2.1 Reminder on strong actions on categories

Let $G$ be a connected reductive group over $\mathbb {C}$ and let $\kappa$ denote an invariant symmetric bilinear form on the Lie algebra $\mathfrak {g}$ of $G$ (when $G$ is simple, the vector space of such bilinear forms is one-dimensional, so we can think of $\kappa$ as an element of $\mathbb {C}$). Then there is a notion of strong action of the group $G_{\mathbf {F}}$ on a category $\mathcal {C}$ of level $\kappa$. We refer the reader to [Reference GaitsgoryGai16] for details of the definition. It will be important for us later that this definition is, in some sense, invariant under integral shifts. Specifically, any category $\mathcal {C}$ endowed with a $G_{\mathbf {F}}$-action of level $\kappa$ has a natural $G_{\mathbf {F}}$-action of level $\kappa +\kappa '$, where $\kappa '$ is another form as above which is, moreover, integral in the sense that the corresponding quadratic form is integral and even on elements of the coweight lattice of $G$. Here are two important examples.

(1) Let $\hat {\mathfrak {g}}_{\kappa }$ denote the central extension of $\mathfrak {g}_{\mathbf {F}}$ associated with the form $\kappa$. Let $\hat {\mathfrak {g}}_{\kappa }$-mod be the category of continuous (with respect to $t$-adic topology) modules over $\hat {\mathfrak {g}}_{\kappa }$ such that the element 1 of the center acts on the module as the identity. Then the adjoint action of $G_{\mathbf {F}}$ on $\hat {\mathfrak {g}}_{\kappa }$ has a natural lift to a strong action of $G_{\mathbf {F}}$ on $\hat {\mathfrak {g}}_{\kappa }$-mod of level $\kappa$. More generally, let $\mathfrak a=\mathfrak a_{\bar 0}\oplus \mathfrak a_{\bar 1}$ be a Lie superalgebra. Make the following assumptions.

1. (i) $G$ acts on $\mathfrak a$.

2. (ii) We are given a map $\iota \colon \mathfrak {g}\to \mathfrak a_{\bar 0}$ such that the corresponding adjoint action of $\mathfrak {g}$ on $\mathfrak a$ is equal to the derivative of the $G$-action from (i).

3. (iii) The algebra $\mathfrak a$ is equipped with an invariant symmetric (in the super sense) bilinear form $\kappa _{\mathfrak a}$. Let $\kappa _{\mathfrak {g}}$ be its pullback to $\mathfrak {g}$.

Associated with the form $\kappa _{\mathfrak a}$, there is a canonical Kac–Moody extension $\hat {\mathfrak a}$ of $\mathfrak a_{\mathbf {F}}$. As before, we denote by $\hat {\mathfrak a}$-mod the category of continuous modules $M$ over $\hat {\mathfrak a}$ such that the element 1 of the center acts on $M$ by the identity. This category comes equipped with a $G_{\mathbf {F}}$-action of level $\kappa _{\mathfrak {g}}$.

We will mostly be interested in the following special case of the above construction. Fix a pair $M,N$, of non-negative integers. Let $\mathfrak a=\mathfrak {gl}(M|N)$, $G=\operatorname{GL}_M\times \operatorname{GL}_N$. Let $\kappa _{\mathfrak a}(x,y)=c\cdot \operatorname {sTr}(xy)$, where $\operatorname {sTr}$ stands for ‘super-trace’ and $c\in \mathbb {C}$.

(2) Assume that the form $\kappa$ is integral and even. Then this form gives rise to a central extension $\hat {G}$ of $G_{\mathbf {F}}$. Let $X$ be an ind-scheme equipped with a $\hat {G}$-equivariant line bundle $\mathcal {L}$. Then, for any $c\in \mathbb {C}$, one has the category $D_c(X)$-mod of $c$-twisted $D$-modules on $X$. This category has a natural strong $G_{\mathbf {F}}$-action of level $c\cdot \kappa$.

2.2 Digression on the local (quantum) geometric Langlands correspondence

Let $G$ and $\kappa$ be as above and assume in addition that the form $\kappa$ is non-degenerate. Let $G^{\vee }$ denote the Langlands dual group. Since $\kappa$ is non-degenerate it gives a similar form $\kappa ^{\vee }$ for $G^{\vee }$. Further, put $\kappa _{\operatorname {crit}}=-\frac 12{\operatorname {Killing}}_{\mathfrak {g}}$, where ${\operatorname {Killing}}_{\mathfrak {g}}$ stands for the Killing form on the Lie algebra ${\mathfrak {g}}$ of $G$.

The local quantum geometric Langlands conjecture is, roughly speaking, an equivalence of 2-categories

\[\{ \text{categories with strong $G_{\mathbf{F}}$-action of level $\kappa+\kappa_{\operatorname{crit}}$}\} \]

and

\[ \{ \text{categories with strong $G^{{\vee}}_{\mathbf{F}}$-action of level $-\kappa^{{\vee}}-\kappa^{{\vee}}_{\operatorname{crit}}$}\}. \]

The form $\kappa _{\operatorname {crit}}$ being integral, the shift by $\kappa _{\operatorname {crit}}$ in the above formulation is not essential. However, it is convenient for many applications to make this shift.

To give a rigorous meaning to the above conjecture one needs, first of all, to replace all ‘categories’ by suitable ‘dg-categories’. This upgrades each side of the equivalence to an $(\infty ,2)$-category. Then, for generic (i.e. non-rational) $\kappa$, the above equivalence is expected to hold as stated. For general $\kappa$, more corrections are necessary but we will not discuss this here since the local geometric Langlands correspondence will only serve as a guiding principle.

There is a ‘limiting version’ of the above conjecture for $\kappa =0$. To explain this, we use the notion of a ‘category over a stack $\mathcal {S}$’; cf. [Reference GaitsgoryGai16, § 6] and references therein. Write $\mathcal {D}^\circ ={\operatorname{Spec}}({\mathbf {F}})$ and let $\operatorname {LocSys}_{G^{\vee }}(\mathcal {D}^\circ )$ be the classifying stack of principal $G^{\vee }$-bundles on $\mathcal {D}^\circ$ equipped with a connection. The ‘classical’ local geometric Langlands conjecture predicts a close relationship between

\[ \{ \text{categories with strong $G_{\mathbf{F}}$-action of level $\kappa_{\operatorname{crit}}$}\} \]

and

\[ \{\text{categories over $\operatorname{LocSys}_{G^{{\vee}}}(\mathcal{D}^\circ)$}\}. \]

Here, $\kappa _{\operatorname {crit}}$ can be replaced by $0$, since $\kappa _{\operatorname {crit}}$ is integral. Again, it is possible to make the informal relation above a rigorous mathematical conjecture; cf. again [Reference GaitsgoryGai16, § 6] for a more detailed discussion.

