2.2.1 Satake category

Let $D=D_{-}=D_{+}=\text{Spec }k[[t]]$ be copies of the formal disk, $D^{\times }=\text{Spec }k((t))\subset D,D_{-},D_{+}$ the punctured formal disk, and $\mathbb{D}={D_{-}\coprod }_{D^{\times }}D_{+}$ the non-separated disk with two zeros $0_{-}\in D_{-},0_{+}\in D_{+}$ .

Let $\text{Bun}_{G}(\mathbb{D})$ be the moduli of $G$ -bundles on $\mathbb{D}$ .

Introduce the Laurent series loop group $G((t))=\text{Maps}(D^{\times },G)$ , with its parahoric arc subgroup $G[[t]]=\text{Maps}(D,G)$ , and affine Grassmannian $\text{Gr}_{G}=G((t))/G[[t]]$ . The gluing presentation $\mathbb{D}={D_{-}\coprod }_{D^{\times }}D_{+}$ induces a double-coset presentation

Let ${\mathcal{H}}_{G}^{\text{sph}}=\mathit{Sh}_{c}(\text{Bun}_{G}(\mathbb{D}))$ be the dg spherical Hecke category of constructible complexes on $\text{Bun}_{G}(\mathbb{D})$ with proper support, or equivalently $G[[t]]$ -equivariant constructible complexes on $\text{Gr}_{G}$ with proper support. Convolution and fusion equips ${\mathcal{H}}_{G}^{\text{sph}}$ with an $E_{3}$ -monoidal structure, which preserves the heart ${\mathcal{H}}_{G,\heartsuit }^{\text{sph}}\subset {\mathcal{H}}_{G}^{\text{sph}}$ with respect to the perverse $t$ -structure. The $E_{3}$ -monoidal structure on ${\mathcal{H}}_{G,\heartsuit }^{\text{sph}}$ naturally lifts to a symmetric monoidal structure. Though we mention it for clarity, we will not need the $E_{3}$ -monoidal structure on ${\mathcal{H}}_{G}^{\text{sph}}$ but only the symmetric monoidal structure on ${\mathcal{H}}_{G,\heartsuit }^{\text{sph}}$ .

The geometric Satake correspondence [Reference Mirković and VilonenMV07, Reference GinzburgGin95] provides a symmetric monoidal equivalence

(2.1)

where ${\mathcal{H}}_{G,\heartsuit ,\unicode[STIX]{x1D70C}^{\vee }}^{\text{sph}}$ denotes the same monoidal category ${\mathcal{H}}_{G,\heartsuit }^{\text{sph}}$ but with its twisted commutativity constraint. There is also a derived geometric Satake correspondence but we will not need it.

2.2.2 Affine Hecke category

Let $\text{Bun}_{G}(\mathbb{D},\{0_{-},0_{+}\})$ be the moduli of $G$ -bundles on $\mathbb{D}$ with $B$ -reductions at the points $0_{-},0_{+}\in \mathbb{D}$ . The natural projection $\text{Bun}_{G}(\mathbb{D},\{0_{-},0_{+}\})\rightarrow \text{Bun}_{G}(\mathbb{D})$ is a ${\mathcal{B}}\times {\mathcal{B}}$ -fibration.

Let $I\subset G[[t]]$ be the Iwahori subgroup given by the inverse image of $B\subset G$ under the evaluation map at $0\in D$ , and $\text{Fl}_{G}=G((t))/I$ the corresponding affine flag variety. The gluing presentation $\mathbb{D}={D_{-}\coprod }_{D^{\times }}D_{+}$ induces a double-coset presentation

Let ${\mathcal{H}}_{G}^{\text{aff}}=\mathit{Sh}_{c}(\text{Bun}_{G}(\mathbb{D},\{0_{-},0_{+}\}))$ be the dg affine Hecke category of constructible complexes on $\text{Bun}_{G}(\mathbb{D},\{0_{-},0_{+}\})$ with proper support, or equivalently $I$ -equivariant constructible complexes on $\text{Fl}_{G}$ with proper support. Convolution equips ${\mathcal{H}}_{G}^{\text{aff}}$ with a monoidal structure.

Recall we write $\unicode[STIX]{x1D707}:\widetilde{{\mathcal{N}}}^{\vee }\rightarrow {\mathcal{N}}^{\vee }$ for the Springer resolution, and identify ${\mathcal{N}}^{\vee }$ with the unipotent elements in $G^{\vee }$ via the exponential map. The Steinberg variety $\mathit{St}_{G^{\vee }}$ is the derived scheme given by the derived fiber product

Passing to adjoint quotients, we have

Let $\text{Coh}^{G^{\vee }}(\mathit{St}_{G^{\vee }})$ be the dg derived category of coherent complexes on $\mathit{St}_{G^{\vee }}/G^{\vee }$ , or equivalently $G^{\vee }$ -equivariant coherent complexes on $\mathit{St}_{G^{\vee }}$ . Convolution equips it with a monoidal structure.

Bezrukavnikov’s theorem [Reference BezrukavnikovBez16, Theorem 1] provides a monoidal equivalence

(2.2)

Example 2.2.4 ( $G=\text{SL}(2)$ ).

The affine Weyl group $W^{\text{aff}}$ can be identified with the infinite dihedral group acting on the real line $\mathbb{R}$ with fundamental domain $[0,1]$ . For $x\in \mathbb{Z}$ , let $r_{x}\in W^{\text{aff}}$ be the reflection with center $x$ , then $W^{\text{aff}}=\langle r_{0},r_{1}\rangle$ . Note that under the usual indexing scheme for affine Coxeter groups, $r_{0}$ (respectively $r_{1}$ ) corresponds to the simple reflection $s_{1}$ (respectively $s_{0}$ ) with respect to the simple root $\unicode[STIX]{x1D6FC}_{1}$ (respectively $\unicode[STIX]{x1D6FC}_{1}$ ) of the affine $\mathfrak{s}\mathfrak{l}(2)$ .

Correspondingly there are two standard monoidal generators $T_{0\ast },T_{1\ast }$ for ${\mathcal{H}}^{\text{aff}}$ given by the $\ast$ -extensions of $\text{}\underline{\mathbb{Q}}_{\text{Fl}^{r_{0}}}[1]$ and $\text{}\underline{\mathbb{Q}}_{\text{Fl}^{r_{1}}}[1]$ . Similarly define $T_{0!}$ and $T_{1!}$ using $!$ -extensions instead of $\ast$ -extensions. Then we have monoidal inverses $T_{0\ast }^{-1}\simeq T_{0!}$ , $T_{1\ast }^{-1}\simeq T_{1!}$ .

For $k\in \mathbb{Z}$ , the Wakimoto sheaf can be expressed as $J_{2k}\simeq (T_{0\ast }T_{1\ast })^{k}$ , which corresponds under $\unicode[STIX]{x1D6F7}^{\text{aff}}$ to the twist of the structure sheaf of the relative diagonal ${\mathcal{O}}_{\widetilde{{\mathcal{N}}}^{\vee }}(2k)$ . This follows from the construction of the functor $\unicode[STIX]{x1D6F7}^{\text{aff}}$ , see [Reference BezrukavnikovBez16, § 4.1].

The finite braid operator $T_{0\ast }$ corresponds under $\unicode[STIX]{x1D6F7}^{\text{aff}}$ to the classical (namely underived) structure sheaf ${\mathcal{O}}_{\mathit{St}_{G^{\vee }}}^{\text{cl}}$ . Its inverse $T_{0\ast }^{-1}\simeq T_{0!}$ corresponds to the twist ${\mathcal{O}}_{\mathit{St}_{G^{\vee }}}^{\text{cl}}(-1,-1)$ . We briefly indicate how to deduce $\unicode[STIX]{x1D6F7}^{\text{aff}}(T_{0\ast })\cong {\mathcal{O}}_{\mathit{St}_{G^{\vee }}}^{\text{cl}}$ from the results of [Reference BezrukavnikovBez16]. By [Reference BezrukavnikovBez16, Theorem 1], there is an equivalence

