localization and the long intertwining operator for...

LOCALIZATION AND THE LONG INTERTWINING OPERATOR

FOR REPRESENTATIONS OF AFFINE KAC-MOODY ALGEBRAS

S. ARKHIPOV AND D. GAITSGORY

Introduction

This text is a raft draft, untouched since January 2009. What prevented its completion atthe time was the absence of the documented formalism of loop groups acting on categories.Once this formalism is written down (hopefully, soon), this draft will be turned into an actualpaper.

Here is an overview of what’s done in this paper:

The first author introduced a certain functor, denoted Φ, from the category of Kac-Moodyrepresentations at the negative level to that at the positive level. We interpret this functor asa composition of two contravariant functors: one is the usual contragredient duality (from thenegative level to itself), and then Verdier duality from the negative level to the positive level(that comes from the fact that the two categories are dual to each other in the sense of Lurie).

According to Kashwara-Tanisaki, the category at the negative level localizes onto the “thin”affine flag space, and the category at the negative level localizes onto the “thick” affine flagspace. We show that in terms of the localization functors, Arkhipov’s functor corresponds tothe “long intertwining operator” from the category of (twisted) D-modules on thin flags to thaton thick flags.

In the process we will need to consider the two version of the category of D-modules onBunG, recently studied in [DrGa], denoted D-mod(BunG) and D-mod(BunG)co. We show thatthe category at the positive level naturally localizes on the former, whereas the category at thenegative level naturally localizes on the latter.

Date: May 15, 2015.

1

2 S. ARKHIPOV AND D. GAITSGORY

1. The finite-dimensional case

1.1. The categories.

1.1.1. Let G be an algebraic group and g its Lie algebra. According to our conventions, weshall denote by g-mod the canonical object of DGCat corresponding to the derived categoryof g-modules, and by H(g-mod) its heart, i.e., the abelian category of g-modules.

By Sect 10.2.3, the category g-mod identifies with its own dual (g-mod)∨, using the anti-automorphism

τ : U(g)x 7→−x, x∈g−→ U(g).

Namely, the pairing g-mod⊗ g-mod→ Vect is given by

(1.1) M,N 7→ 〈M,N〉g := M ⊗U(g)

N ' H•(g,M ⊗N).

Thus, we have an equivalence

Cang : g-mod→ (g-mod)∨,

and in particular,

(g-mod)c → ((g-mod)∨)c ' (g-mod)c,o.

We shall denote the resulting contravariant functor g-modc → g-modc by Dg. Explicitly, it isgiven by

Dg(M) ' Homg-mod(M,U(g)),

viewed as a right, and via τ as a left, g-module.

1.1.2. Let Y be a smooth scheme. Let DY-mod (resp., Let DopY -mod) be the (canonical object

of DGCat corresponding to) the derived category of left (resp., right) D-modules on Y, andlet H(DY-mod) and H(Dop

Y -mod) denote their respective hearts, the corresponding abeliancategories.

By Sect 10.2.4, the categories DY-mod and DopY -mod are naturally dual to each other with

the pairing given by

(1.2) F1,F2 7→ 〈F1,F2〉Y := ΓDR(Y,F1 ⊗OY-mod

F2) ' Γ(Y,F1 ⊗DY-mod

F2),

where F1 ⊗DY-mod

F2 is regarded as a sheaf of k-vector spaces in the Zariski topology.

Thus, we obtain an identification

CanY : (DY-mod)∨ ' Dop

Y -mod,

and in particular

(DY-mod)c →

((Dop

Y -mod)∨)c ' (Dop

Y -mod)c,o

.

We shall denote the resulting contravariant functor (DY-mod)c →

(Dop

Y -mod)c

by DY; this isthe usual Verdier duality functor.

LOCALIZATION AND THE LONG INTERTWINING OPERATOR 3

1.1.3. Let Y be as above and suppose that it is acted on by G. In this case we have a naturalfunctor

Γ(Y,−)l : DY-mod→ g-mod,

and its left adjoint, denoted Locg,Y given by M 7→ DY ⊗U(g)

M .

In addition, we can consider the functor Γ(Y,−)r : DopY -mod→ g-mod.

Proposition 1.1.4. Under the identifications

g-mod ' (g-mod)∨

and DopY -mod ' (DY-mod)∨,

the functors Γ(Y,−)r and Locg,Y are naturally mutually dual.

Proof. The assertion of the proposition is equivalent to the existence of a canonical isomorphism

〈M,Γ(Y,F)〉g ' 〈Locg,Y(M),F〉Y,which follows from the definitions. �

From Lemma 10.2.1, we obtain:

Corollary 1.1.5. The left adjoint of the functor Γ(Y,−)r is the ind-extension of the functor

DY ◦ Locg,Y ◦Dg : (g-mod)c → (g-mod)c → (DY-mod)c →

(Dop

Y -mod)c.

1.1.6. Assume for a moment that the structure sheaf OY admits a right D-module structure,which is moreover, G-equivariant. This defines an identification DY-mod → Dop

Y -mod, which

intertwines the functors Γ(Y,−)l and Γ(Y,−)r. Under these identifications, the isomorphism ofCorollary 1.1.5 can be reformulated as the commutativity of the next diagram

(1.3)

(g-mod)cDg−−−−→ (g-mod)c

Locg,Y

y Locg,Y

y(DY-mod)

c DY−−−−→ (DY-mod)c.

1.1.7. A variant. Suppose that in the above situation Y carries an action of another algebraicgroup, denoted T . In particular, we have a weak action of T on the categories DY-mod andDop

Y -mod. Let DY-modT,w and DopY -modT,w denote the corresponding weakly equivariant cat-

egories. From Lemma 10.4.2, we obtain:

Lemma 1.1.8. The categories DY-modT,w and DopY -modT,w are compactly generated.

Hence, by Sect 10.4.1, we obtain that the duality functor (1.2), which naturally gives rise toa functor

(1.4) DY-modT,w ⊗DopY -modT,w → VectT,w = Rep(T )

InvT→ Vect

(here InvT is the functor of T -invariants) makes the categories DY-modT,w and DopY -modT,w

mutually dual. We shall denote by

CanT,wY : DY-modT,w → DopY -modT,w

the resulting equivalence, and by DT,wY the functor(DY-modT,w

)c→(Dop

Y -modT,w)c.

These functors commute with CanY and DY via the tautological forgetful functors to DY-modand Dop

Y -mod, respectively.


By Sect 10.5.2, the t-structures on DY-mod and DopY -mod induce t-structures on the cat-

egories DY-modT,w and DopY -modT,w, whose hearts are the usual categories of weakly T -

equivariant D-modules. We have:

Lemma 1.1.9. The categories DY-modT,w and DopY -modT,w identify with the derived categories

of their hearts.

1.1.10. Assume now that the action of T on Y commutes with that of G. In this case the functorΓ(Y,−)l naturally lifts to a functor

DY-modT,wΓ(Y,−)l→ g-mod⊗ Rep(T ),

and we define the functor ΓT,w(Y,−)l : DY-modT,w → g-mod as the composition

DY-modT,wΓ(Y,−)l→ g-mod⊗ Rep(T )

InvT→ g-mod.

The functor Locg,Y naturally lifts to a functor g-mod → DY-modT,w and is the left adjointof ΓT,w(Y,−)l.

1.1.11. Let us return to the setting of Sect 1.1.3. The statement and proof of Proposition 1.1.4extends to the present situation, i.e., under the identifications

(g-mod)∨ ' g-mod and

(DY-modT,w

)∨' Dop

Y -modT,w,

the functors Locg,Y and ΓT,w(Y,−)r and mutually dual, and the left adjoint to ΓT,w(Y,−)r isthe left adjoint of the functor

DY ◦ Locg,Y,T ◦Dg : (g-mod)c →(Dop

Y -modT,w)c.

1.2. The equivariant situation. Let B ⊂ G be a subgroup. All of the above categories carrya strong action of G, and in particular, of B.

1.2.1. Let g-modB denote the corresponding (strongly) B-equivariant category. By Sect 10.5.3,

the category g-modB inherits a t-structure from g-mod, whose heart is the abelian category ofHarish-Chandra modules with respect to the pair (g, B). By Sect 10.5.4, we have:

Lemma 1.2.2.

(a) The category g-modB identifies with the derived category of its heart, i.e., g-modB '(g, B)-mod.

(b) The category g-modB is compactly generated, and (g-modB)c is the preimage of g-modunder the forgetful functor.

From Sect 10.4.3, we obtain that the pairing (1.1) gives rise to a functor

g-modB ⊗ g-modB → VectBH•B→ Vect,

(here H•B denotes the functor of equivariant cohomology), which identifies g-modB with its owndual, i.e.,

(1.5) (g-modB)∨ ' g-modB .

We shall denote byCanBg : g-modB → (g-modB)∨,

and by DBg : (g-modB)c → (g-modB)c the resulting functors. They commute with Cang and

Dg via the tautological forgetful functor g-modB → g-mod.


1.2.3. The pairing (1.5) can be viewed in terms of the equivalence given by Lemma 1.2.2 as fol-

lows: By the above lemma, g-modB identifies with (canonical object of DGCat correspondingto) the derived category (g, B)-mod of Harish-Chandra modules with respect to the pair (g, B).Choose a splitting B ←↩ T . Then the pairing

(g, B)-mod⊗ (g, B)-mod→ Vect,

corresponding to (1.5) is given by

(1.6) M,N 7→ 〈M,N〉(g,B) :=

(M ⊗

g;TN

)⊗ det(t[1]).

1.2.4. Let DY-modB and DY-modB;T,w denote the B-equivariant categories corresponding toDY-mod and DY-modT,w, respectively.

Lemma 1.2.5. The categories DY-mod and DY-modT,w are compactly generated.

By Sect 10.4.3, the pairing (1.2) gives rise to a functor

DY-modB ⊗DopY -modB → VectB

H•B→ Vect,

and defines an equivalence

(1.7) CanBY :(DY-modB

)∨' Dop

Y -modB ; DBY :(DY-modB

)c'(Dop

Y -modB)c,

and similarly

(1.8) CanB;T,wY :

(DY-modB;T,w

)∨'(Dop

Y -modB;T,w),

(1.9) DB;T,wY :

(DY-modB;T,w

)c'(Dop

Y -modB;T,w)c

1.2.6. The functors Γ(Y,−)l and Locg,Y are naturally compatible with the strong G-actions,and hence with the B-actions. By Sect 10.4, we obtain the mutually adjoint functors

Locg,Y : DY-modB � g-modB : Γ(Y,−)l,

which are compatible with the dualities in the same sense as in Proposition 1.1.4 and Corol-lary 1.1.5. If the line bundle KY admits a G-equivariant trivialization, then the isomorphismof (1.3) holds for B-equivariant categories as well. Similarly, the assertions of Sect 1.1.11 carryover to the B-equivariant context.

1.3. The enhanced affine space. From now on, G will be a reductive group, B ⊂ G a Borelsubgroup, Y = G/N and T be the Cartan group acting on G/N on the right.

1.3.1. Let χ be a k-point of Spec(Zg), where Zg := Z(U(g)). Let g-modχ be the full subcategoryof g-mod consisting of objects, whose localization off χ is 0, i.e.,

g-modχ ⊗Zg-mod

Zg-modχ,

where Zg-modχ denotes corresponding the full sub-category of Zg-mod. From Sect 10.1.1, weobtain that g-modχ is compactly generated.

By Sect 10.2.2, we obtain that the duality (1.1) restricts to a pairing g-modχ ⊗ g-modτ(χ),which defines an identification

Cang,χ : (g-modχ)∨ ' g-modτ(χ).

By Sect 10.5.1, the t-structure on g-mod induces a t-structure on g-modχ. We have:


Lemma 1.3.2. The natural functor makes g-modχ equivalent to the derived category of itsheart.

The above lemma implies that g-modχ identifies with the (canonical object of DGCatcorresponding to) derived category of g-modules, on which Zg acts with this generalized centralcharacter.

1.3.3. The fact that the weak action of T on DG/N -mod canonically extends to a strong one

implies, by Sect 10.4.4, that DG/N -modT,w is a category over the scheme t∗. Let λ ∈ t∗ be a

weight. Let DG/N -modT,w,λ be the full subcategory of DG/N -modT,w, whose localization off λis zero, i.e.,

DG/N -modT,w,λ := DG/N -modT,w,λ ⊗Sym(t)-mod

Sym(t)-modχ.

From Sect 10.1.1, we obtain that the category DG/N -modT,w,λ is compactly generated.

Note that the canonical line bundle KG/N admits a canonical trivialization as a G-equivariantline bundle, and the failure of this trivialization to be T -equivariant is given by the character−2ρ : T → Gm.

By Sect 10.2.2 and Sect 1.1.11, we obtain that the restriction of the pairing (1.4) identifies:

CanT,w,λG/N :(DG/N -modT,w,λ

)∨' DG/N -modT,w,−λ−2ρ.

We shall denote by DT,w,λG/N the resulting contravariant functor(DG/N -modT,w,λ

)c→(DG/N -modT,w,−λ−2ρ

)c.

By Sect 10.5.1, the t-structure on DG/N -modT,w induces a t-structure on DG/N -modT,w,λ.We have:

Lemma 1.3.4. The category DG/N -modT,w,λ identifies with the derived category of its heart.

1.3.5. Let $ : t∗ → Spec(Zg) be the Harish-Chandra map. We normalize it so that $(λ) =$(w(λ+ ρ)− ρ).

If χ = $(λ), it is easy to see that the functor ΓT,w(G/N,−)l sends DG/N -modT,w,λ to

g-modχ, and that the functor Locg,G/N sends g-modχ to DG/N -modT,w,λ.

We have the following theorem of [BB]:

Theorem 1.3.6. Assume that λ is such that λ+ρ is regular. Then the functors ΓT,w(G/N,−)l

and Locg,G/N define mutually inverse equivalences:

DG/N -modT,w,λ � g-modχ.

If λ + ρ is, moreover, dominant, then the above functors are exact, i.e., compatible with thet-structures.

Combining Theorem 1.3.6 with Proposition 1.1.11 we obtain:

Corollary 1.3.7. The functors ΓT,w(G/N,−)l and Locg,G/N intertwine the functors

DT,w,λG/N :(DG/N -modT,w,λ

)c→(DG/N -modT,w,−λ−2ρ

)cand

Dg,χ : (g-modχ)c → (g-modτ(χ))c.


