2.1.5

We also need the Ran space with a marked point.

Taking union defines a map

(2.1.4)\begin{equation} \cup_x: {\mathop{\operatorname{\rm Ran}}}\times {\mathop{\operatorname{\rm Ran}}}_{x}\to {\mathop{\operatorname{\rm Ran}}}_{x}. \end{equation}

It equips ${\mathop {\operatorname {\rm Ran}}}_{x}$ with a structure of module space over ${\mathop {\operatorname {\rm Ran}}}$.

Let ${\operatorname {Gr}}_{T, {\mathop {\operatorname {\rm Ran}}}_x}$ be ${\mathop {\operatorname {\rm Ran}}}_x\underset {{\mathop {\operatorname {\rm Ran}}}}{\times }{\operatorname {Gr}}_{T, {\mathop {\operatorname {\rm Ran}}}}$. It has a closed sub-prestack $({\operatorname {Gr}}^{\text {neg}}_{T, {\mathop {\operatorname {\rm Ran}}}_x})_{\infty \cdot x}$.

Definition 2.1.7 For any affine scheme $S$, a $S$-point $(\mathcal {I}, \mathcal {P}_T,\alpha )$ of ${\operatorname {Gr}}_{T, {\mathop {\operatorname {\rm Ran}}}_x}$ belongs to the sub-prestack $({\operatorname {Gr}}^{\text {neg}}_{T, {\mathop {\operatorname {\rm Ran}}}_x})_{\infty \cdot x}$, if there exists a $T$-bundle $\mathcal {P}_{T,1}$ on $S \times X$ and an isomorphism

\[ \alpha': \mathcal{P}_{T,1} |_{ S \times (X\setminus x)} \simeq \mathcal{P}_{T} |_{ S \times (X\setminus x)}, \]

such that the resulting point $(\mathcal {I}, \mathcal {P}_{T,1},\alpha \circ \alpha ')$ of ${\operatorname {Gr}}_{T, {\mathop {\operatorname {\rm Ran}}}_x}$ belongs to ${\mathop {\operatorname {\rm Ran}}}_x\mathop {\times }_{{\mathop {\operatorname {\rm Ran}}}}{\operatorname {Gr}}^{\text {neg}}_{T, {\mathop {\operatorname {\rm Ran}}}}$.

2.2.2 Connected components

Connected components of ${\mathop {\operatorname {\rm Conf}}}$ are indexed by $\Lambda ^{\text {neg}}\setminus \{0\}$,

\[ {\mathop{\operatorname{\rm Conf}}}= \mathop{\bigsqcup}_{\Lambda^{\text{neg}}\setminus\{0\}} {\mathop{\operatorname{\rm Conf}}}^\lambda. \]

Here ${\mathop {\operatorname {\rm Conf}}}^\lambda$ denotes the subscheme of ${\mathop {\operatorname {\rm Conf}}}$ where we require the total degree of $D$ in (2.2.1) (i.e. $\sum _k \lambda _k$) to be $\lambda$. If $\lambda =-\sum _i n_i\cdot \alpha _i$, then ${{\mathop {\operatorname {\rm Conf}}}^\lambda }$ is isomorphic to $\prod _i {X^{(n_i)}}$, where ${X^{(n_i)}}$ classifies unordered $n_i$ points in $X$.

Similar to the Ran space, ${\mathop {\operatorname {\rm Conf}}}$ is equipped with a structure of non-unital commutative semi-group. There is a map

(2.2.2)$$\begin{gather} \operatorname{add}_{\mathop{\operatorname{\rm Conf}}}: {\mathop{\operatorname{\rm Conf}}}\times {\mathop{\operatorname{\rm Conf}}}\to {\mathop{\operatorname{\rm Conf}}} \nonumber\\ D_1, D_2\mapsto D_1+D_2. \end{gather}$$

If we restrict this map to the open subscheme $({\mathop {\operatorname {\rm Conf}}}\times {\mathop {\operatorname {\rm Conf}}})_{\rm disj}$ with disjoint support condition, then it is étale.

Configuration space is essentially the same as ${\operatorname {Gr}}^{\text {neg}}_{T, {\mathop {\operatorname {\rm Ran}}}}$. The following lemma is from [Reference Gaitsgory and LysenkoGL19, Lemma 4.6.4].

Lemma 2.2.3 Evaluation on fundamental weights gives rise to a morphism

(2.2.3)\begin{equation} {\operatorname{Gr}}^{\text{neg}}_{T, {\mathop{\operatorname{\rm Ran}}}}\to {\mathop{\operatorname{\rm Conf}}}. \end{equation}

It induces an isomorphism of the sheafifications in the topology generated by finite surjective maps.

In particular, (2.2.3) induces an equivalence between categories of gerbes on ${\operatorname {Gr}}^{\text {neg}}_{T, {\mathop {\operatorname {\rm Ran}}}}$ and ${\mathop {\operatorname {\rm Conf}}}$. Furthermore, it induces an equivalence of corresponding categories of twisted sheaves.

2.2.4

Similarly, we define the configuration space with a marked point.

Definition 2.2.5 Fix $x\in X$. We denote by ${\mathop {\operatorname {\rm Conf}}}_{\infty \cdot x}$ the ind-scheme classifying the colored divisors on $X$ with $\Lambda$-coefficient

(2.2.4)\begin{equation} D= \lambda_x\cdot x+ \mathop{\sum}_k \lambda_k\cdot x_k, \end{equation}

such that $\lambda _k\in \Lambda ^{\text {neg}}$, $\lambda _x\in \Lambda$ and $x_k\neq x$.

Regard ${\mathop {\operatorname {\rm Conf}}}$ as a (non-unital) algebra in the category of prestacks, the addition map

(2.2.5)\begin{equation} \operatorname{add}_{{\mathop{\operatorname{\rm Conf}}}_x}: {\mathop{\operatorname{\rm Conf}}}\times {\mathop{\operatorname{\rm Conf}}}_{\infty\cdot x}\to {\mathop{\operatorname{\rm Conf}}}_{\infty\cdot x} \end{equation}

gives ${\mathop {\operatorname {\rm Conf}}}_{\infty \cdot x}$ a module structure over ${\mathop {\operatorname {\rm Conf}}}$.

If we restrict $\operatorname {add}_{{\mathop {\operatorname {\rm Conf}}}_x}$ to $({\mathop {\operatorname {\rm Conf}}}\times {\mathop {\operatorname {\rm Conf}}}_{\infty \cdot x})_{\rm disj}$, the open ind-scheme with disjoint support condition, then it is étale. In particular, $\operatorname {add}_{{\mathop {\operatorname {\rm Conf}}}_x}^!\simeq \operatorname {add}_{{\mathop {\operatorname {\rm Conf}}}_x}^*$ on $({\mathop {\operatorname {\rm Conf}}}\times {\mathop {\operatorname {\rm Conf}}}_{\infty \cdot x})_{\rm disj}$.