where $I^{0}=I\,\times _{B}N\subset I$ , and $\mathit{St}^{\prime }=\widetilde{\mathfrak{g}^{\vee }}\,\times _{\mathfrak{g}^{\vee }}\widetilde{{\mathcal{N}}}^{\vee }$ ( $\widetilde{\mathfrak{g}^{\vee }}\rightarrow \mathfrak{g}^{\vee }$ is the Grothendieck alteration). By [Reference BezrukavnikovBez16, Example 57], the object $T_{0\ast }$ , viewed as an object in $\mathit{Sh}_{c}(I^{0}\backslash \text{Fl}_{G})$ , corresponds to the structure sheaf of $\unicode[STIX]{x1D6E4}_{s}^{\prime }=\unicode[STIX]{x1D6E4}_{s}\,\cap \,\mathit{St}^{\prime }$ (where $\unicode[STIX]{x1D6E4}_{s}$ is the closure of the graph of the non-trivial element $s\in W$ on the regular locus of $\widetilde{\mathfrak{g}^{\vee }}\times _{\mathfrak{g}^{\vee }}\widetilde{\mathfrak{g}^{\vee }}$ , and $\unicode[STIX]{x1D6E4}_{s}^{\prime }$ is the scheme-theoretic intersection of $\unicode[STIX]{x1D6E4}_{s}$ with $\mathit{St}^{\prime }$ ). Under the natural embedding $i:\mathit{St}_{G^{\vee }}{\hookrightarrow}\mathit{St}^{\prime }$ , one checks that $i_{\ast }{\mathcal{O}}_{\mathit{St}_{G^{\vee }}}^{\text{cl}}\cong {\mathcal{O}}_{\unicode[STIX]{x1D6E4}_{s}^{\prime }}$ . Since the forgetful functor ${\mathcal{H}}_{G}^{\text{aff}}=\mathit{Sh}_{c}(I\backslash \text{Fl}_{G})\rightarrow \mathit{Sh}_{c}(I^{0}\backslash \text{Fl}_{G})$ corresponds to $i_{\ast }:\text{Coh}^{G^{\vee }}(\mathit{St}_{G^{\vee }})\rightarrow \text{Coh}^{G^{\vee }}(\mathit{St}^{\prime })$ , we see that $i_{\ast }\unicode[STIX]{x1D6F7}^{\text{aff}}(T_{0\ast })\cong i_{\ast }{\mathcal{O}}_{\mathit{St}_{G^{\vee }}}^{\text{cl}}$ . Using the right exactness of $i_{\ast }$ , we conclude that $\unicode[STIX]{x1D6F7}^{\text{aff}}(T_{0\ast })\cong {\mathcal{O}}_{\mathit{St}_{G^{\vee }}}^{\text{cl}}$ .

Below we will give more examples of how objects correspond to each other under the equivalence $\unicode[STIX]{x1D6F7}^{\text{aff}}$ . The proof of the matchings follow from the two paragraphs above by easy calculations, which we omit here.

The affine braid operator $T_{1\ast }$ corresponds under $\unicode[STIX]{x1D6F7}^{\text{aff}}$ to the twisted classical structure sheaf ${\mathcal{O}}_{\mathit{St}_{G^{\vee }}}^{\text{cl}}(-1,1)$ . This follows from the fact that $J_{2}=T_{0\ast }T_{1\ast }$ corresponds to ${\mathcal{O}}_{\widetilde{{\mathcal{N}}}^{\vee }}(2)$ , and that $T_{0\ast }$ corresponds to ${\mathcal{O}}_{\mathit{St}_{G^{\vee }}}^{\text{cl}}$ . Similarly $T_{1\ast }^{-1}\simeq T_{1!}$ corresponds to the twist ${\mathcal{O}}_{\mathit{St}_{G^{\vee }}}^{\text{cl}}(-2,0)$ . The conjugate $T_{0\ast }T_{1\ast }T_{0\ast }^{-1}\simeq J_{2}T_{1\ast }J_{2}^{-1}$ corresponds to the twist ${\mathcal{O}}_{\mathit{St}_{G^{\vee }}}^{\text{cl}}(1,-1)$ , and its inverse $T_{0\ast }T_{1!}T_{0\ast }^{-1}\simeq J_{2}T_{1!}J_{2}^{-1}$ to the twist ${\mathcal{O}}_{\mathit{St}_{G^{\vee }}}^{\text{cl}}(0,-2)$ .

Let $\text{Avg}$ be the IC-sheaf of the closure of $\text{Fl}^{r_{0}}$ . Then $\text{Avg}$ corresponds to ${\mathcal{O}}_{\mathbb{P}^{1}\times \mathbb{P}^{1}}(-1,-1)$ under $\unicode[STIX]{x1D6F7}^{\text{aff}}$ . The natural distinguished triangles

correspond to the natural distinguished triangles

2.2.6 Compatibilty

Gaitsgory’s nearby cycles construction [Reference GaitsgoryGai01] provides a central functor

Under the Satake equivalence (2.1) and Bezrukavnikov’s equivalence $\unicode[STIX]{x1D6F7}^{\text{aff}}$ , the central functor becomes the natural functor

where the first functor is the pullback along the projection $\widetilde{{\mathcal{N}}}^{\vee }/G^{\vee }\rightarrow \text{pt}/G^{\vee }$ . Its monodromy automorphism corresponds to the universal unipotent automorphism of the pullback.

2.2.7 Finite and aspherical Hecke categories

Let ${\mathcal{H}}_{G}^{f}=\mathit{Sh}_{c}(B\backslash G/B)$ be the finite Hecke category of $B$ -equivariant constructible complexes on the flag variety ${\mathcal{B}}=G/B$ , with monoidal structure defined by convolution. Pushforward along the closed embedding ${\mathcal{B}}=G/B{\hookrightarrow}G((t))/I=\text{Fl}_{G}$ gives a fully fiathful monoidal functor ${\mathcal{H}}_{G}^{f}\rightarrow {\mathcal{H}}_{G}^{\text{aff}}$ .

Let $\unicode[STIX]{x1D6EF}\in \text{Perv}_{N}({\mathcal{B}})\subset \mathit{Sh}_{c}(G/B)$ be the tilting extension of the shifted constant sheaf $\text{}\underline{\mathbb{Q}}_{{\mathcal{B}}^{w_{0}}}[\dim {\mathcal{B}}]$ on the open $N$ -orbit ${\mathcal{B}}^{w_{0}}\subset {\mathcal{B}}$ . Equivalently, in the abelian category $\text{Perv}_{N}({\mathcal{B}})$ , it is also the projective cover of the skyscraper sheaf on the closed $N$ -orbit.

Consider the functor

where we first forget $B$ -equivariance via the pullback $q^{\ast }:{\mathcal{H}}_{G}^{f}\rightarrow \mathit{Sh}_{c}({\mathcal{B}})$ along $q:G/B\rightarrow B\backslash G/B$ .

The functor $\mathbb{V}$ calculates the vanishing cycles at a generic covector at the closed $N$ -orbit. It is the universal quotient of ${\mathcal{H}}_{G}^{f}$ with the kernel the full monoidal ideal $\langle \text{IC}_{w}|w\not =1\in W_{f}\rangle$ generated by $\text{IC}$ -sheaves of $N$ -orbits ${\mathcal{B}}^{w}\subset {\mathcal{B}}$ , for $w\not =1\in W_{f}$ , that are not closed. It can be equipped with a monoidal structure (for the usual tensor product on $\text{Vect}$ ).

The aspherical affine Hecke category is defined to be the tensor product

$$\begin{eqnarray}{\mathcal{H}}_{G}^{\text{asph}}:={\mathcal{H}}_{G}^{\text{aff}}\otimes _{{\mathcal{H}}_{G}^{f}}\text{Vect},\end{eqnarray}$$

where the ${\mathcal{H}}_{G}^{f}$ -module structure on $\text{Vect}$ is given by $\mathbb{V}$ . It has a natural ${\mathcal{H}}_{G}^{\text{aff}}$ -module structure via convolution on the left.

When the base field has positive characteristic, Bezrukavnikov [Reference BezrukavnikovBez16] realizes ${\mathcal{H}}_{G}^{\text{asph}}$ as the dg category of Iwahori–Whittaker sheaves on the affine flag variety with the help of an Artin–Schreier sheaf. By [Reference BezrukavnikovBez16, Theorem 2], there is an equivalence of dg categories

(2.3)

Moreover, the ${\mathcal{H}}_{G}^{\text{aff}}$ -action on the right-hand side gets intertwined with the $\text{Coh}^{G^{\vee }}(\mathit{St}_{G^{\vee }})$ -action on the left-hand side by left convolution via the equivalence $\unicode[STIX]{x1D6F7}^{\text{aff}}$ .

The above equivalence also holds when the base field is $\mathbb{C}$ . One way to see this is to work with $D$ -modules (where the exponential $D$ -module plays the role of an Artin–Schreier sheaf) to obtain an equivalence between the $\mathbb{C}$ -linearizations of the two sides of (2.3), and then descend it to $\mathbb{Q}$ . Another way is to use a $\mathbb{G}_{m}$ -averaged version of an Artin–Schreier sheaf, as we do when introducing the Whittaker sheaf in § 2.5.2.

2.3.1 Spherical Hecke action

For $x\in X\setminus S$ , we may define the moduli stack $\text{Bun}_{G}(\mathbb{X}_{x},S)$ of $G$ -bundles on $\mathbb{X}_{x}$ with $B$ -reductions at $S$ . We have the following diagram.