1.3.8. Let us now consider the B-equivariant situation. By Sect 10.4.5, the following commu-tative diagrams of categories are in fact a pull-back squares:

g-modBχ −−−−→ g-modBy yg-modχ −−−−→ g-mod

andDG/N -modB;T,w,λ −−−−→ DG/N -modB;T,wy yDG/N -modT,w,λ −−−−→ DG/N -modT,w.

This implies that both g-modBχ and DG/N -modB;T,w,λ are compactly generated, and we have:

(1.10) CanBg,χ :(g-modBχ

)∨' g-modBτ(χ)

and

(1.11) CanB;T,w,λG/N :

(DG/N -modB;T,w,λ

)∨' DG/N -modB;T,w,−λ−2ρ.

By Sect 10.5.3, the category g-modBχ inherits t-structures from g-modχ. By Sect 10.5.4, wehave:

Lemma 1.3.9. The category g-modBχ identifies with the derived category of its heart.

Similarly, the category DG/N -modB;T,w,λ inherits a t-structure from DG/N -modB;T,w. BySect 10.5.5, we have:

Lemma 1.3.10. The category DG/N -modB;T,w,λ identifies with the derived category of its heart.

1.3.11. Finally, from Theorem 1.3.6 we obtain that the functors ΓT,w(G/N,−)l and Locg,G/Ndefine mutually inverse equivalences:

DG/N -modB;T,w,λ � g-modBχ ,

and

(1.12) DB;T,w,λG/N ◦ Locg,G/N ' Locg,G/N ◦DBg,χ

as functors(g-modB

)c→(DG/N -modB;T,w

)cand

(g-modBχ

)c→(DG/N -modB;T,w,−λ−2ρ

)cand

(1.13) DBg,χ ◦ ΓT,w(G/N,−)l ' ΓT,w(G/N,−)l ◦ DB;T,w,λG/N

as functors (DG/N -modB;T,w,λ

)c→(g-modBτ(χ)

)c.

1.4. Contragredient duality and Arkhipov’s functor. Let G be a reductive group andB ⊂ G be the Borel subgroup.


1.4.1. Consider the category g-modB . Let us recall the definition of the contragredient dualityfunctor Contrg : (g-modB) ' (g-modB)∨. It will correspond to a contravariant functor

(1.14) M 7→M∗ : (g-modB)c → (g-modB)c.

We shall use the description of g-modB as (g, B)-mod given by Lemma 1.2.2. So, it issufficient to define the functor (1.14) as an exact contravariant self-equivalence on the categoryof finitely generated (g, B)-modules.

The latter is defined as follows. Let us choose a splitting B ←↩ T . For M ∈ H((g, B)-mod),let M ' ⊕

νM(ν) be the decomposition into weight spaces with respect to T . Since M is

assumed finitely generated, all M(ν) are finite-dimensional. Set M∗ := ⊕νM(ν)∗, where M(ν)∗

denotes the linear dual of M(ν). We define an action of g on M∗ by conjugating the naturalaction by (a choice of the representative of) the element w0 ∈ W . It is clear, however, thatthe construction of the functor M 7→M∗ does not depend, up to a canonical isomorphism, oneither the choice of T or that of a representative of w0.

It is clear also that for M = ∆λ–the Verma module with h.w. λ, we have

(∆λ)∗ = ∇−w0(λ),

where for a weight µ, we denote by ∇µ the dual Verma module with h.w. µ.

This implies that the functor (1.14) sends finitely generated Harish-Chandra modules tofinitely generated ones. The fact that (M∗)∗ 'M implies that an (anti)-self-equivalence.

1.4.2. We define the functor Φ : g-modB → g-modB as the composition

Φ = CanBg ◦Contrg .

We define the functor Ψ as the inverse of Φ, i.e., as the composition in the other orderContrg ◦CanBg . Our goal to describe this functor explicitly.

Recall that for any category C acted on by G and any F ∈ DG-modB,B there exists acanonical functor M 7→ F ?M : CB → CB . Let jw0,!, jw0,∗ denote the standard and co-standard

objects of DG-modB,B , corresponding to the element w0 of the Weyl group.

The main theorem in the finite-dimensional case reads as follows:

Theorem 1.4.3.

Φ⊗ det(b[1]) ' jw0,! ◦ − and Ψ⊗ det(b∗[−1]) ' jw0,∗.

Note that since jw0,! ◦ jw0,∗ ' δB ' jw0,∗ ◦ jw0,!, and hence the functors of convolution withjw0,! and jw0,∗ are mutually inverse, the two assertions in the theorem are equivalent to eachother.

1.4.4. Let us give two tautological reformulations of Theorem 1.4.3. We remark that in theinfinite-dimensional case, Theorem 1.4.3 will not admit a direct analogue, whereas these refor-mulations will.

Consider jw0,! and jw0,∗ as B × B-equivariant D-modules on G. We will not distinguishbetween left and right D-modules on G due to the existence of a G×G-equivariant trivializationof the line bundle KG. Consider the objects

Γ(G, jw0,!),Γ(G, jw0,∗) ∈ (g⊕ g)-modB×B ' g-modB ⊗ g-modB ' (g-modB)∨ ⊗ g-modB .


Corollary 1.4.5. The functor Φ is given by the kernel

Γ(G, jw0,!)⊗ det(b∗[−1])⊗2 ∈ (g-modB)∨ ⊗ g-modB

and the functor Ψ is given by the kernel

Γ(G, jw0,∗) ∈ (g-modB)∨ ⊗ g-modB .

Corollary 1.4.6. The diagrams

DG/N -modB;T,w jw0,!?−

−−−−−→ DG/N -modB;T,w

ΓT,w(G/N,−)ly yΓT,w(G/N,−)l

g-modBΦ⊗det(b[1])−−−−−−−−→ g-modB

and

DG/N -modB;T,w jw0,∗?−−−−−−→ DG/N -modB;T,w

ΓT,w(G/N,−)ly yΓT,w(G/N,−)l

g-modBΨ⊗det(b∗[−1])−−−−−−−−−→ g-modB

commute.

1.5. Proof of Theorem 1.4.3. As a first step, we give the following reformulation of thefunctor Contrg.

Proposition 1.5.1. For M,N ∈ (g-modB)c,

Homg-modB (M,N∗) ' HomDG-modB×B (Locg⊕g,G(M ⊗N), jw0,∗) .

Note that this proposition provides a more comprehensible definition of the contragredientduality functor.

Proof. It is enough it establish the isomorphism at the level of complexes, where both M and Nadmit Verma flags. We claim that in this case both sides are acyclic off cohomological degree 0,and it would be enough to establish a canonical isomorphism between their 0-th cohomologies.Note that acyclicity for the LHS is evident: there are no higher Exts from a Verma module toa dual Verma module.

Let us make the choices B ←↩ T and w0 as in the definition of the functor Contrg. Bydefinition, the LHS is given by the relative Lie algebra cohomology of g, relative to t withcoefficients in Homk(M ⊗N, k), where the action of g on N is twisted by w0 (we shall denotethe resulting module by Nw0). Thus, the LHS calculates to

(1.15) Hom(g,T )-mod(M ⊗Nw0 , k) ' Homk(H•(g;T,M ⊗Nw0), k).

Since jw0,∗ is obtained by averaging with respect to B × B of the the D-module δw0, which

is equivariant with respect to T embedded via t 7→ (t, w0(t)), the RHS is isomorphic to

(1.16) HomVectT ((Locg⊕g,G(M ⊗N))w0 , k) ,

where (Locg⊕g,G(M ⊗N))w0is the fiber of Locg⊕g,G(M ⊗N) regarded as an object of VectT .

We have:

(Locg⊕g,G(M ⊗N))w0' U(g) ⊗

U(g)⊗U(g)(M ⊗Nw0) 'M ⊗

gNw0 .

and we obtain that the RHS in (1.15) is canonically isomorphic to (1.16).�


As a corollary, we obtain that for M,N ∈ (g-modB)c,

Homg-modB (M, jw0,! ? Contrg(N)) ' HomDG-modB×B (Locg⊕g,G(M ⊗N), δB) .

I.e., to prove the theorem, it is sufficient to prove the following:

Proposition 1.5.2. For M,N ∈ (g-modB)c there exists a canonical isomorphism

Homg-modB(M,DBg (N)

)⊗ det(b∗[−1]) ' HomDG-modB×B (Locg⊕g,G(M ⊗N), δB)

Proof. Denote DBg (M) =: M1 and DBg (N) =: N1. The LHS is then 〈M1, N1〉(g,B)⊗ det(b∗[−1])and the RHS, by (1.3), is isomorphic to

HomDG-modB×B (δB [−dim(G)],Locg⊕g,G(M1 ⊗N1)) ,

which can be rewritten as

HomVectB (k,M1 ⊗gN1) ' Hom(g,B)-mod(k,M1 ⊗N1),

and we are done by (1.6). �

2. Representations of the Kac-Moody algebra

2.1. The category of modules.

2.1.1. We fix a level κ : g ⊗ g → k and consider the Kac-Moody extension gκ. The object ofDGCat, denoted gκ-mod, has been introduced in [FG2]. We remind that, by definition, gκ-mod

is generated by compact objects of the form Indgκk (k), where k ⊂ g[[t]] is a lattice subalgebra.

2.1.2. The category gκ-mod is acted on strongly by the group ind-scheme G((t)) at level κ. Inparticular, if K is a pro-unipotent open-compact subgroup of G[[t]] we have a full subcategory

gκ-modK ⊂ gκ-mod of K-equivariant objects.

Lemma 2.1.3. Each of the categories gκ-modK is compactly generated;

(gκ-modK)c = gκ-modK ∩ (gκ-mod)c.

2.1.4. For two open compact subgroups K1 ⊂ K2, let eK2,K1 denote the tautological inclusion

functor gκ-modK2 → gκ-modK1 , and let AvK2,K1be its right adjoint.

By the general formalism of categories acted on by G((t)),

(2.1) gκ-mod ' lim−→K

gκ-modK ,

where the colimit is taken with respect to the functors e, and also

(2.2) gκ-mod ' lim←−K

gκ-modK ,

where the limit is taken with respect to the functors Av.

2.2. Duality in the Kac-Moody case. Let κ′ be the opposite level, i.e., κ′ = −κ − κKil.Our current goal is to construct a pairing

(2.3) 〈−,−〉g : gκ-mod⊗ gκ′-mod→ Vect,

which would identify (gκ-mod)∨ with gκ′ -mod.

The construction of the pairing (2.3) depends on an additional choice of trivializing a certaingerbe. This choice can be made by fixing an open-compact subgroup K0.


2.2.1. The pairing (2.3) is defined as follows. We consider DG pairing

C+(gκ-mod)⊗C+(gκ′ -mod)→ C(Vect)

given by

(2.4) M,N 7→ C∞2 (g, k0,M ⊗N),

where C∞2 is the semi-infinite Chevalley complex taken with respect to the lattice k0 :=

Lie(K0) ⊂ g.

It is easy to see that if either M or N is acyclic, then so is C∞2 (g, k0,M ⊗N).

Since (gκ-mod)c (resp., (gκ′ -mod)c) is a obtained as a quotient category of a subcategory ofC+(gκ-mod) (resp., C+(gκ′ -mod)) by its intersection with the subcategory of acyclic complexes,we obtain that (2.4) gives rise to a pairing

(gκ-mod)c ⊗ (gκ′ -mod)c → Vect,

which we then ind-extend to obtain the pairing (2.3).

Theorem 2.2.2. The pairing (2.3) defines a perfect duality between gκ-mod and gκ′ -mod.

2.2.3. Let K1 ⊂ K2 be two open-compact subgroups. We observe:

Lemma 2.2.4. For M1 ∈ gκ-modK1 , M2 ∈ gκ-modK2 ,

〈eK2,K1(M2),M1〉g ' 〈M2,AvK2,K1

(M1)〉g.

Therefore, to prove Theorem 2.2.2, it suffices to show that the pairing (2.3) restricts to aperfect pairing

(2.5) gκ-modK ⊗ gκ′ -modK → Vect

for a fixed open-compact subgroup K.

2.2.5. We are now going to construct an object

(2.6) RK,G((t))κ,κ′ ∈ gκ-modK ⊗ gκ′ -modK ,

with the property that for M ∈ gκ-modK

(2.7) 〈RK,G((t))κ,κ′ ,M〉g 'M ⊗ det. rel.(k, k0) ∈ gκ-modK ,

and for N ∈ gκ′ -modK

(2.8) 〈N,RK,G((t))κ,κ′ 〉g ' N ⊗ det. rel.(k, k0) ∈ gκ′ -modK .

The existence of such RK,G((t))κ,κ′ will imply the duality assertion.

2.2.6. Recall that Dκ,κ′

G((t))-mod denotes the category of twisted D-modules on G((t)). It has a

natural forgetful functor Γ(G((t)),−) to gκ-mod⊗ gκ′ -mod. The category Dκ,κ′

G((t))-mod is acted

on strongly by the group G((t)) × G((t)) at the level (κ, κ′) and the above forgeftul functor iscompatible with the action. In particular, we obtain a functor

Γ(G((t)),−) : Dκ,κ′

G((t))-modK,K → gκ-modK ⊗ gκ′-modK .

Let δκ,κ′

G((t)),K be the canonical object in Dκ,κ′

G((t))-modK,K , corresponding to distributions on K

inside G((t)). We set

RK,G((t))κ,κ′ := Γ(G((t)), δκ,κ

′

G((t)),K).


Proposition 2.2.7. The object RK,G((t))κ,κ′ satisfies (2.7) and (2.8).

Proof. We shall prove (2.7), since (2.8) is similar. It is enough to consider the case M ∈(gκ-modK)c. By definition, it is enough to construct a functorial quasi-isomorphism of com-plexes

C∞2 (g, k0,M ⊗ RK,G((t))

κ,κ′ ) 'M ⊗ det. rel.(k, k0).

However, the latter has been carried out in [FG1].�

2.2.8. Let Cang denote the resulting equivalence

(gκ-mod)∨ ' gκ′ -mod,

and let us denote by

Dg : (gκ-mod)c → (gκ′ -mod)c

the corresponding contravariant equivalence.

