates St G to the Steinberg object for a Levi subgroup M ⊆ G.
Theorem D. For any parabolic P ⊆ G with Levi M , there is a canonical isomorphism in D(LS P ).
1.4.9. Formula (1.2) allows to compute the !-fibers of St G : where u is the Lie algebra of the unipotent radical of a parabolic P with Levi M .
Example 1.4.10. In particular, St G | σ ≃ when σ is irreducible. This is obvious from the definition, as Example 1.4.11. At the other extreme, for σ triv the trivial G-local system, (1.3) yields where |R| is the number of roots of G. This can also be seen directly: indeed, and the latter can be simplified with the help of some standard Weyl combinatorics.
1.4.12. The functor St G ⊗ − : QCoh(LS G ) → QCoh(LS G ) ss is very far from being an inclusion: as shown above, it annihilates skyscrapers at non-semisimple G-local systems, so it is not even conservative. Thus, the next result comes perhaps as a surprise.
Recall that Coh N (LS G ) denotes the (non-cocomplete) full subcategory of QCoh(LS G ) consisting of coherent sheaves with nilpotent singular support (see [1]). Thus, QCoh(LS G ) ss contains a "hidden" copy of Coh N (LS G ), which we will later denote as Coh St N (LS G ).
Remark 1.4.13. This statement is the Langlands dual of an evident statement on the automorphic side: the fact that the composition of the miraculous and the naive duality is fully faithful when restricted to compact objects. We will explain this, as well as the relation between miraculous duality, Deligne-Lusztig duality and the Steinberg object, in the next section.
1.5. Deligne-Lusztig duality and the proof of Conjecture B.
1.5.1. Let C be a dualizable cocomplete DG category. Recall that functors from C ∨ → C are given by "kernels" in C ⊗ C. In the case C = D(Y) with Y a quasi-compact 3 algebraic stack, the kernel provides a self-duality equivalence Ps- When Y is not quasi-compact, Ps-Id * is not an equivalence, unless the closure of any quasi-compact open of Y is itself quasi-compact (see [12]). In particular, Ps-Id * : D(Bun G ) ∨ → D(Bun G ) is never an equivalence when G is not abelian.
to be the functor determined by the kernel The stack Y is said to be miraculous if Ps-Id ! is an equivalence. By [12], Bun G is miraculous; moreover it contains an exhausing sequence of miraculous quasi-compact opens.
1.5.3. Let us consider the composition of the miraculous and the naive duality, that is, the functor The essential image of T BunG is easy to identify and relevant to our discussion: indeed, in Section 2.1, we will prove that where is the standard adjunction. In short: the functor T Bun G [−d G ] is Langlands dual to the composition of temperization with the action by StǦ. 3 The correct technical condition is QCA, see [14].
1.5.5. Let us explain how this statement ought to follow from the Langlands conjecture. On the automorphic side, it was conjectured that where DL G is the Deligne-Lusztig functor Here, the functors Eis enh P and CT enh P are the enhanced Eisenstein series and constant term functors, see [18,Section 6.3]. We will not need their definition, hence we do not recall it.
Remark 1.5.6. Let us briefly review the history of (1.5). For G = SL 2 , this was conjectured by V. Drinfeld and J. Wang, see [15,Appendix C]. For G arbitrary, we learned the statement from a private communication with D. Gaitsgory; the statement essentially appears in [28, Section 6.6 and especially Remark 6.6.5]. Finally, while the present paper was undergoing the publication process, (1.5) was proven by L. Chen in [9].
In view of [9], the following remark is obsolete; we leave it for reference.
1.5.9. With this theorem proven, the assertion of Conjecture B ′ is a corollary (modulo the geometric Langlands conjecture) of the combination of (1.4) and Theorem C.
Remark 1.5.10. In the course of the proof of (1.4), we will see that, while T BunG is not even conservative, it is nevertheless fully faithful on compact objects. Hence, the same property must be true for DL G and DL spec G . Combining this with the statement of Theorem F led us to the statement of Theorem E.
1.6. Restoring the "duality". Theorem F implies that the Deligne-Lusztig functor DL spec G is not a duality. However, Theorem E suggests a way to modify DL spec G to make it into an equivalence.
is the essential image of the fully faithful functor appearing in Theorem E. We also define This DG category comes with a tautological essentially surjective functor Ψ St : 1.6.2. Theorem E shows that the action of St G yields an equivalence DL spec,enh , which is ought to be Langlands dual to the inverse of the miraculous duality. Likewise, Ψ St is Langlands dual to the naive duality.
Theorem F shows that the square is commutative. Langlands dually (and changing G withǦ), the above commutative diagram ought to read as where the tensor product on the bottom line denotes the action of QCoh(LSǦ) on D(Bun G ) given by the vanishing theorem of [18, Section 4.5].
1.7. Compatibility with Eisenstein series. To conclude the introduction, we ask how the enhanced Deligne-Lusztig duality interacts with Eisenstein series. (This can be skipped by the reader, as it will not be used anywhere in the paper.) In other words, we wish to describe the rightmost vertical functor in the following commutative diagram: To this end, consider the functor 4