Passing to sheaves, and choosing a local coordinate to identify $\text{Bun}_{G}(\mathbb{D}_{x})$ with $G[[t]]\backslash G((t))/G[[t]]$ , one obtains the spherical Hecke modifications

It evidently preserves the full dg subcategory $\mathit{Sh}_{!}(\text{Bun}_{G}(X,S))\subset \mathit{Sh}(\text{Bun}_{G}(X,S))$ .

Natural generalizations of the above constructions provide $\mathit{Sh}(\text{Bun}_{G}(X,S))$ the requisite coherences of an ${\mathcal{H}}_{G}^{\text{sph}}$ -module.

Restricting to the heart of ${\mathcal{H}}_{G}^{\text{sph}}$ , one obtains a tensor action

Remark 2.3.2. It is straightforward to generalize the above from a point $x\in X\setminus S$ to a family of points parametrized by $Y\rightarrow X\setminus S$ to obtain a functor

2.3.3 Affine Hecke action

For $s\in S$ , let $S_{\pm }=S\coprod _{S\setminus \{s\}}S\subset \mathbb{X}_{s}$ . We may similarly define the moduli stack $\text{Bun}_{G}(\mathbb{X}_{s},S_{\pm })$ of $G$ -bundles on $\mathbb{X}_{s}$ with $B$ -reductions at $S_{\pm }$ , and obtain a diagram

(2.4)

Passing to sheaves, and choosing a local coordinate to identify $\text{Bun}_{G}(\mathbb{D}_{s},\{s_{-},s_{+}\})$ with $I\backslash G((t))/I$ , one obtains the affine Hecke modifications

(2.5)

More often, we will use the binary notation $\star _{s}$ to denote the affine Hecke action

$$\begin{eqnarray}{\mathcal{K}}\star _{s}{\mathcal{F}}:=\text{Hecke}_{s}^{\text{aff}}({\mathcal{K}},{\mathcal{F}}).\end{eqnarray}$$

It evidently preserves the full dg subcategory $\mathit{Sh}_{!}(\text{Bun}_{G}(X,S))\subset \mathit{Sh}(\text{Bun}_{G}(X,S))$ .

Natural generalizations of the above constructions provide $\mathit{Sh}(\text{Bun}_{G}(X,S))$ the requisite coherences of an ${\mathcal{H}}_{G}^{\text{aff}}$ -module structure. For different $s\in S$ , the resulting ${\mathcal{H}}_{G}^{\text{aff}}$ -actions on $\mathit{Sh}(\text{Bun}_{G}(X,S))$ commute with each other.

In particular, restricting the action of ${\mathcal{H}}_{G}^{\text{aff}}$ to $\text{Perf}(\widetilde{{\mathcal{N}}}^{\vee }/G^{\vee })$ via the monoidal functor

(2.6)

where $\unicode[STIX]{x1D6E5}:\widetilde{{\mathcal{N}}}^{\vee }/G^{\vee }\rightarrow \mathit{St}_{G^{\vee }}/G^{\vee }$ is the diagonal map, one obtains commuting tensor actions

(2.7)

2.3.4 Compatibility

Let $s\in S$ , and let $U_{s}\subset X\setminus (S\setminus \{s\})$ be a disk around $s$ (in the classical topology). Let $U_{s}^{\times }=U_{s}\setminus \{s\}$ be the punctured disk. Recall the Hecke operators over $U_{s}^{\times }$ are defined as in Remark 2.3.2. By the construction of the central functor $Z$ in [Reference GaitsgoryGai01], there is a natural equivalence of bifunctors ${\mathcal{H}}_{G}^{\text{sph}}\otimes \mathit{Sh}(\text{Bun}_{G}(X,S))\rightarrow \mathit{Sh}(\text{Bun}_{G}(X,S))$

(2.8)

where $\unicode[STIX]{x1D6F9}_{s}:\mathit{Sh}(\text{Bun}_{G}(X,S)\times U_{s}^{\times })\rightarrow \mathit{Sh}(\text{Bun}_{G}(X,S))$ denotes nearby cycles towards the $s$ -fiber of $\text{Bun}_{G}(X,S)\times U_{s}\rightarrow U_{s}$ . Moreover, the monodromy of the central functor $Z$ coincides with the monodromy of $\unicode[STIX]{x1D6F9}_{s}$ .

2.5.1 Twisted $N$ -bundles

Consider the distinguished $T$ -bundle

$$\begin{eqnarray}\unicode[STIX]{x1D714}(S):=\unicode[STIX]{x1D70C}^{\vee }\otimes \unicode[STIX]{x1D714}_{X}(S)\in \text{Bun}_{T}(X)\simeq \unicode[STIX]{x1D6EC}_{T}\otimes _{\mathbb{Z}}\text{Pic}(X).\end{eqnarray}$$

Define the moduli

$$\begin{eqnarray}\text{Bun}_{N}^{\unicode[STIX]{x1D714}(S)}(X,S)=\text{Bun}_{N}^{\unicode[STIX]{x1D714}(S)}(X)\times _{\text{Bun}_{G}(X)}\text{Bun}_{G}(X,S)\end{eqnarray}$$

classifying triples $({\mathcal{E}}_{B},\unicode[STIX]{x1D70F},\{{{\mathcal{F}}_{s}\}}_{s\in S})$ , where ${\mathcal{E}}_{B}$ is a $B$ -torsor, $\unicode[STIX]{x1D70F}:{\mathcal{E}}_{B}/N\rightarrow \unicode[STIX]{x1D714}(S)$ is an isomorphism of $T$ -torsors, and ${\mathcal{F}}_{s}$ is a $B$ -reduction of the fiber ${\mathcal{E}}_{G}|_{s}$ of the $G$ -bundle induced by ${\mathcal{E}}_{B}$ . In other words, the choice of ${\mathcal{F}}_{s}$ is equivalent to the choice of a point of the twisted flag variety ${\mathcal{B}}_{{\mathcal{E}}_{B}|_{s}}={\mathcal{E}}_{B}|_{s}\stackrel{B}{\times }{\mathcal{B}}$ .

Observe that there is an open substack

$$\begin{eqnarray}\text{Bun}_{N}^{\unicode[STIX]{x1D714}(S),\circ }(X,S)\subset \text{Bun}_{N}^{\unicode[STIX]{x1D714}(S)}(X,S),\end{eqnarray}$$

where the $B$ -reductions ${\mathcal{E}}_{B}|_{s}$ and ${\mathcal{F}}_{s}$ of the fiber ${\mathcal{E}}_{G}|_{s}$ are transverse, for each $s\in S$ . If we let ${\mathcal{B}}^{\circ }\subset {\mathcal{B}}$ be the open $B$ -orbit, then the choice of ${\mathcal{F}}_{s}$ is now equivalent to the choice of a point of the twisted open cell

$$\begin{eqnarray}{\mathcal{B}}_{{\mathcal{E}}_{B}|_{x}}^{\circ }\subset {\mathcal{B}}_{{\mathcal{E}}_{B}|_{s}}.\end{eqnarray}$$

Note since $B\backslash {\mathcal{B}}^{\circ }\rightarrow T\backslash pt$ is an equivalence, the choice of such ${\mathcal{F}}_{s}$ is in turn equivalent to a splitting of ${\mathcal{E}}_{B}|_{s}\rightarrow \unicode[STIX]{x1D714}(S)|_{s}$ .

Thus the abelianization map $N\rightarrow N/[N,N]\simeq \prod _{i=1}^{r}\mathbb{G}_{a}$ , where $r$ is the rank, induces a map

(2.16)

where $\text{Bun}_{\mathbb{G}_{a},S}^{\unicode[STIX]{x1D714}_{X}(S)}(X)$ classifies extensions $\unicode[STIX]{x1D714}_{X}(S)\rightarrow {\mathcal{E}}\rightarrow {\mathcal{O}}_{X}$ with a splitting at each $s\in S$ .

Pushout of extensions along the inclusion $\unicode[STIX]{x1D714}_{X}\rightarrow \unicode[STIX]{x1D714}_{X}(S)$ provides a canonical equivalence

(2.17)

since the inclusion ${\mathcal{E}}_{x}\rightarrow {\mathcal{E}}_{x}^{\prime }$ factors through ${\mathcal{E}}_{x}\rightarrow {\mathcal{O}}_{X,x}$ , and hence its image gives a splitting of ${\mathcal{E}}_{x}^{\prime }\rightarrow {\mathcal{O}}_{X,x}$ .