Here is an example of a calculation of this functor. Let M = Indgκk (k). We claim that

D(M) ' Indgκ′k (k)⊗ det. rel.(k, k0). Indeed, we have

Homgκ′ -mod(D(M), N) ' 〈M,N〉g ' H•(k, N)⊗ det. rel.(k, k0) '

' Homgκ′ -mod(Indgκ′k (k), N)⊗ det. rel.(k, k0).

2.2.9. From now on we shall fix K0 to be the Iwahori subgroup I.

Consider now the category gκ-modI , which can be identified with (gκ-modK)I/K for any Kwhich is a normal subgroup of I. We have:

Lemma 2.2.10. The category gκ-modI is compactly generated. Its compact objects are thosewhich become compact under the forgetful functor to gκ-mod.

The pairing (2.3), being G((t))-invariant gives rise to a pairing

gκ-modI ⊗ gκ′ -modI → VectI ,

which, when composed with HI : VectI → Vect defines a pairing

(2.9) gκ-modI ⊗ gκ′ -modI → Vect .

By Sect 10.4.3, the pairing (2.9) is also perfect. We shall denote by CanIg the resultingequivalence

(gκ-modI)∨ ' (gκ′ -modI),

and by DIg the corresponding contravariant functor

(gκ-modI)c ' (gκ′ -modI)c;

these functors are compatible with Cang and Dg via the natural forgetful functors to (gκ-mod)c

and (gκ′ -mod)c,o, respectively.

We remark that the canonical object in gκ-modI ⊗ gκ′ -modI that defines CanIg is

(2.10) RI,G((t))κ,κ′ := Γ(G((t)), δI)

κ,κ′ .

For example, for M = ∆κ,µ–the Verma module with h.w. µ, we have DIg(M) ' ∆κ′,−µ.

2.3. Affine contragredient duality and affine Arkhipov’s functor.


2.3.1. The affine algebra gκ we have been dealing with was the extension of the loop algebrag((t)). In what follows, we shall sometimes add a subscript gκ,0, where 0 ∈ P1 with t being aglobal coordinate on P1.

We shall also consider the affine algebra gκ,0, which is the central extension of the loopalgebra g((t−1)). Of course, one could identify gκ,0 ' gκ,∞, but we prefer not to do that.

2.3.2. From now on, we shall make a distinction positive and negative levels. We shall say thatthe level κ is negative if (on every simple factor of g), κ + κKil

2 = c · κKil, with c /∈ Q≥0. We

shall say that κ is positive if κ + κKil2 = c · κKil, with c /∈ Q≤0. Evidently, κ is negative if

and only if κ′ is positive. We shall say that κ is irrational if c /∈ Q (i.e., irrational=positive ∩negative). We shall say that κ is critical if c = 0 (i.e., κcrit = −κKil2 ).

2.3.3. Let κ be negative. We shall now define the contragredient duality functor

(2.11) Contr0→∞κ→κ′ : gκ,0-modI0 ' (gκ,∞-modI∞)∨.

This will be based on the following:

Lemma 2.3.4. Let κ be negative.

(1) The category H(gκ-modI) is Artinian, i.e., every finitely generated object is of finite length.

(2) The triangulated category (gκ-modI)c is equivalent to Db(H(gκ-modI)f.g.).

As in Sect 1.4.1, it suffices to define the corresponding exact contravariant

(2.12) M 7→M∗ : H(gκ,0-modI0)f.g. ' H(gκ,∞-modI∞)f.g..

For M ∈ H(gκ,0-modI0)f.g., write M ' ⊕µ,n

M(µ, n), where µ refers to the grading by means

of T ⊂ B ⊂ I, and n ∈ k is the degree with respect of the Sugawara L0 operator. Since Mis finitely generated, and κ is negative, each M(µ, n) is finite-dimensional. We set the vectorspace underlying M∗ to be ⊕

µ,nM(µ, n)∗. The action of the affine algebra gκ,∞ is defined as

follows: an element x⊗ tn acts as Adw0(x)⊗ tn; the fact that this action extends continuously

to power series follows from the definitions.

It is clear that (∆κ,µ,0)∗ ' ∇κ,−w0(µ),∞, where the latter denotes the dual affine Vermamodule, which is known to be finitely generated. This implies that (2.12) constructed abovehas the required properties, and in particular, gives rise to a functor (2.11), as required.

2.3.5. Let us again assume that κ is negative. We define the functor

Φ0→∞κ→κ′ : gκ,0-modI0 ' gκ′,∞-modI∞

to be the composition

(2.13) gκ,0-modI0Contr0→∞

κ→κ′−→ (gκ,∞-modI∞)∨CanIg−→ gκ′,∞-modI∞ .

We define the functorΨ∞→0κ′→κ : gκ′ -modI∞ ' gκ,0-modI0

to be the inverse of Φ0→∞κ→κ′ .

The goal of the rest of this paper is describe explicitly the kernels

S0→∞κ′,κ′ ∈ gκ′,0-modI0 ⊗ gκ′,∞-modI∞ and S∞→0

κ,κ ∈ gκ,0-modI0 ⊗ gκ,∞-modI∞

that give rise to the functors Φ0→∞κ→κ′ and Ψ∞→0

κ′→κ via the pairing 〈−,−〉g. In addition, we willshow how the functors Φ0→∞

κ→κ′ and Ψ∞→0κ′→κ are compatible with the localization functors.


2.3.6. Tautologically, we have the following descriptions:

(2.14) S0→∞κ′,κ′ = (Id⊗Φ) (RI0,G((t))

κ′,κ ) and S∞→0κ,κ = (Ψ⊗ Id) (RI∞,G((t−1))

κ′,κ ).

Proposition 2.3.7. The objects S0→∞κ′,κ′ and S∞→0

κ,κ belong to the hearts of the correspondingt-structures.

Proof. We have:

(2.15) Φ0→∞κ→κ′(∇κ,µ,0) = ∆κ′,w0(µ),∞

and, correspondingly,

(2.16) Ψ∞→0κ′→κ(∆κ′,w0(µ),∞) ' ∇κ,µ,0,

to prove the proposition, it suffices to observe that RI0,G((t))κ′,κ , regarded merely as an object of

H(gκ,0-modI0), has a filtration with subquotients isomorphic to dual Verma modules, and that

RI∞,G((t−1))κ′,κ , regarded merely as an object of H(gκ′,∞-modI∞) has a filtration with subquotients

isomorphic to Verma modules.�

3. D-modules on BunG

3.1. The spherical case.

3.1.1. Let X be a smooth complete curve. Let BunG(X) denote the moduli stack of principalG-bundles on X. In this section we shall establish some basic facts about the category ofD-modules on X.

By definition, the category of (left) D-modules, denoted DBunG(X)-mod on BunG(X) islim←−U

DU -mod, where the limit is taken over the partially ordered set of open substacks of finite

type U ⊂ BunG(X) and the functors DU2-mod→ DU1 -mod for j1,2 : U1 ↪→ U2 are j∗1,2 ' j!1,2.

More generally, a choice of a level κ, defines a TDO DκBunG(X) on BunG(X), and we set

DκBunG(X)-mod! := lim

←−U

DκU -mod.

In the sequel we will establish the following result:

Theorem 3.1.2. The category DκBunG(X)-mod! is compactly generated.

From the definitions (and independent of Theorem 3.1.2) we have the following descriptionof compact objects of Dκ

BunG(X)-mod!:

Lemma 3.1.3. An object F ∈ DκBunG(X)-mod! is compact if and only if there exists an open

sub-stack U ⊂ BunG(X) of finite type and FU ∈ (DκU -mod)c, such that F|U = FU and such that

the following equivalent conditions hold:

• For any F1 ∈ DκBunG(X)-mod!, supported on BunG(X)− U , we have Hom(F,F1) = 0.

• For any Uj↪→ U1, we have:

F|U1 ' j!(FU ),

in particular, the object j!(FU ) is defined.

We remark that the proof of Theorem 3.1.2 will provide an even more explicit description ofcompact objects in Dκ

BunG(X)-mod!.


3.1.4. We define the category DκBunG(X)-mod∗ as

lim−→U

DκU -mod,

where for U1 ⊂ U2, the functor DκU1

-mod→ DκU2

-mod is (j1,2)∗.

Observe that for every open substack U ⊂ BunG(X) of finite type, we have:

(3.1) (DκU -mod)

∨ ' Dκ,opU -mod and Dκ,op

U -mod ' Dκ′

U -mod.

Hence, from Lemma 10.1.2 we obtain:

Corollary 3.1.5. There exists a canonical equivalence:(Dκ

BunG(X)-mod∗

)∨' Dκ′

BunG(X)-mod!.

3.1.6. Note that there is a naturally defined functor

(3.2) Q∗→! : DκBunG(X)-mod∗ → Dκ

BunG(X)-mod!.

Indeed, having such a functor amounts to a compatible collection of functors

DκU1

-mod→ DκU2

-mod

for all pairs U1, U2 of open subsets of finite type of BunG(X). The functors in question aredefined by

DκU1

-modj∗12,1→ Dκ

U1∩U2-mod

(j12,2)∗→ DκU2

-mod,

where ji,ij : Ui ∩ Uj ↪→ Ui.

We shall also prove the following:

Proposition 3.1.7. If κ is irrational, then the functor Q∗→! of (3.2) is an equivalence.

In fact, we shall prove a stronger assertion:

Proposition 3.1.8. Let κ be irrational. Then there exists an open substack U ⊂ BunG(X) offinite type, such that no non-zero object of Dκ

BunG(X)-mod! is supported on BunG(X)− U .

3.1.9. Let(Dκ

BunG(X)-mod)c,!∗

denote the small category

lim←−U

(DU -mod)c,

where the limit is taken over the poset of open sub-stacks U of finite type.

By construction, it admits a fully faithful functor into DκBunG(X)-mod!, and receives a fully

faithful functor from(Dκ

BunG(X)-mod!

)c. In addition, the Verdier duality functor defines a

contravariant equivalence:(Dκ

BunG(X)-mod)c,!∗

→(Dκ′

BunG(X)-mod)c,!∗

.


3.1.10. From Theorem 3.1.2 and Corollary 3.1.5, by duality we obtain a fully faithful functor(Dκ

BunG(X)-mod∗

)c↪→(Dκ

BunG(X)-mod)c,!∗

.

It is easy to see that the ind-extension of the composition(Dκ

BunG(X)-mod∗

)c↪→(Dκ

BunG(X)-mod)c,!∗

↪→ DκBunG(X)-mod!,

is the functor Q∗→! defined above.

3.2. Level structure.

3.2.1. Let x := x1, ..., xn be a finite collection of points on X. Consider the group-scheme

G(Ox) := G(Ox1 × ...× Oxn). Let K be a group sub-scheme of G(Ox) of finite codimension.

Let U ⊂ BunG(X) be an open sub-stack of finite type. Let UK be the stack (of infinite type)that classifies pairs (PG, α), where PG is a point of U , and α is a structure of level K on PG atx. Note that for a fixed U and K small enough, UK is a scheme of finite type. If K is normal

in G(Ox), we have an action of G(Ox)/K on UK and (UK)/(G(Ox)/K) ' U .

Set

DκBunG(X,K)-mod! := lim

←−U

DκUK -mod.

Note that unlike the case of K = G(Ox) (and an appropriate Iwahori variant discussedbelow), we do not know whether the category Dκ

BunG(X,K)-mod! is compactly generated or even

dualizable. Whether or not or compact objects generate DκBunG(X,K)-mod!, they are described

by the corresponding version of Lemma 3.1.3.

If K1 is a normal subgroup of K2, we obtain that DκBunG(X,K1)-mod! is acted on strongly by

K2/K1 and (Dκ

BunG(X,K1)-mod!

)K2/K1

' DκBunG(X,K2)-mod!.

In particular, we have the tautological forgetful functor

π!K1,K2

: DκBunG(X,K2)-mod! → Dκ

BunG(X,K1)-mod!,

(where πK1,K2denotes the map BunG(X,K1)→ BunG(X,K2)), and its right adjoint

(πK1,K2)∗ : Dκ

BunG(X,K1)-mod! → DκBunG(X,K2)-mod!.

3.2.2. We define the category DκBunG(X,K)-mod∗ as

lim−→U

DκU -mod,

where for U1 ⊂ U2, the functor DκUK1

-mod → DκUK2

-mod is (jK1,2)∗, where jK1,2 denotes the

corresponding open embedding UK1 ↪→ UK2 .

The equivalence (3.1) continues to hold. In partcular, we obtain a pairing

(3.3) 〈−,−〉BunG(X,K) : DκBunG(X,K)-mod! ⊗Dκ′

BunG(X,K)-mod∗ → Vect .

By Lemma 10.1.2 we obtain:

Corollary 3.2.3. If one of the categories DκBunG(X,K)-mod∗ or Dκ′

BunG(X,K)-mod! is dualizable,

then so is the other one, and in this case (3.3) identies them as each other’s duals.


As in the case of BunG(X) we can introduce the small category, there exists a naturallydefined functor

QK∗→! : DκBunG(X,K)-mod∗ → Dκ

BunG(X,K)-mod!.

However, for general K, the functor QK∗→! is not an equivalence even for irrational κ.

3.2.4. Additionally, as in the case of BunG(X) we can introduce the small category(Dκ

BunG(X,K)-mod)c,!∗

:= lim←−U

(DκUK -mod)c,

equipped with fully faithful functors(Dκ

BunG(X,K)-mod!

)c↪→(Dκ


↪→ DκBunG(X,K)-mod!,

and Verdier duality defines an contravariant equivalence(Dκ


'(Dκ′


.

However, the conclusion of Sect 3.1.10 holds only conditionally:

Lemma 3.2.5. If Dκ′

BunG(X,K)-mod! is compactly generated, and hence the functor(Dκ

BunG(X,K)-mod∗

)c→(Dκ


is well-defined, then the ind-extension of the composition(Dκ

BunG(X,K)-mod∗

)c→(Dκ


↪→ DκBunG(X,K)-mod!

identifies with the functor QK∗→!.