Similar to (2.2.3), there is a map of prestacks

(2.2.6)\begin{equation} ({\operatorname{Gr}}^{\text{neg}}_{T, {\mathop{\operatorname{\rm Ran}}}})_{\infty\cdot x}\to {\mathop{\operatorname{\rm Conf}}}_{\infty\cdot x}. \end{equation}

2.4.1 Gerbe on the Hecke prestack

We denote by $\mathit{\mathcal{H}\kern-.2ex ecke}_G$ the Hecke prestack which classifies the data: $(\mathcal {P}_{G,1}, \mathcal {P}_{G,2}, \alpha )$, where $\mathcal {P}_{G,1}$ and $\mathcal {P}_{G,2}$ are $G$-bundles on $X$ and $\alpha$ is an isomorphism of $\mathcal {P}_{G,1}$ and $\mathcal {P}_{G,2}$ over $X\setminus x$, $\alpha : \mathcal {P}_{G,1}|_{X\setminus x}\simeq \mathcal {P}_{G,2}|_{X\setminus x}$. Then by the $G(\mathcal {O})$-equivariance of $\mathcal {G}^G$ (see [Reference Gaitsgory and LysenkoGL18, § 7.3]), $\mathcal {G}^G$ gives rise to a gerbe on the Hecke prestack. We denote the descent gerbe on $\mathit{\mathcal{H}\kern-.2ex ecke}_G$ by $\mathcal {G}^G_{\mathit{\mathcal{H}\kern-.2ex ecke}}$.

2.4.2 $\omega ^\rho$-twisted prestacks

Fix a square root of the canonical line bundle $\omega$ on $X$ and denote it by $\omega ^{\otimes \frac {1}{2}}$. We define $\omega ^\rho$ as the $T$-bundle induced from $\omega ^{\otimes \frac {1}{2}}$ by the morphism of group schemes

(2.4.1)\begin{equation} 2\rho: \mathbb{G}_m\to T. \end{equation}

Here $\rho$ is the sum of all fundamental coweights.

If we replace the trivial $G$-bundle $\mathcal {P}_G^0$ by the $G$-bundle $\mathcal {P}_G^{\omega }:=\omega ^\rho \overset {T}{\times }G$ in the definition of ${\operatorname {Gr}}_{G, {\mathop {\operatorname {\rm Ran}}}}$, we will obtain the $\omega ^\rho$-twisted Beilinson–Drinfeld affine Grassmannian. Let us denote it by ${\operatorname {Gr}}^{\omega ^\rho }_{G, {\mathop {\operatorname {\rm Ran}}}}$. It is still a factorization prestack over ${\mathop {\operatorname {\rm Ran}}}$. Similarly, we can also define ${\operatorname {Fl}}^{\omega ^\rho }_{G, {\mathop {\operatorname {\rm Ran}}}_x}$, ${\operatorname {Gr}}_{G}^{\omega ^\rho }$, ${\operatorname {Fl}}_{G}^{\omega ^\rho }$, $G(\mathcal {K})^{\omega ^\rho }$, $N(\mathcal {K})^{\omega ^\rho }$, ${\operatorname {Gr}}_{T, {\mathop {\operatorname {\rm Ran}}}}^{\omega ^\rho }$, $({\operatorname {Gr}}_{T, {\mathop {\operatorname {\rm Ran}}}}^{\omega ^\rho })^{\text {neg}}$, $({\operatorname {Gr}}_{T, {\mathop {\operatorname {\rm Ran}}}_x}^{\omega ^\rho })_{\infty \cdot x}^{\text {neg}}$, etc. Similar to the classical affine flags and affine Grassmannian, we have

\[ {\operatorname{Fl}}_{G}^{\omega^\rho}\simeq G(\mathcal{K})^{\omega^\rho}/I^{\omega^\rho}, \]

and

\[ {\operatorname{Gr}}_{G}^{\omega^\rho}\simeq G(\mathcal{K})^{\omega^\rho}/G(\mathcal{O})^{\omega^\rho}. \]

Here $I^{\omega ^\rho }$ and $G(\mathcal {O})^{\omega ^\rho }$ are $\omega ^\rho$-twisted version of $I$ and $G(\mathcal {O})$, respectively.

2.4.3 Gerbes on ${\operatorname {Fl}}^{\omega ^\rho }_G$ and ${\operatorname {Gr}}^{\omega ^\rho }_G$

By definition, taking $\mathcal {P}_{G,2}$ to be $\mathcal {P}_G^{\omega }$ defines a map ${\operatorname {Gr}}_{G, {\mathop {\operatorname {\rm Ran}}}}^{\omega ^\rho }\to \mathit{\mathcal{H}\kern-.2ex ecke}_G$. The pullback of $\mathcal {G}^G_{\mathit{\mathcal{H}\kern-.2ex ecke}}$ along this map is a factorization gerbe on ${\operatorname {Gr}}_{G, {\mathop {\operatorname {\rm Ran}}}}^{\omega ^\rho }$. With some abuse of notation, we denote it by $\mathcal {G}^G$. Its pullback to ${\operatorname {Fl}}_{G,{\mathop {\operatorname {\rm Ran}}}_x}^{\omega ^\rho }$ (respectively, ${\operatorname {Fl}}_{G}^{\omega ^\rho }$, ${\operatorname {Gr}}_{G}^{\omega ^\rho }$, $G(\mathcal {K})^{\omega ^\rho }$, etc.) is also denoted by $\mathcal {G}^G$.

By [Reference Gaitsgory and LysenkoGL18, Proposition 7.2.5], the pullback of $\mathcal {G}^G$ along $G(\mathcal {K})^{\omega ^\rho }\to {\operatorname {Fl}}_{G,{\mathop {\operatorname {\rm Ran}}}_x}^{\omega ^\rho }$ is a multiplicative gerbe, i.e.

\[ m^!(\mathcal{G}^G)\simeq \mathcal{G}^G\boxtimes \mathcal{G}^G. \]

Here $m$ denotes the multiplication map

\[ G(\mathcal{K})^{\omega^\rho}\times G(\mathcal{K})^{\omega^\rho}\to G(\mathcal{K})^{\omega^\rho}. \]

In particular, the gerbes $\mathcal {G}^G$ on ${\operatorname {Fl}}_{G}^{\omega ^\rho }$ and ${\operatorname {Gr}}_{G}^{\omega ^\rho }$ are equivariant with respect to the action of $G(\mathcal {K})^{\omega ^\rho }$ against the gerbe $\mathcal {G}^G$.

2.4.4 Gerbe on ${\mathop {\operatorname {\rm Conf}}}$

Replace $G$ by $B$ in the definition of ${\operatorname {Gr}}_{G, {\mathop {\operatorname {\rm Ran}}}}^{\omega ^\rho }$, one can define a factorization prestack ${\operatorname {Gr}}_{B, {\mathop {\operatorname {\rm Ran}}}}^{\omega ^\rho }$. Consider the following diagram of prestacks.

(2.4.2)

The pullback of $\mathcal {G}^G$ on ${\operatorname {Gr}}_{G,{\mathop {\operatorname {\rm Ran}}}}^{\omega ^\rho }$ along the left morphism gives a factorization gerbe on ${\operatorname {Gr}}_{B, {\mathop {\operatorname {\rm Ran}}}}^{\omega ^\rho }$. By [Reference Gaitsgory and LysenkoGL18, § 5.1], this factorization gerbe descends to a factorization gerbe on ${\operatorname {Gr}}_{T, {\mathop {\operatorname {\rm Ran}}}}^{\omega ^\rho }$. We denote the resulting gerbe by $\mathcal {G}^T$.