Composing (2.16) with the inverse of (2.17) and taking the sum of the canonical evaluations

we obtain the total evaluation

Note that the total evaluation is $\mathbb{G}_{m}$ -equivariant for the action on $\text{Bun}_{N}^{\unicode[STIX]{x1D714}(S),\circ }(X,S)$ induced via $\unicode[STIX]{x1D70C}^{\vee }:\mathbb{G}_{m}\rightarrow T$ from the adjoint $T$ -action and the usual rotation action on $\mathbb{G}_{a}$ . Therefore, it descends to a map

We also have the natural induction map

which descends to a map

where again the $\mathbb{G}_{m}$ -action on $\text{Bun}_{N}^{\unicode[STIX]{x1D714}(S),\circ }(X,S)$ is induced via $\unicode[STIX]{x1D70C}^{\vee }:\mathbb{G}_{m}\rightarrow T$ from the adjoint $T$ -action.

2.5.2 $\mathbb{G}_{m}$ -averaged Artin–Schreier sheaf

Let us write $j:pt=\mathbb{G}_{m}/\mathbb{G}_{m}\rightarrow \mathbb{G}_{a}/\mathbb{G}_{m}$ for the open inclusion. Let

$$\begin{eqnarray}\unicode[STIX]{x1D6F9}:=j_{\ast }\text{}\underline{\mathbb{Q}}_{\text{pt}}[-1]\in D_{\mathbb{G}_{m}}(\mathbb{G}_{a}).\end{eqnarray}$$

This sheaf should be thought of as a $\mathbb{G}_{m}$ -equivariant version of an Artin–Schreier sheaf over $\mathbb{G}_{a}$ if we worked over a base field of finite characteristic, or a $\mathbb{G}_{m}$ -equivariant version of the exponential $D$ -module over $\mathbb{G}_{a}$ if we worked in the $D$ -module setting.

Definition 2.5.3. The Whittaker sheaf is the object

where

$$\begin{eqnarray}d_{S}=\dim B\cdot (g-1)+\langle 2\unicode[STIX]{x1D70C},\unicode[STIX]{x1D70C}^{\vee }\rangle (2g-2+\#S)\end{eqnarray}$$

is the dimension of $\text{Bun}_{B}^{-\unicode[STIX]{x1D714}(S)}(X)$ .

Example 2.5.4 ( $X=\mathbb{P}^{1},S=\{0,\infty \}$ ).

In this case, we have $\unicode[STIX]{x1D714}(S)\simeq {\mathcal{O}}_{\mathbb{P}^{1}}$ , and hence the Whittaker sheaf is supported on the open locus, where the underlying $G$ -bundle is semistable or equivalently trivializable

On the other hand, let ${\mathcal{B}}^{\circ }$ be the open $N$ -orbit in ${\mathcal{B}}$ , then we have

$$\begin{eqnarray}\text{Bun}_{N}^{\unicode[STIX]{x1D714}(S),\circ }(\mathbb{P}^{1},\{0,\infty \})\simeq N\backslash ({\mathcal{B}}^{\circ }\times {\mathcal{B}}^{\circ }).\end{eqnarray}$$

If we choose a point $B^{-}\in {\mathcal{B}}^{\circ }$ represented by a Borel opposite to $B$ , then we have $G\backslash ({\mathcal{B}}\times {\mathcal{B}})\simeq B^{-}\backslash {\mathcal{B}}$ by fixing the first coordinate to be $B^{-}$ ; similarly, we have $N\backslash ({\mathcal{B}}^{\circ }\times {\mathcal{B}}^{\circ })\simeq {\mathcal{B}}^{\circ }$ by fixing the first coordinate to be $B^{-}$ . Under the above isomorphisms, the map $p:\text{Bun}_{N}^{\unicode[STIX]{x1D714}(S),\circ }(\mathbb{P}^{1},\{0,\infty \})\rightarrow \text{Bun}_{G}(\mathbb{P}^{1},\{0,\infty \})$ is the evident composition

Let $\unicode[STIX]{x1D6EF}\in \text{Perv}_{N}({\mathcal{B}})$ be the tilting extension to ${\mathcal{B}}$ of the constant perverse sheaf $\text{}\underline{\mathbb{Q}}_{{\mathcal{B}}^{\circ }}[\dim {\mathcal{B}}^{\circ }]$ . We claim that

(2.18) $$\begin{eqnarray}\text{Wh}_{\{0,\infty \}}\simeq u_{!}q_{!}\unicode[STIX]{x1D6EF}[\dim B].\end{eqnarray}$$

Here $u:B^{-}\backslash {\mathcal{B}}\simeq \text{Bun}_{G}^{\text{triv}}(\mathbb{P}^{1},\{0,\infty \}){\hookrightarrow}\text{Bun}_{G}(\mathbb{P}^{1},\{0,\infty \})$ is the open inclusion. To see this, we only need to note that both sides of (2.18), up to appropriate shifts, corepresent the functor of vanishing cycles at a generic covector at the image of $\text{Bun}_{B}^{0}(\mathbb{P}^{1})\rightarrow \text{Bun}_{G}(\mathbb{P}^{1},\{0,\infty \})$ .

The Whittaker sheaf $\text{Wh}_{S}$ enjoys an asphericity property, as we spell out now. For $s\in S$ and a parabolic subgroup $P\subset G$ , we may define a moduli stack $\text{Bun}_{G}(X,S)_{s,P}$ , where the level structure at $s$ is changed to a $P$ -reduction. We have a proper smooth projection

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{s,P}:\text{Bun}_{G}(X,S)\rightarrow \text{Bun}_{G}(X,S)_{s,P},\end{eqnarray}$$

which induces adjoint functors

Lemma 2.5.5. Let $s\in S$ and $P\subset G$ be a parabolic subgroup, which is not a Borel. Then

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{s,P!}\text{Wh}_{S}\simeq 0.\end{eqnarray}$$

Proof. Let $P_{i}$ be the standard parabolic whose Levi only has simple root $\unicode[STIX]{x1D6FC}_{i}$ . Then each $P$ , which is not a Borel, contains some $P_{i}$ , and $\unicode[STIX]{x1D70B}_{s,P}$ factors as

$$\begin{eqnarray}\text{Bun}_{G}(X,S)\xrightarrow[{}]{\unicode[STIX]{x1D70B}_{s,P_{i}}}\text{Bun}_{G}(X,S)_{s,P_{i}}\rightarrow \text{Bun}_{G}(X,S)_{s,P}.\end{eqnarray}$$

Therefore, it suffices to show that $\unicode[STIX]{x1D70B}_{s,P_{i}!}\text{Wh}_{S}\simeq 0$ , for each $P_{i}$ .

We denote $\text{Bun}_{G}(X,S)_{s,P_{i}}$ simply by $\text{Bun}_{G}(X,S)_{s,i}$ , and denote $\unicode[STIX]{x1D70B}_{s,P_{i}}$ similarly by $\unicode[STIX]{x1D70B}_{s,i}$ , which is a $\mathbb{P}^{1}$ -fibration.

Let us extend the maps in the definition of $\text{Wh}_{S}$ to a commutative (but not Cartesian) diagram

where we denote by $\text{Bun}_{N}^{\unicode[STIX]{x1D714}(S),\circ }(X,S)_{s,i}/\mathbb{G}_{m}$ the moduli, where we replace the $B$ -reduction at $s$ with a $P_{i}$ -reduction in general position with the given $N$ -structure, and $\unicode[STIX]{x1D70B}_{s,i}^{\prime }$ is the natural $\mathbb{A}^{1}$ -fibration, where we forget the $B$ -reduction at $s$ to a $P_{i}$ -reduction.

Now returning to the definition of $\text{Wh}_{S}$ , we have

and so it suffices to show

Fix a point $\unicode[STIX]{x1D709}:\text{pt}\rightarrow \text{Bun}_{N}^{\unicode[STIX]{x1D714}(S),\circ }(X,S)_{s,i}$ , and consider the following base-changed Cartesian diagram.

Then it suffices to show

Finally, observe that $\text{ev}\circ \widetilde{\unicode[STIX]{x1D709}}:\mathbb{A}^{1}\rightarrow \mathbb{G}_{a}$ is an isomorphism of schemes, and so

where $j^{\prime }:U=(\text{ev}\circ \widetilde{\unicode[STIX]{x1D709}})^{-1}(\mathbb{G}_{m}){\hookrightarrow}\mathbb{A}^{1}$ is the complement of one point in $\mathbb{A}^{1}$ . Thus we have the required vanishing

Proof. (i) Since any $w\not =1$ can be written as a product of simple reflections $\unicode[STIX]{x1D70E}_{i}$ , it suffices to show $\text{IC}_{\unicode[STIX]{x1D70E}_{i}}\star _{s}\text{Wh}_{S}\simeq 0$ , for the simple reflections $\unicode[STIX]{x1D70E}_{i}\in W_{f}$ . Let $P_{i}$ be the standard parabolic of $G$ whose Levi has only simple root $\unicode[STIX]{x1D6FC}_{i}$ . Then

$$\begin{eqnarray}\text{IC}_{\unicode[STIX]{x1D70E}_{i}}\star _{s}\text{Wh}_{S}\simeq \unicode[STIX]{x1D70B}_{s,P_{i}}^{\ast }\unicode[STIX]{x1D70B}_{s,P_{i},!}\text{Wh}_{S}[1],\end{eqnarray}$$

which vanishes by Lemma 2.5.5. Therefore, (i) is proved.

