On a strange invariant bilinear form on the space of automorphic forms

Abstract

Let F be a global field and $G:=SL(2)$. We study the bilinear form ${{\mathcal {B}}}$ on the space of K-finite smooth compactly supported functions on $G({\mathbb {A}})/G(F)$ defined by

$$\begin{aligned} {{\mathcal {B}}}(f_1,f_2):={{\mathcal {B}}}_{\mathrm {naive}}(f_1,f_2)-\langle M^{-1}{{\mathrm{{CT}}}}(f_1)\, ,{{\mathrm{{CT}}}}(f_2)\rangle , \end{aligned}$$

where ${{\mathcal {B}}}_{\mathrm {naive}}$ is the usual scalar product, ${{\mathrm{{CT}}}}$ is the constant term operator, and M is the standard intertwiner. This form is natural from the viewpoint of the geometric Langlands program. To justify this claim, we provide a dictionary between the classical and ‘geometric’ theory of automorphic forms. We also show that the form ${{\mathcal {B}}}$ is related to S. Schieder’s Picard–Lefschetz oscillators.

Appendices

Appendix A. Relation to works on the geometric Langlands program

In this appendix we relate this article to [11, 12, 14]. Subsections A.5.4–A.5.5, Subsections A.8–A.9, and Subsections A.11.5–A.11.7 are the nontrivial ones.

In Subsections A.5.4–A.5.5 we motivate the definition of the subspace ${\mathcal {A}}_{ps-c}\subset {\mathcal {A}}$ given in Subsection 4.2. In Subsections A.8–A.9 we motivate the definition of the function b from Subsection 1.3.2 and the definition of the form ${{\mathcal {B}}}$. In Subsections A.11.5–A.11.7 we discuss the relation between the operator ${{\mathrm{{Eis}}}}'$ from Subsection  2.12 and the functor ${{\mathrm{{Eis}}}}_!$ from [12].

1.1 A.1 D-modules, l-adic sheaves, and functions

1.1.1 A.1.1 ‘Left’ and ‘right’ functors

We will consider two different cohomological formalisms:

• (i) Constructible l-adic sheaves on schemes of finite type over a field;

• (ii) D-modules on schemes of finite type over a field of characteristic 0.

In each of them we have two adjoint pairs of functors $(f^*, f_*)$ and $(f_!, f^!)$. We will refer to $f^*$ and $f_!$ as ‘left’ functors and to $f_*$ and $f^!$ as ‘right’ functors (each ‘left’ functor is left adjoint to the ‘right’ functor from the same pair). Caveat: in the D-module setting the ‘left’ functors are, in general, only partially defined (because D-modules are not assumed holonomic). Thus in the D-module setting, we have to consider the ‘right’ functors as the ‘main’ ones. We prefer to do this in the constructible setting as well (then the analogy between the two settings becomes transparent).

Both cohomological formalisms (i) and (ii) exist in the more general setting of algebraic stacks locally of finite type over a field²¹, but the situation with the pushforward functor is subtle (in the D-module setting it is discussed in Subsection A.4 below). However, if a morphism f between algebraic stacks is representable ²² and has finite type²³, then $f_*$ and $f_!$ are as good as in the case of schemes.

1.1.2 A.1.2 Functions–sheaves dictionary (an unusual convention)

Let ${\mathcal {Y}}$ be an algebraic stack locally of finite type over ${\mathbb {F}}_q\,$ and let M be an object of the bounded constructible derived category of $\overline{{\mathbb {Q}}}_l$-sheaves on ${\mathcal {Y}}$. To such a pair we associate a ‘trace function’ on the groupoid²⁴ ${\mathcal {Y}}({\mathbb {F}}_q)$. We do it in an unconventional way: namely, our trace function corresponding to M equals Grothendieck’s trace function corresponding to the Verdier dual ${\mathbb {D}}M$ (in other words, the value of our trace function at $y\in {\mathcal {Y}}({\mathbb {F}}_q)$ is the trace of the arithmetic Frobenius acting on the !-stalk of M at y).

Thus the pullback of functions corresponds to the !-pullback of l-adic complexes, and the pushforward of functions with respect to a morphism f of finite type corresponds²⁵ to the $*$-pushforward of l-adic complexes. In other words, the standard operators between spaces of functions correspond to the ‘right’ functors in the sense of Subsection A.1.1.

Example A.1.3

According to our convention, the constant function 1 corresponds to the dualizing complex of ${\mathcal {Y}}$, which will be denoted by ${\omega }_{{\mathcal {Y}}}\,$.

Example A.1.4

In Subsection 1.3.2 we defined a function b on $({{\mathrm{{Bun}}}}_G\times {{\mathrm{{Bun}}}}_G)({\mathbb {F}}_q)$, where $G:=SL(2)$. According to our new convention, b corresponds to the complex $\Delta _!({\omega }_{{{\mathrm{{Bun}}}}_G})$, where $\Delta :{{\mathrm{{Bun}}}}_G\rightarrow {{\mathrm{{Bun}}}}_G\times {{\mathrm{{Bun}}}}_G$ is the diagonal. The stack ${{\mathrm{{Bun}}}}_G$ is smooth of pure dimension $d=3g_X-3$, where $g_X$ is the genus of the curve X. So the function $q^{-d}\cdot b$ corresponds to $\Delta _!((\overline{{\mathbb {Q}}}_l)_{{{\mathrm{{Bun}}}}_G})$.

Remark A.1.5

Let M be an l-adic complex on ${\mathcal {Y}}$ and f the corresponding function on ${\mathcal {Y}}({\mathbb {F}}_q )$. According to our convention, the function corresponding to M[2](1) equals qf.

1.1.3 A.1.6 Analogy between D-modules and functions

For certain reasons (including serious ones) the works [11, 12, 14] deal with D-modules but not with constructible sheaves. There is no direct relation between D-modules (which live in characteristic 0) and functions on ${\mathcal {Y}}({\mathbb {F}}_q)$, where ${\mathcal {Y}}$ is as in Subsection A.1.2. However there is an analogy between them. It comes from the analogy between the two cohomological formalisms considered in Subsection A.1.1 and the functions–sheaves dictionary as formulated in Subsection A.1.2.

1.2 A.2 Some categories of D-modules on ${{\mathrm{{Bun}}}}_G$ and ${{\mathrm{{Bun}}}}_T\,$

1.2.1 A.2.1

Let k denote an algebraically closed field of characteristic zero and X a smooth complete connected curve over k. Just as in the rest of the article, $G:=SL(2)$ and $T\subset G$ is the group of diagonal matrices.²⁶ Let ${{\mathrm{{Bun}}}}_G$ (resp.  ${{\mathrm{{Bun}}}}_T$) denote the moduli stack of principal G-bundles (resp. T-bundles) on X.

We will say ‘stack’ instead of ‘algebraic stack locally of finite type over k whose k-points have affine automorphism groups’. We will mostly deal with the stacks ${{\mathrm{{Bun}}}}_G$ and ${{\mathrm{{Bun}}}}_T\,$, which are not quasi-compact.

1.2.2 A.2.2

For any stack ${\mathcal {Y}}$ over k, one has the DG category of (complexes of) D-modules on ${\mathcal {Y}}$, denoted by ${\text {D-mod}}({\mathcal {Y}})$. Let ${\text {D-mod}}({\mathcal {Y}})_c\subset {\text {D-mod}}({\mathcal {Y}})$ denote the full subcategory of objects $M\in {\text {D-mod}}({\mathcal {Y}})$ such that for some quasi-compact open $U\overset{j}{\hookrightarrow }{\mathcal {Y}}$ the morphism $M\rightarrow j_*j^*M$ is an isomorphism. Let ${\text {D-mod}}({\mathcal {Y}})_{ps-c}\subset {\text {D-mod}}({\mathcal {Y}})$ denote the full subcategory of objects $M\in {\text {D-mod}}({\mathcal {Y}})$ such that for some quasi-compact open $U\overset{j}{\hookrightarrow }{\mathcal {Y}}$ the object $j_!j^!M$ is defined and the morphism $j_!j^!M\rightarrow M$ is an isomorphism.

Remark A.2.3

In D-module theory the functor $j_!$ is only partially defined, in general. An open quasi-compact substack $U\overset{j}{\hookrightarrow }{\mathcal {Y}}$ is said to be co-truncative if $j_!$ is defined everywhere. ²⁷ A stack ${\mathcal {Y}}$ is said to be truncatable if every quasi-compact open substack of ${\mathcal {Y}}$ is contained in a co-truncative one. The stacks ${{\mathrm{{Bun}}}}_G$ and ${{\mathrm{{Bun}}}}_T$ are truncatable. For ${{\mathrm{{Bun}}}}_T$ this is obvious (because each connected component of ${{\mathrm{{Bun}}}}_T$ is quasi-compact); for ${{\mathrm{{Bun}}}}_G$ this is proved in [11].

1.2.3 A.2.4

Note that ${\text {D-mod}}({{\mathrm{{Bun}}}}_T)_c={\text {D-mod}}({{\mathrm{{Bun}}}}_T)_{ps-c}$ (because each connected component of ${{\mathrm{{Bun}}}}_T$ is quasi-compact). On the other hand,

$$\begin{aligned} {\text {D-mod}}({{\mathrm{{Bun}}}}_G)_c\ne {\text {D-mod}}({{\mathrm{{Bun}}}}_G)_{ps-c}. \end{aligned}$$

Remark A.2.5

The approach of [10–12, 14] is to work only with cocomplete DG categories (i.e., those in which arbitrary inductive limits are representable). The DG category ${\text {D-mod}}({\mathcal {Y}})$ is cocomplete for any stack ${\mathcal {Y}}$. On the other hand, ${\text {D-mod}}({\mathcal {Y}})_c$ and ${\text {D-mod}}({\mathcal {Y}})_{ps-c}$ are not cocomplete if ${\mathcal {Y}}$ equals ${{\mathrm{{Bun}}}}_G$ or ${{\mathrm{{Bun}}}}_T\,$.

The reader may prefer to skip the next remark.

Remark A.2.6

For any cocomplete DG category ${\mathcal {D}}$, let ${\mathcal {D}}'\subset {\mathcal {D}}$ denote the following full subcategory: $M\in {\mathcal {D}}'$ if and only if there exists a finite collection S of compact objects of ${\mathcal {D}}$ such that M belongs to the cocomplete DG subcategory of ${\mathcal {D}}$ generated by S. For any truncatable stack ${\mathcal {Y}}$, one has the following description of ${\text {D-mod}}({\mathcal {Y}})_{ps-c}$ and ${\text {D-mod}}({\mathcal {Y}})_c$ in terms of the DG category ${\text {D-mod}}({\mathcal {Y}})$ and its Lurie dual ${\text {D-mod}}({\mathcal {Y}})^\vee$ (the latter two DG categories are cocomplete):

$$\begin{aligned} {\text {D-mod}}({\mathcal {Y}})_{ps-c}={\text {D-mod}}({\mathcal {Y}})' , \end{aligned}$$

(A.1)

$$\begin{aligned} {\text {D-mod}}({\mathcal {Y}})_c=\left( {\text {D-mod}}({\mathcal {Y}})^\vee \right) '. \end{aligned}$$

(A.2)

To prove (A.1), use [11, Prop. 2.3.7] and the following fact: for any quasi-compact²⁸ stack ${\mathcal {Z}}$, the DG category ${\text {D-mod}}({\mathcal {Z}})$ is generated by finitely many compact objects.²⁹ To prove (A.2), one can use the description of ${\text {D-mod}}({\mathcal {Y}})^\vee$ given in [11, Cor. 4.3.2] or [14, Subsection 1.2].