By constructions similar to (2.2.3) and (2.2.6), we have maps

(2.4.3)\begin{equation} ({\operatorname{Gr}}_{T, {\mathop{\operatorname{\rm Ran}}}}^{\omega^\rho})^{\text{neg}}\to {\mathop{\operatorname{\rm Conf}}}, \end{equation}

and

(2.4.4)\begin{equation} ({\operatorname{Gr}}_{T, {\mathop{\operatorname{\rm Ran}}}_x}^{\omega^\rho})_{\infty\cdot x}^{\text{neg}}\to {\mathop{\operatorname{\rm Conf}}}_{\infty\cdot x}. \end{equation}

By (a tiny modification of) Lemmas 2.2.3 and 2.2.6, (2.4.3) and (2.4.4) induce equivalences of gerbes on the Beilinson–Drinfeld affine Grassmannians and Configuration spaces. Hence, we can descend the factorization gerbe $\mathcal {G}^T$ on $({\operatorname {Gr}}_{T, {\mathop {\operatorname {\rm Ran}}}}^{\omega ^\rho })^{\text {neg}}$ (respectively, $({\operatorname {Gr}}_{T, {\mathop {\operatorname {\rm Ran}}}_x}^{\omega ^\rho })_{\infty \cdot x}^{\text {neg}}$) to a gerbe $\mathcal {G}^\Lambda$ on ${\mathop {\operatorname {\rm Conf}}}$ (respectively, ${\mathop {\operatorname {\rm Conf}}}_{\infty \cdot x}$). Note that the above maps are compatible with the factorization structures, the gerbes on ${\mathop {\operatorname {\rm Conf}}}$ and ${\mathop {\operatorname {\rm Conf}}}_{\infty \cdot x}$ are factorizable.

Following [Reference Gaitsgory and LysenkoGL18, § 4.2], we can get a quadratic form

(2.4.5)\begin{equation} q: \Lambda\to \mathsf{k}/\mathbb{Z} \end{equation}

from a factorization gerbe $\mathcal {G}^\Lambda$ on ${\mathop {\operatorname {\rm Conf}}}$ in the D-module setting. In the Betti setting and the $\ell$-adic setting, $q$ takes value in $\mathsf {e}^{\mathsf {torsion},\times }(-1):= {\rm colim}_n {\mathop {\operatorname {\rm Hom}}}(\mu _n, \mathsf {e}^{\mathsf {torsion},\times })$.

2.4.5 Gerbe on ${\operatorname {Bun}}_G$

Let ${\operatorname {Bun}}_G$ denote the algebraic stack classifying principal $G$-bundles on $X$. It is shown in [Reference Gaitsgory and LysenkoGL18] that any factorization gerbe on ${\operatorname {Gr}}_{G, {\mathop {\operatorname {\rm Ran}}}}$ descends to a gerbe on ${\operatorname {Bun}}_G$. We also denote the resulting gerbe on ${\operatorname {Bun}}_G$ by $\mathcal {G}^G$.

Remark 2.4.6 The pullback of $\mathcal {G}^G$ on ${\operatorname {Bun}}_G$ along

(2.4.6)\begin{equation} {\operatorname{Gr}}_{G,{\mathop{\operatorname{\rm Ran}}}}^{\omega^\rho}\to {\operatorname{Bun}}_G \end{equation}

is the gerbe $\mathcal {G}^G$ defined in § 2.4.3 tensored with the fiber $\mathcal {G}^G|_{\omega ^\rho \in {\operatorname {Bun}}_G}$.

3.1.2

We note that Definition 3.1.1 involves taking invariants with respect to an ind-pro-group scheme, we need to be more precise about this definition.

Write $N(\mathcal {K})^{\omega ^\rho }$ as

(3.1.3)\begin{equation} N(\mathcal{K})^{\omega^\rho}= \bigcup_{k\geq 0}N_k, \end{equation}

where $N_k:= \text {Ad}_{t^{-k\rho }}N(\mathcal {O})^{\omega ^\rho }$.

First, by [Reference BeraldoBer17, § 4.4.3], we have

(3.1.4)\begin{equation} \operatorname{Whit}_{\rm q}({\operatorname{Fl}}^{\omega^\rho}_G)=\operatorname{Shv}_{\mathcal{G}^G}({\operatorname{Fl}}^{\omega^\rho}_G)^{N(\mathcal{K})^{\omega^\rho},\chi} \simeq \mathop{\lim}_{k, \operatorname{oblv}}\operatorname{Shv}_{\mathcal{G}^G}({\operatorname{Fl}}^{\omega^\rho}_G)^{N_k,\chi}. \end{equation}

Fix a natural number $k\geq 0$. By (3.1.4), we only need to define $\operatorname {Shv}_{\mathcal {G}^G}({\operatorname {Fl}}^{\omega ^\rho }_G)^{N_k,\chi }$.

Since ${\operatorname {Fl}}^{\omega ^\rho }_G$ is an ind-scheme, we can write ${\operatorname {Fl}}^{\omega ^\rho }_G$ as a colimit of finite-dimensional schemes $Y_i$. Furthermore, we can assume that each $Y_i$ is $N_k$-invariants. Then by

\[ \operatorname{Shv}_{\mathcal{G}^G}({\operatorname{Fl}}^{\omega^\rho}_G)\simeq \mathop{\lim}_{i,!-\operatorname{pullback}}\operatorname{Shv}_{\mathcal{G}^G}(Y_i), \]

we have

(3.1.5)\begin{equation} \operatorname{Shv}_{\mathcal{G}^G}({\operatorname{Fl}}^{\omega^\rho}_G)^{N_k,\chi}\simeq \mathop{\lim}_{i, !-\operatorname{pullback}} \operatorname{Shv}_{\mathcal{G}^G}(Y_i)^{N_k,\chi}. \end{equation}

We note that $N_k$ is a pro-scheme of finite type, we can write it as

\[ N_k= \mathop{\lim}_{l}N^l_k, \]

such that each $N^l_k$ is a finite-dimensional unipotent group scheme and the action of $N_k$ on $Y_i$ factors through $N^l_k$. Finally, we define

(3.1.6)\begin{equation} \operatorname{Shv}_{\mathcal{G}^G}(Y_i)^{N_k,\chi}:= \operatorname{Shv}_{\mathcal{G}^G}(Y_i)^{N^l_k,\chi}. \end{equation}

Since for any $l'\geq l$, the kernel of $N_k^{l'}\to N_k^{l}$ is unipotent, the above definition is independent of the choice of $N_k^l$.

3.1.3 Averaging functors

Denote by

(3.1.7)\begin{equation} \operatorname{oblv}_{N(\mathcal{K})^{\omega^\rho},\chi}\colon \operatorname{Whit}_{\rm q}({\operatorname{Fl}}^{\omega^\rho}_G)\to \operatorname{Shv}_{\mathcal{G}^G}({\operatorname{Fl}}^{\omega^\rho}_G) \end{equation}

the fully faithful forgetful functor.

It admits a (partially defined) left adjoint functor $\operatorname {Av}_!^{N(\mathcal {K})^{\omega ^\rho },\chi }$. Since $\operatorname {Av}_!^{N(\mathcal {K})^{\omega ^\rho },\chi }$ is a (partially defined) left adjoint functor, it commutes with filtered colimits. In contrast, the right adjoint functor $\operatorname {Av}_*^{N(\mathcal {K})^{\omega ^\rho },\chi }$ of $\operatorname {oblv}_{N(\mathcal {K})^{\omega ^\rho },\chi }$ is discontinuous.