3.3. The Iwahori case.

3.3.1. Let x := x1, ..., xn be a finite collection of points on X. Let BunG(X, Ix) (resp.,

BunG(X,◦Ix)) be the moduli stack of principal G-bundles equipped with a reduction of their

fibers at x1, ..., xn to B (resp., N). The stack BunG(X,◦Ix) is acted on by the torus Tn =

T × ...× T .

By the above, we have the categories DκBunG(X,Ix)-mod! and Dκ

BunG(X,◦Ix)

-mod!, the latter

being acted on strongly by Tn and we have

(Dκ

BunG(X,◦Ix)

-mod!)Tn ' Dκ

BunG(X,Ix)-mod!.

In addition, we can consider the category

Dκ

BunG(X,◦Ix)

-modTn,w

! := (Dκ

BunG(X,◦Ix)

-mod!)Tn,w,

which is easily seen to be equivalent to

lim←−U

DκU ′ -modT

n,w,

where U ′ denotes the preimage of U in BunG(X,◦Ix).


For a collections of weights λ = λ1, ..., λn, we can consider the corresponding monodromic

category (Dκ

BunG(X,◦Ix)

-mod!)Tn,w,λ, and it is easy to see that we have an equivalence:

Dκ

BunG(X,◦Ix)

-modTn,w,λ

! := (Dκ

BunG(X,◦Ix)

-mod!)Tn,w,λ ' lim

←−U

DκU ′ -modT

n,w,λ.

Proceeding as above, we define also the corresponding categories as the corresponding co-limits:

DκBunG(X,Ix)-mod∗, D

κ

BunG(X,◦Ix)

-mod∗, Dκ

BunG(X,◦Ix)

-modTn,w∗ , Dκ

BunG(X,◦Ix)

-modTn,w,λ∗ .

3.3.2. We shall prove the following:

Theorem 3.3.3. The category (Dκ

BunG(X,◦Ix)

-mod!)Tn,w,λ is compactly generated.

By Corollary 3.2.3, we obtain:

Corollary 3.3.4. (Dκ

BunG(X,◦Ix)

-modTn,w,λ∗

)∨' Dκ′

BunG(X,◦Ix)

-modTn,w,−λ

! .

3.3.5. The construction of Sect 3.1.6 renders to the present situation, i.e., we have a naturallydefined functor

QTn,w,λ∗→! : Dκ

BunG(X,◦Ix)

-modTn,w,λ∗ → Dκ

BunG(X,◦Ix)

-modTn,w,λ

! .

We will prove the following generalization of Proposition 3.1.7:

Proposition 3.3.6. Assume that κ is irrational and λ1, ..., λn are rational. Then the functor

QTn,w,λ∗→! is an equivalence.

As in the case of BunG(X), Proposition 3.3.6 follows from the next assertion:

Proposition 3.3.7. Let κ and λ be as in Proposition 3.3.6. Then there exists a sub-stack of

finite type U ⊂ BunG(X), such that no non-zero object of Dκ

BunG(X,◦Ix)

-modTn,w,λ

! is supported

on the complement to U ′ ⊂ BunG(X,◦Ix).

3.3.8. Finally, as in Sect 3.1.9, we can introduce a small category

(Dκ

BunG(X,◦Ix)

-modTn,w,λ

)c,!∗,

equipped with the corresponding functors, and the conclusion of Sect 3.1.10 holds.

3.4. The full level structure.

3.4.1. We are now going to define the category

DκBunG(X,x)

-mod!

of twisted D-modules over ”BunG(X) with a full level structure” at x.

We set

DκBunG(X,x)

-mod! := colim−→K

DκBunG(X,K)-mod!,

where the co-limit is taken with respect to the pull-back functors π!K1,K2

.


By Lemma 10.1.2, we have:

DκBunG(X,x)

-mod! ' lim←−K

DκBunG(X,K)-mod!,

where the limit is taken with respect to the functors (πK1,K2)∗.

The latter implies that we also have the equivalence:

DκBunG(X,x)

-mod! ' lim←−U

DκU x

-mod,

where

DκU x

-mod := colim−→,π!

K

DκUK -mod ' lim

←−,π∗K

DκUK -mod.

Although each of the categories DκU x

-mod is compactly generated, we do not know whether

DκBunG(X,x)

-mod! is.

The category DκBunG(X,x)

-mod! is naturally acted on by G(Ox), and for K as above we have:

(DκBunG(X,x)

-mod!)K ' Dκ

BunG(X,K)-mod!.

3.4.2. We define the category DκBunG(X,x)

-mod∗ as

colim−→U

DκU x

-mod,

with respect to the functors

(jx1,2)∗ : Dκ

U x1-mod→ Dκ

U x2-mod.

By definition, DκBunG(X,x)

-mod∗ is equivalent to

colim−→K

DκBunG(X,K)-mod∗,

with respect to the functors (πK1,K2)!, and also

lim←−K

DκBunG(X,K)-mod∗,

with respect to the functors (πK1,K2)∗.

The assertion of Lemma 3.2.3 and the constructions of Sect 3.2.4 render ditto into the presentcontext.

3.4.3.

Theorem-Construction 3.4.4.

(1) The action of G(Ox) on the categories

DκBunG(X,x)

-mod! and DκBunG(X,x)

-mod∗

naturally extends to an action of the group ind-scheme G(Kx) at level κ.

(2) The pairing

DκBunG(X,x)

-mod! ⊗Dκ′

BunG(X,x)-mod∗ → Vect

is G(Kx)-invariant.


Proof. Fill in.�

4. Localization on BunG

4.1. Definition of localization.

4.1.1. Let gκ,x be the central extension at level κ of

(g⊗ Kx1)⊕ ...⊕ (g⊗ Kxn).

Let gκ,x-mod be the corresponding category of modules. It is naturally acted on by the

group ind-scheme G(Kx) at level κ.


(1) There exists a canonical functor

Locgκ,x,BunG(X,x),! : gκ,x-mod→ DκBunG(X,x)

-mod!,

compatible with the action of G(Ox).

(2) The data compatibility of Locgκ,x,BunG(X,x),! with G(Ox) can be naturally lifted to a data of

compatibility with the action of G(Kx).

4.1.3. Proof of Theorem 4.1.2(1). The data of the functor Locgκ,x,BunG(X,x),! amounts, by def-

inition, to a compatible family of functors

Locgκ,x,U x : gκ,x-mod→ DκU x

-mod.

Recall that (gκ,x-mod)c is a full subcategory in Db(H(gκ,x-mod)). Hence, it is sufficient toconstruct a compatible family of functors

LLocgκ,x,U x : Db(H(gκ,x-mod))→ gκ,x-mod→ DκU x

-mod.

The latter are obtained as the left derived functor of the naive localization functor

H(Locgκ,x,U x) : H(gκ,x-mod)→ H(DκU x

-mod).

Explicitly, let K ⊂ G(Ox) be small enough, so that k ∩ Stabg(Kx)

(u) = 0 for any u ∈ U x.

Then

LLocgκ,x,U x(Indgκ,xk (k)) ' H Locgκ,x,U x(Ind

gκ,xk (k)) ' Dκ

UK ,

where DκUK ∈ Dκ

UK -mod is regarded as an object of DκU x

-mod via the pull-back functor

DκUK -mod-mod→ Dκ

U x-mod.

The compatibility of the functors Locgκ,x,U x among themselves and with the action of G(Ox)

follows from the construction.


4.1.4. In the course of the proof we have shown that for a fixed U , the functor Locgκ,x,U x sends

(gκ,x-modK)c → (DκU x

-modK)c,

and, hence,

(gκ,x-mod)c → (DκU x

-mod)c.

In particular, the restriction of the functor Locgκ,x,BunG(X,x),! to (gκ,x-mod)c factors as

(4.1) (gκ,x-mod)cLocgκ,x,BunG(X,x),!∗

−→(Dκ

BunG(X,x)-mod

)c,!∗↪→ Dκ

BunG(X,x)-mod!,

and

(4.2) (gκ,x-modK)cLocK

gκ,x,BunG(X,x),!∗−→

(Dκ


↪→ DκBunG(X,K)-mod!,

4.1.5. Proof of Theorem 4.1.2(2). Fill in.�

4.2. The functor of sections.

4.2.1. Consider now the category DκBunG(X,x)

-mod∗.


(1) There exists a canonical functor

Γ(BunG(X, x),−)∗ : DκBunG(X,x)

-mod∗ → gκ,x-mod,

compatible with the action of G(Ox).

(2) The data compatibility of Γ(BunG(X, x),−)∗ with G(Ox) can be naturally lifted to a data of

compatibility with the action of G(Kx).

4.2.3. Proof of Theorem 4.2.2(1). The data of a functor Γ(BunG(X, x),−)∗ amounts to a com-patible family of functors

Γ(U x,−) : DκU x

-mod→ gκ,x-mod

for open sub-stacks U ⊂ BunG(X) of finite type.

Each of the latter is determined by the corresponding functor

Γ(U x,−) : (DκU x

-mod)c → gκ,x-mod.

The latter is constructed as the composition of the usual derived functor of global sections

RΓ(U x,−) : (DκU x

-mod)c → Db(H(gκ,x-mod)) ↪→ D+(H(gκ,x-mod)),

and the fully faithful embedding

D+(H(gκ,x-mod)) ↪→ gκ,x-mod

of [FG2].

4.2.4. Proof of Theorem 4.2.2(2). Fill in.�

4.3. Duality.


4.3.1. The functor Locgκ,x,BunG(X,x),! does not in general map (gκ,x-mod)c to the subcategory

of compact objects in DκBunG(X,x)

-mod. Therefore, it does not admit a colimit preserving right

adjoint.

However, we have:

Proposition 4.3.2. The pairings

〈−,−〉BunG(X,x) : DκBunG(X,x)

-mod! ⊗Dκ′

BunG(X,x)-mod∗ → Vect

and

〈−,−〉gx : gκ,x-mod⊗ gκ′,x-mod

are compatible with the functors Locgκ,x,BunG(X,x),! and Γ(BunG(X, x),−)∗.

Proof. The assertion of the proposition reads that for

M ∈ gκ,x-mod and F ∈ Dκ′

BunG(X,x)-mod∗,

we have a functorial isomorphism

(4.3) 〈Locgκ,x,BunG(X,x),!(M),F〉BunG(X,x) ' 〈M,Γ(BunG(X, x),F)∗〉gx .

By the construction of both functors, the assertion reduces to showing that for an opensub-stack U ⊂ BunG(X) of finite type, M ∈ (gκ,x-mod)c and and F ∈ (Dκ′

U x-mod)c, we have a

functorial isomorphism:

(4.4) 〈Locgκ,x,U x(M),F〉U x ' 〈M,Γ(U x,F)〉gx .

The latter isomorphism follows from the definitions.�

4.3.3. Recall the functor

Locgκ,x,BunG(X,x),!∗ : (gκ,x-mod)c →(Dκ

BunG(X,x)-mod

)c,!∗of (4.1).

Let DBunG(X,x) denote the contravariant equivalence(Dκ

BunG(X,x)-mod

)c,!∗→(Dκ′

BunG(X,x)-mod

)c,!∗.

As a formal corollary of Proposition 4.3.2, we obtain:

Corollary 4.3.4. The functors

DBunG(X,x) ◦ Locgκ,x,BunG(X,x),!∗ ◦Dgx and Locgκ′,x,BunG(X,x),!∗

mapping

(gκ′,x-mod)c →(Dκ′

BunG(X,x)-mod

)c,!∗are isomorphic.

5. Finiteness properties of the category of D-modules on BunG

5.1. The spherical case.


5.1.1. The goal of this subsection is to prove Theorem 3.1.2. In fact, we shall prove a strongerassertion, explained to us by V. Drinfeld:

Theorem 5.1.2. The stack BunG(X) can be represented as a union of open sub-stacks of finite

type Uα, such that for every pair Uαjα,β↪→ Uβ, the functor (jα,β)∗ sends(

DκUα -mod

)c → (DκUβ

-mod)c.

Let us show how this theorem implies Theorem 3.1.2. Indeed, by Verdier duality we obtain:

Corollary 5.1.3. For every pair Uαjα,β↪→ Uβ, the functor left adjoint to

(jα,β)! : DκUα -mod→ Dκ

Uβ-mod,

is defined.

From Lemma 10.1.2, we then obtain:

Corollary 5.1.4.

(1) The category DκBunG(X)-mod! can be written as colim

−→α

DκUα

-mod, where for Uα ⊂ Uβ, the

functor DκUα

-mod→ DκUβ

-mod is (jα,β)!.

(2) The category DκBunG(X)-mod! is compactly generated. Its compact objects are of the form

(jα)!(F) for F ∈(DκUα

-mod)c

, and where (jα)! denotes the tautological functor

DκUα -mod→ Dκ

BunG(X)-mod!

from point (1).

5.1.5. For the proof of Theorem 5.1.2, we shall need to recall some reduction theory. Let ΛGdenote the coweight lattice of G, and set ΛQ := Q⊗

ZΛ. Let ψ denote the map

ΛG ↪→ ΛQG ' ΛQ

[G,G] ⊕ ΛQZ0(G) → ΛQ

[G,G] ↪→ ΛQG.

Let ΛposG ⊂ ΛG denote the positive span of positive simple roots, and Λpos,QG ⊂ ΛQG the

corresponding rational span. Let Λ+G (resp., Λ+,Q

G ) denote the dominant cones.

For a parabolic P with Levi factor M consider the quotient ΛG,P := ΛG/Λ[M,M ]sc ' π1(M).Let ψP denote the map

ΛG,P ↪→ ΛQG,P ' (Λ[G,G]/Λ[M,M ])

Q ⊕ ΛQZ0(G) → (Λ[G,G]/Λ[M,M ])

Q ↪→ ΛQG,P .

Let also φP denote the map

ΛG,P ↪→ ΛQG,P ' ΛQ

[G,G]∩Z0(M) ⊕ ΛQZ0(G) → ΛQ

[G,G]∩Z0(M) ↪→ ΛQG.

Let ΛposG,P (resp., Λpos,QG,P ) be the image of ΛposG (resp., Λpos,QG ) in ΛG,P (resp., ΛQG,P ).