1.3 A.3 D-module analogs of ${\mathcal {A}}^K$, ${\mathcal {A}}_c^K$, ${{\mathcal {C}}}^K$, and ${{\mathcal {C}}}_c^K$

Let F be a function field. Then the space ${\mathcal {A}}^K$ (i.e., the subspace of K-invariants in ${\mathcal {A}}$) identifies with the space of all functions on $K\backslash G({\mathbb {A}})/G(F)$, i.e., on the set of isomorphism classes of G-bundles on the smooth projective curve over ${\mathbb {F}}_q$ corresponding to F. So we consider the DG category ${\text {D-mod}}({{\mathrm{{Bun}}}}_G)$ to be an analog of the vector space ${\mathcal {A}}^K$. This DG category was studied in [11, 12, 14] in the spirit of ‘geometric functional analysis’ (with complexes of D-modules playing the role of functions and cocomplete DG categories playing the role of abstract topological vector spaces). Let us note that the D-module analog of the whole space ${\mathcal {A}}$ has not been studied in this spirit, and it is not clear how to do it.

Because of the convention of Subsection A.1.2, we consider ${\text {D-mod}}({{\mathrm{{Bun}}}}_G)_c$ to be an analog of ${\mathcal {A}}_c^K$.

We consider ${\text {D-mod}}({{\mathrm{{Bun}}}}_T)$ to be a D-module analog of the space ${{\mathcal {C}}}^K$ (the reason is clear from Example 2.3.1). We consider ${\text {D-mod}}({{\mathrm{{Bun}}}}_T)_c$ to be an analog of ${{\mathcal {C}}}_c^K$. Similarly to the subspaces ${{\mathcal {C}}}_{\pm }^K\subset {{\mathcal {C}}}$ (see Subsection 2.3) one defines the full subcategories ${\text {D-mod}}({{\mathrm{{Bun}}}}_T)_{\pm }\subset {\text {D-mod}}({{\mathrm{{Bun}}}}_T)$.

There is also a (non-obvious) analogy between ${\text {D-mod}}({{\mathrm{{Bun}}}}_G)_{ps-c}$ and ${\mathcal {A}}_{ps-c}^K\,$. It will be explained in Subsection A.5 below. But first we have to recall some material from [10].

1.4 A.4 Good and bad direct image for D-modules

For any morphism $f:{\mathcal {Y}}'\rightarrow {\mathcal {Y}}$ between quasi-compact algebraic stacks one defines in [10] two pushforward functors³⁰: the ‘usual’ functor $f_*:{\text {D-mod}}({\mathcal {Y}}')\rightarrow {\text {D-mod}}({\mathcal {Y}})$ (which is very dangerous, maybe pathological) and the ‘renormalized direct image’ $f_\blacktriangle :{\text {D-mod}}({\mathcal {Y}}')\rightarrow {\text {D-mod}}({\mathcal {Y}})$ (which is nice). One also defines a canonical morphism $f_\blacktriangle \rightarrow f_*\,$, which is an isomorphism if and only if f is safe. By definition, a quasi-compact morphism f is safe if for any geometric point $y\rightarrow {\mathcal {Y}}$ and any geometric point $\xi : y'\rightarrow {\mathcal {Y}}'_y:={\mathcal {Y}}'\underset{{\mathcal {Y}}}{\times }y$ the neutral connected component of the automorphism group of $\xi$ is unipotent. For instance, any representable morphism is safe. We will use the functor $f_*$ only for safe morphisms f (in which case $f_*=f_\blacktriangle$).

The nice properties of $f_\blacktriangle$ are continuity (i.e., commutation with infinite direct sums) and base change with respect to !-pullbacks. Because of base change, we consider $f_\blacktriangle$ as a ‘right’ functor (in the sense of Subsection A.1.1), even though if f is not safe, then $f_\blacktriangle$ is not right adjoint to the partially defined functor $f^*$.

Base change allows one to define $f_\blacktriangle$ if f is quasi-compact while ${\mathcal {Y}}$ is not. Moreover, one defines $f_\blacktriangle (M)$ if f is not necessarily quasi-compact but $M\in {\text {D-mod}}({\mathcal {Y}}')$ is such that $M=j_*j^*M$ for some open substack $U\overset{j}{\hookrightarrow }{\mathcal {Y}}'$ quasi-compact over ${\mathcal {Y}}$: namely, one sets $f_\blacktriangle (M):=(f\circ j)_\blacktriangle (j^*M)$ for any U with the above property.

Finally, if ${\mathcal {Y}}={{\mathrm{{Spec}}}}k$ and $M\in {\text {D-mod}}({\mathcal {Y}}')$, then one writes $\Gamma _{{\text {ren}}} ({\mathcal {Y}}', M)$ instead of $f_\blacktriangle (M)$. The functor $\Gamma _{{\text {ren}}}$ is called renormalized de Rham cohomology.

1.5 A.5 D-module analogs of ${\mathcal {A}}_{ps-c}^K$ and ${{\mathrm{{CT}}}}^K$

We consider ${\text {D-mod}}({{\mathrm{{Bun}}}}_G)_{ps-c}$ to be an analog of ${\mathcal {A}}_{ps-c}^K\,$. The goal of this subsection is to justify this.

Recall that the subspace ${\mathcal {A}}_{ps-c}\subset {\mathcal {A}}$ is defined in Subsection 4.2 in terms of the operator ${{\mathrm{{CT}}}}:{\mathcal {A}}\rightarrow {{\mathcal {C}}}\,$. So the first step is to define a D-module analog of the corresponding operator ${{\mathrm{{CT}}}}^K:{\mathcal {A}}^K\rightarrow {{\mathcal {C}}}^K$. We will do this in Subsection A.5.3 using the diagram

(A.3)

that comes from the diagram of groups $G\hookleftarrow B\twoheadrightarrow T$ (as usual, $B\subset G$ is the subgroup of upper-triangular matrices).

Remark A.5.1

Diagram (A.3) is closely related to diagram (2.4). The relation is as follows. Suppose for a moment that X is a curve over ${\mathbb {F}}_q$ (rather than over a field of characteristic 0). Then the quotient of diagram (2.4) by the action of the maximal compact subgroup $K\subset G({\mathbb {A}})$ identifies with the diagram

$$\begin{aligned} {{\mathrm{{Bun}}}}_G({\mathbb {F}}_q)\leftarrow {{\mathrm{{Bun}}}}_B({\mathbb {F}}_q)\rightarrow {{\mathrm{{Bun}}}}_T({\mathbb {F}}_q) \end{aligned}$$

corresponding to (A.3).

Remark A.5.2

Unipotence of ${{\mathrm{{Ker}}}}(B\twoheadrightarrow T)$ easily implies that the morphism

$$\begin{aligned} {\mathsf {q}}:{{\mathrm{{Bun}}}}_B\rightarrow {{\mathrm{{Bun}}}}_T \end{aligned}$$

is safe in the sense of Subsection A.4 (although ${\mathsf {q}}$ is not representable).

1.5.1 A.5.3 The functor ${{\mathrm{{CT}}}}_*$ as a D-module analog of the operator ${{\mathrm{{CT}}}}^K$

Following [12], consider the functor

$$\begin{aligned} {{\mathrm{{CT}}}}_*:{\text {D-mod}}({{\mathrm{{Bun}}}}_G)\rightarrow {\text {D-mod}}({{\mathrm{{Bun}}}}_T),\quad {{\mathrm{{CT}}}}_*:={\mathsf {q}}_*\circ {\mathsf {p}}^!. \end{aligned}$$

Note that by Remark A.5.2 and Subsection A.4, the functor ${\mathsf {q}}_*$ equals ${\mathsf {q}}_\blacktriangle \,$, so it is not pathological.

Recall that the operator ${{\mathrm{{CT}}}}:{\mathcal {A}}\rightarrow {{\mathcal {C}}}$ is the pull–push along diagram (2.4) (see Subsection 2.5). So Remark A.5.1 allows us to consider the functor ${{\mathrm{{CT}}}}_*$ as a D-module analog³¹ of the operator ${{\mathrm{{CT}}}}^K$.

1.5.2 A.5.4 The functor ${{\mathrm{{CT}}}}_!$ and its relation to ${{\mathrm{{CT}}}}_*\,$

In [12] one defines another functor

$$\begin{aligned} {{\mathrm{{CT}}}}_!:{\text {D-mod}}({{\mathrm{{Bun}}}}_G)\rightarrow {\text {D-mod}}({{\mathrm{{Bun}}}}_T) \end{aligned}$$

by the formula

$$\begin{aligned} {{\mathrm{{CT}}}}_!:={\mathsf {q}}_!\circ {\mathsf {p}}^*\, , \end{aligned}$$

(A.4)

which has to be understood in a subtle sense. The subtlety is due to the fact that the r.h.s. of (A.4) involves ‘left’ functors. Because of that, the r.h.s. of (A.4) is, a priori, a functor ${\text {D-mod}}({{\mathrm{{Bun}}}}_G)\rightarrow {\text {Pro}} ({\text {D-mod}}({{\mathrm{{Bun}}}}_T))$, where ‘Pro’ stands for the DG category of pro-objects. However, the main theorem of [12] says that the essential image of this functor is contained in ${\text {D-mod}}({{\mathrm{{Bun}}}}_T)\subset {\text {Pro}} ({\text {D-mod}}({{\mathrm{{Bun}}}}_T))$. It also says that one has a canonical isomorphism

$$\begin{aligned} {{\mathrm{{CT}}}}_!\simeq \iota ^*\circ {{\mathrm{{CT}}}}_*\, , \end{aligned}$$

(A.5)

where $\iota ^*:{\text {D-mod}}({{\mathrm{{Bun}}}}_T)\buildrel {\sim }\over {\longrightarrow }{\text {D-mod}}({{\mathrm{{Bun}}}}_T)$ is the pullback along the inversion map $\iota :{{\mathrm{{Bun}}}}_T\buildrel {\sim }\over {\longrightarrow }{{\mathrm{{Bun}}}}_T$ .

1.5.3 A.5.5 Why ${\text {D-mod}}({{\mathrm{{Bun}}}}_G)_{ps-c}$ is an analog of ${\mathcal {A}}_{ps-c}^K\,$

By Lemma 4.1.2,

$$\begin{aligned} {\mathcal {A}}_c^K= \left\{ f\in {\mathcal {A}}^K\,|\, {{\mathrm{{CT}}}}(f)\in {\mathcal {C}}_{-}^K\right\} . \end{aligned}$$

A similar easy argument shows that

$$\begin{aligned} {\text {D-mod}}({{\mathrm{{Bun}}}}_G)_c= & {} \left\{ {\mathcal {F}}\in {\text {D-mod}}\left( {{\mathrm{{Bun}}}}_G\right) \,|\, {{\mathrm{{CT}}}}_* \left( {\mathcal {F}}\right) \in {\text {D-mod}}\left( {{\mathrm{{Bun}}}}_T\right) _-\right\} , \nonumber \\ {\text {D-mod}}\left( {{\mathrm{{Bun}}}}_G\right) _{ps-c}= & {} \left\{ {\mathcal {F}}\in {\text {D-mod}}\left( {{\mathrm{{Bun}}}}_G\right) \,|\, {{\mathrm{{CT}}}}_! \left( {\mathcal {F}}\right) \in {\text {D-mod}}\left( {{\mathrm{{Bun}}}}_T\right) _-\right\} .\qquad \quad \end{aligned}$$

(A.6)

Now combining (A.5) and (A.6), we see that

$$\begin{aligned} {\text {D-mod}}\left( {{\mathrm{{Bun}}}}_G\right) _{ps-c}=\left\{ {\mathcal {F}}\in {\text {D-mod}}\left( {{\mathrm{{Bun}}}}_G\right) \,|\, {{\mathrm{{CT}}}}_* \left( {\mathcal {F}}\right) \in {\text {D-mod}}\left( {{\mathrm{{Bun}}}}_T\right) _+\right\} .\qquad \end{aligned}$$

(A.7)

Formula (A.7) makes clear the analogy between ${\text {D-mod}}({{\mathrm{{Bun}}}}_G)_{ps-c}$ and the space

$$\begin{aligned} {\mathcal {A}}_{ps-c}^K:=\left\{ f\in {\mathcal {A}}^K\,|\, {{\mathrm{{CT}}}}(f)\in {\mathcal {C}}_{+}^K\right\} \end{aligned}$$

introduced in Subsection 4.2.