Theorem D shows that such functor sends Coh
Ind-completing, we obtain a functor Eis St P that makes the square commutative by inspection.
1.8. Structure of the paper. The rest of the paper is devoted to proving our main results, in a different order than the one presented in the introduction: Theorem A in Section 2, Theorem C in Section 6, Theorem D in Section 4, Theorem E in Section 5 and Theorem F in Section 3.
1.9. Conventions and notation. We will mostly use the conventions of [5, Section 2] and [6]. 4 As usual, the notation F denotes the quasi-coherent sheaf underlying the D-module F.

Divergence at infinity on Bun G
In this section, we give details on the phenomenon of divergence at infinity on the stack Bun G and prove Theorem A.

It is established in [12] that any quasi-compact open substack of Bun G is contained in a quasi-compact
open substack U with the following remarkable property: the !-pushforward (j U ) ! along the open embedding Quasi-compact opens of Bun G with this property are called cotruncative. The actual construction of such open substacks is not important for us: we refer to [12] for details. We denote by Cotrnk the 1-category of cotruncative open substacks of Bun G ; any finite union of cotruncative substacks is cotruncative, so that Cotrnk is filtered.
2.1.2. Another property of Bun G of similar kind is the fact that the functor (p Bun G ) ! : D(Bun G ) → Vect is well-defined. This follows from the contractibility of the space of rational maps into G, together with the ind-properness of the Beilinson-Drinfeld Grassmannian (see [20,Corollary 5.3.2] for details).

Terminology. When we say that
Without loss of generality, we can assume such U to be cotruncative. The term * -extension is used accordingly.

It is clear that
Moreover, any compact object of D(Bun G ) is of the form (j U ) ! (F U ) for some U ∈ Cotrnk and some compact 2.1.6. Recall now the miraculous duality of Bun G and the functor T Bun G := Ps-Id * • Ps-Id −1 ! . It is proven in [12,Lemma 4.5.7] that any cotruncative open substack of Bun G is also miraculous. For any QCA stack Y, the functor Ps-Id Y, * is an equivalence: this is our standard way to identify D(Y) with its dual. Hence, for U ∈ Cotrnk (and in fact for any miraculous QCA stack), we regard the functor Ps-Id U,! as a self equivalence of D(U ).
2.1.7. Thanks to [12,Lemma 4.4.12], for any U ∈ Cotrnk, we have It follows that T BunG is fully faithful on !-extensions (in particular: on compact objects), and thus, by taking colimits in the first variable, fully faithful on pairs (any, !-ext). The latter means that, for any F ∈ D(Bun G ) and any * -extension (j U ) ! (F U ), the functor T BunG yields an isomorphism Remark 2.1.8. On the other hand, T Bun G is not fully faithful on the entire D(Bun G ). In fact, it is not even conservative, as To show this, follow the argument of [19] and invoke [5, Corollary 1.4.2] when proving that ω Gr G is infinitely connective.
Proof. Any object of D(Bun G ) is a colimit of !-extensions from cotruncative (hence miraculous) open substacks: (2.1) then shows that the essential image is contained in D(Bun G ) * -gen . By the same formula, any * -extension belongs to the essential image of T Bun G . It remains to show that such essential image is closed under colimits. In other words, we need to show that, for any index ∞-category I and any functor Without loss of generality, we can assume that each U i is cotruncative. Then the assertion follows from the fully faithfulness of T Bun G on !-extensions.