Since $\mathbb{V}:{\mathcal{H}}_{G}^{f}\rightarrow \text{Vect}$ is monoidal and the universal quotient functor with the kernel the monoidal ideal $\langle \text{IC}_{w}|w\not =1\in W_{f}\rangle$ , (ii) follows from (i).☐

2.5.7 Wakimoto action on Whittaker sheaf

For $s\in S$ , we have an action of $\text{Perf}(\widetilde{{\mathcal{N}}}^{\vee }/G^{\vee })$ on $\mathit{Sh}_{!}(\text{Bun}_{G}(X,S))$ as the restriction of the affine Hecke action at $s$ , see (2.6) and (2.7). By acting on $\text{Wh}_{S}$ , we obtain a functor

(2.19)

such that line bundles go to translations of $\text{Wh}_{S}$ by Wakimoto operators

Proof. By Corollary 2.5.6, the $\star _{s}$ -action of ${\mathcal{H}}_{G}^{\text{aff}}$ on the object $\text{Wh}_{S}$ factors through the aspherical quotient ${\mathcal{H}}_{G}^{\text{asph}}$ , or in other words, we have a functor

and a canonical equivalence ${\mathcal{K}}\star _{s}\text{Wh}_{S}\simeq \unicode[STIX]{x1D6FC}_{s}^{\prime }(\overline{{\mathcal{K}}})$ , where ${\mathcal{K}}\in {\mathcal{H}}_{G}^{\text{aff}}$ , and $\overline{{\mathcal{K}}}\in {\mathcal{H}}_{G}^{\text{asph}}$ is its image. By construction, the functor $\unicode[STIX]{x1D6FC}_{s}^{\prime }$ is a ${\mathcal{H}}_{G}^{\text{aff}}$ -module map.

Now we claim that $\unicode[STIX]{x1D6FC}_{s}$ and $\unicode[STIX]{x1D6FC}_{s}^{\prime }$ are the same functors under the equivalence $\unicode[STIX]{x1D6F7}^{\text{asph}}$ . By the construction in [Reference BezrukavnikovBez16], $\unicode[STIX]{x1D6F7}^{\text{asph}}$ is the composition $\text{Perf}(\widetilde{{\mathcal{N}}}^{\vee }/G^{\vee })\rightarrow {\mathcal{H}}_{G}^{\text{aff}}\rightarrow {\mathcal{H}}_{G}^{\text{asph}}$ given by ${\mathcal{K}}\mapsto \overline{\unicode[STIX]{x1D6E5}_{\ast }{\mathcal{K}}}$ . Thus we have canonical equivalences

Finally, since $\unicode[STIX]{x1D6F7}^{\text{asph}}$ intertwines the ${\mathcal{H}}_{G}^{\text{aff}}$ -action and the $\text{Coh}^{G^{\vee }}(\mathit{St}_{G^{\vee }})$ -action via the monoidal equivalence $\unicode[STIX]{x1D6F7}^{\text{aff}}$ , and $\unicode[STIX]{x1D6FC}_{s}^{\prime }$ is a ${\mathcal{H}}_{G}^{\text{aff}}$ -module map, the lemma follows.☐

3.1.1 $G=\text{PGL}(2)$

By the exact sequence $1\rightarrow \text{GL}(1)\rightarrow \text{GL}(2)\rightarrow \text{PGL}(2)\rightarrow 1$ and the vanishing of the Brauer group of a curve over $\mathbb{C}$ , we have an isomorphism

Thus we can represent $\text{PGL}(2)$ -bundles by rank-two vector bundles up to tensoring with a line bundle. There is a disjoint union decomposition

given by the parity of the degree of a rank-two vector bundle.

The stack $\text{Bun}_{\text{PGL}(2)}(\mathbb{P}^{1},S)$ is the moduli of $\text{PGL}(2)$ -bundles on $\mathbb{P}^{1}$ with $B$ -reductions at the points of $S=\{0,1,\infty \}$ . We can represent objects of $\text{Bun}_{\text{PGL}(2)}(\mathbb{P}^{1},S)$ by $({\mathcal{E}},\{{\ell _{s}\}}_{s\in S})$ , where ${\mathcal{E}}$ is a rank-two vector bundles on $\mathbb{P}^{1}$ up to tensoring with a line bundle, and $\ell _{s}$ is a line in the fiber ${\mathcal{E}}_{s}$ for each $s\in S$ .

Let us list the isomorphism classes of objects of $\text{Bun}_{\text{PGL}(2)}(\mathbb{P}^{1},S)$ . For each isomorphism class of ${\mathcal{E}}\in \text{Bun}_{\text{PGL}(2)}(\mathbb{P}^{1})$ , we describe the poset of points in $\text{Bun}_{\text{PGL}(2)}(\mathbb{P}^{1},S)$ over it, where an arrow $x\rightarrow y$ means $y$ lies in the closure of $x$ .

(i) ${\mathcal{E}}={\mathcal{O}}_{\mathbb{P}^{1}}\oplus {\mathcal{O}}_{\mathbb{P}^{1}}$ , $\text{Aut}({\mathcal{E}})\simeq \text{PGL}(2)$ , with the poset of configurations of lines

where $c_{0}(R)$ denotes the locus where two lines $\ell _{s}$ and $\ell _{s^{\prime }}$ at distinct points $s,s^{\prime }\in S$ are equal if and only if $s,s^{\prime }\in R$ .

(ii) ${\mathcal{E}}={\mathcal{O}}_{\mathbb{P}^{1}}(1)\oplus {\mathcal{O}}_{\mathbb{P}^{1}}$ , $\text{Aut}({\mathcal{E}})\simeq \mathbb{G}_{m}\ltimes \mathbb{G}_{a}^{2}$ , with the poset of configurations of lines

where $c_{1}(R)$ denotes where the lines $\ell _{r}$ lie in the summand ${\mathcal{O}}_{\mathbb{P}^{1}}(1)$ , for $r\in R\subset S$ , and in the summand ${\mathcal{O}}_{\mathbb{P}^{1}}$ , for $r\not \in R\subset S$ . The generic configuration $c_{1}(\ast )$ denotes where none of the lines $\ell _{s}$ lie in ${\mathcal{O}}_{\mathbb{P}^{1}}(1)$ , for $s\in S$ , and also they do not all lie in the image of any map ${\mathcal{O}}_{\mathbb{P}^{1}}\rightarrow {\mathcal{E}}$ (as in the configuration $c_{1}(\varnothing )$ ).

(iii) $k\geqslant 2$ , ${\mathcal{E}}={\mathcal{O}}_{\mathbb{P}^{1}}(k)\oplus {\mathcal{O}}_{\mathbb{P}^{1}}$ , and we have an exact sequence

The poset of configurations of lines is the product

with automorphisms

Let us denote by $c_{k}(R)$ where the lines $\ell _{r}$ lie in the summand ${\mathcal{O}}_{\mathbb{P}^{1}}(k)$ , for $r\in R\subset S$ , and in the summand ${\mathcal{O}}_{\mathbb{P}^{1}}$ , for $r\not \in R\subset S$ .

3.1.2 $G=\text{SL}(2)$

Note that $1\rightarrow \text{SL}(2)\rightarrow \text{GL}(2)\rightarrow \text{GL}(1)\rightarrow 1$ allows us to represent $\text{SL}(2)$ -bundles by rank-two vector bundles with trivialized determinant.

Let $\text{Bun}_{\text{SL}(2)}(\mathbb{P}^{1},S)$ denote the moduli of $\text{SL}(2)$ -bundles on $\mathbb{P}^{1}$ with $B$ -reductions at the points of $S=\{0,1,\infty \}$ . We can represent objects of $\text{Bun}_{\text{SL}(2)}(\mathbb{P}^{1},S)$ by $({\mathcal{E}},\unicode[STIX]{x1D70F},\{{\ell _{s}\}}_{s\in S})$ where ${\mathcal{E}}$ is a rank-two vector bundle on $\mathbb{P}^{1}$ , $\unicode[STIX]{x1D70F}:{\mathcal{O}}_{\mathbb{P}^{1}}\stackrel{{\sim}}{\rightarrow }\det ({\mathcal{E}})$ , and $\ell _{s}$ is a line in the fiber ${\mathcal{E}}_{s}$ for $s\in S$ .