In the remainder of this section we will pretend that both quantum and classical geometric Langlands conjectures hold as stated. We will write $\mathcal {C}\mapsto \mathcal {C}^{\vee }$ for the resulting correspondence.

An important example of such a correspondence is as follows. Fix a non-degenerate form $\kappa$ and let $\mathcal {C}=D_{\kappa +\kappa _{\operatorname {crit}}}({\mathbf {Gr}}_G)$-mod be the category of $(\kappa -\kappa _{\operatorname {crit}})$-twisted $D$-modules on the affine Grassmannian ${\mathbf {Gr}}_G$ of $G$. It is expected that in this case the category $\mathcal {C}^{\vee }$ is the category $D_{-\kappa ^{\vee }-\kappa ^{\vee }_{\operatorname {crit}}}({\mathbf {Gr}}_{G^{\vee }})$-mod. In the limiting case $\kappa =0$ category $\mathcal {C}^{\vee }$ is expected to be a pushforward of category $\operatorname {QCoh}({\operatorname {pt}}/G^{\vee })$ under the natural map ${\operatorname {pt}}/G^{\vee }\to \operatorname {LocSys}_{G^{\vee }}(\mathcal {D}^\circ )$ induced by the embedding of the trivial local system. These expectations would have the following implication.

1. (i) If $\kappa$ is non-degenerate, then $\mathcal {C}^{G_{\mathbf {O}}}\simeq (\mathcal {C}^{\vee })^{G^{\vee }_{\mathbf {O}}}$ (here $\mathcal {C}^{G_{\mathbf {O}}}$ denotes the category of $G_{\mathbf {O}}$-equivariant objects in $\mathcal {C}$).

2. (ii) If $\kappa =0$ then $\mathcal {C}^{G_{\mathbf {O}}}$ is equivalent to the pullback of the category $\mathcal {C}^{\vee }$ under the map ${\operatorname {pt}}/G^{\vee }\to \operatorname {LocSys}_{G^{\vee }}(\mathcal {D}^\circ )$ corresponding to the trivial local system.

2.3 Whittaker category and the fundamental local equivalence

In this subsection we discuss another important example of Langlands dual categories. We refer to [Reference GaitsgoryGai16, Reference GaitsgoryGai20] for details.

Let $U$ be a maximal unipotent subgroup of $G$ and let $\chi _0\colon U\to \mathbb {G}_a$ be a non-degenerate character. We define a character $\chi \colon U_{\mathbf {F}}\to \mathbb {G}_a$ by

\[ \chi(u(t))=\operatorname{Res}_{t=0} \chi_0(u(t))\,dt. \]

Given a (cocomplete, dg) category $\mathcal {C}$ with a strong $G_{\mathbf {F}}$-action of a fixed level, one can define a category $\operatorname {Whit}(\mathcal {C})$ of $(U_{\mathbf {F}},\chi )$-equivariant objects in $\mathcal {C}$.

We consider the category $D_{\kappa }(G_{\mathbf {F}})$-mod of $\kappa$-twisted $D$-modules on $G_{\mathbf {F}}$. According to [Reference Arkhipov and GaitsgoryAG02], this category has a natural action of $G_{\mathbf {F}}$ of level $\kappa$ that comes from left translations, and also another action of $G_{\mathbf {F}}$ of level $-\kappa +2\kappa _{\operatorname {crit}}$ that comes from right translations.

Let $\operatorname {Whit}^r_{\kappa }(G_{\mathbf {F}})$ denote its Whittaker category with respect to the right action. This category inherits the left action of level $\kappa$. One could ask what is its Langlands dual category.

For simplicity, below we will only consider either the case where $\kappa$ is not rational (i.e. the value of the corresponding quadratic form on any coroot is not a rational number) or the case $\kappa =0$.

Assume first that $\kappa$ is non-degenerate and not rational. Then it is expected (cf. [Reference GaitsgoryGai16]) that

(2.3.1)\begin{equation} \text{$\operatorname{Whit}^r_{\kappa-\kappa_{\operatorname{crit}}}(G_{\mathbf{F}})^{{\vee}}$ is the category $\widehat{\mathfrak{g}^{{\vee}}}\!\!\!_{-\kappa^{{\vee}}+\kappa^{{\vee}}_{\operatorname{crit}}}$-mod.} \end{equation}

In the case $\kappa =0$ the expected answer is the category $\operatorname {QCoh}(\operatorname {LocSys}_{G^{\vee }}(\mathcal {D}^\circ ))$.

Let $\operatorname {Whit}_{\kappa }({\mathbf {Gr}}_G)$ denote the category of $\kappa$-twisted $D$-modules on ${\mathbf {Gr}}_G$ and let $\operatorname {KL}_{\kappa }(\mathfrak {g})=(\hat {\mathfrak {g}}_{\kappa }\text {-mod})^{G_{\mathbf {O}}}$. Then, statements (i) and (ii) of § 2.2 imply the following result, which has been proved rigorously in [Reference GaitsgoryGai20, Reference GaitsgoryGai08] (it goes under the name ‘fundamental local equivalence’).

Theorem 2.3.1 Assume that $\kappa$ is non-degenerate and not rational. Then

(2.3.2)\begin{gather} \operatorname{Whit}_{\kappa+\kappa_{\operatorname{crit}}}({\mathbf{Gr}}_G)\simeq \operatorname{KL}_{\kappa^{{\vee}}+\kappa^{{\vee}}_{\operatorname{crit}}}(\mathfrak{g}^{{\vee}}), \end{gather}

(2.3.3)\begin{gather} \operatorname{Whit}({\mathbf{Gr}}_G)\simeq {\operatorname{Rep}}(G^{{\vee}}). \end{gather}

Moreover, these equivalences hold at the level of abelian categories.

2.4 Gaiotto conjectures: geometric Langlands form in the case $N>M$

There is one more series of examples which are relevant to the subject of this paper. Fix two non-negative integers $N$ and $M$ such that $N\geq M$ and set $G_{M,N}=\operatorname{GL}_M\times \operatorname{GL}_N$. Note that $G_{M,N}$ is isomorphic to $G_{M,N}^{\vee }$.