1.6 A.6 D-module analog of E

In Subsection 1.1.2 we fixed a field E; according to our convention, all functions take values in E. Thus E is the space of functions on a point.

So the D-module analog of E is the DG category ${{\mathrm{{Vect}}}}:={\text {D-mod}}({{\mathrm{{Spec}}}}k)$, which is just the DG category of complexes of vector spaces over k.

1.7 A.7 D-module analog of ${{\mathcal {B}}}_{\mathrm {naive}}^K\,$

Recall that ${{\mathcal {B}}}_{\mathrm {naive}}$ denotes the usual pairing between ${\mathcal {A}}_c$ and ${\mathcal {A}}$. Let ${{\mathcal {B}}}_{\mathrm {naive}}^K$ denote the restriction of ${{\mathcal {B}}}_{\mathrm {naive}}$ to K-invariant functions.

In Subsection A.6 we defined the DG category ${{\mathrm{{Vect}}}}$. The D-module analog of ${{\mathcal {B}}}_{\mathrm {naive}}^K$ is the functor

$$\begin{aligned} {\text {D-mod}}({{\mathrm{{Bun}}}}_G)_c\times {\text {D-mod}}({{\mathrm{{Bun}}}}_G)\rightarrow {{\mathrm{{Vect}}}}, \quad \quad (M_1,M_2)\mapsto \Gamma _{{\text {ren}}} (M_1\otimes M_2). \end{aligned}$$

Here $\Gamma _{{\text {ren}}}$ is the renormalized de Rham cohomology (see Subsection A.4) and $\otimes$ stands for the !-tensor product, i.e., $M_1\otimes M_2:=\Delta ^! (M_1\boxtimes M_2)$, where $\Delta :{{\mathrm{{Bun}}}}_G\times {{\mathrm{{Bun}}}}_G\rightarrow {{\mathrm{{Bun}}}}_G$ is the diagonal.

1.8 A.8 D-module analogs of ${{\mathcal {B}}}^K\,$ and $L^K$

1.8.1 A.8.1 The pseudo-identity functor

Let ${\mathcal {Y}}$ be a stack. Let ${\text {pr}}_1,{\text {pr}}_2:{\mathcal {Y}}\times {\mathcal {Y}}\rightarrow {\mathcal {Y}}$ denote the projections and $\Delta :{\mathcal {Y}}\rightarrow {\mathcal {Y}}\times {\mathcal {Y}}$ the diagonal morphism. Any ${\mathcal {F}}\in {\text {D-mod}}({\mathcal {Y}}\times {\mathcal {Y}})$ defines functors

$$\begin{aligned}&{\text {D-mod}}({\mathcal {Y}})_c\rightarrow {\text {D-mod}}({\mathcal {Y}}),\quad M\mapsto ({\text {pr}}_1)_\blacktriangle ({\mathcal {F}}\otimes {\text {pr}}_2^!M), \end{aligned}$$

(A.8)

$$\begin{aligned}&{\text {D-mod}}({\mathcal {Y}})_c\times {\text {D-mod}}({\mathcal {Y}})_c\rightarrow {{\mathrm{{Vect}}}}, \nonumber \\&(M_1,M_2)\mapsto \Gamma _{{\text {ren}}} ({\mathcal {Y}}\times {\mathcal {Y}},{\mathcal {F}}\otimes (M_1\boxtimes M_2)), \end{aligned}$$

(A.9)

where $({\text {pr}}_1)_\blacktriangle$ is the renormalized direct image and $\Gamma _{{\text {ren}}}$ is the renormalized de Rham cohomology (see Subsection A.4).

For example, if ${\mathcal {F}}=\Delta _*\omega _{{\mathcal {Y}}}$, then (A.8) is the identity functor and the ‘pairing’ (A.9) takes $(M_1,M_2)$ to $\Gamma _{{\text {ren}}} ({\mathcal {Y}},M_1\otimes M_2)$.

Now let $k_{{\mathcal {Y}}}$ denote the Verdier dual ${\mathbb {D}}\omega _{{\mathcal {Y}}}$ (a.k.a. the constant sheaf³²). Following [11, 14], we define the pseudo-identity functor

$$\begin{aligned} ({{\mathrm{{Ps-Id}}}})_{{\mathcal {Y}},!}:{\text {D-mod}}({\mathcal {Y}})_c\rightarrow {\text {D-mod}}({\mathcal {Y}}) \end{aligned}$$

(A.10)

to be the functor (A.8) corresponding to ${\mathcal {F}}=\Delta _!(k_{{\mathcal {Y}}})$. We will also consider the pairing (A.9) corresponding to ${\mathcal {F}}=\Delta _!(k_{{\mathcal {Y}}})$.

The functor (A.10) has the following important property, which one can check straightforwardly: for any open $U\overset{j}{\hookrightarrow }{\mathcal {Y}}$, one has

$$\begin{aligned} ({{\mathrm{{Ps-Id}}}})_{{\mathcal {Y}},!}\circ j_*=j_!\circ ({{\mathrm{{Ps-Id}}}})_{U,!}. \end{aligned}$$

(A.11)

Remark A.8.2

Suppose that ${\mathcal {Y}}$ is a truncatable stack (this notion was defined in Remark A.2.3). Then ${\text {D-mod}}({\mathcal {Y}})_c$ is a full subcategory of ${\text {D-mod}}({\mathcal {Y}})^\vee$, see formula (A.2). In fact, the functor (A.8) uniquely extends to a continuous functor

$$\begin{aligned} {\text {D-mod}}({\mathcal {Y}})^\vee \rightarrow {\text {D-mod}}({\mathcal {Y}}), \end{aligned}$$

(A.12)

and the pairing (A.9) to a continuous pairing

$$\begin{aligned} {\text {D-mod}}({\mathcal {Y}})^\vee \times {\text {D-mod}}({\mathcal {Y}})^\vee \rightarrow {{\mathrm{{Vect}}}}; \end{aligned}$$

see [11, §4.4.8] or [14, §3.1.1].

Remark A.8.3

Suppose that ${\mathcal {Y}}$ is smooth of pure dimension d and that the morphism $\Delta :{\mathcal {Y}}\rightarrow {\mathcal {Y}}\times {\mathcal {Y}}$ is separated.³³ Then one has a canonical morphism

$$\begin{aligned} \Delta _!(k_{\mathcal {Y}})\rightarrow \Delta _*(k_{\mathcal {Y}})=\Delta _*(\omega _{\mathcal {Y}})[-2d], \end{aligned}$$

which induces a canonical morphism

$$\begin{aligned} ({{\mathrm{{Ps-Id}}}})_{{\mathcal {Y}},!}(M)\rightarrow M[-2d], \quad M\in {\text {D-mod}}({\mathcal {Y}})_c. \end{aligned}$$

(A.13)

1.8.2 A.8.4 D-module analogs of ${{\mathcal {B}}}^K\,$ and $L^K$

On ${\mathcal {A}}_c$ we have the bilinear form ${{\mathcal {B}}}$; its restriction to ${\mathcal {A}}_c^K$ will be denoted by ${{\mathcal {B}}}^K$. In Subsection 3.2 we defined the operator $L:{\mathcal {A}}_c\rightarrow {\mathcal {A}}\,$; it induces an operator $L^K:{\mathcal {A}}_c^K\rightarrow {\mathcal {A}}^K$. Recall that

$$\begin{aligned} {{\mathcal {B}}}(f_1,f_2)={{\mathcal {B}}}_{\mathrm {naive}}(f_1,Lf_2) \end{aligned}$$

for any $f_1,f_2\in {\mathcal {A}}_c\,$; in particular, this is true for $f_1\, ,f_2\in {\mathcal {A}}_c^K$.

According to Theorem 1.3.4, the ‘matrix’ of the bilinear form ${{\mathcal {B}}}^K$ is the function b defined in Subsection 1.3.2. According to Example A.1.4, the D-module $\Delta _!(k_{{{\mathrm{{Bun}}}}_G})$ is an analog of the function $q^{-d}\cdot b$, where

$$\begin{aligned} d:=\dim {{\mathrm{{Bun}}}}_G=3g_X-3\,. \end{aligned}$$

So we consider the pairing (A.9) corresponding to ${\mathcal {Y}}={{\mathrm{{Bun}}}}_G$ and ${\mathcal {F}}=\Delta _!(k_{{\mathcal {Y}}})$ to be the D-module analog of the bilinear form $q^{-d}\cdot {{\mathcal {B}}}^K$. Accordingly, we consider the functor $({{\mathrm{{Ps-Id}}}})_{{{\mathrm{{Bun}}}}_G,!}:{\text {D-mod}}({{\mathrm{{Bun}}}}_G)_c\rightarrow {\text {D-mod}}({{\mathrm{{Bun}}}}_G)$ defined in Subsection A.8.1 to be the D-module analog of the operator $q^{-d}\cdot L^K:{\mathcal {A}}_c^K\rightarrow {\mathcal {A}}^K$.

1.9 A.9 Miraculous duality and a D-module analog of Corollary 4.3.2

Formula (A.11) implies that for any stack ${\mathcal {Y}}$, the functor $({{\mathrm{{Ps-Id}}}})_{{\mathcal {Y}},!}$ maps ${\text {D-mod}}({\mathcal {Y}})_c$ to the full subcategory ${\text {D-mod}}({\mathcal {Y}})_{ps-c}\subset {\text {D-mod}}({\mathcal {Y}})$, so one gets a functor

$$\begin{aligned} ({{\mathrm{{Ps-Id}}}})_{{\mathcal {Y}},!}: {\text {D-mod}}({\mathcal {Y}})_c\rightarrow {\text {D-mod}}({\mathcal {Y}})_{ps-c}. \end{aligned}$$

(A.14)

Now suppose that ${\mathcal {Y}}={{\mathrm{{Bun}}}}_G\,$. Then the main result of [14] (namely, Theorem 0.1.6) says that the functor (A.12) corresponding to ${\mathcal {F}}=\Delta _!(k_{{\mathcal {Y}}})$ is an equivalence.³⁴ By [11, Lemma 4.5.7], this implies that the functor (A.14) is an equivalence. This is an analog of the part of Corollary 4.3.2 that says that the operator $L^K:{\mathcal {A}}_c\rightarrow {\mathcal {A}}$ induces an isomorphism ${\mathcal {A}}_c^K\buildrel {\sim }\over {\longrightarrow }{\mathcal {A}}_{ps-c}^K\,$.

Corollary 4.3.2 also explicitly describes the inverse isomorphism

$$\begin{aligned} {\mathcal {A}}_{ps-c}^K\buildrel {\sim }\over {\longrightarrow }{\mathcal {A}}_c^K. \end{aligned}$$

This suggests a conjectural description of the functor inverse to (A.14). The conjecture is formulated in Appendix C.