Let us list the isomorphism classes of objects of $\text{Bun}_{\text{SL}(2)}(\mathbb{P}^{1},S)$ according to the isomorphism type of the underlying rank-two bundles.

(i) ${\mathcal{E}}={\mathcal{O}}_{\mathbb{P}^{1}}\oplus {\mathcal{O}}_{\mathbb{P}^{1}}$ $\text{Aut}({\mathcal{E}})\simeq \text{SL}(2)$ , with the poset of configurations of lines

where $c_{0}(R)$ denotes where the lines $\ell _{r}$ coincide, for $r\in R\subset S$ .

(ii) $k\geqslant 1$ , ${\mathcal{E}}={\mathcal{O}}_{\mathbb{P}^{1}}(k)\oplus {\mathcal{O}}_{\mathbb{P}^{1}}(-k)$ , and we have an exact sequence

The poset of of configurations of lines is product

with automorphisms

Let us denote by $c_{2k}(R)$ where the lines $\ell _{r}$ lie in the summand ${\mathcal{O}}_{\mathbb{P}^{1}}(k)$ , for $r\in R\subset S$ , and in the summand ${\mathcal{O}}_{\mathbb{P}^{1}}(-k)$ , for $r\not \in R\subset S$ .

3.2.1 Atkin–Lehner modifications for $G=\text{PGL}(2)$

Atkin–Lehner modifications provide involutions exchanging the two connected components of $\text{Bun}_{\text{PGL}(2)}(\mathbb{P}^{1},S)$ . For $r\in S$ , define the involution

where ${\mathcal{E}}^{\prime }\subset {\mathcal{E}}$ is the lower modification at $r\in \mathbb{P}^{1}$ so that $\ell _{r}\subset {\mathcal{E}}_{r}$ factors through ${\mathcal{E}}_{r}^{\prime }\subset {\mathcal{E}}_{r}$ , the resulting map ${\mathcal{E}}\rightarrow {\mathcal{E}}_{r}$ induces an isomorphism

and $\ell _{r}^{\prime }\subset {\mathcal{E}}_{r}^{\prime }$ is the image of the map ${\mathcal{E}}(-r)_{r}\rightarrow {\mathcal{E}}_{r}^{\prime }$ , and the other lines are unchanged $\ell _{s}^{\prime }=\ell _{s}\subset {\mathcal{E}}_{s}^{\prime }={\mathcal{E}}_{s}$ , for $s\not =r\in S$ . Note the involution $AL_{r}$ exchanges the open points

The Atkin–Lehner modifications generate a group $(\mathbb{Z}/2\mathbb{Z})^{S}$ of order 8. For $R\subset S$ of even size, the Atkin–Lehner modifications $AL_{R}=\prod _{r\in R}AL_{r}$ preserve the two connected components, and generate a subgroup $(\mathbb{Z}/2\mathbb{Z})^{S,ev}$ of order 4.

3.2.2 Central automorphisms for $G=\text{SL}(2)$

The inclusion $\unicode[STIX]{x1D707}_{2}\simeq Z(\text{SL}(2))\subset \text{SL}(2)$ of the center induces an automorphisms of the identity functor of $\text{Bun}_{\text{SL}(2)}(\mathbb{P}^{1},S)$ .

3.3.1 $G=\text{PGL}(2)$

Recall the points of $\text{Bun}_{\text{PGL}(2)}(\mathbb{P}^{1},S)$ are discretely parameterized and their automorphism groups are connected. We have the corresponding generating set of objects of $\mathit{Sh}_{!}(\text{Bun}_{\text{PGL}(2)}(\mathbb{P}^{1},S))$ given by the respective extensions by zero of constant sheaves (in the following $j$ denotes the inclusion of $c_{k}(R)$ into $\text{Bun}_{\text{PGL}(2)}(\mathbb{P}^{1},S)$ ):

Another generating set for $\mathit{Sh}_{!}(\text{Bun}_{\text{PGL}(2)}(\mathbb{P}^{1},S))$ is defined as follows. For $k\geqslant 0$ , let $i_{k}:B_{k}{\hookrightarrow}\text{Bun}_{\text{PGL}(2)}(\mathbb{P}^{1},S)$ be the locally closed substack where the underlying bundle is isomorphic to ${\mathcal{O}}_{\mathbb{P}^{1}}(k)\oplus {\mathcal{O}}_{\mathbb{P}^{1}}$ . Let $j_{k}^{R}:c_{k}(R){\hookrightarrow}B_{k}$ be the inclusion map. We define

(3.1) $$\begin{eqnarray}\text{IC}_{k}(R):=i_{k,!}j_{k,!\ast }^{R}\text{}\underline{\mathbb{Q}}_{c_{k}(R)}[-\!\dim \text{Aut}(c_{k}(R))]\end{eqnarray}$$

to be the IC-sheaf of the closure of $c_{k}(R)$ in $B_{k}$ , extended by zero off of $B_{k}$ .

The decomposition into connected components

provides a direct sum decomposition

The above basis of objects ${\mathcal{F}}_{k}(R)$ belongs to $\mathit{Sh}_{!}(\text{Bun}_{\text{PGL}(2)}^{\overline{k}}(\mathbb{P}^{1},S))$ , where $\overline{k}=k\hspace{0.6em}{\rm mod}\hspace{0.2em}2$ . For $r\in S$ , note the Atkin–Lehner involution $AL_{r}$ exchanges the basis elements

3.3.2 Whittaker sheaf for $G=\text{PGL}(2)$

Let us record the form of the Whittaker sheaf. Consider the open substacks of the odd component

classifying respectively bundles ${\mathcal{E}}\simeq {\mathcal{O}}_{\mathbb{P}^{1}}(1)\,\oplus \,{\mathcal{O}}_{\mathbb{P}^{1}}$ with generic lines $\ell _{0},\ell _{1},\ell _{\infty }$ , and more generally, lines $\ell _{0},\ell _{1},\ell _{\infty }$ with none contained within ${\mathcal{O}}_{\mathbb{P}^{1}}(1)$ . Then the Whittaker sheaf is given by

Note the twist in the definition of $\text{Wh}_{S}$ disappears because $d_{S}=-1$ in this situation.

3.3.3 $G=\text{SL}(2)$

Recall the points of $\text{Bun}_{\text{SL}(2)}(\mathbb{P}^{1},S)$ are discretely parameterized and their automorphism groups are connected except for the configuration $c_{0}(\varnothing )$ with $\text{Aut}\simeq Z(\text{SL}(2))\simeq \unicode[STIX]{x1D707}_{2}$ . Let $\mathbb{Q}_{c_{0}(\varnothing )}^{\text{alt}}$ denote the rank-one local system on $c_{0}(\varnothing )$ , where the automorphism group $\unicode[STIX]{x1D707}_{2}$ acts by the sign character. We have the corresponding basis of objects of $\mathit{Sh}_{!}(\text{Bun}_{\text{PGL}(2)}(\mathbb{P}^{1},S))$ given by the respective extensions by zero of constant sheaves and one additional sheaf $\mathbb{Q}_{c_{0}(\varnothing )}^{\text{alt}}$ :

The canonical automorphisms of the identity functor of $\text{Bun}_{\text{SL}(2)}(\mathbb{P}^{1},S)$ given by $\unicode[STIX]{x1D707}_{2}\simeq Z(\text{SL}(2))$ provides a direct sum decomposition

determined by whether the induced action of $\unicode[STIX]{x1D707}_{2}\simeq Z(\text{SL}(2))$ on sheaves is trivial or alternating.

The second summand admits an equivalence

since all of its objects are finite complexes built out of shifts of ${\mathcal{F}}_{0}(\varnothing )^{\text{alt}}$ whose automorphisms are scalars.

3.3.4 Relation between $G=\text{PGL}(2)$ and $G=\text{SL}(2)$

The natural map $\text{SL}(2)\rightarrow \text{PGL}(2)$ induces a map

which sends $c_{2k}(R)\in \text{Bun}_{\text{SL}(2)}(\mathbb{P}^{1},S)$ to the same-named point $c_{2k}(R)$ in $\text{Bun}_{\text{PGL}(2)}^{\overline{0}}(\mathbb{P}^{1},S)$ , for any $k\geqslant 0$ and $R\subset S$ .

Pullback provides an equivalence

(3.2)

that acts on the above basis by

Thus using the prior decompositions and Atkin–Lehner involutions, we see that to understand any of the above categories, it suffices to understand, for example, $Sh_{!}(\text{Bun}_{\text{PGL}(2)}^{\overline{1}}(\mathbb{P}^{1},S))$ . We prefer the odd component of $\text{Bun}_{\text{PGL}(2)}(\mathbb{P}^{1},S)$ since it supports the Whittaker sheaf.