Proof of Theorem
A. The following result shows that the inclusion D(Bun G ) * -gen ⊆ D(Bun G ) is actually very strict (for G non-abelian): any object of D(Bun G ) * -gen has no de Rham cohomology with compact supports.
Proof. We proceed in six steps.
Step 1. Without loss of generality, we may assume that U is cotruncative. By adjunction, we need to show that Hom D(BunG) ((j U ) * F U , ω BunG ) ≃ 0 for any F U ∈ D(U ) cpt . Tautologically, we have where j U→U ′ : U ֒→ U ′ is the structure inclusion.
Step 2. Now, note that the functor (j U→U ′ ) * : D(U ) → D(U ′ ) admits a continuous right adjoint, which will be denoted by (j U→U ′ ) ? . This follows from the definition of cotruncativeness: indeed, the functor (j U→U ′ ) ! is clearly defined and (j U0→U ) ? is tautologically its dual (under the standard self dualities of the DG category of D-modules on a QCA stack, see [14]).
Step 3. Hence, Thus, the theorem is equivalent to proving that, for any U , we have: Step 4. Let k Bun G be the constant sheaf on Bun G , that is, the Verdier dual of ω Bun G . By smoothness, we This is immediate from the discussion of Section 2.1.7 and the remark following it.
Step 5. Starting from (2.2), we obtain that The objects appearing on the RHS are all coherent: hence, we can apply Verdier duality to obtain Step 6. By adjunction (using cotruncativeness), we rewrite the LHS as and further as which is what we were looking for.
2.2.2. As a corollary of the vanishing of (p Bun G ) ! • j * , we deduce that, for any F ∈ D(Bun G ) and any Z = Bun G −U with U cotruncative, we have This means that F and any of its "tails" have the same cohomology with compact support. In particular, taking F = ω Bun G in the above formula and dualizing, we obtain that pullback in de Rham cohomology yields the isomorphism

Proof of Theorem F
Since from now on we only consider the spectral side of geometric Langlands, let us switchǦ with G and consider the endo-functor DL spec G of IndCoh N (LS G ). First, we need to show that such functor annihilates the subcategory of IndCoh N (LS G ) right orthogonal to QCoh(LS G ). This will already imply that DL spec G factors as where the middle arrow is the action by a D-module on LS G . Second, we will identify such D-module with the Steinberg D-module St G .
3.1. Singular support and enhanced Eisenstein series. We assume familiarity with the theory of singular support for coherent sheaves on quasi-smooth stacks, see [1] and [2]. We also assume some familiarity with the theory of H, as developed in [4] and in [6]. The latter two references are not strictly necessary for the proof, but they help streamline the argument.
3.1.1. As the stack LS G is quasi-smooth, ind-coherent sheaves on it get assigned a singular support in Sing(LS G ). Recall that Sing(LS G ) parametrizes pairs (σ, A) where σ is a G-local system and A a horizontal section of the flat vector bundle g * σ . Let N ⊂ Sing(LS G ) denote the global nilpotent cone, that is, the closed conical locus defined by the requiring that A be nilpotent.
These notations all agree with the ones used in [2].
3.1.3. We need to recall the rule of propagation of singular support under pushforwards, see [1,Section 7]. Given f : X → Y as above, we have two natural maps at the level of the spaces of singularities: s f is the singular codifferential, while t f is simply the projection to the second component. Now, let M ⊆ Sing(X) and N ⊆ Sing(Y) be closed conical subsets.
Proposition 3.1.4. With the above notation, assume further that Then: Proof. The first item is [1, Lemma 8. 3.1.6. The enhanced constant term functor CT enh,spec P is, by definition, the right adjoint to Eis enh,spec P . Tautologically, it can be expressed as the composition CT enh,spec where the rightmost functor is the natural projection (right adjoint to the obvious inclusion).
3.1.7. By adjunction, the assignment P Eis enh,spec P • CT enh,spec P upgrades to a functor By adjunction again, we obtain a natural arrow colim P ∈Par ′ Eis enh,spec whose cone is by definition the functor DL spec G .

3.2.
Proof of Theorem F. The proof rests on a contractibility statement proven in [2], to which we reduce via a "microlocal" argument as in [6, Section 2.2-2.3].
3.2.1. By construction, for any P ∈ Par, the functor Eis enh,spec P • CT enh,spec P commutes with the action of H(LS G ), so the same holds true for DL spec G . Hence, we expect the functor to be given by the action of an object F DL ∈ D(N) ⇒ : indeed, we conjecture that

3.2.2.
To work around this conjecture, we work on the smooth atlas LS x G ։ LS G obtained by choosing a point x ∈ X and by considering G-local systems with a trivialization at x. By [1, Section 10.6], we know that LS x G is a global complete intersection scheme. The reason this fact is useful is that, for Y a global complete intersection scheme, we know that D(Sing(Y )) ⇒ acts on IndCoh(Y ), see [6, Section 2.2]. Let us denote such action by * .