We are going to produce an example of Langlands dual categories for $G_{M,N}$; for $M=0$ we will recover (2.3.1). First, we describe the analogue of the right-hand side. Let $c\in \mathbb {C}$. Then the category in question will be the category $\widehat {\mathfrak {gl}}(M|N)_c$-mod of modules over the affine Lie superalgebra $\widehat {\mathfrak {gl}}(M|N)$ of level $c\cdot \kappa _{M,N}+\frac 12{\operatorname {Killing}}_{M,N}$, where

1. (1) $\kappa _{M,N}(x,y)=\operatorname {sTr}(x y)$,

2. (2) ${\operatorname {Killing}}_{M,N}$ is the restriction of the Killing form of $\mathfrak {gl}(M|N)$ to the even part (note that it is degenerate if $M=N$).

This category has a natural action of the group $G_{M,N}({\mathbf {F}})$ of a certain level which is an integral shift of $(c\cdot \kappa _M,-c\cdot \kappa _N)$ (here $\kappa _N$ denotes the standard invariant bilinear form on the Lie algebra $\mathfrak {gl}_N$ equal to $\operatorname {Tr}(X\cdot Y)$). As has been explained above, one can twist the action of $G_{M,N}({\mathbf {F}})$ on this category so that the twist becomes equal to $(c\cdot \kappa _M,-c\cdot \kappa _N)-\kappa _{\operatorname {crit}}$ (here by $\kappa _{\operatorname {crit}}$ we mean the critical bilinear form for the Lie algebra $\mathfrak {g}_{M,N}$). Hence it makes sense to consider its Langlands dual. This should be a category with a strong action of $G_{M,N}({\mathbf {F}})$ of level $c^{-1}\cdot (\kappa _M,-\kappa _N)-\kappa _{\operatorname {crit}}$. However, we can again twist the action and think of it as a category with an action of $G_{M,N}({\mathbf {F}})$ of level $c^{-1}\cdot (\kappa _M,-\kappa _N)$. Let us give a conjectural description of the Langlands dual category according to a prediction of D. Gaiotto.

Next, let $M< N$. We define a unipotent subgroup $U_{M,N}$ of $\operatorname{GL}_N$ as follows. If $M=N-1$ this subgroup is trivial, and if $M=0$ it is the group $U_N$ of unipotent upper-triangular matrices. In the general case, $U_{M,N}$ is a subgroup of $U_N$ defined as follows.

Let $e_{M,N}\in \mathfrak {gl}_N$ be the standard upper-triangular Jordan block of size $N-M$, that is,

\[ e_{M,N}=\sum_{i=1}^{N-M-1} E_{i,i+1}, \]

where $E_{ij}$ stands for the matrix whose $(i,j)$th entry is equal to 1 and all other entries are equal to 0. The element $e_{M,N}$ is part of an $\mathfrak {sl}_2$-triple $(e_{M,N},h_{M,N},f_{M,N})$. Here $f_{M,N}=\sum _{i=1}^{N-M-1} i(N-M-i) E_{i+1,i}$ and $h_{M,N}$ is the diagonal matrix which has diagonal entries $(N-M-1,N-M-3,\ldots ,-N+M+1,0,\ldots ,0)$. For any integer $l$ we let $\mathfrak {g}_l$ denote the $l$-eigenspace of the adjoint action of $h_{M,N}$ on $\mathfrak {gl}_N$. We set

\[ \mathfrak{u}_{M,N}=\bigoplus_{l\geq 2} \mathfrak{g}_l\oplus \mathfrak{g}_1^+, \]

where $\mathfrak {g}_1^+$ is the intersection of $\mathfrak {g}_1$ with the Lie algebra of upper-triangular matrices. We define a Lie algebra homomorphism $\chi ^0_{M,N}\colon \mathfrak {u}_{M,N}\to \mathbb {C}$ by sending a matrix $(u_{ij})$ to $\sum _{i=1}^{N-M-1} u_{i,i+1}$. Let $U_{M,N}$ be a unipotent subgroup of $\operatorname{GL}_N$ with Lie algebra $\mathfrak {u}_{M,N}$ and let $U_{M,N}\to \mathbb {G}_a$ be the homomorphism induced by $\chi _{M,N}^0$ (which we denote by the same symbol $\chi _{M,N}^0$).

We embed the group $\operatorname{GL}_M$ into the centralizer of the element $h_{M,N}$ in $\operatorname{GL}_N$ (this is the block-diagonal embedding corresponding to rows ${N-M+1},\ldots ,N$). It is easy to see that the group $\operatorname{GL}_M$ normalizes the subgroup $U_{M,N}$, and the homomorphism $\chi ^0_{M,N}$ is fixed by the $\operatorname{GL}_M$-action on $U_{M,N}$ by conjugation.

Remark 2.4.1 $U_{M,N}$ is conjugate to the subgroup $U'_{M,N}$ formed by the block-upper-triangular matrices

\[ \begin{pmatrix}U_r & {*} & {*}\\0 & 1_{M+1} & {*}\\0 & 0 & U_s\end{pmatrix},\quad \text{where } r=\lceil(N-M-1)/2\rceil,\ s=\lfloor(N-M-1)/2\rfloor; \]

in particular, $r+s=N-M-1$. Here $U_p$ stands for an arbitrary unipotent upper-triangular matrix in $\operatorname{GL}_p$, and the notation ‘$*$’ is used for arbitrary matrices of an appropriate size. Moreover, the conjugation can be chosen so that the character $\chi _{M,N}^0$ corresponds to the character $\chi _{M,N}^{(r,s)}$ on $\mathfrak u_{M,N}^{(r,s)}:=\operatorname {Lie} U_{M,N}^{(r,s)}$ given by $(u_{ij})\mapsto \sum _{i=1}^{r-1} u_{i,i+1} + u_{rk} + u_{k,N-s+1} + \sum _{i=N-s+1}^{N-1} u_{i,i+1}$ for any choice of $k\in \{r+1,\ldots ,N-s\}$. The subgroup $U_{M,N}^{(r,s)}\subset \operatorname{GL}_N$ and the character $\chi _{M,N}^{(r,s)}$ are defined for an arbitrary pair $(r,s)\in {\mathbb {N}}^2:={\mathbb {Z}}_{\geq 0}^2$ with $r+s=N-M-1$. (In the two extreme cases $\{r,s\}=\{0,N-M-1\}$, one of the middle terms in the formula for $\chi _{M,N}^{(r,s)}$ is undefined and should be omitted.) Moreover, $\chi _{M,N}^{(r,s)}$ can be replaced by an arbitrary representative of the open $N_{\operatorname{GL}_N}(U_{M,N}^{(r,s)})$-orbit in $\mathfrak u_{M,N}^{(r,s)*}$.