With the ind-pro-group scheme presentation (3.1.3) of $N(\mathcal {K})^{\omega ^\rho }$, we may write $\operatorname {Av}_!^{N(\mathcal {K})^{\omega ^\rho },\chi }$ more precisely,

(3.1.8)\begin{equation} \operatorname{Av}_!^{N(\mathcal{K})^{\omega^\rho},\chi}= \mathop{{\rm colim}}_{k, !-\operatorname{averaging}} \operatorname{Av}_!^{N_k,\chi}. \end{equation}

If $\operatorname {Av}_!^{N_k,\chi }(\mathcal {F})$ can be defined for any $k$, then $\operatorname {Av}_!^{N(\mathcal {K})^{\omega ^\rho },\chi }(\mathcal {F})$ can be defined by taking colimit. In particular, $\operatorname {Av}_!^{N(\mathcal {K})^{\omega ^\rho },\chi }$ can be defined in the $\ell$-adic setting and for ind-holonomic D-modules in the D-module setting.

3.2.1 Mixed representation category

Let ${\rm Vect}_{\rm q}^\Lambda$ denote ${\operatorname {Rep}}_{\rm q}(\check {T})$, the braided monoidal category of $\mathsf {e}$-representations of the quantum torus $\check {T}$. We denote by $U_{\rm q}^{\rm L}(\check {\mathfrak {n}}){\operatorname {-mod}}^{\mathsf {loc.nil}}$ the ind-completion of the derived category of finite-dimensional $U_{\rm q}^{\rm L}(\check {\mathfrak {n}})$-modules in ${\rm Vect}_{\rm q}^\Lambda$.

3.2.4 Verma modules

We denote by $\operatorname {ind}_{L\to Dr}$ the left adjoint functor of the forgetful functor from ${\operatorname {Rep}}_{\rm q}^{{\mathop {\operatorname {\rm mxd}}}}(\check {G})$ to $U_{\rm q}^{\rm L}(\check {\mathfrak {n}}){\operatorname {-mod}}^{\mathsf {loc.nil}}$, and by $\operatorname {coind}_{\rm DK\to Dr}$ the right adjoint functor of the forgetful functor from ${\operatorname {Rep}}_{\rm q}^{{\mathop {\operatorname {\rm mxd}}}}(\check {G})$ to $U_{\rm q}^{\rm DK}(\check {\mathfrak {n}}^-){\operatorname {-mod}}$. For $\lambda \in \Lambda$, we define Verma modules and co-Verma modules in ${\operatorname {Rep}}_{\rm q}^{{\mathop {\operatorname {\rm mxd}}}}(\check {G})$ as

(3.2.1)\begin{align} V_\lambda^{{\mathop{\operatorname{\rm mxd}}}} &:= \operatorname{ind}_{L\to Dr}(\mathsf{e}^\lambda), \end{align}

(3.2.2)\begin{align} V_\lambda^{{\mathop{\operatorname{\rm mxd}}}, \vee} &:= \operatorname{coind}_{\rm DK\to Dr}(\mathsf{e}^\lambda). \end{align}

Here $\mathsf {e}^\lambda$ denotes the one-dimensional representation of $U_{\rm q}^{\rm L}(\check {\mathfrak {n}})$ (or $U_{\rm q}^{\rm DK}(\check {\mathfrak {n}}^-)$) in ${\rm Vect}_{\rm q}^\Lambda$, where the action of the torus corresponds to $\lambda$ and the action of the unipotent group is trivial.

Following [Reference GaitsgoryGai21a, 5.3.2], we have

\[ \mathcal{H}om_{{\operatorname{Rep}}_{\rm q}^{{\mathop{\operatorname{\rm mxd}}}}(\check{G})}(V_\lambda^{{\mathop{\operatorname{\rm mxd}}}}, V_\mu^{{\mathop{\operatorname{\rm mxd}}}, \vee})= \left\{\begin{aligned}\! \mathsf{e},\quad \text{if } \lambda=\mu,\\ 0,\quad \text{if } \lambda\neq\mu. \end{aligned}\right. \]

Furthermore, by construction, the objects $V_\lambda ^{{\mathop {\operatorname {\rm mxd}}}}$ are compact and compactly generate ${\operatorname {Rep}}_{\rm q}^{{\mathop {\operatorname {\rm mxd}}}}(\check {G})$. Thus, there is a highest weight structure of ${\operatorname {Rep}}_{\rm q}^{{\mathop {\operatorname {\rm mxd}}}}(\check {G})$ with standard objects $V_\lambda ^{{\mathop {\operatorname {\rm mxd}}}}$ and costandard objects $V_\mu ^{{\mathop {\operatorname {\rm mxd}}}, \vee }$.

3.3.1 Avoid small torsion

A quadratic form $q$ avoids small torsion ([Reference GaitsgoryGai21b, 1.1.5], [Reference LusztigLus10, 35.1.2 (a)]) if for any long coroot $\alpha _l$ of a simple factor of $G$, there is

\[ \text{ord}(q(\alpha_l)) \geq d_G + 1, \]

where $d_G = 1, 2, 3$ is the lacing number (i.e. the maximal number of edges in the Dynkin diagram).

Theorem 3.3.2 In the setting of D-modules, when $q$ avoids small torsion, there exists a $t$-exact equivalence of highest weight categories

(3.3.1)\begin{equation} \operatorname{Whit}_{\rm q}({\operatorname{Fl}}^{\omega^\rho}_G)\simeq {\operatorname{Rep}}_{\rm q}^{{\mathop{\operatorname{\rm mxd}}}}(\check{G}). \end{equation}

3.3.3 Highest weight category structure

In § 5.5, we define a collection of standard objects $\Delta _\lambda$ in $\operatorname {Whit}_{\rm q}({\operatorname {Fl}}^{\omega ^\rho }_G)$ indexed by $\Lambda$. They are given by $!$-averaging the Wakimoto sheaves. In addition, we define a collection of costandard objects $\nabla _\lambda \in \operatorname {Whit}_{\rm q}({\operatorname {Fl}}_G^{\omega ^\rho })$. We show that Verma modules $V_\lambda ^{{\mathop {\operatorname {\rm mxd}}}}$ and $\Delta _\lambda$ match under the equivalence (3.3.1) and similarly for costandards.

3.3.4 t-structure

The equivalence (3.3.1) is an equivalence at the derived level. We define a t-structure on $\operatorname {Whit}_{\rm q}({\operatorname {Fl}}^{\omega ^\rho }_G)$ in § 5.6 (see Definition 5.6.2) using $\Delta _\lambda$, and we show that it is compatible with the tautological t-structure on ${\operatorname {Rep}}^{{\mathop {\operatorname {\rm mxd}}}}_{\rm q}(\check {G})$ under the equivalence.

4.1.2

Given a factorization algebra $\Omega$ on ${\mathop {\operatorname {\rm Conf}}}$, we can consider its module category on ${\mathop {\operatorname {\rm Conf}}}_{\infty \cdot x}$.

Definition 4.1.3 We call a twisted sheaf $\mathcal {M}\in \operatorname {Shv}_{\mathcal {G}^\Lambda }({\mathop {\operatorname {\rm Conf}}}_{\infty \cdot x})$ factorization module over $\Omega$ if it is compatible with the factorization structure of $\Omega$, i.e. there is an isomorphism

(4.1.2)\begin{equation} \operatorname{add}_{{\mathop{\operatorname{\rm Conf}}}_x}^!(\mathcal{M})|_{({\mathop{\operatorname{\rm Conf}}}\times {\mathop{\operatorname{\rm Conf}}}_{\infty\cdot x})_{\rm disj}}\simeq \Omega\boxtimes\mathcal{M}|_{({\mathop{\operatorname{\rm Conf}}}\times {\mathop{\operatorname{\rm Conf}}}_{\infty\cdot x})_{\rm disj}}, \end{equation}

with higher homotopy coherence.

We denote by $\Omega \text {-}{\operatorname {FactMod}}$ the category of factorization modules over $\Omega$.