A notational convention.
For Z a space over LS G , denote by Z x := Z × LSG LS x G its pullback along the atlas. Similarly, for a map f : Z 1 → Z 2 of spaces over LS G , denote by f x : Z x 1 → Z x 2 the base-changed map. For instance, is the natural induction map.

Now consider the DG category IndCoh
, which is acted upon by D(N x ) ⇒ . By "pulling-back" the constructions of Sections 3.1.5-3.1.6 to LS x G , we obtain comonads Eis enh,spec,x P • CT enh,spec,x P . Indeed, observe that the rule of propagation of singular support yields a functor Accordingly, we have a resulting functor DL spec,x 3.2.5. We fix a G-equivariant identification g * ≃ g once and for all, so that A will be always regarded as a horizontal section of the adjoint bundle. Consider the stack N P ⊆ LS P × LS G Sing(LS G ) consisting of pairs (σ P , A P ∈ H 0 dR (X, p σP )) for which A P is nilpotent. Note that the base-change of p P along Sing(LS G ) → LS G restricts to a map p Sing P : N P → N.
is given, up to shift of grading, by the object Proof. The comonad in question is the result of a general construction that takes the map (3.1) and the sets N x P,M , N x as inputs. Since (3.1) is a proper map of quasi-smooth schemes, with the target a global complete intersection, we will be able to use the theory developed in [6, Section 2].
Here is the general paradigm that we will apply. Let f : X → Y be a proper map of quasi-smooth schemes, with Y a global complete intersection. Let Explicitly, this is given by the composition We use "microlocality" (i.e., the following equivalences, proven partly in [2, Section 3] and partly in [6, Under these equivalences, the adjunction in question is tensored up (up to a shift of grading) from N → N is the obvious (proper) projection.
Coming back to our case, we immediately 6 see that Clearly, this object is the pullback along LS x G → LS G of an object F DL ∈ D(N), obtained by removing the decorations "x" in the above formula.
where i : LS G ֒→ N is the inclusion of the zero section. By base-change, the latter simplifies to the Steinberg object : St G := cone colim P ∈Par ′ (p P ) * ,dR (ω LS P ) −→ ω LS G ∈ D(LS G ). 6 One needs to unravel the effect of the identification g * ≃ g: under such identification, the P -representation g * × p * m * corresponds to g × g/u (p/u) ≃ p, the adjoint P -representation.

Proof of Theorem D
In this section, we use some Weyl combinatorics to prove the main property of St G , that is, Theorem D.
Let us recall the statement: for P 0 a parabolic subgroup of G with Levi M 0 , we need to construct a canonical isomorphism is the natural map and rk denotes the semisimple rank of a reductive group. We will later deduce Theorem D ′ which describes the geometric fibers of St G .

4.1.
Preliminaries. We will use some standard Weyl combinatorics: we refer to [2, Sections 8.1-8.2] for a handy review and for the notation we use. In particular, R (respectively, R + ) denotes the set of roots of G (respectively, positive roots with respect to the chosen B). For a parabolic P ⊆ G, let J P be the subset of the Dynkin diagram associated to P (e.g., J B = ∅).

4.1.1.
In the proof that follows, we assume that P 0 is a proper standard parabolic. If P 0 is not standard, the strategy is the same, up to multiplying w ′ 0 by an appropriate element of W . Below, we abuse notation and write J 0 in place of J P0 .

Let
The quotient stack P 0 \G/P has strata indexed by W ′ P := {w ∈ W ′ : w(J P ) ⊆ R + }. For w ∈ W ′ (but not necessarily in W ′ P ), the notations (P 0 \G/P ) ≤w and (P 0 \G/P ) <w have their evident meanings. We also set