1.10 A.10 The functor ${{\mathrm{{Ps-Id}}}}_{{\mathcal {Y}},!}$ for ${\mathcal {Y}}={{\mathrm{{Bun}}}}_T\,$

The material of this subsection will be used in Subsection A.11.7.

1.10.1 A.10.1 D-module setting

The morphism $\Delta :{{\mathrm{{Bun}}}}_T\rightarrow {{\mathrm{{Bun}}}}_T\times {{\mathrm{{Bun}}}}_T$ factors as

$$\begin{aligned} {{\mathrm{{Bun}}}}_T\overset{\pi }{\longrightarrow }{\mathcal {Z}}\overset{i}{\hookrightarrow }{{\mathrm{{Bun}}}}_T\times {{\mathrm{{Bun}}}}_T\, , \end{aligned}$$

where i is a closed embedding and $\pi :{{\mathrm{{Bun}}}}_T\rightarrow {\mathcal {Z}}$ is a ${\mathbb {G}}_m$-torsor. So $\Delta _! (k_{{{\mathrm{{Bun}}}}_T})=\Delta _* (k_{{{\mathrm{{Bun}}}}_T})[-1]$. The stack ${{\mathrm{{Bun}}}}_T$ is smooth and has pure dimension $g_X-1$, where $g_X$ is the genus of X. So $k_{{{\mathrm{{Bun}}}}_T}=\omega _{{{\mathrm{{Bun}}}}_T}[2-2g_X]$. Thus

$$\begin{aligned} \Delta _! (k_{{{\mathrm{{Bun}}}}_T})=\Delta _* (\omega _{{{\mathrm{{Bun}}}}_T})[1-2g_X]. \end{aligned}$$

Therefore the functor

$$\begin{aligned} {{\mathrm{{Ps-Id}}}}_{{{\mathrm{{Bun}}}}_T,!}:{\text {D-mod}}({{\mathrm{{Bun}}}}_T )_c\rightarrow {\text {D-mod}}({{\mathrm{{Bun}}}}_T )_{ps-c}={\text {D-mod}}({{\mathrm{{Bun}}}}_T )_c \end{aligned}$$

equals ${\mathop {{\text {Id}}}}[1-2g_X]$.

1.10.2 A.10.2 l-adic setting

In the l-adic setting the formulas are similar to those from Subsection A.10.1, but now we have to take the Tate twists into account:

$$\begin{aligned} \Delta _! \left( \left( \overline{{\mathbb {Q}}}_l\right) _{{{\mathrm{{Bun}}}}_T}\right)= & {} \Delta _* \left( \left( \overline{{\mathbb {Q}}}_l\right) _{{{\mathrm{{Bun}}}}_T}\right) [-1]= \Delta _*\left( {\omega }_{{{\mathrm{{Bun}}}}_T}\right) \left[ 1-2g_X\right] (1-g_X),\nonumber \\ {{\mathrm{{Ps-Id}}}}_{{{\mathrm{{Bun}}}}_T,!}= & {} {\mathop {{\text {Id}}}}\left[ 1-2g_X\right] \left( 1-g_X\right) . \end{aligned}$$

(A.15)

1.10.3 A.10.3 Analog at the level of functions

We consider the vector space ${{\mathcal {C}}}_c^K$ to be an analog of ${\text {D-mod}}({{\mathrm{{Bun}}}}_T )_c\,$. We consider the operator

$$\begin{aligned} -q^{1-g_X}\cdot {\mathop {{\text {Id}}}}\in {{\mathrm{{End}}}}\left( {{\mathcal {C}}}_c^K\right) \end{aligned}$$

(A.16)

to be an analog of the functor ${{\mathrm{{Ps-Id}}}}_{{{\mathrm{{Bun}}}}_T,!}:{\text {D-mod}}({{\mathrm{{Bun}}}}_T )_c\rightarrow {\text {D-mod}}({{\mathrm{{Bun}}}}_T )_c\,$. This is justified by formula (A.15); in particular, the minus sign in (A.16) is due to the fact that the number $1-2g_X$ from (A.15) is odd. In Subsection A.11.7 we will see that this minus sign is closely related to the minus sign in Proposition 3.2.2(ii).

1.11 A.11 Eisenstein functors

The operators

$$\begin{aligned} {{\mathrm{{Eis}}}}:{{\mathcal {C}}}_+\rightarrow {\mathcal {A}}, \quad {{\mathrm{{Eis}}}}' :{{\mathcal {C}}}_-\rightarrow {\mathcal {A}}\end{aligned}$$

induce operators ${{\mathrm{{Eis}}}}^K :{{\mathcal {C}}}_+^K\rightarrow {\mathcal {A}}^K$ and $({{\mathrm{{Eis}}}}')^K :{{\mathcal {C}}}_-^K\rightarrow {\mathcal {A}}^K$. In Subsections A.11.3–A.11.4 we will discuss the functor ${{\mathrm{{Eis}}}}_*\,$, which is a D-module analog of the operator ${{\mathrm{{Eis}}}}^K$. In Subsections A.11.5–A.11.7 we will discuss the functor ${{\mathrm{{Eis}}}}_!\,$, whose analog at the level of functions is closely related to $({{\mathrm{{Eis}}}}')^K$, see formula (A.22). In Subsection A.11.8 we briefly discuss the compactified Eisenstein functor ${{\mathrm{{Eis}}}}_{!*}$ and the enhanced Eisenstein functor ${{\mathrm{{Eis}}}}^{{{\mathrm{{enh}}}}}\,$.

Both ${{\mathrm{{Eis}}}}_*\,$ and ${{\mathrm{{Eis}}}}_!\,$ are defined using the diagram of stacks

which was already used in Subsection A.5. We will need the following remarks.

Remark A.11.1

The morphism ${\mathsf {p}}:{{\mathrm{{Bun}}}}_B\rightarrow {{\mathrm{{Bun}}}}_G$ is representable, i.e., its fibers are algebraic spaces (in fact, schemes).

Remark A.11.2

The morphism ${\mathsf {p}}:{{\mathrm{{Bun}}}}_B\rightarrow {{\mathrm{{Bun}}}}_G$ is not quasi-compact. But the restriction of ${\mathsf {p}}$ to the substack ${\mathsf {q}}^{-1}({{\mathrm{{Bun}}}}_T^{\ge a} )$ is quasi-compact for any $a\in {\mathbb {Z}}$. Here

$$\begin{aligned} {{\mathrm{{Bun}}}}_T^{\ge a}\subset {{\mathrm{{Bun}}}}_T={{\mathrm{{Bun}}}}_{{\mathbb {G}}_m} \end{aligned}$$

is the stack of ${\mathbb {G}}_m$-bundles of degree $\ge a$.

1.11.1 A.11.3 The functor ${{\mathrm{{Eis}}}}_*\,$ as a D-module analog of the operator ${{\mathrm{{Eis}}}}^K$

Denote by ${\text {D-mod}}({{\mathrm{{Bun}}}}_T)_+$ the full subcategory formed by those $M\in {\text {D-mod}}({{\mathrm{{Bun}}}}_T)$ whose support is contained in ${{\mathrm{{Bun}}}}_T^{\ge a}$ for some $a\in {\mathbb {Z}}$. Define a functor

$$\begin{aligned} {{\mathrm{{Eis}}}}_*:{\text {D-mod}}({{\mathrm{{Bun}}}}_T)_+\rightarrow {\text {D-mod}}({{\mathrm{{Bun}}}}_G) \end{aligned}$$

(A.17)

by ${{\mathrm{{Eis}}}}_*:={\mathsf {p}}_*\circ {\mathsf {q}}^!$. Since we consider ${\text {D-mod}}({{\mathrm{{Bun}}}}_T)_+$ rather than ${\text {D-mod}}({{\mathrm{{Bun}}}}_T)\,$, Remarks A.11.1-A.11.2 ensure that taking ${\mathsf {p}}_*$ does not lead to pathologies. It is easy to check that

$$\begin{aligned} {{\mathrm{{Eis}}}}_*({\text {D-mod}}({{\mathrm{{Bun}}}}_T)_c)\subset {\text {D-mod}}({{\mathrm{{Bun}}}}_G)_c\,. \end{aligned}$$

(A.18)

The functor ${{\mathrm{{Eis}}}}_*$ is a D-module analog of the operator ${{\mathrm{{Eis}}}}^K :{{\mathcal {C}}}_+^K\rightarrow {\mathcal {A}}^K$, and formula (A.18) is similar to the inclusion ${{\mathrm{{Eis}}}}^K ({{\mathcal {C}}}_c^K)\subset {\mathcal {A}}_c^K$. This is clear from Remark A.5.1.

The reader may prefer to skip the next subsection and go directly to Subsection A.11.5.

1.11.2 A.11.4 Relation to the notation of [14]

The functor (A.17) is the restriction of the functor

$$\begin{aligned} {{\mathrm{{Eis}}}}_*:{\text {D-mod}}({{\mathrm{{Bun}}}}_T)\rightarrow {\text {D-mod}}({{\mathrm{{Bun}}}}_G) \end{aligned}$$

(A.19)

defined in [14, §1.1.9].

On the other hand, the DG category ${\text {D-mod}}({{\mathrm{{Bun}}}}_G)_c$ is a full subcategory of the Lurie dual ${\text {D-mod}}({{\mathrm{{Bun}}}}_G)^\vee$, see Remark A.2.6 and especially formula (A.2). The DG category ${\text {D-mod}}({{\mathrm{{Bun}}}}_G)^\vee$ has a realization introduced in [11, §4.3.3] (or [14, §1.2.2]) and denoted there by ${\text {D-mod}}({{\mathrm{{Bun}}}}_G)_{{\text {co}}}\,$; for us, ${\text {D-mod}}({{\mathrm{{Bun}}}}_G)_{{\text {co}}}$ is a synonym of ${\text {D-mod}}({{\mathrm{{Bun}}}}_G)^\vee$. We have that the functor ${{\mathrm{{Eis}}}}_*:{\text {D-mod}}({{\mathrm{{Bun}}}}_T)_c\rightarrow {\text {D-mod}}({{\mathrm{{Bun}}}}_G)_c$ is the restriction of the functor

$$\begin{aligned} {\text {D-mod}}({{\mathrm{{Bun}}}}_T)\rightarrow {\text {D-mod}}({{\mathrm{{Bun}}}}_G)^\vee ={\text {D-mod}}({{\mathrm{{Bun}}}}_G)_{{\text {co}}} \end{aligned}$$

(A.20)

introduced in [14] and denoted there by $({{\mathrm{{CT}}}}_* )^\vee$ or ${{\mathrm{{Eis}}}}_{{\text {co}},*}$ (the notation $({{\mathrm{{CT}}}}_* )^\vee$ is introduced in [14, §0.1.7] and the synonym ${{\mathrm{{Eis}}}}_{{\text {co}},*}$ in [14, §1.4.1]).

Let us note that the relation between the functors (A.19) and (A.20) is described in [14, Prop. 2.1.7].