4.1.4 Structure of $\Omega {\text {-}}{\operatorname {FactMod}}$

Given $\lambda \in \Lambda$, let ${\mathop {\operatorname {\rm Conf}}}_{=\lambda \cdot x}$ be the locally closed subscheme of ${\mathop {\operatorname {\rm Conf}}}_{\infty \cdot x}$ consisting of the points $D= \lambda \cdot x+\sum \lambda _i\cdot x_i$ such that $\lambda _i\in \Lambda ^{\text {neg}}\ \text {and}\ x_i\neq x\ \forall i$.

The restriction of (2.2.5) to ${\mathop {\operatorname {\rm Conf}}}\times {\mathop {\operatorname {\rm Conf}}}_{=\lambda \cdot x}$ induces a map

\[ \operatorname{add}_{{\mathop{\operatorname{\rm Conf}}}_x}:({\mathop{\operatorname{\rm Conf}}}\times {\mathop{\operatorname{\rm Conf}}}_{=\lambda\cdot x})_{\rm disj}\longrightarrow {\mathop{\operatorname{\rm Conf}}}_{=\lambda\cdot x}. \]

We call ${\check {\mathcal {M}}}{}\in \operatorname {Shv}_{{\mathcal {G}}^\Lambda }({\mathop {\operatorname {\rm Conf}}}_{=\lambda \cdot x})$ a factorization module over $\Omega$, if there is

\[ \operatorname {add}_{{\mathop {\operatorname {\rm Conf}}}_x}^!(\mathcal {M}) |_{({\mathop {\operatorname {\rm Conf}}}\times {\mathop {\operatorname {\rm Conf}}}_{=\lambda \cdot x})_{\rm disj}}\simeq \Omega \boxtimes \mathcal {M}|_{({\mathop {\operatorname {\rm Conf}}}\times {\mathop {\operatorname {\rm Conf}}}_{=\lambda \cdot x})_{\rm disj}}. \]

We denote by $\Omega {\text {-}}{\operatorname {FactMod}}_{=\lambda }$ the category of factorization modules on ${\mathop {\operatorname {\rm Conf}}}_{=\lambda \cdot x}$ over $\Omega$.

Let $\mathcal {G}^\Lambda |_{\lambda \cdot x}$ be the fiber of $\mathcal {G}^\Lambda$ at $\lambda \cdot x$. The following lemma is from [Reference Gaitsgory and LysenkoGL19, Lemma 5.3.5].

Lemma 4.1.5 Taking $!$-stalks at $\lambda \cdot x$ defines a $t$-exact equivalence

(4.1.3)\begin{equation} i_{\lambda}^!: \Omega{\text{-}}{\operatorname{FactMod}}_{=\lambda}\simeq {\rm Vect}_{\mathcal{G}^\Lambda|_{\lambda\cdot x}}. \end{equation}

Here ${\rm Vect}_{\mathcal {G}^\Lambda |_{\lambda \cdot x}}$ denotes the category of $\mathcal {G}|_{\lambda \cdot x}$-twisted vector spaces.

Definition 4.1.6 Assume that $\Omega$ is in the heart of $\operatorname {Shv}_{\mathcal {G}^\Lambda }({\mathop {\operatorname {\rm Conf}}})$. Consider the perverseFootnote ⁴ generator of $\Omega {\text {-}}{\operatorname {FactMod}}_{=\lambda }$, and we denote its $*$ (respectively, $!$)-pushforward along the locally closed embedding

\[ {\mathop{\operatorname{\rm Conf}}}_{=\lambda\cdot x}\to {\mathop{\operatorname{\rm Conf}}}_{\infty\cdot x} \]

by $\nabla _{\lambda, \Omega }$ (respectively, $\Delta _{\lambda, \Omega }$).

Here $\nabla _{\lambda, \Omega }$ is called the costandard object of $\Omega {\text {-}}{\operatorname {FactMod}}$ and $\Delta _{\lambda, \Omega }$ is called the standard object.

By definition, standard objects $\Delta _{\lambda, \Omega }$ are compact and generate $\Omega {\text {-}}{\operatorname {FactMod}}$. Standard objects $\Delta _{\lambda, \Omega }$ and costandard objects $\nabla _{\lambda, \Omega }$ are perverse.

4.1.7

The Verdier duality functor is well-defined for both $\Delta _{\lambda, \Omega }$ and $\nabla _{\lambda, \Omega }$. Furthermore, we have

(4.1.4)\begin{equation} \mathbb{D}^{{\mathop{\operatorname{\rm Verdier}}}}(\Delta_{\lambda, \Omega})\simeq \nabla_{\lambda, \mathbb{D}^{{\mathop{\operatorname{\rm Verdier}}}}(\Omega)}\quad \forall \lambda\in \Lambda. \end{equation}

Here $\Delta _{\lambda, \Omega }\in \operatorname {Shv}_{\mathcal {G}^\Lambda }({\mathop {\operatorname {\rm Conf}}}_{\infty \cdot x})$ and $\nabla _{\lambda, \mathbb {D}^{{\mathop {\operatorname {\rm Verdier}}}}(\Omega )}\in \operatorname {Shv}_{(\mathcal {G}^\Lambda )^{-1}}({\mathop {\operatorname {\rm Conf}}}_{\infty \cdot x})$.

By definition, $\Delta _{\lambda, \Omega }$ and $\nabla _{\mu, \Omega }$ satisfy the following orthogonality property.

Lemma 4.1.8 For $\lambda,\mu \in \Lambda$, we have

\[ \mathcal{H}om_{\Omega{\text{-}}{\operatorname{FactMod}}}(\Delta_{\lambda,\Omega}, \nabla_{\mu, \Omega})= \left\{\begin{aligned} \mathsf{e},\quad \text{if}\ \lambda=\mu,\\ 0, \quad \text{if}\ \lambda\neq\mu. \end{aligned}\right. \]

4.1.9

The tautological t-structure on $\operatorname {Shv}_{\mathcal {G}^\Lambda }({\mathop {\operatorname {\rm Conf}}}_{\infty \cdot x})$ gives a t-structure on $\Omega {\text {-}}{\operatorname {FactMod}}$. In fact, by [Reference Gaitsgory and LysenkoGL19, Proposition 5.4.2], we can describe this t-structure on $\Omega {\text {-}}{\operatorname {FactMod}}$ by $\Delta _{\lambda,\Omega }$. To be more precise, an $\Omega$-factorization module $\mathcal {M}$ is coconnective as a twisted sheaf if and only if for any $\lambda \in \Lambda$,

(4.1.5)\begin{equation} {\mathop{\operatorname{\rm Hom}}}_{\Omega{\text{-}}{\operatorname{FactMod}}}(\Delta_{\lambda,\Omega}[k], \mathcal{M})=0,\quad \text{if}\ k> 0. \end{equation}

4.2.1

By factorization (i.e. (4.1.1)), we only need to indicate how to extend $\Omega ^{\rm L}_{\rm q}$ from its restriction on ${\mathop {\operatorname {\rm Conf}}}^\lambda \setminus X$ to ${\mathop {\operatorname {\rm Conf}}}^\lambda$, for any $\lambda \in \Lambda ^{\text {neg}}\setminus \{0\}$. Here $X$ embeds into ${\mathop {\operatorname {\rm Conf}}}^\lambda$ by assigning $x$ to $\lambda \cdot x$.

We denote by $\jmath _\lambda$ the open embedding from ${\mathop {\operatorname {\rm Conf}}}^{\lambda }\setminus X$ to ${\mathop {\operatorname {\rm Conf}}}^\lambda$ and by $l$ the length function of the Weyl group.