4.1.3.
Recall that W ′ has a unique longest element w ′ 0 , characterized by the fact that w ′ 0 (R + ) ∩ R + = R + J0 . Alternatively: w ′ 0 is the product w 0,P0 · w 0 , where w 0,P0 and w 0 are the longest elements of W M0 and W respectively. From this expression, it is clear that (w ′ 0 ) −1 sends the simple roots of S P0 to simple roots; we Consequently, 4.1.4. Consider the mapping stack Y P := Maps(X dR , P 0 \G/P ) and its closed substacks Y P,≤w := Maps(X dR , (P 0 \G/P ) ≤w ).
Tautologically, we have: Define also Y P,<w and Y P,w is a similar way. For instance, we have Denote by π P : Y P −→ LS P0 π P,≤w : Y P,≤w −→ LS P0 π P,<w : Y P,<w −→ LS P0 π P,w : Y P,w −→ LS P0 the obvious maps.
Example 4.1.5. We have seen above that Y P,w ≃ LS P w ∩P0 whenever w(J P ) ⊆ R + . In this case, π P,w is the induction map i P w ∩P0→P0 .
4.2. The proof. We are now ready to construct the natural isomorphism appearing in (4.1).
Proof. It suffices to show that the map colim P ∈Par ′ (π P,<w ) * ω YP,<w −→ colim P ∈Par ′ (π P,≤w ) * ω Y P,≤w is an isomorphism in D(LS P0 ). As argued in [2, Lemma 6.1.7], this can be checked at the level of geometric points, that is, after pulling back to a P 0 -local system σ P0 → LS P0 . Observe that On the other hand, we tautologically have , the open-closed fiber sequence, combined with the discussion of Section 4.1, yields 4.2.5. To conclude our proof, it remains to simplify the RHS of . The proof of this latter claim amounts to applying the following general lemma to the functor where U in the unipotent radical of a parabolic with Levi M and u = Lie(U ).
Proof. The map σ : pt → LS G factors as pt σP − − → LS P pP − − → LS G , where σ P is the P -local system induced by σ M . Then base change yields is the DG scheme of M -reductions of σ M × M P . The classical scheme underlying Y is isomorphic to the vector space H 0 (X dR , u σM ). In particular, Y cl is homologically contractible and smooth of dimension h 0 (X dR , u σM ). The assertion follows.
Viceversa, suppose that σ is not semisimple: this means that σ ≃ σ P × P G for some P ∈ Par ′ and some P -local system σ P which is not M -reducible. Then St G | σ = 0 by Theorem D.

Consider the functor
In this section, we will prove Theorem E, which states that such functor is fully faithful when restricted to Coh N (LS G ). As a key tool, we apply the second adjunction (an instance of Braden's theorem) in the context of D(LS G ).

5.1.
Braden's theorem and contraction principle for local systems. In this section, we render some of the material of [8], [11], [13] to the setting of G-local systems. defined by (p P ) * ,dR • (q P ) !,dR . Note that de Rham push-forward (p P ) * ,dR is continuous since the map p P is schematic. Our goal is to prove that Eis D P, * admits a left adjoint. Such left adjoint is at least partially defined: it is given by the formula

Consider the Eisenstein series functor
The question is then to show that this functor is defined on the entire category D(LS G ). The push-forward (q P ) * ,dR is continuous because the map q P is safe in the terminology of [14]. Proof. The proof is an instance of Braden's theorem. For instance, one might copy the one given in [11] for G-bundles.

5.1.4.
Let us also record the following consequence of the contraction principle. For an appropriate cocharacter γ : G m → Z(M ), the resulting G m -action on LS P is contracting (and trivializable), with fixed locus LS M . This implies that (i P ) * ,dR is fully faithful, with left adjoint isomorphic to (i P ) * ,dR ≃ (q P ) * ,dR . Similarly, (q P ) !,dR is fully faithful, with left adjoint isomorphic to (i P ) !,dR . For the proofs, see [11, Section 4.1.6].

D-module functoriality. This is a quick reminder of the basic D-module functors on QCA algebraic
stacks. Recall the conventions of Section 1.9.
5.2.1. We denote by (ind R , oblv R ) the induction/forgetul functors for right D-modules. Recall that ind R is dual (as well as left adjoint) to oblv R , with respect to the standard self dualities of D(Y) and IndCoh(Y). The forgetful functor oblv R intertwines the two types of !-pullbacks. By duality, ind R intertwines IndCohpushforwards with renormalized de Rham push-forwards, see [14].

5.2.2.
We also have the induction/forgetful adjunction (ind L , oblv L ) for left D-modules. This adjunction is valid only for bounded (that is, eventually coconnective) stacks; we are not in danger, as we will only apply it to quasi-smooth stacks. The forgetful functor oblv L intertwines * -pullbacks of quasi-coherent sheaves with !-pullbacks of D-modules.