1.11.3 A.11.5 The functor ${{\mathrm{{Eis}}}}_!\,$

Define a functor

$$\begin{aligned} {{\mathrm{{Eis}}}}_!:{\text {D-mod}}({{\mathrm{{Bun}}}}_T)\rightarrow {\text {D-mod}}({{\mathrm{{Bun}}}}_G) \end{aligned}$$

by ${{\mathrm{{Eis}}}}_!:={\mathsf {p}}_!\circ {\mathsf {q}}^*$. According to [11, Cor. 2.3], the functor ${{\mathrm{{Eis}}}}_!$ is defined everywhere.³⁵

Note that by Remark A.11.2, the restriction of ${{\mathrm{{Eis}}}}_!$ to ${\text {D-mod}}({{\mathrm{{Bun}}}}_T)_+$ preserves holonomicity. It is easy to check that

$$\begin{aligned} {{\mathrm{{Eis}}}}_!({\text {D-mod}}({{\mathrm{{Bun}}}}_T)_c)\subset {\text {D-mod}}({{\mathrm{{Bun}}}}_G)_{ps-c}\,. \end{aligned}$$

(A.21)

1.11.4 A.11.6 The analog of ${{\mathrm{{Eis}}}}_!$ at the level of functions

First, let us define a certain automorphism of the space ${{\mathcal {C}}}^K$. As explained in Example 2.3.1, ${{\mathcal {C}}}^K$ identifies with the space of functions on ${{\mathrm{{Pic}}}}X$, where X is the smooth projective curve over ${\mathbb {F}}_q$ corresponding to the global field F. So the inversion map $\iota :{{\mathrm{{Pic}}}}X\rightarrow {{\mathrm{{Pic}}}}X$ induces an operator $\iota ^*:{{\mathcal {C}}}^K\rightarrow {{\mathcal {C}}}^K$, which interchanges the subspaces ${{\mathcal {C}}}_+^K$ and ${{\mathcal {C}}}_-^K\,$.

Now we claim that the functor ${{\mathrm{{Eis}}}}_!:{\text {D-mod}}({{\mathrm{{Bun}}}}_T)_+\rightarrow {\text {D-mod}}({{\mathrm{{Bun}}}}_G)$ is a D-module analog of the operator

$$\begin{aligned} q^{2-2g_X}\cdot ({{\mathrm{{Eis}}}}')^K\circ \iota ^* :{{\mathcal {C}}}_+^K\rightarrow {\mathcal {A}}^K\,, \end{aligned}$$

(A.22)

where ${{\mathrm{{Eis}}}}':{\mathcal {C}}_{-}\rightarrow {\mathcal {A}}$ is the operator defined in Subsection 2.12. This claim is justified by Theorem B.2.1 of Appendix B. In Subsection A.11.7 below we show that this claim agrees with Theorem 4.1.2 of [14]; this gives another justification.

Note that by Proposition 4.3.3, the operator (A.22) maps ${{\mathcal {C}}}_c^K$ to ${\mathcal {A}}_{ps-c}^K\,$. This is similar to the inclusion (A.21).

1.11.5 A.11.7 Comparison with [14, Theorem 4.1.2]

Theorem 4.1.2 of [14] tells us³⁶ that the functor

$$\begin{aligned} {{\mathrm{{Eis}}}}_!\circ {{\mathrm{{Ps-Id}}}}_{{{\mathrm{{Bun}}}}_T,!}:{\text {D-mod}}({{\mathrm{{Bun}}}}_T)_c\rightarrow {\text {D-mod}}({{\mathrm{{Bun}}}}_G)_{ps-c} \end{aligned}$$

is isomorphic to the functor

$$\begin{aligned} {{\mathrm{{Ps-Id}}}}_{{{\mathrm{{Bun}}}}_G,!}\circ {{\mathrm{{Eis}}}}_*\circ \, \iota ^* :{\text {D-mod}}({{\mathrm{{Bun}}}}_T)_c\rightarrow {\text {D-mod}}({{\mathrm{{Bun}}}}_G)_{ps-c}\,, \end{aligned}$$

(A.23)

where $\iota :{{\mathrm{{Bun}}}}_T\buildrel {\sim }\over {\longrightarrow }{{\mathrm{{Bun}}}}_T$ is the inversion map. By Subsections A.8.4 and A.11.3, the analog of (A.23) at the level of functions is the operator

$$\begin{aligned} q^{3-3g_X}\cdot L^K\circ {{\mathrm{{Eis}}}}^K\circ \, \iota ^*:{{\mathcal {C}}}_c^K\rightarrow {\mathcal {A}}_{ps-c}^K\,. \end{aligned}$$

This operator equals $-q^{3-3g_X}\cdot ({{\mathrm{{Eis}}}}')^K\circ \iota ^*$ by Proposition 3.2.2(ii). By Subsection A.10.3, the analog of the functor ${{\mathrm{{Ps-Id}}}}_{{{\mathrm{{Bun}}}}_T,!}$ at the level of functions is the operator of multiplication by $-q^{1-g_X}$. So we see that the claim made in Subsection A.11.6 agrees with Theorem 4.1.2 of [14].

1.11.6 A.11.8 Other Eisenstein functors

The functor ${{\mathrm{{Eis}}}}_*$ has an ‘enhanced’ version ${{\mathrm{{Eis}}}}^{{{\mathrm{{enh}}}}}={{\mathrm{{Eis}}}}_B^{{{\mathrm{{enh}}}}}$, see Subsection C.1. Both ${{\mathrm{{Eis}}}}_*$ and ${{\mathrm{{Eis}}}}^{{{\mathrm{{enh}}}}}$ are D-module analogs of the operator ${{\mathrm{{Eis}}}}^K$.

One also has the compactified Eisenstein functor ${{\mathrm{{Eis}}}}_{!*}\,$, see Subsection B.6(i). In Appendix B we work with slightly different functors $\overline{{{\mathrm{{Eis}}}}}$ and $\underline{{{\mathrm{{Eis}}}}}$ (see Subsection B.3.1), which are good enough for $G=SL(2)$. Formulas (B.12)–(B.13) from Corollary B.4.2 describe the analogs of $\overline{{{\mathrm{{Eis}}}}}$ and $\underline{{{\mathrm{{Eis}}}}}$ at the level of functions. In terms of Eisenstein series (rather than Eisenstein operators) the functor ${{\mathrm{{Eis}}}}_{!*}$ corresponds to the product of the Eisenstein series by a normalizing factor, which is essentially an L-function in the case $G=SL(2)$ and a product of L-functions in general (see [25, Theorem 3.3.2] and [6, Subsection 2.2] for more details).

Appendix B. Relation between the functor ${{\mathrm{{Eis}}}}_!$ and the operator $({{\mathrm{{Eis}}}}')^K$

In this section we work over ${\mathbb {F}}_q\,$. Our main goal is to prove Theorem B.2.1, which justifies the claim made in Subsection A.11.6.

1.1 B.1 Notation and conventions

1.1.1 B.1.1

We will say ‘stack’ instead of ‘algebraic stack locally of finite type over ${\mathbb {F}}_q\,$’.

1.1.2 B.1.2

Let X be a smooth complete geometrically connected curve over ${\mathbb {F}}_q\,$. Just as in the rest of the article, $G:=SL(2)$, $T\subset G$ is the group of diagonal matrices, and $B\subset G$ is the subgroup of upper-triangular matrices. Let ${{\mathrm{{Bun}}}}_G$ (resp.  ${{\mathrm{{Bun}}}}_T$) denote the moduli stack of principal G-bundles (resp. T-bundles) on X.

1.1.3 B.1.3

We fix a prime l not dividing q and an algebraic closure $\overline{{\mathbb {Q}}}_l$ of ${\mathbb {Q}}_l\,$. For any stack ${\mathcal {Y}}$, one has the bounded constructible derived category of $\overline{{\mathbb {Q}}}_l$-sheaves, denoted by ${\mathscr {D}}({\mathcal {Y}})$.

To any ${\mathcal {F}}\in {\mathscr {D}}({\mathcal {Y}})$ we associate a function $f_{\mathcal {F}}:{\mathcal {Y}}({\mathbb {F}}_q)\rightarrow \overline{{\mathbb {Q}}}_l$ using the (nonstandard) convention of Subsection A.1.2: namely, the value of $f_{\mathcal {F}}$ at $y\in {\mathcal {Y}}({\mathbb {F}}_q)$ is the trace of the arithmetic Frobenius acting on the !-stalk of ${\mathcal {F}}$ at y. So the standard operators between spaces of functions correspond to the ‘right’ functors in the sense of Subsection A.1.1.

1.1.4 B.1.4

If a stack ${\mathcal {Y}}$ is quasi-compact, let ${{\mathrm{\mathbf {K}}}}({\mathcal {Y}})$ denote the Grothendieck group of ${\mathscr {D}}({\mathcal {Y}})$. In general, let ${{\mathrm{\mathbf {K}}}}({\mathcal {Y}})$ denote the projective limit of the groups ${{\mathrm{\mathbf {K}}}}(U)$ corresponding to quasi-compact open substacks $U\subset {\mathcal {Y}}$. We equip ${{\mathrm{\mathbf {K}}}}({\mathcal {Y}})$ with the projective limit topology.

The assignment ${\mathcal {F}}\mapsto f_{\mathcal {F}}$ from Subsection B.1.3 clearly yields a group homomorphism from ${{\mathrm{\mathbf {K}}}}({\mathcal {Y}})$ to the space of functions ${\mathcal {Y}}({\mathbb {F}}_q)\rightarrow \overline{{\mathbb {Q}}}_l$. This homomorphism is still denoted by ${\mathcal {F}}\mapsto f_{\mathcal {F}}\,$.

1.1.5 B.1.5

Just as in Subsection A.11, we consider the diagram of stacks

(B.1)

that comes from the diagram of groups $G\hookleftarrow B\twoheadrightarrow T$. The morphism ${\mathsf {p}}:{{\mathrm{{Bun}}}}_B\rightarrow {{\mathrm{{Bun}}}}_G$ is representable. It is not quasi-compact, but the restriction of ${\mathsf {p}}$ to the substack ${\mathsf {q}}^{-1}({{\mathrm{{Bun}}}}_T^{\ge a} )$ is quasi-compact for any $a\in {\mathbb {Z}}$. So we have functors

$$\begin{aligned}&{{\mathrm{{Eis}}}}_*:{\mathscr {D}}({{\mathrm{{Bun}}}}_T)_+\rightarrow {\mathscr {D}}({{\mathrm{{Bun}}}}_G), \quad {{\mathrm{{Eis}}}}_*:={\mathsf {p}}_*\circ {\mathsf {q}}^!, \end{aligned}$$

(B.2)

$$\begin{aligned}&{{\mathrm{{Eis}}}}_!:{\mathscr {D}}({{\mathrm{{Bun}}}}_T)_+\rightarrow {\mathscr {D}}({{\mathrm{{Bun}}}}_G), \quad {{\mathrm{{Eis}}}}_!:={\mathsf {p}}_!\circ {\mathsf {q}}^*, \end{aligned}$$

(B.3)

where ${\mathscr {D}}({{\mathrm{{Bun}}}}_T)_+$ denotes the full subcategory formed by those ${\mathcal {F}}\in {\mathscr {D}}({{\mathrm{{Bun}}}}_T)$ whose support is contained in ${{\mathrm{{Bun}}}}_T^{\ge a}$ for some $a\in {\mathbb {Z}}$.

1.1.6 B.1.6

Let ${{\mathrm{\mathbf {K}}}}({{\mathrm{{Bun}}}}_T)_+$ denote the direct limit of ${{\mathrm{\mathbf {K}}}}({{\mathrm{{Bun}}}}_T^{\ge a})$, $a\in {\mathbb {Z}}$. The functors (B.2)-(B.3) induce group homomorphisms

$$\begin{aligned} {{\mathrm{{Eis}}}}_*:{{\mathrm{\mathbf {K}}}}({{\mathrm{{Bun}}}}_T)_+\rightarrow {{\mathrm{\mathbf {K}}}}({{\mathrm{{Bun}}}}_G), \quad {{\mathrm{{Eis}}}}_!:{{\mathrm{\mathbf {K}}}}({{\mathrm{{Bun}}}}_T)_+\rightarrow {{\mathrm{\mathbf {K}}}}({{\mathrm{{Bun}}}}_G). \end{aligned}$$

1.1.7 B.1.7

Let F denote the field of rational functions on X and ${\mathbb {A}}$ its adele ring. Recall that $K\subset G({\mathbb {A}})$ denotes the standard maximal compact subgroup.