As before, we define a homomorphism $\chi _{M,N}\colon U_{M,N}({\mathbf {F}})\to \mathbb {G}_a$ to be $\operatorname {Res}_{t=0}\chi ^0_{M,N}$.

For any $c'\in \mathbb {C}$, we now consider the category $D_{c'}(\operatorname{GL}(N,{\mathbf {F}}))^{U_{M,N}({\mathbf {F}}),\chi _{M,N}}$. By definition this is the derived category of $D$-modules on $\operatorname{GL}(N,{\mathbf {F}})$ twisted by $c'\cdot \kappa _N$ that are equivariant on the left with respect to $(U_{M,N}({\mathbf {F}}),\chi _{M,N})$. This category has a natural action of $\operatorname{GL}(M,{\mathbf {F}})$ of level $c'\cdot \kappa _M$ coming from left multiplication and an action of $\operatorname{GL}(N,{\mathbf {F}})$ of level $-c'\cdot \kappa _M-\operatorname {Killing}_{{\mathfrak {gl}}_N}$ coming from right multiplication (recall that $\operatorname {Killing}_{{\mathfrak {gl}}_N}$ denotes the Killing form on $\mathfrak {gl}_N$). As before, we can twist the second action to make it an action of level $-c'\cdot \kappa _M$.

We now take $c'=1/c$. Then, according to a conjecture of D. Gaiotto, the category $D_{1/c}(\operatorname{GL}(N,{\mathbf {F}}))^{U_{M,N}({\mathbf {F}}),\chi _{M,N}}$ is Langlands dual to $\widehat {\mathfrak {gl}}(M|N)_c$-mod.

We can also consider the limit $c\to \infty$. To simplify the discussion we will not do it now, but we will discuss it later when we turn to $G_{\mathbf {O}}$-equivariant objects.

2.5 Gaiotto conjectures: geometric Langlands form for $N=M$

Let us also discuss the case $N=M$. In this case on the left we again take the same category $\widehat {\mathfrak {gl}}(N|N)_c$-mod of modules over the affine Lie superalgebra $\widehat {\mathfrak {gl}}(N|N)$ of level $c\cdot \kappa _{N,N}-\frac 12{\operatorname {Killing}}_{N,N}$ (note that the Killing form is zero for ${N=M}$). The Langlands dual category (according to Gaiotto) is the derived category $D_{1/c}(\operatorname{GL}(N,{\mathbf {F}})\times {\mathbf {V}})$ of $1/c$-twisted $D$-modules on $\operatorname{GL}(N,{\mathbf {F}})\times {\mathbf {V}}$ where

1. (1) the twisting is with respect to the first factor,

2. (2) the two actions of $\operatorname{GL}(N,{\mathbf {F}})$ come from the diagonal action coming from the left action of $\operatorname{GL}(N,{\mathbf {F}})$ on the first factor and the natural action of $\operatorname{GL}(N,{\mathbf {F}})$ on the second factor, and the right multiplication action $\operatorname{GL}(N,{\mathbf {F}})$ on the first factor.

2.6 Gaiotto conjectures: ‘simple-minded’ form

The above statements are not really well-formulated mathematical conjectures, since the local geometric Langlands duality is not known at present. However, we can turn them into precise conjectures by using (i) at the end of § 2.2, that is, we are going to take $G_{M,N}({\mathbf {O}})$-invariants on both sides. We get the following conjectures.

We now want to take the limit $c\to \infty$. In this case $1/c$ goes to $0$ and the category of $1/c$-twisted $D$-modules in Conjecture 2.6.1 just becomes the category of usual $D$-modules. The $c\to \infty$ limit of the category $\operatorname {KL}_c(\widehat {\mathfrak {gl}}(M|N))$ is not canonically defined: one has to choose some nice extension of the corresponding family of categories from ${\mathbb{A}}^1$ to $\mathbb {P}^1$; cf. [Reference ZhaoZha17, § 6]. Naively, one might think that the correct extension is just the category of representations of the supergroup $\operatorname{GL}(M|N)$. However, it turns out that this is not the right choice. Instead, one needs to consider the category of representations of the group $\underline{\operatorname{GL}}(M|N))$ defined in § 1.5. With these conventions one gets the following conjecture.

In the present work we prove Conjecture 2.6.2 for $M=N$ and $M=N-1$.

3.1 Setup and notation

We follow the notation of [Reference Finkelberg, Ginzburg and TravkinFGT09]. Recall that ${\mathbf {Gr}}={\mathbf {Gr}}_{\operatorname{GL}_N}={\mathbf {G}}_{\mathbf {F}}/{\mathbf {G}}_{\mathbf {O}}=\operatorname{GL}(N,{\mathbf {F}})/\operatorname{GL}(N,{\mathbf {O}})$, where ${\mathbf {F}}={\mathbb {C}}(\!(t)\!)\supset {\mathbb {C}}[\![t]\!]={\mathbf {O}}$. We consider a complex vector space $V$ with a basis $e_1,\ldots ,e_N$. We set ${\mathbf {V}}=V\,{\otimes}\, {\mathbf {F}}\supset V\,{\otimes}\, {\mathbf {O}}={\mathbf {V}}_0$.

Recall that the ${\mathbf {G}}_{\mathbf {F}}$-orbits in ${\mathbf {Gr}}\times {\mathbf {Gr}}$ (respectively, in ${\mathbf {Gr}}\times {\mathbf {Gr}}\times {\vphantom {j^{X^2}}\smash {\overset {\circ }{\vphantom {\rule {0pt}{0.55em}}\smash {\mathbf {V}}}}}$) are numbered in [Reference Finkelberg, Ginzburg and TravkinFGT09, § 3.1] by signaturesFootnote ⁵ (respectively, by bisignatures, i.e. pairs of signatures) in such a way that the ${\mathbf {G}}_{\mathbf {F}}$-orbits on ${\mathbf {Gr}}\times {\mathbf {Gr}}$ numbered by partitions ${\boldsymbol {\nu }}=(\nu _1\geq \cdots \geq \nu _N\geq 0)$ correspond to the pairs of lattices $L^1\subset L^2$. More precisely, the orbit corresponding to a bisignature $({\boldsymbol {\lambda }},{\boldsymbol {\mu }})$ contains a point

\[ \bigg(L^1={\mathbf{O}}\langle e_1,e_2,\ldots,e_N\rangle,\ L^2={\mathbf{O}}\langle t^{-\lambda_1-\mu_1}e_1, t^{-\lambda_2-\mu_2}e_2,\ldots,t^{-\lambda_N-\mu_N}e_N\rangle,\ v=\sum_{i=1}^Nt^{-\lambda_i}e_i\bigg). \]