The following lemma is indicated in [Reference Gaitsgory and LysenkoGL19, § 29], see [Reference Chen and FuCF21, Theorem 1.2.1] for a precise statement and proof.

Lemma 4.2.3 In the D-module setting and the Betti setting, there is

(4.2.2)\begin{equation} {\operatorname{Rep}}_{\rm q}^{{\mathop{\operatorname{\rm mxd}}}}(\check{G}):= Z_{{\rm Dr}, {\rm Vect}_{\rm q}^\Lambda}(U_{\rm q}^{\rm L}(\check{\mathfrak{n}}) {\operatorname{-mod}}^{\mathsf{loc.nil}})\simeq \Omega_{q}^{\rm L}\text{-}{\operatorname{FactMod}}. \end{equation}

Furthermore, under the above equivalence, $V_\lambda ^{{\mathop {\operatorname {\rm mxd}}}}$ (respectively, $V_\lambda ^{{\mathop {\operatorname {\rm mxd}}},\vee }$) corresponds to $\Delta _{\lambda, \Omega _{\rm q}^{\rm L}}$ (respectively, $\nabla _{\lambda, \Omega _{\rm q}^{\rm L}}$).

Proof. By Lefschetz principle and Riemann–Hilbert correspondence, we only need to prove the claim in the setting of constructible sheaves with coefficients in $\mathbb {C}$ in classical topology.

Following [Reference Gaitsgory and LysenkoGL19, § 29.5.1], one can associate a factorization algebra with a Hopf algebra $A$. Furthermore, the relative Drinfeld center of $A$-mod is equivalent to the category of factorization modules over the corresponding factorization algebra. Applying it to our case, there is an equivalence of categories

(4.2.3)\begin{equation} {\operatorname{Rep}}_{\rm q}^{\mathop{\operatorname{\rm mxd}}}(\check{G})\simeq \Omega_{q, \text{quant}}^{\rm L}\text{-}{\operatorname{FactMod}}. \end{equation}

Here $\Omega _{q, \text {quant}}^{\rm L}$ denotes the $\mathcal {G}^\Lambda$-twisted factorization algebra associated with $U_{\rm q}^{\rm L}(\check {\mathfrak {n}})$ for a certain factorization algebra.

However, by the Verdier dual of [Reference GaitsgoryGai21b, § 2.3.8, Theorem 3.6.2], there is $\Omega _{q, \text {quant}}^{\rm L}\simeq \Omega ^{\rm L}_{\rm q}$.

5.2.1

Let $I^0$ be the unipotent radical of the $\omega ^\rho$-twisted Iwahori subgroup $I^{\omega ^\rho }$, and let $T^{\omega ^\rho }$ be $I^{\omega ^\rho }/I^0$. When there is no twisting, ${{J}}_\lambda$ is Iwahori-equivariant. However, there are ‘much fewer’ Iwahori-equivariant objects in the twisted case.

Indeed, since $I^0$ is pro-unipotent, there is a unique (up to a non-canonical isomorphism) $I^0$-equivariant trivialization of the gerbe $\mathcal {G}^G$ on any Iwahori orbit

\[ {\operatorname{Fl}}_G^{\widetilde{w}}:= I^{\omega^\rho}\cdot \widetilde{w} \cdot I^{\omega^\rho}/I^{\omega^\rho} \]

of ${\operatorname {Fl}}^{\omega ^\rho }_G$. In the twisted case, the $I^0$-equivariant trivialization of $\mathcal {G}^G$ on ${\operatorname {Fl}}_G^{\widetilde {w}}$ is not necessarily Iwahori-equivariant. Instead, it is equivariant with respect to a certain character of $T^{\omega ^\rho }$. For example, when the quadratic form $q$ is generic, there is no $I^{\omega ^\rho }$-equivariant sheaf on ${\operatorname {Fl}}_G^{t^\lambda w}$ unless $\lambda =0$.

The solution to fix this problem is to consider $I^0$-equivariant sheaves on $\widetilde {{\operatorname {Fl}}}:=G(\mathcal {K})^{\omega ^\rho }/I^0$, rather than Iwahori-equivariant sheaves on ${\operatorname {Fl}}_G$.

5.2.2

The exact sequence

\[ 1\to I^0\to I^{\omega^\rho}\to T^{\omega^\rho}\to 1 \]

is split. Hence, we may consider the right action of $T^{\omega ^\rho }$ on $\widetilde {{\operatorname {Fl}}}:=G(\mathcal {K})^{\omega ^\rho }/I^0$. In order to define BMW sheaves in the twisted case, we need to consider sheaves on $\widetilde {{\operatorname {Fl}}}$ which are right $T^{\omega ^\rho }$-equivariant against a character. This idea appears in [Reference BezrukavnikovBez16] and [Reference Lusztig and YunLY20].

Let $b(-,-)$ be the symmetric bilinear form associated with the quadratic form $q$, i.e. in the D-module setting,

(5.2.1)\begin{equation} b(\lambda_1,\lambda_2):= q(\lambda_1+\lambda_2)- q(\lambda_1)-q(\lambda_2); \end{equation}

in the Betti setting and the $\ell$-adic setting,

(5.2.2)\begin{equation} b(\lambda_1,\lambda_2):= q(\lambda_1+\lambda_2)\cdot q(\lambda_1)^{-1}\cdot q(\lambda_2)^{-1}. \end{equation}

In the D-module setting (respectively, Betti setting and $\ell$-adic setting), we denote by

(5.2.3)\begin{equation} b_\lambda:=b(\lambda,-)\colon\ \Lambda\to \mathsf{k}/\mathbb{Z}\ (\text{respectively,}\ \mathsf{e}^{\mathsf{torsion},\times}(-1)), \end{equation}

the associated character of $T^{\omega ^\rho }$. With some abuse of notation, we denote by $b_\lambda$ the associated Kummer sheaf on $T^{\omega ^\rho }$. Its pullback to $I^{\omega ^\rho }$ is also denoted by $b_\lambda$.

Definition 5.2.3 We define

\[ \operatorname{Shv}_{\mathcal{G}^G}(\widetilde{{\operatorname{Fl}}})^{I,\lambda}_\mu:= \operatorname{Shv}_{\mathcal{G}^G}(I^{\omega^\rho},b_\lambda\backslash \widetilde{{\operatorname{Fl}}}/ T^{\omega^\rho},b_\mu) \]

as the category of left $(I^{\omega ^\rho },b_\lambda )$-equivariant and right $(T^{\omega ^\rho },b_\mu )$-equivariant twisted sheaves on $\widetilde {{\operatorname {Fl}}}$.

Similarly, we define

\begin{align*} \operatorname{Shv}_{\mathcal{G}^G}(\widetilde{{\operatorname{Fl}}})^{I^0}_\mu:=\operatorname{Shv}_{\mathcal{G}^G} (I^0\backslash \widetilde{{\operatorname{Fl}}}/ T^{\omega^\rho},b_\mu) \end{align*}

as the category of left $I^0$-equivariant and right $(T^{\omega ^\rho },b_\mu )$-equivariant twisted sheaves on $\widetilde {{\operatorname {Fl}}}$.

When $\mu =0$, we omit $\mu$ in the notation. In this case, we can realize right $T^{\omega ^\rho }$-equivariant objects on $\widetilde {{\operatorname {Fl}}}$ as objects on ${\operatorname {Fl}}_G^{\omega ^\rho }$.