5.1.6. Consider the stack BunP (X) and let pP and qP denote its natural projections to BunG(X)and BunM (X), respectively. Recall that the group ΛG,P enumerates the connected componentsof the stacks BunM (X) and BunP (X) (the latter are in bijection via the projection pQ). For

λP ∈ ΛG,P we shall denote by BunλPP (X) the corresponding connected component.

Recall that a G-bundle PG ∈ BunG(X) is called ”semi-stable” if for every parabolic P , such

that PG = pP (PP ) with PP ∈ BunλPP (X), then ψP (λP ) ∈ −Λpos,QG,P .

Equivalently, PG is semi-stable if for every reduction PB of PG to the Borel B, we have

PB ∈ BunλB with ψ(λ) ∈ −Λpos,QG .

It is known that semi-stable bundles form an open sub-stack of finite type, BunssG (X). LetBunssM (X) be the corresponding open sub-stack of BunM (X), and let BunssP (X) be the pre-image of BunssM (X) in BunssP (X).

5.1.7. For a parabolic P let λP be an element ΛG,P . We say that is dominant if φP (λP ) ∈ Λ+,QG .

Let Λ+G,P denote the cone of dominant elements. We say that λP ∈ Λ+

G,P is regular if it lies off

the walls of Λ+G,P , i.e., if 〈φP (λP ), αi〉 > 0 for every simple root αi /∈ [M,M ].

Theorem 5.1.8.

(1) Let λP ∈ ΛG/Λ[M,M ] be dominant and regular. Then the map pP defines an isomorphism

between BunλP ,ssP (X) and a locally closed sub-stack of finite type in BunG(X). (We shall denote

this sub-stack BunP,λPG (X).)

(2) The sub-stacks BunP,λPG (X) for various pairs (P, λP ) are pairwise non-intersecting, and their

union covers BunG(X), i.e., every geometric point of BunG belongs to exactly one BunP,λPG (X).

(3) If PG ∈ BunP,λPG (X) admits a reduction to a parabolic P1 with parameter λP1, then the

element φP (λP )− φP1(λP1) ∈ Λpos,QG .

(4) Substacks of the form ⋃(P,λP ), φP (λP )−λ0∈Λpos,QG

BunP,λPG (X),

for some fixed λ0 ∈ Λ+,Q are closed.

5.1.9. Let P be a parabolic and P− an opposite parabolic. We shall identify their Levi factorsvia embedding M ' P ∩ P− into both P and P−.

Lemma 5.1.10. There exists an element d ∈ Q≥0, which depends on the genus of X, with thefollowing property:

For every parabolic P and λP ∈ ΛG,P such that αi(ϕP (λP )) ≥ d for every simple root αi /∈[M,M ], and for every PM ∈ BunλP ,ssM (X) the following holds:

• (i) H1(X, (NP )PM ) = 0, where NP is the unipotent radical of P .• (ii) H1(X,Lie((NP ))PM ) = 0.

We call a stratum BunP,λPG (X) with λ satisfying the assumption of the lemma ”deep enough”.

The first of these conditions implies that the map BunλP ,ssM (X) → BunλP ,ssP (X), given by theembedding is smooth and surjective. The second conditions implies that the map

p−P : BunλP ,ssP− (X)→ BunG(X)

is smooth.


5.1.11. Finally, we are ready to prove Theorem 5.1.2. We take the open subsets Uα to be open

subsets that are unions of the strata BunP,λPG (X), which include all strata that are not deepenough. To prove the theorem, it is enough to show that if U is an open subset of finite type of

BunG(X) and V := BunλP ,ssG (X) is a stratum, which is contained and is closed in U , and F is

an object of(DκU−V -mod

)c, then the ∗-extension of F to the entire U will belong to (Dκ

U -mod)c.

Let us denote the morphisms U − V ↪→ U and V ↪→ U by j and i respectively. Since theimage of (Dκ

U -mod)c

under j∗ generates(DκU−V -mod

)c, it sufficient to show that the functor i!

sends (DκU -mod)

cto (Dκ

V -mod)c.

Consider the morphism p−P : BunλP ,ssP− (X) → BunG(X). By assumption, it is smooth. Let

U and V be the preimages in BunλP ,ssP− (X) of U and V , respectively. Let j and i denote thecorresponding morphisms.

Lemma 5.1.12. For any P and dominant regular λP , the natural morphism

BunλP ,ssM (X)→ BunλP ,ssP− (X) ×BunG(X)

BunλP ,ssP (X)

is an isomorphism.

By the ”deep enough” assumtion, the map π : U → U is smooth, and its restriction to V ,

i.e., the map V → V is a tower of fibrations into affine spaces. Hence, it is enough to show that

i!(F) ∈(DκV

-mod)c,

where F is the pull-back of F to U − V .

5.1.13. However, the latter situation falls into the following paradigm: we have a base stack S

(in our case S = BunλP ,ssM (X)), and another stack S′ mapping to S (in our case, the fiber of

S′ over PM ∈ BunλP ,ssM (X) is H1(X, (UP )PM )), in such a way that the morphism S′ → S isaffine, and there exists a Gm-action on S′ that preserves S and which contracts S′ to a sectioni : S → S′ (in our case, the Gm action is given by a regular dominant co-character of Z0(M)).

In this case, the functor i! sends(DS′-modGm

)cto (DS-mod)

c.

5.1.14. Proof of Proposition 3.1.8. We claim that the open sub-stack in question is any open

union of the strata BunλP ,ssG (X), which contains all those that are not sufficiently deep. In fact,we claim that for any sufficiently deep stratum, the category Dκ

BunλP ,ss

G (X)-mod is zero.

The stratum in question is isomorphic, as a stack, to BunλP ,ssP (X). The group Z0(M) actson it via its adjoint action on P . Moreover, the twisting given by κ is equivariant with respectto Z0(M) against the character of Lie(Z0(M)) equal to −κ(λP ,−), which is non-integral. Tofinish the proof, it suffices to notice that the ”sufficiently deep” assumption implies that the

above action of Z0(M) on BunλP ,ssP (X) is in fact trivial.

5.2. The Iwahori case.


5.2.1. As in the spherical case, Theorem 3.3.3 follows from the next result:

Theorem 5.2.2. For Uα, Uβ as in Theorem 5.1.2 with α ”deep enough”, the functor

(j′α,β)∗ : DκU ′α

-modTn,w,λ → Dκ

U ′β-modT

n,w,λ

sends (DκU ′α

-modTn,w,λ)c to (Dκ

U ′β-modT

n,w,λ)c.

This theorem admits the same corollaries as in the spherical case:

Corollary 5.2.3.

(1) The functor (j′α,β)!, left adjoint to (j′α,β)! : DκU ′β

-modTn,w,λ → Dκ

U ′α-modT

n,w,λ is defined.

(2) The category Dκ

BunG(X,◦Ix)

-modTn,w,λ

! is compactly generated and is equivalent to

lim−→α

DκU ′α

-modTn,w,λ,

under the functors (j′α,β)!.

(3) The compact objects of Dκ

BunG(X,◦Ix)

-modTn,w,λ

! are of the form (j′α)!(F), where F is an

object of(DκU ′α

-modTn,w,λ

)c, and where (j′α)! denotes the tautological functor Dκ

U ′α-mod →

Dκ

BunG(X,◦Ix)

-modTn,w,λ

! .

5.2.4. Proof of Theorem 5.2.2. Let (P, λP ) be as in Sect 5.1.7. Let BunλP ,ssG (X, Ix) (resp.,

BunλP ,ssG (X,◦Ix)) denote the preimage of the corresponding stratum in BunG(X, Ix) (resp.,

BunG(X,◦Ix)).

In the present situation we modify the definition of pair (P, λP ) to be deep enough, bymaking it slightly stronger. Namely, we shall require that

• (i) H1(X, (NP )′PM ) = 0, where (NP )′ is the sub-sheaf of Maps(X,NP ) equal to thekernel of the homomorphism of the latter to Nn

P , corresponding to evaluation at x =x1, ..., xn.• (ii) H1(X,Lie((NP ))PM (−x)) = 0.

As in the BunG(X) case, it is easy to see that all but finitely many strata are ”deep enough”.The open sub-stacks Uα that appear in Theorem 5.2.2 are open unions of strata that containall strata that are not deep enough in the new sense. As in the proof of Theorem 5.1.2, for theproof of Theorem 5.2.2, it is sufficient to show that for an open sub-stack U ⊂ BunG(X) of

finite type that contains a stratum BunλP ,ssG (X), the !-pullback functor sends

(DκU ′ -modT

n,w,λ)c → (Dκ

BunλP ,ss

G (X,◦Ix)

-modTn,w,λ)c.

To prove this, we shall need to refine our stratification. By definition, we have:

BunλP ,ssG (X, Ix) ' BunλP ,ssP (X) ×(pt /P )n

(B\G/P )n,

where the map

BunP → (pt /P )n

corresponds to taking the fiber of PP at the points x1, ..., xn.


Let w := w1, ..., wn be an n-tuple of elements of the set W/WM . Let (B\G/P )w be the corre-

sponding Schubert stratum in (B\G/P )n, and let BunλP ,ss,wG (X, Ix) (resp., BunλP ,ss,wG (X,◦Ix))

be its preimage in BunλP ,ssG (X, Ix) (resp., BunλP ,ssG (X,◦Ix)).

It is sufficient to show that for an open sub-stack of finite type U ⊂ BunG(X,◦Ix), which

contains a deep enough stratum BunλP ,ssG (X), and F ∈ (DκU ′ -modT

n,w,λ)c, the !-restriction of

F to the stratum BunλP ,ss,wG (X,◦Ix) belongs to (Dκ

BunλP ,ss,w

G (X,◦Ix)

-modTn,w,λ)c.

5.2.5. Let P− be the opposite parabolic. Consider the Cartesian products

BunλP ,ssP− (X) ×BunG(X)

BunG(X, Ix),

and

BunλP ,ssP− (X) ×BunG(X)

BunG(X,◦Ix),

and let BunλP ,ss,wP− (X, Ix) (resp., BunλP ,ss,wP− (X,◦Ix)) be their locally closed sub-stacks, respec-

tively, corresponding to the condition that the relative position of the reductions of the G-torsor(PG)xi to B and P− is wi. (Our conventions are such that given a G-torsor with transversalreductions to P and P−, there exists exactly one reduction to B, which is in position w withrespect to P and P−.)

By the ”deep enough” assumption, the projections

BunλP ,ss,wP− (X, Ix)→ BunP−(X, Ix) and BunλP ,ss,wP− (X,◦Ix)→ BunP−(X,

◦Ix)

are smooth, and their images contain the strata BunλP ,ss,wG (X, Ix) and BunλP ,ss,wG (X,◦Ix), re-

spectively.

Consider the Cartesian product

(5.1) BunλP ,ss,wP− (X,◦Ix) ×

BunG(X,◦Ix)

BunλP ,ss,wG (X,◦Ix).

By Lemma 5.1.12, the stack (5.1) identifies with BunλP ,ssM (X,◦Ix) and its projection onto

BunλP ,ss,wG (X,◦Ix) is a tower of affine fibrations.

Hence, we need to show that for

F ∈

(Dκ

BunλP ,ss,w

P−(X,◦Ix)

-modTn,w,λ

)c,

its !-restriction to the sub-stack (5.1) is a compact object of the corresponding category.

Let i denote the corresponding closed embedding

i : BunλP ,ssM (X,◦Ix) ↪→ BunλP ,ss,wP− (X,

◦Ix).

In addition, we have a natural projection

(5.2) π : BunλP ,ss,wP− (X,◦Ix) � BunλP ,ssM (X,

◦Ix).

By the ”sufficiently deep” assumption, the morphism π is representable and affine. The twostacks appearing in (5.2) are acted on by Z0(M)n (via the embedding Z0(M) ↪→ T and the


action of the latter on G/N), and another copy of Z0(M) induced by its adjoint action on P−.Consider the action of Z0(M) given by

(5.3) z 7→ (w−11 (z), ..., w−1

n (z); z−1).

It acts trivially on BunλP ,ssM (X,◦Ix) and contracts BunλP ,ss,wP− (X,

◦Ix) to the image of the section

i. Hence, we find ourselves in the paradigm of Sect 5.1.13 (in the monodromic rather thanequivariant situation, with the conclusion being the same).

5.2.6. Proof of Proposition 3.3.7. We will show that the category Dκ

BunλP ,ss,w

G (X,◦Ix)

-modTn,w,λ

is zero for any (P, λP ), which is deep enough and any w.

By the above, it is sufficient to show that the corresponding category

Dκ

BunλP ,ss

M (X,◦Ix)

-modTn,w,λ

is zero. For that it is sufficient that the corresponding category of twisted D-modules

Dκ,λ

BunλP ,ss

M (X,Ix)-mod

on BunλP ,ssM (X, Ix) is zero.

Consider again the action of Z0(M) given by (5.3). The required assertion follows from the

fact that the resulting twisting on BunλP ,ssM (X, Ix) is Z0(M)-equivariant against the character

w−11 (λ1) + ...+ w−1

n (λn) + κ(λP ,−), which is non-integral.

6. Finiteness properties of the localization functor

6.1. Localization in the Iwahori case.

6.1.1. Let us return to the set-up of Sect 4.1. Consider the group-scheme K =◦Ix :=

◦Ix1×...×

◦Ix.

We obtain a functor

(6.1) Loc◦Ixgκ,x,BunG(X,x),!

: gκ,x-mod◦Ix → Dκ

BunG(X,◦Ix)

-mod!,

and the corresponding functors

(6.2) LocIxgκ,x,BunG(X,x),!

: gκ,x-modIx → DκBunG(X,Ix)-mod!,

(6.3) Loc◦Ix,T

n,w

gκ,x,BunG(X,x),!: gκ,x-mod

◦Ix,T

n,w := (gκ,x-mod◦Ix)T

n,w → Dκ

BunG(X,◦Ix)

-modTn,w

! ,

(6.4) Loc◦Ix,T

n,w,λ

gκ,x,BunG(X,x),!: gκ,x-mod

◦Ix,T

n,w,λ :=

(gκ,x-mod◦Ix)T

n,w,λ → DκBunG(X,Ix)-modT

n,w,λ! ,

for a collection λ = λ1, ..., λn of weights.