Irreducible representations of $\operatorname{GL}_N=\operatorname{GL}(V)$ are also numbered by the signatures, so that, for example, the determinant character $\det V$ corresponds to $(1^N)$. To a signature ${\boldsymbol {\nu }}=(\nu _1\geq \cdots \geq \nu _N)$ we associate an irreducible representation $V_{\boldsymbol {\nu }}$ with the highest weight ${\boldsymbol {\nu }}$. The geometric Satake equivalence takes the irreducible perverse sheaf ${\operatorname {IC}}_{\boldsymbol {\nu }}$ to the irreducible representation $V^*_{\boldsymbol {\nu }}$. Thus if ${\boldsymbol {\nu }}$ is a partition (respectively, a negative partition $(0\geq \nu _1\geq \cdots \geq \nu _N)$), then $V^*_{\boldsymbol {\nu }}$ is an antipolynomial (respectively, polynomial) representation of $\operatorname{GL}_N$ (a polynomial functor in $V^*$, $V$, respectively).

A word of apology for our weird convention is in order. The numbering of ${\mathbf {G}}_{\mathbf {O}}$-orbits in ${\mathbf {Gr}}$ such that the orbits of sublattices $L\subset {\mathbf {V}}_0$ are numbered by partitions goes back at least to [Reference LusztigLus81]. We choose the numbering such that the orbits of sublattices are numbered by negative partitions since under this numbering the adjacency order of ${\mathbf {G}}_{\mathbf {O}}$-orbits in ${\mathbf {Gr}}\times {\vphantom {j^{X^2}}\smash {\overset {\circ }{\vphantom {\rule {0pt}{0.55em}}\smash {\mathbf {V}}}}}$ goes to Shoji's order ([Reference ShojiSho04], [Reference Finkelberg, Ginzburg and TravkinFGT09, Proposition 12]) on the set of bisignatures. Furthermore, we choose the Satake equivalence ${\operatorname {IC}}_{\boldsymbol {\nu }}\mapsto V^*_{\boldsymbol {\nu }}$ (as opposed to ${\operatorname {IC}}_{\boldsymbol {\nu }}\mapsto V_{\boldsymbol {\nu }}$) since it makes the statement of our main result Theorem 3.6.1 neater.

3.2 Constructible mirabolic category and convolutions

The triangulated category $D_{{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\vphantom {j^{X^2}}\smash {\overset {\circ }{\vphantom {\rule {0pt}{0.55em}}\smash {\mathbf {V}}}}})$ is defined as in [Reference Finkelberg, Ginzburg and TravkinFGT09, § 2.6]. We will denote it by $D_{*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$. Recall that an object ${\mathcal {F}}$ of $D_{*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ is supported on ${\mathbf {Gr}}\times t^m{\mathbf {V}}_0$ for certain $m\in {\mathbb {Z}}$, and there exist $n>m$ and a ${\mathbf {G}}_{\mathbf {O}}$-equivariant sheaf ${\mathcal {F}}_n$ on ${\mathbf {Gr}}\times (t^m{\mathbf {V}}_0/t^n{\mathbf {V}}_0)$ such that ${\mathcal {F}}=p_n^*{\mathcal {F}}_n$, where

\[ p_n\colon{\mathbf{Gr}}\times t^m{\mathbf{V}}_0\to{\mathbf{Gr}}\times(t^m{\mathbf{V}}_0/t^n{\mathbf{V}}_0) \]

is the natural projection. In other words, ${\mathcal {F}}$ is a collection of ${\mathbf {G}}_{\mathbf {O}}$-equivariant sheaves ${\mathcal {F}}_{n'}$ on ${\mathbf {Gr}}\times (t^m{\mathbf {V}}_0/t^{n'}{\mathbf {V}}_0)$ for $n'\geq n$ along with a compatible system of isomorphisms $p_{n''/n'}^*{\mathcal {F}}_{n'}{\buildrel \sim \over \longrightarrow} {\mathcal {F}}_{n''}$ for $n''\geq n'$, where

\[ p_{n''/n'}\colon {\mathbf{Gr}}\times t^m{\mathbf{V}}_0/t^{n''}{\mathbf{V}}_0\twoheadrightarrow{\mathbf{Gr}}\times t^m{\mathbf{V}}_0/t^{n'}{\mathbf{V}}_0 \]

are the natural projections.

If in the above definition we replace $p_n^*$ by $p_n^!$ and $p_{n''/n'}^*$ by $p_{n''/n'}^!$, then we obtain a triangulated category $D_{!{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$. Note that $p_{n''/n'}$ is a smooth morphism of relative dimension $N(n''-n')$, so we have a canonical isomorphism $p_{n''/n'}^!\cong p_{n''/n'}^*[2N(n''-n')]$. We also consider the intermediate version

\[ p_{n''/n'}^{!*}:=p_{n''/n'}^*[N(n''-n')]=p_{n''/n'}^![{-}N(n''-n')], \]

exact for the perverse $t$-structure. The corresponding triangulated category will be denoted $D_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$.

We will make use of an identification ${\mathbf {Gr}}\times {\mathbf {V}}={\mathbf {G}}_{\mathbf {F}}\stackrel {{\mathbf {G}}_{\mathbf {O}}}{\times }{\mathbf {V}}=({\mathbf {G}}_{\mathbf {F}}\times {\mathbf {V}})/{\mathbf {G}}_{\mathbf {O}}$ (quotient with respect to the diagonal right-left action). We will denote the orbit of $(g,v)\in {\mathbf {G}}_{\mathbf {F}}\times {\mathbf {V}}$ by $[g,v]$. Note that the left diagonal action of ${\mathbf {G}}_{\mathbf {O}}$ on ${\mathbf {Gr}}\times {\mathbf {V}}$ in terms of the above identification is $h\cdot [g,v]=[hg,v]$. We consider the following convolution diagram:

(3.2.1)

\[ ([g_1,g_2v],[g_2,v])\xleftarrow{{\mathbf{p}}}(g_1,[g_2,v])\xrightarrow{{\mathbf{q}}}[g_1,[g_2,v]] \xrightarrow{{\mathbf{m}}}[g_1g_2,v]. \]