Let $\widetilde {{\operatorname {Fl}}}^{\widetilde {w}}$ be the preimage of ${{\operatorname {Fl}}}^{\widetilde {w}}_G$ in $\widetilde {{\operatorname {Fl}}}$. We have equivalences

\begin{align*} \operatorname{Shv}_{\mathcal{G}^G}(\widetilde{{\operatorname{Fl}}}^{\widetilde{w}})_{\mu}^{I^0}&\simeq \operatorname{Shv}_{\mathcal{G}^G}((\widetilde{w}I^{\omega^\rho}/I^0)/T^{\omega^\rho}, b_\mu)\simeq \operatorname{Shv}_{\mathcal{G}^G}(\widetilde{w}I^{\omega^\rho}/I^{\omega^\rho}, b_\mu)\\&\simeq \operatorname{Shv}_{\mathcal{G}^G}(\widetilde{w}T({\mathcal{O}})^{\omega^\rho}/T({\mathcal{O}})^{\omega^\rho}, b_\mu). \end{align*}

Given an identification of $\operatorname {Shv}_{\mathcal {G}^G}(\widetilde {w}T({\mathcal {O}})^{\omega ^\rho }/T({\mathcal {O}})^{\omega ^\rho }, b_\mu )$ with ${\rm Vect}$ which preserves the cohomological degree of $!$-fiber, we denote by $(c_{\widetilde {w}})_{\mu }$ the twisted sheaf on $\widetilde {{\operatorname {Fl}}}^{\widetilde {w}}$ corresponding to $\mathsf {e}\in {\rm Vect}$. Different identifications will change the resulting twisted sheaf $(c_{\widetilde {w}})_{\mu }$ by tensoring by a line in cohomological degree $0$, so we can regard $(c_{\widetilde {w}})_{\mu }$ as an object defined up to tensoring by a line.

Definition 5.2.5 Let $({{J}}_{\widetilde {w},!})_{\mu }$ (respectively, $({{J}}_{\widetilde {w},*})_{\mu })$ be the ! (respectively, *)-extension of $(c_{\widetilde {w}})_{\mu }[{-}l(\widetilde {w})]$ along the locally closed embedding

\[ \widetilde{{\operatorname{Fl}}}^{\widetilde{w}}\to \widetilde{{\operatorname{Fl}}}. \]

5.7.2

The functor $\operatorname {Av}_*^{\rm ren}$ (see (5.5.3)) maps all morphisms of the form (5.7.1) to isomorphisms. In particular, it induces a functor $\operatorname {Av}_*^{\rm ren}$ from $\operatorname {Whit}_{\rm q}({\operatorname {Fl}}_G^{\omega ^\rho })_{co}$ to $\operatorname {Whit}_{\rm q}({\operatorname {Fl}}_G^{\omega ^\rho })$. The following lemma is proved in [Reference RaskinRas21, Theorem 2.1.1].

Lemma 5.7.3 The functor

(5.7.2)\begin{equation} \operatorname{Av}_*^{\rm ren}: \operatorname{Whit}_{\rm q}({\operatorname{Fl}}_G^{\omega^\rho})_{co}\to \operatorname{Whit}_{\rm q}({\operatorname{Fl}}_G^{\omega^\rho}) \end{equation}

is an equivalence of categories.

5.7.4

For a DG-category $\mathcal {C}$, we denote by $\mathcal {C}^\vee$ the dual category of $\mathcal {C}$ if it is dualizable. By definition, it is given by the DG-category of functors ${\mathop {\operatorname {\rm Funct}}}(\mathcal {C}, {\rm Vect})$. Since $\operatorname {Whit}_{\rm q}({\operatorname {Fl}}_G^{\omega ^\rho })$ is compactly generated and any compactly generated category is dualizable, $\operatorname {Whit}_{\rm q}({\operatorname {Fl}}_G^{\omega ^\rho })$ is dualizable. By Lemma 5.7.3, $\operatorname {Whit}_{\rm q}({\operatorname {Fl}}_G^{\omega ^\rho })_{co}$ is also dualizable.

By definition, $\operatorname {Whit}_{\rm q}({\operatorname {Fl}}_G^{\omega ^\rho })_{co}^\vee$ is the full subcategory of $\operatorname {Shv}_{\mathcal {G}^G}({\operatorname {Fl}}_G^{\omega ^\rho })^\vee$ spanned by the functors

(5.7.3)$$\begin{gather} \operatorname{Shv}_{\mathcal{G}^G}({\operatorname{Fl}}_G^{\omega^\rho})\to {\rm Vect} \nonumber\\ \mathcal{F}\mapsto \langle \mathcal{F}, \mathcal{F}_0\rangle, \end{gather}$$

such that for any $\mathcal {F}\in \operatorname {Shv}_{\mathcal {G}^G}({\operatorname {Fl}}_G^{\omega ^\rho })$ and $k$,

\[ \langle \operatorname{Av}_*^{N_k,\chi}(\mathcal{F}), \mathcal{F}^0 \rangle \overset{\sim}{\longrightarrow} \langle\mathcal{F}, \mathcal{F}^0 \rangle, \]

that is,

\[ \langle \mathcal{F}, \operatorname{Av}_*^{N_k,\chi}(\mathcal{F}^0) \rangle \overset{\sim}{\longrightarrow} \langle\mathcal{F}, \mathcal{F}^0 \rangle. \]

In other words, we require $\operatorname {Av}_*^{N_k,\chi }(\mathcal {F}^0)\simeq \mathcal {F}^0$ for any $k$. Hence, the duality functor

(5.7.4)\begin{equation} \begin{aligned} \operatorname{Shv}_{(\mathcal{G}^G)^{-1}}({\operatorname{Fl}}_G^{\omega^\rho}) & \simeq \operatorname{Shv}_{\mathcal{G}^G}({\operatorname{Fl}}_G^{\omega^\rho})^\vee,\\ \mathcal{F}_0 & \mapsto (\mathcal{F}\mapsto \langle \mathcal{F}, \mathcal{F}_0\rangle). \end{aligned} \end{equation}

induces an equivalence of full subcategories

(5.7.5)\begin{align} \operatorname{Whit}_{\rm q}({\operatorname{Fl}}_G^{\omega^\rho})_{co}^\vee\simeq \operatorname{Whit}_{q^{-1}}({\operatorname{Fl}}_G^{\omega^\rho}). \end{align}

Definition 5.7.5 The Verdier duality functor for Whittaker sheaves is defined as the composition of functors

(5.7.6)\begin{equation} \mathbb{D}^{{\mathop{\operatorname{\rm Verdier}}}}: \operatorname{Whit}_{\rm q}({\operatorname{Fl}}_G^{\omega^\rho})^\vee \underset{(\operatorname{Av}_*^{\rm ren})^\vee}{\overset{\sim}{\longrightarrow}} \operatorname{Whit}_{\rm q}({\operatorname{Fl}}_G^{\omega^\rho})^\vee_{co} \underset{\text{(5.7.5)}}{\overset{\sim}{\longrightarrow}} \operatorname{Whit}_{q^{-1}}({\operatorname{Fl}}_G^{\omega^\rho}). \end{equation}

In particular, it defines an equivalence of subcategories generated by compact objects.

(5.7.7)\begin{equation} \mathbb{D}^{{\mathop{\operatorname{\rm Verdier}}}}: (\operatorname{Whit}_{\rm q}({\operatorname{Fl}}_G^{\omega^\rho})^c)^{op} {\overset{\sim}{\longrightarrow}} \operatorname{Whit}_{q^{-1}}({\operatorname{Fl}}_G^{\omega^\rho})^c. \end{equation}

The dual of the standard object $\Delta _\lambda$ can be described by the dual BMW sheaf ${{J}}_\lambda ^{\mathbb {D}}$.