Let ${\mathcal {A}}$ and ${{\mathcal {C}}}$, ${\mathcal {C}}_{+}$, ${\mathcal {C}}_{-}$ be the functional spaces defined in Subsections 1.1.2 and 2.3; we take $\overline{{\mathbb {Q}}}_l$ as the field in which our functions take values.

As explained in Example 2.3.1, we identify ${\mathcal {C}}^K$ (i.e., the subspace of K-invariant functions in ${\mathcal {C}}$) with the space of all $\overline{{\mathbb {Q}}}_l$-valued functions on ${{\mathrm{{Bun}}}}_T({\mathbb {F}}_q)={{\mathrm{{Bun}}}}_{{\mathbb {G}}_m}({\mathbb {F}}_q)$. We identify ${\mathcal {A}}^K$ with the space of all functions ${{\mathrm{{Bun}}}}_G({\mathbb {F}}_q)\rightarrow \overline{{\mathbb {Q}}}_l\,$.

1.1.8 B.1.8

The inversion map $\iota :{{\mathrm{{Bun}}}}_T\rightarrow {{\mathrm{{Bun}}}}_T$ induces an operator $\iota ^*:{{\mathcal {C}}}^K\rightarrow {{\mathcal {C}}}^K$, which interchanges the subspaces ${{\mathcal {C}}}_+^K$ and ${{\mathcal {C}}}_-^K\,$.

The operators ${{\mathrm{{Eis}}}}:{\mathcal {C}}_{+}\rightarrow {\mathcal {A}}$ and ${{\mathrm{{Eis}}}}':{\mathcal {C}}_{-}\rightarrow {\mathcal {A}}$ defined in Subsections 2.6 and 2.12 induce operators ${{\mathrm{{Eis}}}}^K:{\mathcal {C}}_{+}^K\rightarrow {\mathcal {A}}^K$ and $({{\mathrm{{Eis}}}}')^K:{\mathcal {C}}_{-}^K\rightarrow {\mathcal {A}}^K$. Since $\iota ^* ({{\mathcal {C}}}^K_+)={{\mathcal {C}}}^K_-$ we get an operator

$$\begin{aligned} ({{\mathrm{{Eis}}}}')^K\circ \iota ^* :{{\mathcal {C}}}_+^K\rightarrow {\mathcal {A}}^K\,. \end{aligned}$$

(B.4)

1.2 B.2 Formulation of the theorem

For any ${\mathcal {F}}\in {{\mathrm{\mathbf {K}}}}({{\mathrm{{Bun}}}}_T)_+$, one has the functions

$$\begin{aligned} f_{\mathcal {F}}\in {\mathcal {C}}_{+}^K, \quad f_{{{\mathrm{{Eis}}}}_*{\mathcal {F}}}\in {\mathcal {A}}^K, \quad f_{{{\mathrm{{Eis}}}}_!{\mathcal {F}}}\in {\mathcal {A}}^K \end{aligned}$$

defined as explained in Subsections B.1.3–B.1.4. Since formula (B.2) involves only ‘right’ functors, it is clear that

$$\begin{aligned} f_{{{\mathrm{{Eis}}}}_*{\mathcal {F}}}={{\mathrm{{Eis}}}}^K(f_{\mathcal {F}}), \quad {\mathcal {F}}\in {{\mathrm{\mathbf {K}}}}({{\mathrm{{Bun}}}}_T)_+ \; . \end{aligned}$$

The next theorem expresses $f_{{{\mathrm{{Eis}}}}_!{\mathcal {F}}}$ in terms of $f_{\mathcal {F}}$ and the operator (B.4).

Theorem B.2.1

For any ${\mathcal {F}}\in {{\mathrm{\mathbf {K}}}}({{\mathrm{{Bun}}}}_T)_+$, one has

$$\begin{aligned} f_{{{\mathrm{{Eis}}}}_!{\mathcal {F}}}= q^{2-2g_X}\cdot \left( ({{\mathrm{{Eis}}}}')^K\circ \iota ^*\right) \left( f_{\mathcal {F}}\right) \, , \end{aligned}$$

(B.5)

where $g_X$ is the genus of X.

A proof is given in Subsections B.3–B.5 below. To make it self-contained, we used an approach which is somewhat barbaric (as explained in Subsection B.6).

Remark B.2.2

Using [7, Cor. 4.5], one can express $f_{{{\mathrm{{Eis}}}}_!{\mathcal {F}}}$ in terms of $f_{\mathcal {F}}$ for any reductive group G (at least, in the case of principal Eisenstein series).

1.3 B.3 The compactified Eisenstein functors

We will need the ‘compactified Eisenstein’ functors $\overline{{{\mathrm{{Eis}}}}}, \underline{{{\mathrm{{Eis}}}}}:{\mathscr {D}}({{\mathrm{{Bun}}}}_T)_+\rightarrow {\mathscr {D}}({{\mathrm{{Bun}}}}_G)$, which go back to [25].

1.3.1 B.3.1 Definition of $\overline{{{\mathrm{{Eis}}}}}$ and $\underline{{{\mathrm{{Eis}}}}}$

Consider the diagram

in which $\overline{{{\mathrm{{Bun}}}}}_B$ denotes³⁷ the stack of rank 2 vector bundles ${\mathcal {L}}$ on X with trivialized determinant equipped with an invertible subsheaf ${\mathcal {M}}\subset {\mathcal {L}}\,$ (the open substack ${{\mathrm{{Bun}}}}_B\subset \overline{{{\mathrm{{Bun}}}}}_B$ is defined by the condition that ${\mathcal {M}}$ is a subbundle). Note that the morphism $\overline{{\mathsf {p}}}:\overline{{{\mathrm{{Bun}}}}}_B\rightarrow {{\mathrm{{Bun}}}}_G$ is representable and its restriction to the substack $\overline{{\mathsf {q}}}^{-1}({{\mathrm{{Bun}}}}_T^{\ge a} )\subset \overline{{{\mathrm{{Bun}}}}}_B$ is proper for any $a\in {\mathbb {Z}}$.

Now define the functor $\overline{{{\mathrm{{Eis}}}}}:{\mathscr {D}}({{\mathrm{{Bun}}}}_T)_+\rightarrow {\mathscr {D}}({{\mathrm{{Bun}}}}_G)$ and the group homomorphism $\overline{{{\mathrm{{Eis}}}}}:{{\mathrm{\mathbf {K}}}}({{\mathrm{{Bun}}}}_T)_+\rightarrow {{\mathrm{\mathbf {K}}}}({{\mathrm{{Bun}}}}_G)$ by

$$\begin{aligned} \overline{{{\mathrm{{Eis}}}}}:=\overline{{\mathsf {p}}}_*\circ \overline{{\mathsf {q}}}^!. \end{aligned}$$

Similarly, define the functor $\underline{{{\mathrm{{Eis}}}}}:{\mathscr {D}}({{\mathrm{{Bun}}}}_T)_+\rightarrow {\mathscr {D}}({{\mathrm{{Bun}}}}_G)$ and the group homomorphism $\underline{{{\mathrm{{Eis}}}}}:{{\mathrm{\mathbf {K}}}}({{\mathrm{{Bun}}}}_T)_+\rightarrow {{\mathrm{\mathbf {K}}}}({{\mathrm{{Bun}}}}_G)$ by

$$\begin{aligned} \underline{{{\mathrm{{Eis}}}}}:=\overline{{\mathsf {p}}}_!\circ \overline{{\mathsf {q}}}^*. \end{aligned}$$

1.3.2 B.3.2 Relation between $\overline{{{\mathrm{{Eis}}}}}$ and $\underline{{{\mathrm{{Eis}}}}}$

Recall that the restriction of $\overline{{\mathsf {p}}}:\overline{{{\mathrm{{Bun}}}}}_B\rightarrow {{\mathrm{{Bun}}}}_G$ to the substack $\overline{{\mathsf {q}}}^{-1}({{\mathrm{{Bun}}}}_T^{\ge a} )\subset \overline{{{\mathrm{{Bun}}}}}_B$ is proper, so $\overline{{\mathsf {p}}}_!=\overline{{\mathsf {p}}}_*\,$. On the other hand, the following (well known) fact implies that $\overline{{\mathsf {q}}}^*$ differs from $\overline{{\mathsf {q}}}^!$ only by a cohomological shift and a Tate twist.

Proposition B.3.3

As before, assume that $G=SL(2)$. Then

• (i) the morphism $\overline{{\mathsf {q}}}:\overline{{{\mathrm{{Bun}}}}}_B\rightarrow {{\mathrm{{Bun}}}}_T$ is smooth.

• (ii) the fiber of $\overline{{\mathsf {q}}}$ over ${\mathcal {M}}\in {{\mathrm{{Bun}}}}_T$ has pure dimension $-\chi ({\mathcal {M}}^{\otimes 2})=g_X-1-2 \deg {\mathcal {M}}\,$.

We skip the proof because it is quite similar to that of [24, Cor. 2.10].

Corollary B.3.4

One has

$$\begin{aligned} \underline{{{\mathrm{{Eis}}}}}({\mathcal {F}})=\overline{{{\mathrm{{Eis}}}}} ({\mathcal {F}}[2m](m)),\quad {\mathcal {F}}\in {\mathscr {D}}({{\mathrm{{Bun}}}}_T)_+\, , \end{aligned}$$

where $m:{{\mathrm{{Bun}}}}_T\rightarrow {\mathbb {Z}}$ is the locally constant function whose value at ${\mathcal {M}}\in {{\mathrm{{Bun}}}}_T$ equals $2 \deg {\mathcal {M}}+1-g_X\,$.

1.3.3 B.3.5 Expressing $\overline{{{\mathrm{{Eis}}}}}$ and $\underline{{{\mathrm{{Eis}}}}}$ in terms of ${{\mathrm{{Eis}}}}_*\,$ and ${{\mathrm{{Eis}}}}_!\,$

The next proposition describes the relation between $\overline{{{\mathrm{{Eis}}}}}$ and ${{\mathrm{{Eis}}}}_*$ and a similar relation between $\underline{{{\mathrm{{Eis}}}}}$ and ${{\mathrm{{Eis}}}}_!$ at the level of Grothendieck groups. To formulate it, we need some notation.

Let ${{\mathrm{{Sym}}}}X$ denote the scheme parametrizing all effective divisors on X; in other words, ${{\mathrm{{Sym}}}}X$ is the disjoint union of ${{\mathrm{{Sym}}}}^nX$ for all $n\ge 0$. Note that ${{\mathrm{{Sym}}}}X$ is a monoid with respect to addition. The morphism

$$\begin{aligned} {\text {act}}:{{\mathrm{{Sym}}}}X\times {{\mathrm{{Bun}}}}_T\rightarrow {{\mathrm{{Bun}}}}_T\, , \quad (D,{\mathcal {M}})\mapsto {\mathcal {M}}(-D) \end{aligned}$$

(B.6)

defines an action of the monoid ${{\mathrm{{Sym}}}}X$ on ${{\mathrm{{Bun}}}}_T\,$. Let

$$\begin{aligned} {\text {pr}}:{{\mathrm{{Sym}}}}X\times {{\mathrm{{Bun}}}}_T\rightarrow {{\mathrm{{Bun}}}}_T \end{aligned}$$

denote the projection.