4.1.1 General definition

We start with a reductive group $G^{\vee }$ over $\mathbb{Q}$ . Let $\text{Loc}_{G^{\vee }}(\mathbb{P}^{1},S)$ be the moduli of (Betti) $G^{\vee }$ -local systems on $\mathbb{P}^{1}\setminus S$ with $B^{\vee }$ -reductions near $S$ with trivial induced $T^{\vee }$ -monodromy. By choosing a point $u_{0}$ in $\mathbb{P}^{1}\setminus S$ and a based loop $\unicode[STIX]{x1D6FE}_{s}$ around $s\in S$ for each $s\in S$ such that $\unicode[STIX]{x1D6FE}_{0}\unicode[STIX]{x1D6FE}_{1}\unicode[STIX]{x1D6FE}_{\infty }=1$ in $\unicode[STIX]{x1D70B}_{1}(\mathbb{P}^{1}\setminus S,u_{0})$ , we obtain the presentation

Here, $(\widetilde{{\mathcal{N}}}^{\vee })^{S,\prod =1}$ is the derived fiber of $1$ of the map

(4.1)

and the map ‘mult’ takes $(A_{0},A_{1},A_{\infty })$ to $A_{0}A_{1}A_{\infty }$ .

4.1.2 $G^{\vee }=\text{SL}(2)$

In this case, $(\widetilde{{\mathcal{N}}}^{\vee })^{S,\prod =1}$ is the derived subscheme of $(\widetilde{{\mathcal{N}}}^{\vee })^{S}$ classifying triples of pairs $(A_{s},\ell _{s})_{s\in S}$ consisting of a matrix $A_{s}\in \text{SL}(2)$ and an eigenline $A_{s}(\ell _{s})\subset \ell _{s}$ with trivial eigenvalue $A_{s}|_{\ell _{s}}=1$ , and the matrices satisfy the equation $A_{0}A_{1}A_{\infty }=1$ inside of $\text{SL}(2)$ .

To write explicit local equations for $(\widetilde{{\mathcal{N}}}^{\vee })^{S,\prod =1}$ , we may apply the $\text{SL}(2)$ -symmetry to assume without loss of generality that $\ell _{\infty }=[1:0]$ , $\ell _{0}=[1:x]$ and $\ell _{1}=[1:y]$ . Then the three matrices take the form

such that $A_{\infty }$ is of the form $(\!\begin{smallmatrix}1 & \ast \\ 0 & 1\end{smallmatrix}\!)$ . Since $\det (A_{\infty })=1$ , we need only impose the equations

These in turn are equivalent to the equations

We conclude that $(\widetilde{{\mathcal{N}}}^{\vee })^{S,\prod =1}$ is a lci classical scheme (i.e., not derived) with five irreducible components.

1. (i) $A_{0}=A_{1}=A_{\infty }=1$ . This component is isomorphic to $\mathbb{P}^{1}\times \mathbb{P}^{1}\times \mathbb{P}^{1}$ . Local equation:

   $$\begin{eqnarray}a=b=0.\end{eqnarray}$$

2. (ii) $A_{0}=1$ (hence $A_{1}=A_{\infty }^{-1}$ ) and $\ell _{1}=\ell _{\infty }$ . This component is isomorphic to $\mathbb{P}^{1}\times \widetilde{{\mathcal{N}}}^{\vee }$ . Local equation:

   $$\begin{eqnarray}a=0,\quad y=0.\end{eqnarray}$$

3. (iii) $A_{1}=1$ (hence $A_{0}=A_{\infty }^{-1}$ ) and $\ell _{0}=\ell _{\infty }$ . This component is isomorphic to $\mathbb{P}^{1}\times \widetilde{{\mathcal{N}}}^{\vee }$ . Local equation:

   $$\begin{eqnarray}b=0,\quad x=0.\end{eqnarray}$$

4. (iv) $A_{\infty }=1$ (hence $A_{0}=A_{1}^{-1}$ ) and $\ell _{0}=\ell _{1}$ . This component is isomorphic to $\mathbb{P}^{1}\times \widetilde{{\mathcal{N}}}^{\vee }$ . Local equation:

   $$\begin{eqnarray}a+b=0,\quad x=y.\end{eqnarray}$$

5. (v) $A_{0},A_{1},A_{\infty }$ all lie in a single Borel. Note this does not mean that $\ell _{0},\ell _{1}$ , $\ell _{\infty }$ are the same; in fact, this component is non-reduced since $A_{0},A_{1},A_{\infty }$ fix $\ell _{0},\ell _{1}$ , $\ell _{\infty }$ respectively. Local equation:

   $$\begin{eqnarray}x^{2}=0,\quad y^{2}=0,\quad xy=0,\quad ax+by=0.\end{eqnarray}$$
   Note for $a,b$ not both zero, there is a unique infinitesimal direction for $(x,y)$ . The reduced structure of this component is isomorphic to the total space of ${\mathcal{O}}(-2)\,\oplus \,{\mathcal{O}}(-2)$ over $\mathbb{P}^{1}$ , and we denote it as $\widetilde{{\mathcal{N}}}_{\unicode[STIX]{x1D6E5}}^{\vee }$ .

If we view $(\widetilde{{\mathcal{N}}}^{\vee })^{S}$ as the cotangent bundle of $(\mathbb{P}^{1})^{S}$ , the five components listed above, after passing to reduced structures, are exactly the conormal bundles of various partial diagonals in $(\mathbb{P}^{1})^{S}$ . For this reason, we introduce the following notation. For a subset $R\subset S$ with $\#R\neq 1$ , we denote by $\unicode[STIX]{x1D6E5}_{R}$ the partial diagonal of $(\mathbb{P}^{1})^{S}$ where the $R$ -components are equal. For example, $\unicode[STIX]{x1D6E5}_{\varnothing }=(\mathbb{P}^{1})^{S}$ . Let $\unicode[STIX]{x1D6EC}_{R}\subset T^{\ast }(\mathbb{P}^{1})^{S}\simeq (\widetilde{{\mathcal{N}}}^{\vee })^{S}$ be the conormal bundle of $\unicode[STIX]{x1D6E5}_{R}$ . Then the reduced structure of the five components of $((\widetilde{{\mathcal{N}}}^{\vee })^{S})^{\prod =1}$ are, in the order listed above, $\unicode[STIX]{x1D6EC}_{\varnothing },\unicode[STIX]{x1D6EC}_{1,\infty },\unicode[STIX]{x1D6EC}_{0,\infty },\unicode[STIX]{x1D6EC}_{0,1}$ and $\unicode[STIX]{x1D6EC}_{S}=\widetilde{{\mathcal{N}}}_{\unicode[STIX]{x1D6E5}}^{\vee }$ .

4.1.3 $G^{\vee }=\text{PGL}(2)$

The stack $\text{Loc}_{\text{PGL}(2)}(\mathbb{P}^{1},S)$ has two connected components. In fact, for $G^{\vee }=\text{PGL}(2)$ , the Springer resolution $\widetilde{{\mathcal{N}}}^{\vee }$ and the unipotent variety ${\mathcal{N}}^{\vee }$ are the same as those of $\text{SL}(2)$ . Therefore, the map ‘mult’ in (4.1) factorizes as

Hence, according to whether the product of three elements in ${\mathcal{N}}^{\vee }$ is $1$ or $-1$ in $\text{SL}(2)$ , we have a decomposition of $\text{Loc}_{\text{PGL}(2)}(\mathbb{P}^{1},S)$

(4.2)

where

The natural map $\text{SL}(2)\rightarrow \text{PGL}(2)$ induces a map

(4.3)

which in turn induces an equivalence

The odd component of $\text{Loc}_{\text{PGL}(2)}(\mathbb{P}^{1},S)$ actually reduces to a single point.