Proposition 5.7.6 For any $\lambda \in \Lambda$, $\Delta _\lambda \in \operatorname {Whit}_{\rm q}({\operatorname {Fl}}_G^{\omega ^\rho })$, we have

(5.7.8)\begin{equation} \mathbb{D}^{{\mathop{\operatorname{\rm Verdier}}}}(\Delta_\lambda)\simeq \operatorname{Av}_*^{\rm ren}({{J}}_\lambda^{\mathbb{D}}). \end{equation}

Proof. The object $\Delta _\lambda$ corresponds to the functor

(5.7.9)\begin{equation} \begin{aligned} \operatorname{Whit}_{\rm q}({\operatorname{Fl}}_G^{\omega^\rho}) & \to {\rm Vect}\\ \mathcal{F} & \mapsto \mathcal{H}om_{\operatorname{Whit}_{\rm q}({\operatorname{Fl}}_G^{\omega^\rho})}(\Delta_\lambda, \mathcal{F}) \end{aligned} \end{equation}

in $\operatorname {Whit}_{\rm q}({\operatorname {Fl}}_G^{\omega ^\rho })^\vee$. By definition, its image under $(\operatorname {Av}_*^{\rm ren})^\vee$ is the functor

(5.7.10)\begin{equation} \begin{aligned} \operatorname{Whit}_{\rm q}({\operatorname{Fl}}_G^{\omega^\rho})_{co} & \to {\rm Vect}\\ \mathcal{F}' & \mapsto \mathcal{H}om_{\operatorname{Whit}_{\rm q}({\operatorname{Fl}}_G^{\omega^\rho})}(\Delta_\lambda, \operatorname{Av}_*^{\rm ren}(\mathcal{F}')). \end{aligned} \end{equation}

Using the adjointness, there is

\[ \mathcal{H}om_{\operatorname{Whit}_{\rm q}({\operatorname{Fl}}_G^{\omega^\rho})}(\Delta_\lambda, \operatorname{Av}_*^{\rm ren}(\mathcal{F}'))\simeq \mathcal{H}om_{\operatorname{Shv}_{\mathcal{G}^G}({\operatorname{Fl}}_G^{\omega^\rho})}({{J}}_\lambda, \operatorname{Av}_*^{\rm ren}(\mathcal{F}')). \]

Note that ${{J}}_\lambda$ goes to ${{J}}_\lambda ^{\mathbb {D}}$ under the equivalence $(\operatorname {Shv}_{\mathcal {G}^G}({\operatorname {Fl}}_G^{\omega ^\rho }))^\vee \simeq \operatorname {Shv}_{(\mathcal {G}^G)^{-1}}({\operatorname {Fl}}_G^{\omega ^\rho })$, we have $\mathcal {H}om_{\operatorname {Shv}_{\mathcal {G}^G}({\operatorname {Fl}}_G^{\omega ^\rho })}({{J}}_\lambda, \operatorname {Av}_*^{\rm ren}(\mathcal {F}')) \simeq \langle \operatorname {Av}_*^{\rm ren}(\mathcal {F}'), {{J}}_{\lambda }^{\mathbb {D}}\rangle$.

Thus, by regarding $\operatorname {Whit}_{\rm q}({\operatorname {Fl}}_G^{\omega ^\rho })_{co}^\vee$ as a subcategory of $\operatorname {Shv}_{\mathcal {G}^G}({\operatorname {Fl}}_G^{\omega ^\rho })^\vee$, the image of $\Delta _\lambda$ in $\operatorname {Shv}_{\mathcal {G}^G}({\operatorname {Fl}}_G^{\omega ^\rho })^\vee$ can be realized as

(5.7.11)\begin{equation} \begin{aligned} \operatorname{Shv}_{\mathcal{G}^G}({\operatorname{Fl}}_G^{\omega^\rho}) & \to {\rm Vect}\\ \mathcal{F} & \mapsto \langle \operatorname{Av}_*^{\rm ren}(\mathcal{F}), {{J}}_{\lambda}^{\mathbb{D}}\rangle. \end{aligned} \end{equation}

Since there is an isomorphism

\[ \langle \operatorname{Av}_*^{\rm ren}(\mathcal{F}), {{J}}_{\lambda}^{\mathbb{D}}\rangle\simeq \langle \mathcal{F}, \operatorname{Av}_*^{\rm ren}({{J}}_{\lambda}^{\mathbb{D}})\rangle, \]

$(\operatorname {Av}_*^{\rm ren})^\vee (\Delta _\lambda )$ is the functor

(5.7.12)\begin{equation} \begin{aligned} \operatorname{Shv}_{\mathcal{G}^G}({\operatorname{Fl}}_G^{\omega^\rho}) & \to {\rm Vect}\\ \mathcal{F} & \mapsto \langle \mathcal{F}, \operatorname{Av}_*^{\rm ren}({{J}}_{\lambda}^{\mathbb{D}})\rangle. \end{aligned} \end{equation}

By the construction of (5.7.5), the image of $(\operatorname {Av}_*^{\rm ren})^\vee (\Delta _\lambda )$ under (5.7.5) is $\operatorname {Av}_*^{\rm ren}({{J}}_{\lambda }^{\mathbb {D}})$.

6.1.1

Recall that we defined the Beilinson–Drinfeld affine flags ${\operatorname {Fl}}_{G, {\mathop {\operatorname {\rm Ran}}}_x}^{\omega ^\rho }$ in §§ 2.3 and 2.4.2. The idea of the construction of the functor $F^{\rm L}$ is to regard a twisted Whittaker sheaf on ${\operatorname {Fl}}^{\omega ^\rho }_G$ as a twisted Whittaker sheaf on ${\operatorname {Fl}}_{G, {\mathop {\operatorname {\rm Ran}}}_x}^{\omega ^\rho }$ (and ${\operatorname {Fl}}_{G, {\mathop {\operatorname {\rm Conf}}}_{\infty \cdot x}}^{\omega ^\rho }\!$), and then pushforward along the projection to ${\mathop {\operatorname {\rm Conf}}}_{\infty \cdot x}$. Let us start by explaining the definition of Whittaker sheaves on ${\operatorname {Fl}}^{\omega ^\rho }_{G, {\mathop {\operatorname {\rm Ran}}}_x}$.

Let $N(\mathcal {K})^{\omega ^\rho }_{{\mathop {\operatorname {\rm Ran}}}}$ (respectively, $N(\mathcal {O})^{\omega ^\rho }_{{\mathop {\operatorname {\rm Ran}}}}\!$) be Ran-ified loop group of $N$. It is the prestack classifying the data $(\mathcal {I}, \alpha )$, where $\mathcal {I}\in {\mathop {\operatorname {\rm Ran}}}(S)$ and $\alpha$ is an automorphism of $\omega ^\rho \overset {T}{\times } B$ on $\overset {\circ }{\mathcal {D}}_{\mathcal {I}}$ (respectively, ${\mathcal {D}}_{\mathcal {I}}$), which is compatible with the identification of $\omega ^\rho$. Similarly, one can define $N(\mathcal {K})^{\omega ^\rho }_{{\mathop {\operatorname {\rm Ran}}}_x}$ and $N(\mathcal {O})^{\omega ^\rho }_{{\mathop {\operatorname {\rm Ran}}}_x}$.