5.2.3.
It remains to discuss the interaction between ind L and (QCoh, * )-pushforwards. First off, we have For Y a Gorenstein (for example, quasi-smooth) stack, we write L Y for the shifted line bundle Ψ(ω Y ) ∈ QCoh(Y). Abusing notation, for H an affine algebraic group, we set L H := L LS H .
Lemma 5.2.4. Let f : Y → Z be a map between Gorenstein QCA stacks. Then Proof. To check the first formula, let us pass to dual functors on both sides: we need to establish a functorial isomorphism or equivalently (thanks to oblv R = Υoblv L ), The assertion is now manifest, as Ψ Z Υ Z = L Z ⊗ −. The second formula is proven in exactly the same way.
Corollary 5.2.5. Let f : Y → Z be a proper (in particular, schematic) map between Gorenstein QCA stacks. Then, for Q ∈ QCoh(Y) and F ∈ D(Z), there is a natural isomorphism

5.3.
Setting up the proof.
5.3.1. It will be actually convenient to slightly reformulate the result. Let us introduce the following terminology: we say that a functor F : C → D is fully faithful on a pair (c, c ′ ) ∈ C × C iff it induces an isomorphism

It is clear that following theorem implies (and in fact it is equivalent to) Theorem E.
Theorem 5.3.3. The functor is fully faithful on pairs of the form (c, c ′ ) ∈ QCoh(LS G ) × Coh N (LS G ).
We will prove this theorem by induction on the semisimple rank of G. For T , the assertion is obvious: this is the base of the induction. We henceforth assume that the theorem is true for any proper Levi subgroup of G.
is an isomorphism for F ′ as above and F arbitrary. Let us distinguish two cases: P = G (to be treated next, in Section 5.4) and P = G (to be treated later, in Section 5.5).
5.4. The first case: P = G.
5.4.1. Let P be a proper parabolic. We need to show that, for F ∈ QCoh(LS G ) and F P ∈ Perf(LS P ), the natural map is an isomorphism. By adjunction, we have: Thus, the assertion reduces to the following one. is fully faithful.
Proof. It suffices to prove that the map is an isomorphism for any F ∈ Perf(LS P ), where act ⊗ denotes the action of QCoh on IndCoh.
Step 1. Let us start manipulating the RHS. By adjunction and then projection formula, it is isomorphic to Let us now recall that, by the contraction principle, the functor (i P ) * is fully faithful. Hence, the RHS of (5.4) is isomorphic to Step 2. Our next goal is to eliminate the two occurrencies of St M from the Hom space above. This will be done by a diagram chase, together with the induction hypothesis. Consider the following cartesian square: Base-change along this diagram, together with the (ind R , oblv R ) adjunction, yields The two ind-coherent sheaves appearing on the RHS belong to the full subcategory Υ(QCoh(LS M )): this is obvious for the right one; as for the left one, it suffices to notice that ξ IndCoh * sends QCoh((LS P ) ∧ LSM ) to QCoh(LS M ) since q P is quasi-smooth. Hence, we can use the induction hypothesis (that is, Theorem 5.3.3 for the group M ) to obtain which is in turn isomorphic to Hom D(LSM ) i ! P (ind R (ΥF)), ω LS M by reasoning backwards.
Step 3. Recall that, by the contraction principle again, the functor (q P ) ! : D(LS P ) → D(LS M ) is well-defined and isomorphic to i ! P . We conclude that The RHS is now manifestly isomorphic to Hom QCoh(LSP ) (F, O LS P ), as desired.
5.5. The second case: P = G.
5.5.1. The next case is the one with P = G, so that F ′ is perfect (while F is still arbitrary). We need to show that the map 5.5.2. Without loss of generality, we may assume that F ′ ≃ O LS G . Thus, we need to prove that the arrow is an isomorphism for arbitrary F. It suffices to do this for F running through a fixed collection of generators of QCoh(LS G ). Thus we assume that: • either F = j * (F 0 ), with j : LS irred G ֒→ LS G the open substack of irreducible G-local systems and • or F = (p P ) * (F P ) with P ∈ Par ′ and F P ∈ Perf(LS P ).
We treat these two subcases separately.
formula. Hence, we just need to show that the map is an isomorphism. Equivalently, we need to show that Hom QCoh(LSG) j * (F 0 ), colim P ∈Par ′ p P, * ω LSP ≃ 0. This fact is a consequence of the next lemma.
Then we need to show that any object of D(LS irred G ) is left orthogonal to Eis D P, * (ω LSM ) ≃ (p P ) * ω LSP . This follows immediately from the "second adjunction", that is, Theorem 5.1.3. 5.5.5. Finally, let us assume that F = (p P ) * (F P ) in (5.5). We need to show: Proposition 5.5.6. For any F P ∈ Perf(LS P ), the functor St G ⊗ − yields an isomorphism Proof. By adjunction, this is equivalent to checking that St G ⊗ − yields an isomorphism Thanks to (5.1), which in our case looks like