Proposition B.3.6

• (i) The map $\overline{{{\mathrm{{Eis}}}}}:{{\mathrm{\mathbf {K}}}}({{\mathrm{{Bun}}}}_T )_+\rightarrow {{\mathrm{\mathbf {K}}}}({{\mathrm{{Bun}}}}_B )$ is equal to ${{\mathrm{{Eis}}}}_*\circ {\text {pr}}_*\circ {\text {act}}^!$.

• (ii) The map $\underline{{{\mathrm{{Eis}}}}}:{{\mathrm{\mathbf {K}}}}({{\mathrm{{Bun}}}}_T )_+\rightarrow {{\mathrm{\mathbf {K}}}}({{\mathrm{{Bun}}}}_B )$ equals ${{\mathrm{{Eis}}}}_!\circ {\text {pr}}_!\circ {\text {act}}^*$.

Remark B.3.7

${{\mathrm{{Sym}}}}^n X$ is proper for each n, so ${\text {pr}}_!={\text {pr}}_*\,$. On the other hand, the morphism (B.6) is smooth, so ${\text {act}}^*$ only slightly differs from ${\text {act}}^!\,$; more precisely, for any ${\mathcal {F}}\in {\mathscr {D}}({{\mathrm{{Bun}}}}_T)_+$ the restrictions of ${\text {act}}^*({\mathcal {F}})$ and ${\text {act}}^!({\mathcal {F}})[-2n](-n)$ to ${{\mathrm{{Sym}}}}^nX\times {{\mathrm{{Bun}}}}_T$ are canonically isomorphic.

Proof of Proposition B.3.6

The proof we give below is straightforward because statement (i) involves only ‘right’ functors and statement (ii) only ‘left’ ones.

First, let us recall the standard stratification of $\overline{{{\mathrm{{Bun}}}}}_B\,$. If ${\mathcal {L}}$ is a rank 2 vector bundle on X with trivialized determinant, ${\mathcal {M}}\subset {\mathcal {L}}$ is a line sub-bundle, and $D\subset X$ is an effective divisor of degree n, then the pair $({\mathcal {L}},{\mathcal {M}}(-D))$ defines an ${\mathbb {F}}_q$-point of $\overline{{{\mathrm{{Bun}}}}}_B\,$. This construction works for S-points instead of ${\mathbb {F}}_q$-points. It defines a locally closed immersion

$$\begin{aligned} i_n:{{\mathrm{{Sym}}}}^nX\times {{\mathrm{{Bun}}}}_B\hookrightarrow \overline{{{\mathrm{{Bun}}}}}_B. \end{aligned}$$

The substacks $i_n({{\mathrm{{Sym}}}}^nX\times {{\mathrm{{Bun}}}}_B)$ form a stratification of $\overline{{{\mathrm{{Bun}}}}}_B\,$.

Now let us prove (i). We have to check the equality

$$\begin{aligned} \overline{{\mathsf {p}}}_*\circ \overline{{\mathsf {q}}}^!={\mathsf {p}}_*\circ {\mathsf {q}}^!\circ {\text {pr}}_*\circ {\text {act}}^!\, , \end{aligned}$$

(B.7)

in which both sides are maps ${{\mathrm{\mathbf {K}}}}({{\mathrm{{Bun}}}}_T)_+\rightarrow {{\mathrm{\mathbf {K}}}}({{\mathrm{{Bun}}}}_G)$. For any ${\mathcal {F}}\in {{\mathrm{\mathbf {K}}}}(\overline{{{\mathrm{{Bun}}}}}_B )$, one has

$$\begin{aligned} {\mathcal {F}}=\sum _{n=0}^{\infty }(i_n)_*\circ i_n^! ({\mathcal {F}}) \end{aligned}$$

(the sum converges in the topology of ${{\mathrm{\mathbf {K}}}}(\overline{{{\mathrm{{Bun}}}}}_B )$ defined in Subsection B.1.4) So

$$\begin{aligned} \overline{{\mathsf {p}}}_*\circ \overline{{\mathsf {q}}}^!=\sum _{n=0}^{\infty }(\overline{{\mathsf {p}}}\circ i_n)_*\circ (\overline{{\mathsf {q}}}\circ i_n)^!\, , \end{aligned}$$

(B.8)

To see that the right-hand sides of (B.7) and (B.8) are equal, it suffices to apply base change to the expression ${\mathsf {q}}^!\circ {\text {pr}}_*$ from the r.h.s. of (B.7).

We have proved (i). Statement (ii) can be either proved similarly or deduced from (i) by Verdier duality. $\square$

1.4 B.4 Passing from sheaves to functions

Recall that we think of ${\mathcal {C}}^K$ as the space of $\overline{{\mathbb {Q}}}_l$-valued functions on ${{\mathrm{{Bun}}}}_T({\mathbb {F}}_q)={{\mathrm{{Bun}}}}_{{\mathbb {G}}_m}({\mathbb {F}}_q)$ (see Example 2.3.1).

Lemma B.4.1

As before, let ${\text {pr}}:{{\mathrm{{Sym}}}}X\times {{\mathrm{{Bun}}}}_T\rightarrow {{\mathrm{{Bun}}}}_T$ denote the projection and ${\text {act}}:{{\mathrm{{Sym}}}}X\times {{\mathrm{{Bun}}}}_T\rightarrow {{\mathrm{{Bun}}}}_T$ the morphism $(D,{\mathcal {M}})\mapsto {\mathcal {M}}(-D)$.

• (i) One has commutative diagrams

  

  (B.9)

  in which each vertical arrow is the map ${\mathcal {F}}\mapsto f_{{\mathcal {F}}}$ and the operator ${\mathfrak {L}}_n: {\mathcal {C}}_{+}^K\rightarrow {\mathcal {C}}_{+}^K$ is defined by

  $$\begin{aligned} ({\mathfrak {L}}_n \varphi )({\mathcal {M}})=\sum _{D\ge 0} q^{n\cdot \deg D}\varphi ({\mathcal {M}}(-D)), \quad \varphi \in {\mathcal {C}}_{+}^K \end{aligned}$$

  (B.10)

  (summation over all effective divisors on X).

• (ii) All horizontal arrows in diagrams (B.9) are invertible.

Proof

Statement (i) is clear (in the case of ${\text {pr}}_!\circ {\text {act}}^*$ use Remark B.3.7).

Let us prove that the operator ${\mathfrak {L}}_n:{\mathcal {C}}_{+}^K\rightarrow {\mathcal {C}}_{+}^K$ is invertible (invertibility of the upper horizontal arrows is proved similarly). The space ${\mathcal {C}}_{+}^K$ is complete with respect to the filtration formed by the subspaces ${{\mathcal {C}}}^K_{\ge {\mathcal {N}}}\subset {\mathcal {C}}_{+}^K$, ${\mathcal {N}}\in {\mathbb {Z}}$. The operator ${\mathfrak {L}}_n$ is compatible with the filtration and acts as identity on the successive quotients. So ${\mathfrak {L}}_n$ is invertible. $\square$

Recall that one has the operator ${{\mathrm{{Eis}}}}^K:{\mathcal {C}}_{+}^K\rightarrow {\mathcal {A}}^K$.

Corollary B.4.2

For any ${\mathcal {F}}\in {{\mathrm{\mathbf {K}}}}({{\mathrm{{Bun}}}}_T)_+$, one has

$$\begin{aligned} f_{{{\mathrm{{Eis}}}}_*{\mathcal {F}}}= & {} {{\mathrm{{Eis}}}}^K\left( f_{\mathcal {F}}\right) , \quad {\mathcal {F}}\in {{\mathrm{\mathbf {K}}}}\left( {{\mathrm{{Bun}}}}_T\right) _+ \; , \end{aligned}$$

(B.11)

$$\begin{aligned} f_{\overline{{{\mathrm{{Eis}}}}}\,{\mathcal {F}}}= & {} \left( {{\mathrm{{Eis}}}}^K\circ {\mathfrak {L}}_0\right) \left( f_{\mathcal {F}}\right) , \quad {\mathcal {F}}\in {{\mathrm{\mathbf {K}}}}\left( {{\mathrm{{Bun}}}}_T\right) _+, \end{aligned}$$

(B.12)

$$\begin{aligned} f_{\underline{{{\mathrm{{Eis}}}}}\, {\mathcal {F}}}= & {} \left( {{\mathrm{{Eis}}}}^K\circ {\mathfrak {L}}_0\circ Q\right) \left( f_{\mathcal {F}}\right) , \quad {\mathcal {F}}\in {{\mathrm{\mathbf {K}}}}\left( {{\mathrm{{Bun}}}}_T\right) _+ \;, \end{aligned}$$

(B.13)

$$\begin{aligned} f_{{{\mathrm{{Eis}}}}_!{\mathcal {F}}}= & {} \left( {{\mathrm{{Eis}}}}^K\circ {\mathfrak {L}}_0\circ {\mathfrak {L}}_1^{-1}\circ Q\right) \left( f_{\mathcal {F}}\right) , \quad {\mathcal {F}}\in {{\mathrm{\mathbf {K}}}}\left( {{\mathrm{{Bun}}}}_T\right) _+\; , \end{aligned}$$

(B.14)

where ${\mathfrak {L}}_0\, ,{\mathfrak {L}}_1:{\mathcal {C}}_{+}^K\rightarrow {\mathcal {C}}_{+}^K$ are defined by formula  (B.10) and $Q:{\mathcal {C}}_{+}^K\rightarrow {\mathcal {C}}_{+}^K$ is the operator of multiplication by the function

$$\begin{aligned} {\mathcal {M}}\mapsto q^{2\deg {\mathcal {M}}+1-g_X}, \quad {\mathcal {M}}\in {{\mathrm{{Bun}}}}_T \left( {\mathbb {F}}_q\right) . \end{aligned}$$

(B.15)

Proof

Formula (B.11) is clear because the definition of ${{\mathrm{{Eis}}}}_*$ involves only ‘right’ functors. Let us prove (B.14) (the proof of (B.12)–(B.13) is similar but easier).

By Proposition B.3.6 and Corollary B.3.4, the map ${{\mathrm{{Eis}}}}_!:{{\mathrm{\mathbf {K}}}}({{\mathrm{{Bun}}}}_T)_+\rightarrow {{\mathrm{\mathbf {K}}}}({{\mathrm{{Bun}}}}_G)$ equals ${{\mathrm{{Eis}}}}_*\circ \widetilde{\mathfrak {L}}_0\circ \widetilde{Q}\circ (\widetilde{\mathfrak {L}}_{-1})^{-1}$, where $\widetilde{\mathfrak {L}}_0:={\text {pr}}_*\circ {\text {act}}^!$, $\widetilde{\mathfrak {L}}_{-1}:={\text {pr}}_!\circ {\text {act}}^*$, and $\widetilde{Q}({\mathcal {F}}):={\mathcal {F}}[2m](m)$ (here m is as in Corollary B.3.4); note that $\widetilde{\mathfrak {L}}_{-1}$ is invertible by Lemma B.4.1(ii). Lemma B.4.1(i) and formula (B.11) imply that

$$\begin{aligned} f_{{{\mathrm{{Eis}}}}_!{\mathcal {F}}}= \left( {{\mathrm{{Eis}}}}^K\circ {\mathfrak {L}}_0\circ Q\circ \left( {\mathfrak {L}}_{-1}\right) ^{-1}\right) \left( f_{\mathcal {F}}\right) . \end{aligned}$$

This is equivalent to (B.14) because $Q^{-1}\circ {\mathfrak {L}}_1\circ Q={\mathfrak {L}}_{-1}\,$. $\square$

1.5 B.5 Proof of Theorem B.2.1

Theorem B.2.1 involves the operator

$$\begin{aligned} q^{2-2g_X}\cdot ({{\mathrm{{Eis}}}}')^K\circ \iota ^*:{\mathcal {C}}_{+}^K\rightarrow {\mathcal {A}}^K. \end{aligned}$$

(B.16)

By definition, $({{\mathrm{{Eis}}}}')^K={{\mathrm{{Eis}}}}^K\circ (M^K)^{-1}$. Formulas (5.11)–(5.12) tell us that

$$\begin{aligned} \left( M^K\right) ^{-1}\circ \iota ^*=q^{2g_X-2}\cdot {\mathfrak {L}}_0\circ {\mathfrak {L}}_1^{-1}\circ Q, \end{aligned}$$

where $Q:{\mathcal {C}}_{+}^K\rightarrow {\mathcal {C}}_{+}^K$ is the operator of multiplication by the function (B.15). So the operator (B.16) is equal to the operator ${{\mathrm{{Eis}}}}^K\circ {\mathfrak {L}}_0\circ {\mathfrak {L}}_1^{-1}\circ Q$, which appears in formula (B.14).