By Sect 4.1.4, each of the above functors, when restricted to the corresponding subcategoryof compact objects, factors as

Loc◦Ixgκ,x,BunG(X,x),!∗

: (gκ,x-mod◦Ix)c → (Dκ

BunG(X,◦Ix)

-mod)c,!∗,

LocIxgκ,x,BunG(X,x),!∗

: (gκ,x-modIx)c → (DκBunG(X,Ix)-mod)c,!∗,


Loc◦Ix,T

n,w

gκ,x,BunG(X,x),!∗: (gκ,x-mod

◦Ix,T

n,w)c → (Dκ

BunG(X,◦Ix)

-modTn,w)c,!∗,

Loc◦Ix,T

n,w,λ


◦Ix,T

n,w,λ)c → (Dκ

BunG(X,◦Ix)

-modTn,w,λ)c,!∗.

6.1.2. In addition, we obtain the functors

(6.5) Γ(BunG(X, x),−)∗ : Dκ

BunG(X,◦Ix)

-mod∗ → gκ,x-mod◦Ix ,

and the corresponding functors

(6.6) Γ(BunG(X, x),−)∗ : DκBunG(X,Ix)-mod∗ → gκ,x-modIx ,

(6.7) ΓTn,w(BunG(X, x),−)∗ : Dκ

BunG(X,◦Ix)

-modTn,w∗ → gκ,x-mod

◦Ix,T

n,w,

(6.8) ΓTn,w(BunG(X, x),−)∗ : Dκ

BunG(X,◦Ix)

-modTn,w,λ∗ → gκ,x-mod

◦Ix,T

n,w,λ.

6.1.3. By Proposition 4.3.2 and Theorem 3.3.3, we have:

Corollary 6.1.4. Under the equivalences(Dκ

BunG(X,Ix)-modTn,w,λ

!

)∨' Dκ′

BunG(X,Ix)-modTn,w,−λ

!

and (gκ,x-mod

◦Ix,T

n,w,λ

)∨' gκ′,x-mod

◦Ix,T

n,w,−λ,

the functors Loc◦Ix,T

n,w,λ

gκ,x,BunG(X,x),!and ΓT

n,w(BunG(X, x),−)∗ become the duals of one another.

Corollary 6.1.5. The functors

DBunG(X,x) ◦ Locgκ,x,BunG(X,x),!∗ ◦Dgx and Locgκ′,x,BunG(X,x),!∗

mapping

(gκ′,x-mod◦I,Tn,w,−λ)c →

(Dκ′

BunG(X,◦Ix)

-modTn,w,−λ

)c,!∗are isomorphic.

6.1.6. The main result of the present section is the following:

Theorem 6.1.7. Assume that κ is positive. Then the functor Loc◦Ix,T

n,w,λ

gκ,x,BunG(X,x)of (6.4) sends(

gκ,x-mod◦Ix,T

n,w,λ

)c→(Dκ

BunG(X,Ix)-modTn,w,λ

!

)c.

By Sect 3.3.8, this theorem can be reformulated as follows:

Theorem 6.1.8. Assume that κ is positive. Then functor

Loc◦Ix,T

n,w,λ


◦Ix,T

n,w,λ)c → (Dκ

BunG(X,◦Ix)

-modTn,w,λ)c,!∗

has its image in

(Dκ

BunG(X,◦Ix)

-modTn,w,λ

! )c ⊂ (Dκ

BunG(X,◦Ix)

-modTn,w,λ)c,!∗.


Further, by Corollary 6.1.5, we can reformulate Theorem 6.1.9 as follows:

Theorem 6.1.9. Assume that κ is negative. Then functor

Loc◦Ix,T

n,w,λ


◦Ix,T

n,w,λ)c → (Dκ

BunG(X,◦Ix)

-modTn,w,λ)c,!∗

has its image in

(Dκ

BunG(X,◦Ix)

-modTn,w,λ∗ )c ⊂ (Dκ

BunG(X,◦Ix)

-modTn,w,λ)c,!∗.

I.e., for M ∈ (gκ,x-mod◦Ix,T

n,w,λ)c, the !-stalks of Loc◦Ix,T

n,w,λ

gκ,x,BunG(X,x),!(M) are zero away from an

open sub-stack of finite type.

As a formal corollary of Theorems 6.1.7 and 6.1.9, we obtain:

Corollary 6.1.10.

(1) Let κ be negative. Then there exists a functor

Loc◦Ix,T

n,w,λ

gκ,x,BunG(X,x),∗: gκ,x-mod

◦Ix,T

n,w,λ → Dκ

BunG(X,◦Ix)

-modTn,w,λ∗ ,

which is the left adjoint to ΓTn,w(BunG(X, x),−)∗. We have:

Q◦Ix,T

n,w,λ∗→! ◦ Loc

◦Ix,T

n,w,λ

gκ,x,BunG(X,x),∗' Loc

◦Ix,T

n,w,λ

gκ,x,BunG(X,x),!.

(2) Let κ be positive. Then there exists a functor

ΓTn,w(BunG(X, x),−)! : Dκ

BunG(X,◦Ix)

-modTn,w,λ∗ → gκ,x-mod

◦Ix,T

n,w,λ,

right adjoint to Loc◦Ix,T

n,w,λ

gκ,x,BunG(X,x),!. We have:

ΓTn,w(BunG(X, x),−)! ◦Q

◦Ix,T

n,w,λ∗→! ' ΓT

n,w(BunG(X, x),−)∗.

6.2. Proof Theorem 6.1.7.

6.2.1. By definition, we have to show that for M among a set of compact generators of

gκ,x-mod◦Ix,T

n,w,λ there exists an open sub-stack of finite type UM ⊂ BunG(X) such that

for any F′ ∈ Dκ

BunG(X,◦Ix)

-modTn,w,λ supported outside of UM , we have

(6.9) HomDκ

BunG(X,◦Ix)

-modTn,w,λ

(Loc

◦Ix,T

n,w,λ

gκ,x,BunG(X,x),!(M),F′

)= 0.

It is sufficient to take M to be the Verma module, i.e., ∆κ,µ,x, where µ = µ1, ..., µn is ann-tuple of weights such that µi − λi ∈ ΛG. The object

Loc◦Ix,T

n,w,λ

gκ,x,BunG(X,x),!(∆κ,µ,x)


is the induced twisted D-module, i.e., for F′ ∈ Dκ

BunG(X,◦Ix)

-modTn,w,λ, we have:

HomDκ

BunG(X,◦Ix)

-modTn,w,λ

(Loc

◦Ix,T

n,w,λ

gκ,x,BunG(X,x),!(∆κ,µ,x),F′

)'

' Homtn

(kµ,Γ(BunG(X,

◦Ix),F′)

),

where kµ is the corresponding 1-dimensional representation of tn.

6.2.2. For (P, λP , w) let ιλP ,w denote the locally closed embedding of the corresponding stratum

into BunG(X,◦Ix). It is sufficient to take F′ from (6.9) to be of the form ιλP ,w∗ (F) for some

(6.10) F ∈ Dκ

BunλP ,ss,w

G (X,◦Ix)

-modTn,w,λ,

and to show that the vanishing occurs for all but finitely many strata. I.e., we have to showthat for all but finitely many strata

(6.11) Homtn

(kµ,Γ(BunG(X,

◦Ix), ιλP ,w∗ (F))

)= 0.

We can assume that F belongs to the heart of the t-structure. Moreover, we can assumethat F is an object of the corresponding category

Dκ,λ

BunλP ,ss,w

G (X,Ix)-mod

of twisted D-modules on BunλP ,ss,wG (X, Ix). Multiplying by the line bundle on BunG(X, Ix),

corresponding to λ− µ, we are reduced to calculating

(6.12) Γ(

BunG(X, Ix), ιλP ,w∗ (F))

for

(6.13) F ∈ Dκ,µ

BunλP ,ss,w

G (X,Ix)-mod,

and showing that it vanishes for all but finitely many strata.

6.2.3. With no restriction of generality we can assume that (P, λP ) is ”deep enough”. Recallthe projection

q−P : BunλP ,ssM (X, Ix)→ BunλP ,ss,wG (X, Ix).

It realizes BunλP ,ss,wG (X, Ix) as a quotient of BunλP ,ssM (X, Ix) with respect to the trivial actionof the unipotent group-scheme U, whose fiber at PM ∈ BunM (X) is

Maps(X,N(P )wPM ),

where N(P )w ⊂ N(P ) is the sub-sheaf, consisting of sections

{x 7→ n(x) |wi(n(xi)) ∈ n+}.

Let u denote the sheaf of Lie algebras corresponding to U.

Consider the pull-back of normal bundle to BunλP ,ssG (X) inside BunG(X); let E1 denote its

pull-back to BunλP ,ssM (X, Ix). This is a vector bundle, whose fiber at PM ∈ BunM (X) is thevector space H1(X, (g/p)PM ).


Consider now the normal bundle to BunλP ,ss,wG (X, Ix) insider

BunG(X, Ix) ×BunG(X)

BunλP ,ssG (X).

Let us denote by E2 its pull-back BunλP ,ssM (X, Ix). This is a vector bundle, whose fiber atPM ∈ BunM (X) is the vector space

⊕i=1,...,n

(g/n(P ))wi(PM )xi,

where (g/n(P ))w ⊂ g/n(P ) is the subspace , consisting of elements

{g |w0 · w(g) ∈ n+}.

For F in Sect 6.13, let F denote its pull-back to BunλP ,ssM (X, Ix). To prove the vanishingof (6.12), it is sufficient to show that for any integers k1, k2 and k3 the space of sections over

BunλP ,ssM (X, Ix) of the quasi-coherent sheaf

(6.14) F ⊗ Λtop(E1)⊗ Λtop(E2)⊗ Symk1(E1)⊗ Symk2(E2)⊗ Λk3(u∗)

vanishes.

We shall again use the torus Z0(M) that acts trivially on BunλP ,ssM (X,◦Ix), but with respect

to which the quasi-coherent sheaf (6.14) has an equivariant structure. Let us calculate thecharacters of this action.

6.2.4. The action of Z0(M) on E2 is trivial, and hence, so is the action on Λtop(E2)⊗Symk1(E2).

The action of Z0(M) on F is given by the character

w−11 (µ1) + ...+ w−1

1 (µn) + κ(λ,−),

if this character is integral (for otherwise, F = 0).

The action of Z0(M) on E1 consists of the characters α ∈ n(P ), each coming with multiplicity

equal to (1− g) + 〈α, λ〉. Therefore, the characters η1 on Symk1(E1) are sums of α’s from n(P ).

In particular, the action of Z0(M) on the line bundle Λtop(E1) is given by the character(Σ

α∈n(P )〈α, λ〉 · α

)+ 2(1− g) · ρP ,

where the first term is easily seen to be equal to κKil2 (λ,−), and where 2 · ρP := Σ

α∈n(P )α.

The action of Z0(M) on Λk3(u∗) has characters η3 equal to sums of α’s from n(P ).

6.2.5. Let ν : Gm → Z0(M) be a regular dominant co-character, i.e., 〈α, ν〉 > 0 for anyα ∈ n(P ). The resulting character of Gm on the space of sections of (6.14) is the sum of threeterms:

(1) (κ− κKil2 )(λ, ν)

(2) 〈η1, ν〉+ 〈η2, ν〉(3) 2(1− g)〈ρP , ν〉+ Σ

i=1,...,n〈w−1

i (µi), ν〉.

We obtain that the integers on the third line are bounded in a way independent of λP ; theintegers on the second line are positive, and the integer on the first line is arbitrarily large withλP as long as the form (κ− κKil

2 ) is positive definite.�


7. Arkhipov’s functor in the affine case

7.1. The kernel for Arkhipov’s functor.

7.1.1. Let κ be a negative level and recall the objects

S∞→0κ,κ ∈ gκ,0-modI0 ⊗ gκ,∞-modI∞ ' gκ,0,∞-modI0,I∞ ,

which defines the functor

Ψ∞→0κ′→κ,

and the object

S0→∞κ′,κ′ ∈ gκ′,0-modI0 ⊗ gκ′,∞-modI∞ ' gκ′,0,∞-modI0,I∞ ,

which defines the functor

Φ∞→0κ→κ′ ,

The goal of this section is to describe S∞→0κ,κ and S0→∞

κ′,κ′ explicitly.

7.1.2. From now on we shall fix the global curve X to be P1 with two marked points x1 = 0and x2 =∞. We fix a coordinate t on P1 so that t(0) = 0.

Consider the stacks BunG(P1), BunG(P1, I0, I∞). Let BunssG (P1) ⊂ BunG(P1) be the semi-stable locus, which identifies with pt /G. Let BunssG (P1, I0, I∞) ⊂ BunG(P1, I0, I∞) be itspreimage under the forgetful map BunG(P1, I0, I∞)→ BunG(P1). We have an isomorphism:

BunssG (P1, I0, I∞) ' (G/B)× (G/B)/G.

Let

Bunss,genG (P1, I0, I∞) ⊂ BunssG (P1, I0, I∞)

be the open sub-stack corresponding to the open G-orbit on (G/B)× (G/B). We let

: Bunss,genG (P1, I0, I∞) ↪→ BunG(P1, I0, I∞)

denote the corresponding morphism.

Note that the TDO on BunG(P1), corresponding to any κ, trivializes over BunssG (P1). In par-ticular, the corresponding TDO on BunG(P1, I0, I∞) trivializes also over Bunss,genG (P1, I0, I∞).Let

OBunss,genG (P1,I0,I∞) ∈ DκBunss,genG (P1,I0,I∞)-mod

denote the corresponding canonical object. Consider the corresponding object

(7.1) ∗(OBunss,genG (P1,I0,I∞)) ∈ (DκBunG(P1,I0,I∞)-mod∗)

c.

Additionally, we shall consider its Verdier dual, i.e.,

(7.2) !(OBunss,genG (P1,I0,I∞)) ∈ (Dκ′

BunG(P1,I0,I∞)-mod!)c.


7.1.3. The following should be regarded as the main result of the present paper:

Theorem 7.1.4. We have a canonical isomorphism in gκ,0,∞-modI0,I∞ :

S∞→0κ,κ ' Γ

(BunG(P1, 0, ∞), ∗(OBunss,genG (P1,I0,I∞))

)∗.