the LHS becomes
Similarly, the RHS side becomes Then we are back to the statement of Theorem 5.4.2.

Proof of Theorem C
We wish to show that St G and St G are generators of the monoidal ideals QCoh(LS G ) ss ⊆ QCoh(LS G ) and D(LS G ) ss ⊆ D(LS G ), respectively. The claim for St G , treated in Section 6.2, will follow tautologically from the claim for St G , treated immediately below.
6.1. The D-module case. The goal of this section is to prove Theorem C ′ , which states that any object of D(LS G ) ss is "divisible" by St G . In other words:  Note that Div G is defined, with the same formula, on the entire D(LS G ). However, we are only interested in it as a functor out of D(LS G ) ss .
Theorem 6.1.5 (Divisibility by the Steinberg D-module). The functor Proof. First, let us reiterate our usual notational convention: since we are only dealing with D-modules, we omit the decoration "dR" on pullback and pushforward functors.
Step 0. The theorem states that any F ∈ D(LS G ) ss is isomorphic to St G ! ⊗ Div G (F). To prove this, it suffices to exhibit an isomorphism Step 2. By the second adjunction CT D P − ,! ≃ CT D P, * , the latter is isomorphic to In the next two steps, we use the assumption that F ∈ D(LS G ) ss to simplify this expression.
Now recall that F ∈ D(LS G ) ss , so that (p P ) ! (F) ≃ (i P ) * (CT D P, * (F)). It follows that the monad (i P ) * (q P ) * acts as the identity on (p P ) ! (F). We conclude that Combining this and the above step, we obtain that (6.2) simplifies as cone colim Step 5. Unwinding the constructions, we obtain that the LHS of (6.1) is isomorphic to the tensor product of F with the object Thus, to obtain an isomorphism as in (6.1), it suffices to exhibit an isomorphism (6.3) V ≃ colim P ∈Par ′ (p P ) ! (ω LS P ) [1]. The construction of this isomorphism (to be performed in the next two steps) will be compatible with the functor St ! ⊗ ǫ, as requested in Step 0; we leave the details to the reader.
Step 6. Denote by φ : Par → D(LS G ) the functor P (p P ) ! (ω LS P ). In the spirit of Lemma 4.2.6, consider the poset P ′ (I ⊔ ∞) of proper subsets I ⊔ ∞. Here, I is the set of nodes of the Dynkin diagram of G and ∞ is an extra node. For any P ∈ Par, corresponding to the subset J P ⊆ I, we define Step 7. Note, in passing, that colim φ G ≃ St G by definition. Similarly, by Lemma 4.2.6, we obtain that colim φ P ≃ cone colim This allows to rewrite V simply as V ≃ lim P ∈(Par ′ ) op colim φ P .
Step 8. In a stable ∞-category, finite limits commute with finite colimits, whence and let C be the cocompletion of its essential image. We claim that the obvious fully faithful embedding C ⊆ QCoh(LS G ) ss is an equivalence. This follows from the fact that Ind(Perf(LS G )) ≃ QCoh(LS G ), i.e., any object of QCoh(LS G ) is a filtered colimit of perfect objects.
Hence, it suffices to prove the lemma under the assumption that each F a belongs to the essential image of (6.4). Then, for any a ∈ A, there exist a perfect object P a and an isomorphism St G ⊗ P a ≃ F a . The fully faithfulness result of Theorem E implies that the given functor ψ determines a functor φ : A → Perf(LS G ) ⊆ QCoh(LS G ), a P a and a natural equivalence ψ ≃ St G ⊗ φ. Taking colimits (and using the fact that colimits commute with tensor products), we finally obtain F = colim ψ ≃ colim(St G ⊗ φ) ≃ St G ⊗ colim(φ), as desired.