1.6 B.6 Concluding remarks

The above proof of Theorem B.2.1 is self-contained. On the other hand, it is barbaric for the following reasons.

• (i) We heavily used smoothness of the morphism $\overline{{\mathsf {q}}}:\overline{{{\mathrm{{Bun}}}}}_B\rightarrow {{\mathrm{{Bun}}}}_T\,$, which is a specific feature of the case $G=SL(2)$. If G is an arbitrary reductive group, then instead of $\overline{{{\mathrm{{Eis}}}}}$ and $\underline{{{\mathrm{{Eis}}}}}$, one should work with the functor ${{\mathrm{{Eis}}}}_{!*}:{\mathscr {D}}({{\mathrm{{Bun}}}}_T)_+\rightarrow {\mathscr {D}}({{\mathrm{{Bun}}}}_G)$ introduced by A. Braverman and D. Gaitsgory [6, Subsection 2.1] (they denote it simply by ${{\mathrm{{Eis}}}}$; the notation ${{\mathrm{{Eis}}}}_{!*}$ is taken from [14]). In the case $G=SL(2)$ the functor ${{\mathrm{{Eis}}}}_{!*}$ is the ‘geometric mean’ of our functors $\overline{{{\mathrm{{Eis}}}}}$ and $\underline{{{\mathrm{{Eis}}}}}$.

• (ii) Our Proposition B.3.6 is a statement at the level of ${{\mathrm{\mathbf {K}}}}$-groups (and so is the more general Corollary 4.5 from [7]). However, as explained to me by D. Gaitsgory, there is a way to relate the functors ${{\mathrm{{Eis}}}}_{!*}\,$, ${{\mathrm{{Eis}}}}_{!}\,$, and ${{\mathrm{{Eis}}}}_{*}$ themselves (not merely the corresponding homomorphisms of ${{\mathrm{\mathbf {K}}}}$-groups). His formulation of the relation involves the factorization algebras $\Upsilon$ and $\Omega$ introduced in [7, Subsections 3.1 and 3.5]. One can think of these algebras as geometrizations of the operators ${\mathfrak {L}}_0^{-1}$ and ${\mathfrak {L}}_1^{-1}$, where ${\mathfrak {L}}_n$ is defined by (B.10). To make the analogy more precise, one should think of ${\mathfrak {L}}_n$ not as an operator but as an element of the algebra A from Subsection 5.4.1. The fact that $\Upsilon$ and $\Omega$ are factorization algebras is related to the Euler product expression for ${\mathfrak {L}}_n$ in formula (5.9).

Appendix C. A conjectural D-module analog of formula (4.1)

Recall that according to Corollary 4.3.2, in the case of function fields the operator $L:{\mathcal {A}}_c\rightarrow {\mathcal {A}}_{ps-c}$ is invertible and its inverse is given by formula (4.1). In Subsections A.8–A.9 we defined a functor

$$\begin{aligned} ({{\mathrm{{Ps-Id}}}})_{{{\mathrm{{Bun}}}}_G,!}: {\text {D-mod}}({{\mathrm{{Bun}}}}_G)_c\rightarrow {\text {D-mod}}({{\mathrm{{Bun}}}}_G)_{ps-c}\, ; \end{aligned}$$

as explained in Subsections A.8.4, this functor is a D-module analog of the operator

$$\begin{aligned} q^{-d}L^K:{\mathcal {A}}_c^K\rightarrow {\mathcal {A}}_{ps-c}^K\; , \quad d:=\dim {{\mathrm{{Bun}}}}_G. \end{aligned}$$

As explained in Subsection A.9, the main theorem of [14] implies that this functor is invertible. Conjecture C.2.1 below gives a description of the inverse functor, which is inspired by formula (4.1). Before formulating the conjecture, we have to define a certain endofunctor of ${\text {D-mod}}({{\mathrm{{Bun}}}}_G)$, which can be considered as a D-module analog of the operator $1-{{\mathrm{{Eis}}}}\circ {{\mathrm{{CT}}}}$ from the r.h.s of formula (4.1).

1.1 C.1 An endofunctor of ${\text {D-mod}}({{\mathrm{{Bun}}}}_G)$

Conjecture C.2.1 involves the DG category ${{\mathrm{{I}}}}(G,B)$ defined in [15, Section 6] and the adjoint pair of functors

$$\begin{aligned} {{\mathrm{{Eis}}}}_B^{{{\mathrm{{enh}}}}}:{{\mathrm{{I}}}}(G,B)\rightarrow {\text {D-mod}}({{\mathrm{{Bun}}}}_G), \quad {{\mathrm{{CT}}}}_B^{{{\mathrm{{enh}}}}}:{\text {D-mod}}({{\mathrm{{Bun}}}}_G)\rightarrow {{\mathrm{{I}}}}(G,B) \end{aligned}$$

defined in [15, Subsection 6.3]; here ‘enh’ stands for ‘enhanced’. The ideas behind these definitions are explained in [15, Subsection 1.4]. More details regarding ${{\mathrm{{I}}}}(G,B)$, ${{\mathrm{{Eis}}}}_B^{{{\mathrm{{enh}}}}}$, and ${{\mathrm{{CT}}}}_B^{{{\mathrm{{enh}}}}}$ are contained in [1, Subsections 7.1, 7.3.5, 8.2.4].

One can think of the DG category ${{\mathrm{{I}}}}(G,B)$ as a ‘refined version’ of the DG category ${\text {D-mod}}({{\mathrm{{Bun}}}}_T )$. More precisely, ${{\mathrm{{I}}}}(G,B)$ has a filtration indexed by integers whose associated graded equals ${\text {D-mod}}({{\mathrm{{Bun}}}}_T )$ (the grading on ${\text {D-mod}}({{\mathrm{{Bun}}}}_T )$ comes from the degree map ${{\mathrm{{Bun}}}}_T\rightarrow {\mathbb {Z}}$).³⁸ Both ${{\mathrm{{I}}}}(G,B)$ and ${\text {D-mod}}({{\mathrm{{Bun}}}}_T )$ are D-module analogs³⁹ (in the sense of Subsection A.1.6) of the vector space ${{\mathcal {C}}}^K$.

According to [1, §8.2.4], the functor ${{\mathrm{{Eis}}}}_B^{{{\mathrm{{enh}}}}}:{{\mathrm{{I}}}}(G,B)\rightarrow {\text {D-mod}}({{\mathrm{{Bun}}}}_G)$ is left adjoint to ${{\mathrm{{CT}}}}_B^{{{\mathrm{{enh}}}}}:{\text {D-mod}}({{\mathrm{{Bun}}}}_G)\rightarrow {{\mathrm{{I}}}}(G,B)$. Let $\epsilon :{{\mathrm{{Eis}}}}_B^{{{\mathrm{{enh}}}}}\circ {{\mathrm{{CT}}}}_B^{{{\mathrm{{enh}}}}}\rightarrow {\mathop {{\text {Id}}}}_{{\text {D-mod}}({{\mathrm{{Bun}}}}_G)}$ denote the counit of the adjunction. We need its cone, which is a functor

$$\begin{aligned} {{\mathrm{{Cone}}}}(\epsilon ) :{\text {D-mod}}({{\mathrm{{Bun}}}}_G )\rightarrow {\text {D-mod}}({{\mathrm{{Bun}}}}_G ). \end{aligned}$$

(C.1)

Remark C.1.1

We think of ${{\mathrm{{Eis}}}}_B^{{{\mathrm{{enh}}}}}$ and ${{\mathrm{{CT}}}}_B^{{{\mathrm{{enh}}}}}$ as D-module analogs of the operators ${{\mathrm{{Eis}}}}^K$ and ${{\mathrm{{CT}}}}^K$. We think of ${{\mathrm{{Cone}}}}(\epsilon )$ as a D-module analog of the operator $1-{{\mathrm{{Eis}}}}^K\circ {{\mathrm{{CT}}}}^K$.

1.2 C.2 The conjecture

Consider the composition

$$\begin{aligned} {\text {D-mod}}({{\mathrm{{Bun}}}}_G)_{ps-c}\rightarrow {\text {D-mod}}({{\mathrm{{Bun}}}}_G)_c\hookrightarrow {\text {D-mod}}({{\mathrm{{Bun}}}}_G)\, , \end{aligned}$$

(C.2)

where the first arrow is the functor $(({{\mathrm{{Ps-Id}}}})_{{{\mathrm{{Bun}}}}_G,!})^{-1}:{\text {D-mod}}({{\mathrm{{Bun}}}}_G)_{ps-c}\rightarrow {\text {D-mod}}({{\mathrm{{Bun}}}}_G)_c\,$. The following conjecture expresses this composition in terms of the functor (C.1).

Conjecture C.2.1

The composition (C.2) is isomorphic to the restriction of the functor ${{\mathrm{{Cone}}}}(\epsilon )[2d]$ to ${\text {D-mod}}({{\mathrm{{Bun}}}}_G)_{ps-c}\,$, where $d:=\dim {{\mathrm{{Bun}}}}_G\,$.

Remark C.2.2

One can prove that the functor (C.1) indeed maps the subcategory ${\text {D-mod}}({{\mathrm{{Bun}}}}_G)_{ps-c}\subset {\text {D-mod}}({{\mathrm{{Bun}}}}_G)$ to ${\text {D-mod}}({{\mathrm{{Bun}}}}_G)_c$ (as would follow from the conjecture).

Remark C.2.3

Let us compare the above conjecture with formula (4.1). The functor $({{\mathrm{{Ps-Id}}}})_{{{\mathrm{{Bun}}}}_G,!}$ is a D-module analog of the operator $q^{-d}L^K$. Formula (4.1) tells us that $(q^{-d}L^K)^{-1}=q^d (1-{{\mathrm{{Eis}}}}^K\circ {{\mathrm{{CT}}}}^K)$. This agrees with Conjecture C.2.1 by Remarks C.1.1 and A.1.5.

Remark C.2.4

Conjecture C.2.1 implies that for any $N\in {\text {D-mod}}({{\mathrm{{Bun}}}}_G )_{ps-c}$, one has a canonical morphism $N[2d] \rightarrow (({{\mathrm{{Ps-Id}}}})_{{{\mathrm{{Bun}}}}_G,!})^{-1}(N)$. This agrees with Remark A.8.3, which says that for any $M\in {\text {D-mod}}({{\mathrm{{Bun}}}}_G )_c$, one has a canonical morphism

$$\begin{aligned} ({{\mathrm{{Ps-Id}}}})_{{{\mathrm{{Bun}}}}_G,!}(M)\rightarrow M[-2d]. \end{aligned}$$