In addition, we propose the following conjecture:

Conjecture 7.1.5. We have a canonical isomorphism in gκ′,0,∞-modI0,I∞ :

S0→∞κ′,κ′ ' Γ

(BunG(P1, 0, ∞), !(OBunss,genG (P1,I0,I∞))

)!.

Theorem 7.1.4 and Conjecture 7.1.5 can be regarded as affine analogs of Theorem 1.4.3 inits incarnation given by Corollary 1.4.5. Later on we shall establish an affine analog of thecommutative diagram of Corollary 1.4.6.

7.2. Proof of Theorem 7.1.4.

7.2.1. The main step in the proof of the theorem is the following proposition, which is an affineanalog of Proposition 1.5.1:

Proposition 7.2.2. For M ∈ (gκ,0-modI0)c, N ∈ (gκ,∞-modI∞)c there exists a canonicalisomorphism:

Homgκ,0-modI0 (M,N∗) '

' HomDκBunG(P1,I0,I∞)

-mod∗

(LocI0,I∞

gκ,0,∞,BunG(P1,0,∞),∗(M ⊗N), ∗(OBunss,genG (P1,I0,I∞))).

7.2.3. Proof of Proposition 7.2.2. As in the finite-dimensional case, it is enough to establishan isomorphism on the level of complexes, which consists of modules with a standard. Wewill show that if both M and N admit a standard flag, then both sides of the equation in theproposition are acyclic off degree 0 and are isomorphic to one another.

Indeed, since (∆κ,∞,µ)∗ ' ∇κ,0,−w0(µ), the acyclicity assertion is true for the LHS. Its 0-thcohomology for any M and N is given by

Hom(

(M ⊗Nw0)g[t,t−1] , k).

Hence, the full expression is given by

(7.3) Hom(H•(g[t, t−1];T,M ⊗Nw0), k),

since the vector space in (7.3) is acyclic off degree 0 if M and N admit standard flags, becausein this case the tensor product M ⊗N is free over both Lie sub-algebras

t · g[[t]]⊕ n and t−1 · g[[t−1]]⊕ n−.

Remark. The vector space in (7.3) identifies, up to a twist by w0 with the space of conformalblocks on P1 with coefficients in M and N . So, the above assertion is the well-known statementthat the Hom into the contragredient module can be calculated as conformal blocks.

The RHS of Proposition 7.2.2, by adjunction, is given by

HomDκBun

ss,genG

(P1,I0,I∞)-mod

(LocI0,I∞

gκ,0,∞,Bunss,genG (P1,0,∞),∗(M ⊗N),OBunss,genG (P1,I0,I∞)

).

However,

Bunss,genG (P1, I0, I∞) ' pt /T,


and the stabilizer in g((t)) ⊕ g((t−1)) of the (unique) point in the corresponding T -torsor isg[t, t−1]. Hence the RHS of Proposition 7.2.2 is also given by (7.3).

�

7.2.4. Proposition 7.2.2 ⇒ Theorem 7.1.4. By definition, we need to show that for M ∈(gκ,0-modI0)c N ∈ (gκ′,∞-modI∞)c we have a canonical isomrphism

(7.4) Homgκ,0-modI0

(M, (Dg(N))∗

)'

' Homgκ,0-modI0

(M, 〈Γ


)∗, N〉g∞

).

By Proposition 7.2.2 , the LHS can be rewritten as

HomDκBunG(P1,I0,I∞)

-mod∗

(LocI0,I∞

gκ,0,∞,BunG(P1,0,∞),∗(M ⊗ Dg∞(N)), ∗(OBunss,genG (P1,I0,I∞))),

and, by adjunction, further as

Homgκ,0,∞-modI0,I∞

(M ⊗ Dg∞(N),Γ


)∗

).

The latter is, however, tautologically isomorphic to the RHS of (7.4).�

8. Localization onto ”thick” flags

8.1. The ”thick” affine flag scheme.

8.1.1. By definition, the thick affine scheme (resp., enhanced thick affine scheme) is FlthickG :=

BunG(P1,◦I0, ∞) (resp., Fl thickG := BunG(P1, I0, ∞)). We will be concerned with the categories

of equivariant D-modules on it, i.e., one of the categories

Dκ′

FlthickG

-mod!, Dκ′

FlthickG

-modT,w! , Dκ′

FlthickG

-modT,w,λ! ,

defined as in Sect 3.4.

Viewing FlthickG as endowed with an action of G(K∞), following the recipe of Sect 4.1, wehave a functor

Locgκ′,∞,FlthickG ,! : gκ′,∞-mod→ Dκ′

FlthickG

-modT,w! .

Lemma 8.1.2. Under the equivalence:

Dκ′

FlthickG

-modT,w! ' Dκ′

BunG(P1,0,∞)-mod

◦I0

! ,

the functor Locgκ′,∞,FlthickGidentifies with the composition

gκ′,∞-mod→ gκ,0-mod◦I0 ⊗ gκ′,∞-mod

Loc◦I0,T,w

gκ′,0,∞,BunG(P1,0,∞),!

−→ Dκ′

BunG(P1,0,∞)-mod

◦I0,T,w! ,

where the first arrow is the functor M 7→ Indgκ,0

Lie(◦I0)

(k)⊗M .


8.1.3. We shall consider a particular case of the above set up, restricting ourselves to the I∞-equivariant situation. Namely, we will consider the category

Dκ′

FlthickG

-mod(T,w),I∞! .

Proposition 8.1.4. Every object F ∈ Dκ′

FlthickG

-mod(T,w),I∞! canonically splits as a direct sum

F0 ⊕ F 6=0, where

F0 ∈ Dκ′

FlthickG

-mod(T,w,0),I∞! ,

and

F 6=0 ∈ ⊕λ 6=0

Dκ′

FlthickG

-mod(T,w,λ),I∞! .

Proof. By the definition of Dκ′

FlthickG

-mod(T,w),I∞! , we have to prove the corresponding assertion

over each T -stable open sub-stack of finite type

U ⊂ (FlthickG )/I∞ ' BunG(P1,◦I0, I∞).

However, each such sub-stack is a finite union of locally-closed sub-stacks Y, each of theform pt /U for some unipotent group U. It is sufficient to prove the assertion for each of the

categories Dκ′

Y -modT,w. However, the category Dκ′

Y -mod is equivalent to Vect, with the actionof T being trivial as a weak action, and the strong action deviating the standard one by acharacter. Hence, the assertion of the proposition amounts to splitting an object of Rep(T )into isotypic components.

�

8.2. The Kashiwara-Tanisaki equivalence.

8.2.1. Let κ be negative, and hence κ′ be positive. Consider the functor

Loc(T,w,0),I∞

gκ′,∞,FlthickG ,!: gκ′,∞-modI∞ → Dκ′

FlthickG

-mod(T,w,0),I∞!

equal to the composition of

LocI∞gκ′,∞,FlthickG ,!

: gκ′,∞-modI∞ → Dκ′

FlthickG

-mod(T,w),I∞! ,

followed by the functor F 7→ F0, given by Proposition 8.1.4.

Proposition 8.2.2. The above functor Loc(T,w,0),I∞

gκ′,∞,FlthickG ,!sends

(gκ′,∞-modI∞)c → (Dκ′

FlthickG

-mod(T,w),I∞! )c.

Proof. By Proposition 8.1.4, for M ∈ (gκ′,∞-modI∞)c the restriction of Loc(T,w,0),I∞

gκ′,∞,FlthickG ,!(M) to

every T -stable open sub-stack U ⊂ (FlthickG )/I∞ of finite type, belongs to (Dκ′

U -mod(T,w),I∞! )c.

We have to show that there exists a large enough open sub-stack of finite type UM such that

HomDκ′

FlthickG

-mod(T,w),I∞!

(Loc

(T,w,0),I∞

gκ′,∞,FlthickG ,!(M),F

)= 0

for any

F ∈ Dκ′

FlthickG

-mod(T,w),I∞!


supported off U . With restriction of generality, we can take F to be T -equivariant, i.e., anobject of

Dκ′

Fl thickG-modI∞

! ' Dκ′

BunG(P1,I0,I∞).

In this case we have:

HomDκ′

FlthickG

-mod(T,w),I∞!

(Loc

(T,w,0),I∞


)'

' HomDκ′BunG(P1,I0,I∞)

-mod!

(LocI0,I∞

gκ′,0,∞,BunG(P1,0,∞),!(Ind

gκ′,0Lie(I0)(k)⊗M),F

).

Since Indgκ′,0Lie(I0)(k)⊗M ∈ (gκ′,0,∞-modI0,I∞)c, our assertion follows from Theorem 6.1.7.

�

8.2.3. Consider now the functor

ΓT,w(FlthickG ,−)! : Dκ′

FlthickG

-mod(T,w,0),I∞! → gκ′,∞-modI∞ ,

equal to the composition

ΓT,w(BunG(P1, 0, ∞),−)! : Dκ′

FlthickG

-mod(T,w,0),I∞! '

' Dκ′

BunG(P1,0,∞)-mod

(◦I0,T,w,0),I∞

! → gκ,0-mod◦I0,T,w,0 ⊗ gI∞κ′,∞,

(see Corollary 6.1.10) followed by the functor

gκ,0-mod◦I0,T,w,0 ⊗ gI∞κ′,∞ → gI∞κ′,∞

that sends M 7→M◦I0,T .

Proposition 8.2.4. The functors Loc(T,w,0),I∞

gκ′,∞,FlthickG ,!and ΓT,w(BunG(P1, 0, ∞),−)! are mutually

adjoint.

Proof. By definition, we have to establish a functorial isomorphism for M ∈∈ (gκ′,∞-modI∞)c

and F ∈ Dκ′

FlthickG

-mod(T,w),I∞!

(8.1) HomDκ′

FlthickG

-mod(T,w),I∞!

(Loc

(T,w,0),I∞


)'

' HomgI∞κ′,∞

(M,ΓT,w(FlthickG ,F)!

).

The LHS in (8.1) is by definition

(8.2) HomDκ′

BunG(P1,◦I0,I∞)

-modT,w!

(Loc

(◦I0,T,w),I∞

gκ′,0,∞,BunG(P1,0,∞),!(Ind

gκ′,0Lie(I0)(k)⊗M),F

),

which by Proposition 8.2.2 can be rewritten as

(8.3) colim−→j

HomDκ′

BunG(P1,◦I0,I∞)

-modT,w,0!

(Loc

(◦I0,T,w),I∞

gκ′,0,∞,BunG(P1,0,∞),!(Ind

gκ′,0Lie(I0)(k)⊗M)/tj ,F

),

where tj denotes the j-th power of the maximal ideal in Sym(t).


Consider the modules Indgκ′,0Lie(I0)(k)/tj ∈ gκ′,0-mod

◦I0,T,w,0 for j = 0, 1, .... For any N ∈

gκ′,0-mod◦I0,T,w,0 we have:

Homgκ′,0-mod

◦I0,T,w

(Ind

gκ′,0Lie(I0)(k), N

)' colim

−→j

Homgκ′,0-mod

◦I0,T,w

(Ind

gκ′,0Lie(I0)(k)/tj , N

).

Hence, the RHS in (8.1) can be rewritten as

Homg

(◦I0,T,w)I∞κ′,0,∞

(Ind

gκ′,0Lie(I0)(k)⊗M,ΓT,w(FlthickG ,F)!

)'

' colim−→j

Homgκ′,0,∞-mod(

◦I0,T,w,0),

◦I∞

(Ind

gκ′,0Lie(I0)(k)/tj ⊗M,ΓT,w(FlthickG ,F)!

),

which by adjunction can be further rewritten as

(8.4) colim−→j

HomDκ′

BunG(P1,◦I0,I∞)

-modT,w,0!

(Loc

(◦I0,T,w,0),I∞

gκ′,0,∞,BunG(P1,0,∞),!(Ind

gκ′,0Lie(I0)(k)/tj ⊗M),F

),

which is the same as the expression in (8.3).�

8.2.5. The localization theorem at the positive level, due to Kashiwara-Tanisaki reads as follows:

Theorem 8.2.6. The functor ΓT,w(BunG(P1, 0, ∞),−)! realizes Dκ′

FlthickG

-mod(T,w,0),I∞! as a

direct summand of gκ′,∞-modI∞ .

We will show how deduce theorem from another localization result (also due to Kashiwaraand Tanisaki), but which takes place at the negative level.

9. Arkhipov’s functor and localization

9.1. From thin to thick flags.

9.1.1. Let FlthinG (resp., FlthinG ) be the ”thin” affine flag scheme (resp., the ”thin” enhancedaffine flag scheme), attached to point 0 ∈ P1, i.e.,

FlthinG := G(K0)/I0, G(K0)/◦I0.

Note that unlike the corresponding ”thick” spaces, the ”thin” ones are local, i.e., only dependon the formal neighborhood of 0 ∈ P1.

For a level κ we can consider the corresponding categories of twisted D-modules

DκFlthinG

-mod and DκFlthinG

-mod.

The ind-schemes FlthinG and FlthinG are acted on by G(K0), and the above categories carry astrong action of this group ind-scheme at level κ.

9.1.2. The rest of this sub-section is devoted to the proof of the following result:

Theorem-Construction 9.1.3. There exists a canonical equivalence

Φthin→thick : DκFlthinG

-modI0,(T,w,0) ' Dκ′

FlthickG

-mod(T,w,0),I∞!


9.1.4. To construct the functor Φthink→thick we recall that the involution g 7→ g−1 defines anequivalence

DκFlthinG

-modI0,(T,w,0) ' Dκ′

Fl thinG-mod

◦I0,T,w,0.

The action of G(K0) on the category Dκ′

BunG(X,0,I∞)-mod! defines the convolution functor

(9.1) Dκ′

FlthinG-modI0,T,w,0 ⊗Dκ′

BunG(X,0,I∞)-modI0

! → Dκ′

BunG(X,0,I∞)-mod

◦I0,T,w,0! '

' Dκ′

FlthickG

-mod(T,w,0),I∞! .

The sought-for functor Φthin→thick is given by (9.1) by inserting the object

!(OBunss,genG (P1,I0,I∞)) ∈ Dκ′

BunG(P1,I0,I∞)-mod! ' Dκ′

BunG(X,0,I∞)-modI0

! .