Given ${\mathcal {F}}_1,{\mathcal {F}}_2\in D_{!{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$, we define ${\mathcal {F}}_1{\stackrel {!}\circledast} {\mathcal {F}}_2:={\mathbf {m}}_*({\mathcal {F}}_1{\mathop \boxtimes \limits^{\sim}}^!{\mathcal {F}}_2)$, where ${\mathcal {F}}_1{\mathop \boxtimes \limits^{\sim}}^!{\mathcal {F}}_2\in D_{!{\mathbf {G}}_{\mathbf {O}}}\big ({\mathbf {G}}_{\mathbf {F}}\overset {{\mathbf {G}}_{\mathbf {O}}}\times ({\mathbf {Gr}}\times {\mathbf {V}})\big )$ is the canonical descent of ${\mathbf {p}}^!({\mathcal {F}}_1\boxtimes {\mathcal {F}}_2)$ along ${\mathbf {q}}$. Similarly, given ${\mathcal {F}}_1,{\mathcal {F}}_2\in D_{*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$, we define ${\mathcal {F}}_1{\stackrel {*}\circledast} {\mathcal {F}}_2:={\mathbf {m}}_*({\mathcal {F}}_1{\mathop \boxtimes \limits^{\sim}}^*{\mathcal {F}}_2)$, where ${\mathcal {F}}_1{\mathop \boxtimes \limits^{\sim}}^*{\mathcal {F}}_2\in D_{*{\mathbf {G}}_{\mathbf {O}}}\big ({\mathbf {G}}_{\mathbf {F}}\overset {{\mathbf {G}}_{\mathbf {O}}}{\times }({\mathbf {Gr}}\times {\mathbf {V}})\big )$ is the canonical descent of ${\mathbf {p}}^*({\mathcal {F}}_1\boxtimes {\mathcal {F}}_2)$ along ${\mathbf {q}}$, a unique sheaf such that ${\mathbf {p}}^*({\mathcal {F}}_1\boxtimes {\mathcal {F}}_2)={\mathbf {q}}^*({\mathcal {F}}_1{\mathop \boxtimes \limits^{\sim}}^*{\mathcal {F}}_2)$.

We also consider another convolution diagram:

(3.2.2)

\[ ([g_1,v_1],[g_2,v_2])\xleftarrow{{\mathfrak{p}}}(g_1,v_1,[g_2,v_2])\xrightarrow{{\mathfrak{q}}}[g_1,[g_2,g_2^{{-}1}v_1,v_2]] \xrightarrow{{\mathfrak{m}}}[g_1g_2,g_2^{{-}1}v_1+v_2]. \]

Given ${\mathcal {F}}_1,{\mathcal {F}}_2\in D_{!{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$, we define ${\mathcal {F}}_1{\stackrel {!}*}{\mathcal {F}}_2:={\mathfrak {m}}_!({\mathcal {F}}_1{\mathop \boxtimes \limits^{\sim}}^!{\mathcal {F}}_2)$, where ${\mathcal {F}}_1{\mathop \boxtimes \limits^{\sim}}^!{\mathcal {F}}_2\in D_{!{\mathbf {G}}_{\mathbf {O}}}\big ({\mathbf {G}}_{\mathbf {F}}\overset {{\mathbf {G}}_{\mathbf {O}}}{\times }({\mathbf {Gr}}\times {\mathbf {V}}\times {\mathbf {V}})\big )$ is the canonical descent of ${\mathfrak {p}}^!({\mathcal {F}}_1\boxtimes {\mathcal {F}}_2)$ along ${\mathfrak {q}}$. Similarly, given ${\mathcal {F}}_1,{\mathcal {F}}_2\in D_{*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$, we define ${\mathcal {F}}_1{\stackrel {*}*}{\mathcal {F}}_2:={\mathfrak {m}}_*({\mathcal {F}}_1{\mathop \boxtimes \limits^{\sim}}^*{\mathcal {F}}_2)$, where ${\mathcal {F}}_1{\mathop \boxtimes \limits^{\sim}}^*{\mathcal {F}}_2\in D_{*{\mathbf {G}}_{\mathbf {O}}}\big ({\mathbf {G}}_{\mathbf {F}}\overset {{\mathbf {G}}_{\mathbf {O}}}{\times }({\mathbf {Gr}}\times {\mathbf {V}}\times {\mathbf {V}})\big )$ is the canonical descent of ${\mathfrak {p}}^*({\mathcal {F}}_1\boxtimes {\mathcal {F}}_2)$ along ${\mathfrak {q}}$.

If we formally put $v_1=0$ in (3.2.2), we obtain the convolution diagram

(3.2.3)

\[ ([g_1],[g_2,v])\xleftarrow{{\mathfrak{p}}_{\operatorname{left}}}(g_1,[g_2,v])\xrightarrow{{\mathfrak{q}}}[g_1,[g_2,v]] \xrightarrow{{\mathfrak{m}}}[g_1g_2,v], \]

and if we put $v_2=0$, we obtain

(3.2.4)

\[ ([g_1,v],[g_2])\xleftarrow{{\mathfrak{p}}_{\operatorname{right}}}(g_1,v,[g_2])\xrightarrow{{\mathfrak{q}}}[g_1,[g_2,g_2^{{-}1}v]] \xrightarrow{{\mathfrak{m}}}[g_1g_2,g_2^{{-}1}v]. \]

Given ${\mathcal {P}}\in D_{{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}})$ and ${\mathcal {F}}\in D_{?{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ (where $?=!,*,!*$), we define ${\mathcal {P}}*{\mathcal {F}}:={\mathfrak {m}}_*({\mathcal {P}}{\stackrel {\sim}\boxtimes} {\mathcal {F}})$, where ${\mathcal {P}}{\stackrel {\sim}\boxtimes} {\mathcal {F}}\in D_{?{\mathbf {G}}_{\mathbf {O}}}({\mathbf {G}}_{\mathbf {F}}\overset {{\mathbf {G}}_{\mathbf {O}}}{\times }({\mathbf {Gr}}\times {\mathbf {V}})\big )$ is the canonical descent of ${\mathfrak {p}}_{\operatorname {left}}^*({\mathcal {P}}\boxtimes {\mathcal {F}})$ along ${\mathfrak {q}}$. We also define ${\mathcal {F}}*{\mathcal {P}}:={\mathfrak {m}}_*({\mathcal {F}}{\stackrel {\sim}\boxtimes} {\mathcal {P}})$, where ${\mathcal {F}}{\stackrel {\sim}\boxtimes} {\mathcal {P}}\in D_{?{\mathbf {G}}_{\mathbf {O}}}({\mathbf {G}}_{\mathbf {F}}\overset {{\mathbf {G}}_{\mathbf {O}}}{\times }({\mathbf {Gr}}\times {\mathbf {V}})\big )$ is the canonical descent of ${\mathfrak {p}}_{\operatorname {right}}^*({\mathcal {F}}\boxtimes {\mathcal {P}})$ along ${\mathfrak {q}}$. Both the left and right convolutions are biexact for the perverse $t$-structures on $D_{{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}})$ and $D_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$; see [Reference Finkelberg, Ginzburg and TravkinFGT09, § 3.9].