Lemma 4.1.4. The derived scheme $(\widetilde{{\mathcal{N}}}^{\vee })^{S,\widetilde{\prod }=-1}$ is a trivial torsor for $\text{PGL}(2)$ . In particular,

$$\begin{eqnarray}\text{Loc}_{\text{PGL}(2)}^{\overline{1}}(\mathbb{P}^{1},S)\cong \text{Spec }\mathbb{Q}.\end{eqnarray}$$

Proof. Let $(A_{s},\ell _{s})_{s\in S}$ be a point of $(\widetilde{{\mathcal{N}}}^{\vee })^{S,\widetilde{\prod }=-1}$ . We view $A_{s}$ as unipotent elements in $\text{SL}(2)$ , then $A_{0}A_{1}A_{\infty }=-1\in \text{SL}(2)$ . It is easy to see that none of $A_{s}$ can be $1$ , hence each line $\ell _{s}$ is determined by $A_{s}$ . It is also easy to see that no two lines are equal, hence using the $\text{PGL}(2)$ -action we may arrange $\ell _{0}=[1:0],\ell _{1}=[0:1]$ and using the remaining $T^{\vee }$ -conjugacy we may arrange uniquely

Then we have

which is unipotent if and only if $c=-4$ . This shows that $(\widetilde{{\mathcal{N}}}^{\vee })^{S,\widetilde{\prod }=-1}$ is a torsor for $\text{PGL}(2)$ with a rational point.☐

4.2.1 Linearized version

Let $\sum _{\text{SL}(2)}(\mathbb{P}^{1},S)$ denote the linearized version of $\text{Loc}_{\text{SL}(2)}(\mathbb{P}^{1},S)$ defined by the presentation

where we regard $\widetilde{{\mathcal{N}}}^{\vee }$ as the Springer resolution of the nilpotent cone in $\mathfrak{g}^{\vee }=\mathfrak{s}\mathfrak{l}(2)$ , and impose that the sum of the Lie algebra elements be zero. Thus a point of $\sum _{\text{SL}(2)}(\mathbb{P}^{1},S)$ is a triple of pairs $(B_{s},\ell _{s})_{s\in S}$ consisting of a matrix $B_{s}\in \mathfrak{s}\mathfrak{l}(2)$ and an eigenline $B_{s}(\ell _{s})\subset \ell _{s}$ with trivial eigenvalue $B_{s}|_{\ell _{s}}=0$ , and the matrices satisfy the equation $B_{0}+B_{1}+B_{\infty }=0$ inside of $\mathfrak{s}\mathfrak{l}(2)$ .

The local equations for $(\widetilde{{\mathcal{N}}}^{\vee })^{S,\sum =0}$ are exactly the same as those derived above for $(\widetilde{{\mathcal{N}}}^{\vee })^{S,\prod =1}$ except now $B_{0},B_{1}$ , $B_{\infty }$ are nilpotent rather than unipotent matrices

with the requirement that $B_{\infty }$ is of the form $(\!\begin{smallmatrix}0 & & \ast \\ 0 & & 0\end{smallmatrix}\!)$ imposing the equations

Thus we can construct an $\text{SL}(2)$ -equivariant isomorphism

by the assignment

Note that $A_{\infty }-1\neq 2-A_{0}-A_{1}$ as they differ in local coordinates by

$$\begin{eqnarray}\left(\begin{array}{@{}cc@{}}0 & 0\\ ab(x-y) & 0\end{array}\right)\end{eqnarray}$$

though nevertheless $(2-A_{0}-A_{1})\ell _{\infty }=0$ .

We could just as well choose either of the alternative isomorphisms given by the assignments

They give different isomorphisms reflecting the fact that $(\widetilde{{\mathcal{N}}}^{\vee })^{S,\prod =1}$ has automorphisms that infinitesimally move points in its non-reduced component.

4.2.2 de Rham moduli

Let $\text{Conn}_{\text{SL}(2)}(\mathbb{P}^{1},S)$ denote the de Rham version of $\text{Loc}_{\text{SL}(2)}(\mathbb{P}^{1},S)$ classifying data $({\mathcal{E}},\unicode[STIX]{x1D70F},\{{\ell _{s}\}}_{s\in S},\unicode[STIX]{x1D6FB})$ , where ${\mathcal{E}}$ is a rank-two vector bundle on $\mathbb{P}^{1}$ equipped with a line $\ell _{s}\subset {\mathcal{E}}_{s}$ at each $s\in S$ , and a meromorphic connection

with regular singularity at each $s\in S$ , whose residue $\text{Res}_{s}\unicode[STIX]{x1D6FB}$ is trivial when restricted to $\ell _{s}$ , and

is a $\unicode[STIX]{x1D6FB}$ -flat trivialization of the determinant.

Lemma 4.2.3. There is canonical isomorphism from the de Rham moduli to linearized moduli

Proof. First, for any $(({\mathcal{E}},\unicode[STIX]{x1D70F},\{{\ell _{s}\}}_{s\in S},\unicode[STIX]{x1D6FB})\in \text{Conn}_{\text{SL}(2)}(\mathbb{P}^{1},S))$ , we have ${\mathcal{E}}\simeq {\mathcal{O}}_{\mathbb{P}^{1}}^{2}$ . Otherwise, there is an embedding ${\mathcal{O}}_{\mathbb{P}^{1}}(n){\hookrightarrow}{\mathcal{E}}$ with quotient ${\mathcal{O}}_{\mathbb{P}^{1}}(-n)$ , for some $n>0$ . The composition

is ${\mathcal{O}}_{\mathbb{P}^{1}}$ -linear, hence must be zero since $n>1-n$ . Thus $\unicode[STIX]{x1D6FB}$ restricts to a connection on ${\mathcal{O}}_{\mathbb{P}^{1}}(n)$ without poles (because the residues of $\unicode[STIX]{x1D6FB}$ are nilpotent), which is impossible since $n\neq 0$ .

Next, fix an isomorphism $({\mathcal{E}},\unicode[STIX]{x1D70F})\simeq ({\mathcal{O}}_{\mathbb{P}^{1}}^{2},\unicode[STIX]{x1D70F}_{0})$ with the trivial bundle (such choices form an $\text{SL}(2)$ -torsor). The trivial bundle carries the de Rham connection $d$ , and any $(({\mathcal{E}},\unicode[STIX]{x1D70F},\{{\ell _{s}\}}_{s\in S},\unicode[STIX]{x1D6FB})\in \text{Conn}_{\text{SL}(2)}(\mathbb{P}^{1},S))$ is equivalent to one of the form $({\mathcal{O}}_{\mathbb{P}^{1}}^{2},\{{\ell _{s}\}}_{s\in S},\unicode[STIX]{x1D70F}_{0},\unicode[STIX]{x1D6FB}\,=\,d\,+\,\unicode[STIX]{x1D711})\,\in \,\text{Conn}_{\text{SL}(2)}(\mathbb{P}^{1},S)$ , where $\unicode[STIX]{x1D711}:{\mathcal{O}}_{\mathbb{P}^{1}}^{2}\rightarrow {\mathcal{O}}_{\mathbb{P}^{1}}^{2}\,\otimes \,\unicode[STIX]{x1D6FA}_{\mathbb{P}^{1}}(S)$ is a traceless ${\mathcal{O}}$ -linear map whose restriction to $\ell _{s}$ , for each $s\in S$ , is trivial.

Now, define the sought-after isomorphism by the $\text{SL}(2)$ -equivariant assignment

whose image lies in $(\widetilde{{\mathcal{N}}}^{\vee })^{S,\sum =0}$ thanks to the residue theorem for curves

4.3.1 Affine Hecke action

Let $G^{\vee }$ be any reductive group over $\mathbb{Q}$ . Fix $s\in S$ , then the monoidal category $\text{Coh}^{G^{\vee }}(\mathit{St}_{G^{\vee }})$ acts on $\text{Coh}(\text{Loc}_{G^{\vee }}(\mathbb{P}^{1},S))$ as follows. Recall the curve $\mathbb{X}_{s}=\mathbb{P}_{-}^{1}\coprod _{\mathbb{P}^{1}\setminus \{s\}}\mathbb{P}_{+}^{1}$ in § 2.3 with the point $s$ doubled. The moduli stack $\text{Loc}_{G^{\vee }}(\mathbb{X}_{x},S_{\pm })$ can be similarly defined as $\text{Loc}_{G^{\vee }}(\mathbb{P}^{1},S)$ , with $B^{\vee }$ -reductions at both $s_{-}$ and $s_{+}$ . The Steinberg stack $\mathit{St}_{G^{\vee }}/G^{\vee }$ can be identified with the moduli stack $\text{Loc}_{G^{\vee }}(\mathbb{D}_{s},\{s_{-},s_{+}\})$ of $G^{\vee }$ -local systems on the doubled disk $\mathbb{D}_{s}$ with unipotent monodromy and $B^{\vee }$ -reductions at $s_{-}$ and $s_{+}$ . We have the following diagram.

Passing to quasi-coherent sheaves, one obtains the affine Hecke action

which preserves the subcategory $\text{Coh}(\text{Loc}_{G^{\vee }}(\mathbb{P}^{1},S))$ because $p_{+}$ is proper.

Natural generalizations of the above constructions provide $\text{Coh}(\text{Loc}_{G^{\vee }}(\mathbb{P}^{1},S))$ the requisite coherences of a $\text{Coh}^{G^{\vee }}(\mathit{St}_{G^{\vee }})$ -module structure.