9.1.5. Note that the functor Φthin→thick admits also the following interpretation. We claimthat there exists a canonical map

FlthinGr→ FlthickG ,

compatible with the twistings. Here are two (of course, equivalent) ways to view it:

By definiton, FlthinG classifies the data (PG, β, α0), where PG is a principal G-bundle on P1,β is its trivialization on P1 − 0, and α is a reduction of (PG)0 to N .

By definition, FlthickG classifies the data (PG, γ, α0), where PG is a principal G-bundle on P1,β is its trivialization on the formal neighborhood D∞ of ∞ ∈ P1, and α is a reduction of (PG)0

to N .

The map r acts identically on the data of (PG, α0), and attaches to a datum of β its restrictionto D∞ ⊂ P1 − 0, twisted by w0.

The group-theoretic interpretation (although non-rigorous) is as follows:

FlthinG ' G(K0)/◦I0

id×w0↪→ G[t, t−1]\

(G(K0)/

◦I0 ×G(K∞)

)' G′out∞\G(K∞) ' FlthickG ,

where Gout∞ = G[t], and G′out∞ ⊂ Gout∞ is the sub-group of maps whose value at 0 belongsto N .

Let p denote the canonical projection

FlthickG → FlthickG /I∞ ' BunG(P1,◦I0, I∞).

Let q denote the composition

p ◦ r : FlthinG → BunG(P1,◦I0, I∞).

Since FlthinG is ind-proper, it is easy to see that the functor Φthin→thick is canonically iso-morphic to q!.


9.1.6. Let us show that Φthin→thick is an equivalence.

First, it is easy to see that Φthin→thick sends

(DκFlthinG

-modI0,(T,w,0))c → (Dκ′

FlthickG

-mod(T,w,0),I∞! )c.

Secondly, from the affine Bruhat decomposition, it is easy to see that the compact generators

of (Dκ′

FlthickG

-mod(T,w,0),I∞! are in the essential image of (Dκ

FlthinG

-modI0,(T,w,0))c.

Hence, it remains to show that Φthin→thick is fully faithful on (DκFlthinG

-modI0,(T,w,0))c. For

that end, it is enough to show that the corresponding functor

Φthin→thick : DκFl thinG

-modI0 → Dκ′

Fl thickG-modI∞

!

is fully faithful.

For F1,F2 ∈ (DκFl thinG

-modI0)c, by [FG1], we have a commutative diagram

Hom(F1,F2) −−−−→ Hom(Φthin→thick(F1),Φthin→thick(F2))

∼y ∼

yHom(δ1

FlthinG

,F∗1 ? F2) −−−−→ Hom(Φthin→thick(δ1FlthinG

),Φthin→thick(F∗1 ? F2)),

so we can assume that F1 = δ1FlthinG

. In the latter case, the required assertion reduces to the

statement that for F ∈ DκFl thinG

-modI0 , the map

(9.2) Hδ1FlthinG

(FlthinG ,F)→ Hc(Flthin,genG ,F)

is an isomorphism, where Flthin,genG ⊂ FlthinG is the big affine Bruhat cell. The fact that (9.2) isanother instance of Sect 5.1.13.

9.2. The basic commutative diagram.

9.2.1. Assume again that κ is negative. Recall the functor

ΓT,w : DκFlthinG

-mod(T,w,0) → gκ,0-mod.

It admits a right adjoint, denoted Locgκ,0,FlthinG.

The following theorem is due to Kashiwara-Tanisaki (see, e.g., [FG3] for a proof):

Theorem 9.2.2. The functor

ΓT,w : DκFlthinG

-modI0,(T,w,0) → gκ,0-modI0

realizes the former category as a direct summand of the latter.

The right adjoint to the functor in Theorem 9.2.2 is denoted LocI0,(T,w,0)

gκ,0,FlthinG

. It is isomorphic

to the composition of

LocI0gκ,0,FlthinG

: gκ,0-modI0 → DκFlthinG

-modI0 ,

followed by the functor

DκFlthinG

-modI0 → DκFlthinG

-modI0,(T,w,0)

defined as the projection on the corresponding direct summand, in complete analogy withSect 8.2.1.


9.2.3. The goal of this section is to prove the following theorem:

Theorem 9.2.4. We have a commutative diagram of functors:

DκFlthinG

-modI0,(T,w,0) Φthin→thick−−−−−−−−→ Dκ′

FlthickG

-mod(T,w,0),I∞!

LocI0,(T,w,0)

gκ,0,FlthinG

x Loc(T,w,0),I∞gκ,0,Flthick

G,!

xgκ,0-modI0

Φ0→∞κ→κ′−−−−→ gκ′,∞-modI∞ .

Of course, as the horizontal arrows appearing in the theorem are equivalences, its statementis equivalent to the following one:

Theorem 9.2.5. We have a commutative diagram of functors:

DκFlthinG

-modI0,(T,w,0) Ψthick→thin←−−−−−−−− Dκ′

FlthickG

-mod(T,w,0),I∞!

LocI0,(T,w,0)

gκ,0,FlthinG

x Loc(T,w,0),I∞gκ,0,Flthick

G,!

xgκ,0-modI0

Ψ∞→0κ′→κ←−−−− gκ′,∞-modI∞ ,

where Ψthick→thin is the inverse functor to Φthin→thick.

9.2.6. By Theorem 9.2.2, the functor LocI0,(T,w,0)

gκ,0,FlthinG

is also the left adjoint to

ΓT,w : DκFlthinG

-modI0,(T,w,0) → gκ,0-modI0 .

Combining this with Theorems 9.2.4 and 9.2.5, we obtain that the following:

Corollary 9.2.7. We have the following commutative diagrams of functors:

DκFlthinG

-modI0,(T,w,0) Φthin→thick−−−−−−−−→ Dκ′

FlthickG

-mod(T,w,0),I∞!

ΓT,w(FlthinG ,−)

y ΓT,w(FlthickG ,−)

ygκ,0-modI0

Φ0→∞κ→κ′−−−−→ gκ′,∞-modI∞ .

and

DκFlthinG

-modI0,(T,w,0) Ψthick→thin←−−−−−−−− Dκ′

FlthickG

-mod(T,w,0),I∞!

ΓT,w(FlthinG ,−)

y ΓT,w(FlthickG ,−)

ygκ,0-modI0

Ψ∞→0κ′→κ←−−−− gκ′,∞-modI∞ .

The above corollary theorem should be regarded as an affine analog of Corollary 1.4.6.

Remark. Note that the first commutative diagram of Corollary 9.2.7 follows tautologically fromConjecture 7.1.5. In its turn, the commutativity of this diagram would imply Theorems 9.2.4and 9.2.5 by Theorem 9.2.2.

9.2.8. Proof of Theorem 8.2.6. This follows as a combination of Corollary 9.2.7 and Theo-rem 9.2.2.

9.3. Proof of Theorem 9.2.5.


9.3.1. Let us describe the functor Ψthick→thin explicitly. As Φthin→thick is an equivalence,Ψthick→thin equals its right adjoint.

Here is a geometric description of Ψthick→thin in terms of Sect 9.1.5. It equals the compositionof

q! : Dκ′

FlthickG

-mod(T,w,0),I∞! → Dκ

FlthinG

-modT,w,0,

followed by the averaging functor

AvI0 : DκFlthinG

-modT,w,0 → DκFlthinG

-modI0,(T,w,0),

right adjoint to the forgetful functor

DκFlthinG

-modI0,(T,w,0) → DκFlthinG

-modT,w,0.

However, for the purpose of proof of Theorem 9.2.5, we shall need a description of the functor

Ψthick→thin in terms of the action of G(K0) on the category Dκ′

BunG(X,0,I∞)-mod! as in Sect 9.1.4.

9.3.2. Consider the following general paradigm. Let C be a category acted on by G(K0) at level

κ′. For two open-compact subgroups K ′,K ′′ ⊂ G(K0) we have the functors

ConvK′,K′′ : Dκ′

G(K0)-modK

′,K′′ ⊗ CK′′→ CK

′

and

co-ConvK′,K′′ : CK

′→ CK

′′⊗Dκ′

G(K0)-modK

′,K′′ ' CK′′⊗Dκ′

G(K0)/K′′-modK

′.

Let C∨ denote the category of all (co-limit preserving) functors C→ Vect, whether or not C

is dualizable. For F∨ ∈ (CK′′)∨ consider the functor

co-ConvK′,K′′

F∨ : CK′→ Dκ′

G(K0)/K′′-modK

′

equal to the composition of co-ConvK′,K′′ , followed by the functor

CK′′⊗Dκ′

G(K0)/K′′-modK

′ F∨(−)⊗Id−→ Dκ′

G(K0)/K′′-modK

′.

In what follows we will need the following properties of these functors. First, let A : C1 → C2

be a functor between categories acted on by G(K0), and let A∨ : C∨2 → C∨1 be its dual.

Lemma 9.3.3. For F∨ ∈ (CK′′

2 )∨ the composition

CK′′

1A→ CK

′′

2

co-ConvK′,K′′

F∨−→ Dκ′

G(K0)/K′′-modK

′

is canonically isomorphic to

co-ConvK′,K′′

A∨(F∨) : CK′′

1 → Dκ′

G(K0)/K′′-modK

′.

Secondly, let us assume that G(K0)/K ′′ and ind-proper, and that F is a compact object of

CK′′. Let F∨ be the corresponding object of C∨, i.e., Hom(F,−).

Lemma 9.3.4. Under the above circumstances, the functor ConvK′,K′′(−,F) sends compact

objects to compact ones, and its right adjoint is given by co-ConvK′,K′′

F∨ .


9.3.5. Returning to our situation, we obtain that the functor Ψthick→thin is given by

co-Conv(◦I0,T,w,0),I0∗(OBun

ss,genG

(P1,I0,I∞)),

which maps

Dκ′

FlthickG

-mod(T,w,0),I∞! ' Dκ′

BunG(X,◦I0,I∞)

-modT,w,0! ' Dκ′

BunG(X,0,I∞)-mod

◦I0,T,w,0! →

→ Dκ′

G(K0)/I0-mod

◦I0,T,w,0 ' Dκ′

FlthinG-mod

◦I0,T,w,0 ' Dκ

FlthinG

-modI0,(T,w,0).

By Theorem 7.1.4, the assertion of Theorem 9.2.5 reduces to the commutativity of thefollowing diagram of functors:

DκFlthinG

-modI0,(T,w,0)


ss,genG

(P1,I0,I∞))

←−−−−−−−−−−−−−−−−−−−− Dκ′

FlthickG

-mod(T,w,0),I∞!xLoc

I0,(T,w,0)

gκ,0,FlthinG

Loc(T,w,0),I∞gκ,0,Flthick

G,!

xgκ,0-modI0

〈Γ(

BunG(P1,0,∞),∗(OBunss,genG

(P1,I0,I∞)))∗,−〉g∞

←−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− gκ′,∞-modI∞ .

However, it is easy to see that it is sufficient to check the commutativity of the followingdiagram instead:

(9.3)

DκFlthinG

-modI0,(T,w)


ss,genG

(P1,I0,I∞))

←−−−−−−−−−−−−−−−−−−−− Dκ′

FlthickG

-mod(T,w),I∞!xLoc

I0,(T,w)

gκ,0,FlthinG

Loc(T,w),I∞gκ,0,Flthick

G,!

xgκ,0-modI0

〈Γ(

BunG(P1,0,∞),∗(OBunss,genG

(P1,I0,I∞)))∗,−〉g∞

←−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− gκ′,∞-modI∞ .

9.3.6. Note that the counter-clockwise circuit in (9.3) is the composition

gκ′,∞-modI∞

Indgκ′,0

Lie(◦I0)

(k)⊗−

−→ gκ′,0-modI0 ⊗ gκ′,∞-modI∞Loc

I0,I∞gκ′,0,∞,BunG(X,0,∞),!

−→

−→ Dκ′

BunG(X,0,I∞)-mod

◦I0,T,w!


ss,genG

(P1,I0,I∞))

−→ DκFlthinG

-modI0,(T,w).

By Lemma 9.3.3, this can be rewritten as

gκ′,∞-modI∞

Indgκ′,0

Lie(◦I0)

(k)⊗−

−→ gκ′,0-modI0 ⊗ gκ′,∞-modI∞ −→

co-Conv(◦I0,T,w,0),I0

Γ

(BunG(P1,0,∞),∗(OBun

ss,genG

(P1,I0,I∞))

)−→ Dκ

FlthinG

-modI0,(T,w).


9.3.7. The commutativity of (9.3) follows now from the following observation. Let us return

to the paradigm of Sect 9.3.2. Let C := gκ′,0-mod. Let K ′,K ′′ ⊂ G(K0) be open-compact

subgroups (in our situation, we take K ′ = I0, K ′′ =◦I0).

Consider the functor

gκ,0-modK′′'(gκ′,0-modK

′′)∨→ Dκ′

G(K0)/K′′-modK

′

given by

M 7→ co-ConvK′,K′′

M (Indgκ′,0k′′ (k)).

Lemma 9.3.8. The above functor is canonically isomorphic to the localization functor

LocK′′

gκ,0,G(K0)/K′: gκ,0-modK

′′→ Dκ

G(K0)/K′-modK

′′' Dκ′

G(K0)/K′′-modK

′.


10. Appendix: DG categories

10.1.

10.1.1.

Lemma 10.1.2.

10.2.

Lemma 10.2.1.

10.2.2.

10.2.3.

10.2.4.

10.3.

10.4.

10.4.1.

Lemma 10.4.2.

10.4.3.

10.4.4.

10.4.5.

10.5.

10.5.1.

10.5.2.

10.5.3.

10.5.4.

10.5.5.


References

[AG] S. Arkhipov, D. Gaitsgory, Differential operators on the loop group via chiral algebras.[BB] A. Beilinson, J. Bernstein, La localisation des D-modules.

[FG1]

[FG2] E. Frenkel, D. Gaitsgory, D-modules on the affine flag scheme and representations of the Kac-Moody Liealgebra.

[FG3]

[DrGa] V. Drinfeld and D. Gaitsgory, Compact generation of the category of D-modules on the stack of G-bundles on a curve, arXiv:1112.2402

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