3.3 Fusion

Let $X$ be a smooth curve. For any integer $k>0$, and a collection $x=(x_i)_{i=1}^k$ of $S$-points of $X$, we denote by ${\mathcal {D}}_x$ the formal neighborhood of the union of graphs $|x| := {\bigcup} _{i=1}^k \Gamma _{x_i}\subset S\times X$, and we denote by ${\mathcal {D}} _x^\circ :={\mathcal {D}}_x \,{\smallsetminus}\, |x|$ the punctured formal neighborhood. The mirabolic version of the Beilinson–Drinfeld Grassmannian is the ind-scheme ${\mathbf {Gr}}^{\operatorname {mir}}_{BD,k}$ over $X^k$ parametrizing the following collections of data:

\[ (x_i)_{i=1}^k,\ {\mathcal{E}},\ \phi\colon {\mathcal{E}}_{\operatorname{triv}}|_{{\mathcal{D}}_x^\circ}{\buildrel \sim \over \longrightarrow}{\mathcal{E}}|_{{\mathcal{D}}_x^\circ},\ v\in \Gamma({\mathcal{D}}_x^\circ,{\mathcal{E}}) , \]

where ${\mathcal {E}}$ is a rank-$N$ vector bundle on ${\mathcal {D}}_x$. If $X={\mathbb {A}}^1$, we have over the complement to the diagonals a canonical isomorphism

\[ ({\mathbb{A}}^k\,{\smallsetminus}\,\Delta)\times_{{\mathbb{A}}^k}{\mathbf{Gr}}^{\operatorname{mir}}_{BD,k}\cong({\mathbb{A}}^k\,{\smallsetminus}\,\Delta)\times({\mathbf{Gr}}\times{\mathbf{V}})^k. \]

We denote the projection $({\mathbb {A}}^k\,{\smallsetminus}\, \Delta )\times ({\mathbf {Gr}}\times {\mathbf {V}})^k\to ({\mathbf {Gr}}\times {\mathbf {V}})^k$ by $\operatorname {pr}_2$. Given ${\mathcal {F}}_1,{\mathcal {F}}_2\in D_{?{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ (where $?={}!,*,!*$) we take $k=2$ and define the fusion

\[ {\mathcal{F}}_1\star{\mathcal{F}}_2:={\operatorname{pr}}_{2*}\psi_{x-y}\operatorname{pr}_2^*({\mathcal{F}}_1\boxtimes{\mathcal{F}}_2)[1], \]

where $x,y$ are coordinates on ${\mathbb {A}}^2$ (so that $x-y=0$ is the equation of the diagonal $\Delta \subset {\mathbb {A}}^2$), and $\psi _{x-y}$ is the nearby cycles functor for the pullback of the function $x-y$ to ${\mathbf {Gr}}^{\operatorname {mir}}_{BD,2}$, normalized so as to preserve the perverse $t$-structure. Note that the leftmost occurrence of ${\operatorname {pr}}_2$ in the above definition projects ${\mathbb {A}}^1\times {\mathbf {Gr}}\times {\mathbf {V}}$ to ${\mathbf {Gr}}\times {\mathbf {V}}$, while the rightmost occurrence of ${\operatorname {pr}}_2$ projects $({\mathbb {A}}^2\,{\smallsetminus}\, \Delta )\times ({\mathbf {Gr}}\times {\mathbf {V}})^2$ to $({\mathbf {Gr}}\times {\mathbf {V}})^2$.

3.4 Coherent mirabolic category and convolutions

We write $\Pi E$ for an odd vector space obtained from a vector space $E$ by reversing the parity.

We fix a pair of $N$-dimensional vector spaces $V_1\simeq {\mathbb {C}}^N\simeq V_2$. We consider the Lie superalgebra ${\mathfrak {g}}{\mathfrak {l}}(N|N)={\mathfrak {g}}{\mathfrak {l}}(V_1\oplus \Pi V_2)$. We have ${\mathfrak {g}}{\mathfrak {l}}(N|N)={\mathfrak {g}}_{\bar 0}\oplus {\mathfrak {g}}_{\bar 1}$, where ${\mathfrak {g}}_{\bar 1}=\Pi {\operatorname{Hom}}(V_1,V_2)\oplus \Pi {\operatorname{Hom}}(V_2,V_1)$, and ${\mathfrak {g}}_{\bar 0}=\operatorname{End}(V_1)\oplus \operatorname{End}(V_2)$. We set $G_{\bar 0}=\operatorname{GL}(V_1)\times \operatorname{GL}(V_2)$. We consider the dg-algebraFootnote ⁶ ${\mathfrak {G}}_{1,1}^\bullet ={\operatorname{Sym}}({\mathfrak {g}}_{\bar 1}[-1])$ with zero differential, and the triangulated category $D^{G_{\bar 0}}_{\operatorname{perf}}({\mathfrak {G}}_{1,1}^\bullet )$ obtained by localization (with respect to quasi-isomorphisms) of the category of perfect $G_{\bar 0}$-equivariant dg-${\mathfrak {G}}_{1,1}^\bullet$-modules. The category $D^{G_{\bar 0}}_{\operatorname{perf}}({\mathfrak {G}}_{1,1}^\bullet )$ is monoidal with respect to ${\mathcal {M}},{\mathcal {M}}'\mapsto {\mathcal {M}}\,{\otimes} _{{\mathfrak {G}}_{1,1}^\bullet }\ {\mathcal {M}}'$; see § 3.7 below.

We will also need two more versions of ${\mathfrak {G}}_{1,1}^\bullet$, namely

\begin{gather*} {\mathfrak{G}}_{0,2}^\bullet{=}{\operatorname{Sym}}({{\operatorname{Hom}}}(V_1,V_2))\,{\otimes}\,{\operatorname{Sym}}({{\operatorname{Hom}}}(V_2,V_1)[{-}2]),\\ {\mathfrak{G}}_{2,0}^\bullet{=}{\operatorname{Sym}}({{\operatorname{Hom}}}(V_1,V_2)[{-}2])\,{\otimes}\,{\operatorname{Sym}}({{\operatorname{Hom}}}(V_2,V_1)), \end{gather*}

and the corresponding triangulated categories $D^{G_{\bar 0}}_{\operatorname{perf}}({\mathfrak {G}}_{0,2}^\bullet )$ and $D^{G_{\bar 0}}_{\operatorname{perf}}({\mathfrak {G}}_{2,0}^\bullet )$. We will now define the monoidal structures on $D^{G_{\bar 0}}_{\operatorname{perf}}({\mathfrak {G}}_{0,2}^\bullet )$ and $D^{G_{\bar 0}}_{\operatorname{perf}}({\mathfrak {G}}_{2,0}^\bullet )$.

We consider the variety ${\mathcal {Q}}^A$ (respectively, ${\mathcal {Q}}^B$) of sextuples

\begin{gather*} A\in{{\operatorname{Hom}}}(V_1,V_2),\quad B\in{{\operatorname{Hom}}}(V_2,V_1),\quad A'\in{{\operatorname{Hom}}}(V'_1,V_2),\\ B'\in{{\operatorname{Hom}}}(V_2,V'_1),\quad A''\in{{\operatorname{Hom}}}(V_1,V'_1),\quad B''\in{{\operatorname{Hom}}}(V'_1,V_1), \end{gather*}

such that

\[ A=A'A'',\ B'=A''B,\ B''=BA'\ ( \operatorname{respectively},\ B=B''B',\ A'=AB'',\ A''=B'A) \]

(here $V'_1$ is a copy of $V_1$). Clearly, ${\mathcal {Q}}^A\simeq {\operatorname{Hom}}(V_1,V'_1)\times {\operatorname{Hom}}(V'_1,V_2)\times {\operatorname{Hom}}(V_2,V_1)$, and ${\mathcal {Q}}^B\simeq {\operatorname{Hom}}(V'_1,V_1)\times {\operatorname{Hom}}(V_2,V'_1)\times {\operatorname{Hom}}(V_1,V_2)$. We have

(3.4.1)

We denote ${\operatorname{Hom}}(V_1,V_2)\times {\operatorname{Hom}}(V_2,V_1)\times {\operatorname{Hom}}(V'_1,V_2)$ $\times {\operatorname{Hom}}(V_2,V'_1)\times {\operatorname{Hom}}(V_1,V'_1)\times {\operatorname{Hom}}(V'_1,V_1)$ by ${\mathcal {H}}$. We have the natural projections

\begin{gather*} {\operatorname{pr}}_{12}\colon{\mathcal{H}}\to{{\operatorname{Hom}}}(V_1,V_2)\times{{\operatorname{Hom}}}(V_2,V_1),\quad {\operatorname{pr}}_{1'2}\colon{\mathcal{H}}\to {{\operatorname{Hom}}}(V'_1,V_2)\times{{\operatorname{Hom}}}(V_2,V'_1),\\ {\operatorname{pr}}_{11'}\colon{\mathcal{H}}\to{{\operatorname{Hom}}}(V_1,V'_1)\times{{\operatorname{Hom}}}(V'_1,V_1). \end{gather*}

The group $G_{\mathcal {Q}}:=\operatorname{GL}(V_1)\times \operatorname{GL}(V'_1)\times \operatorname{GL}(V_2)$ naturally acts on ${\mathcal {H}}$:

\[ (g_1,g'_1,g_2)(A,A',A'',B,B',B'')= (g_2Ag_1^{{-}1},g_2A'g'^{{-}1}_1,g'_1A''g_1^{{-}1},g_1Bg_2^{{-}1},g'_1B'g_2^{{-}1},g_1B''g'^{{-}1}_1). \]

The projections ${\operatorname {pr}}_{12},{\operatorname {pr}}_{1'2},{\operatorname {pr}}_{11'}$ are equivariant with respect to the same-named projections from $G_{\mathcal {Q}}$ to $\operatorname{GL}(V_1)\times \operatorname{GL}(V_2)$, $\operatorname{GL}(V'_1)\times \operatorname{GL}(V_2)$, $\operatorname{GL}(V_1)\times \operatorname{GL}(V'_1)$.

Given ${\mathcal {M}}_{1'2}\in {\mathsf {Coh}}^{\operatorname{GL}(V'_1)\times \operatorname{GL}(V_2)}\big ({\operatorname{Hom}}(V'_1,V_2)\times {\operatorname{Hom}}(V_2,V'_1)\big )$ and ${\mathcal {M}}_{11'}\in {\mathsf {Coh}}^{\operatorname{GL}(V_1)\times \operatorname{GL}(V'_1)}$ $\big ({\operatorname{Hom}}(V_1,V'_1)\times {\operatorname{Hom}}(V'_1,V_1)\big )$, we set

\begin{align*} {\mathcal{M}}_{11'}{\stackrel {A}*}{\mathcal{M}}_{1'2}&:= {\operatorname{pr}}_{12*}({\operatorname{pr}}_{11'}^*{\mathcal{M}}_{11'}\,{\otimes}_{{\mathbb{C}}[{\mathcal{H}}]}\ {\mathbb{C}}[{\mathcal{Q}}^A]\,{\otimes}_{{\mathbb{C}}[{\mathcal{H}}]}\ {\operatorname{pr}}_{1'2}^*{\mathcal{M}}_{1'2})^{{{\operatorname{GL}}}(V'_1)}\\ &\quad \in {\mathsf{Coh}}^{{{\operatorname{GL}}}(V_1)\times{{\operatorname{GL}}}(V_2)}\big({{\operatorname{Hom}}}(V_1,V_2)\times{{\operatorname{Hom}}}(V_2,V_1)\big),\end{align*}

\begin{align*} {\mathcal{M}}_{11'}{\stackrel {B}*}{\mathcal{M}}_{1'2}&:= {\operatorname{pr}}_{12*}({\operatorname{pr}}_{11'}^*{\mathcal{M}}_{11'}\,{\otimes}_{{\mathbb{C}}[{\mathcal{H}}]}\ {\mathbb{C}}[{\mathcal{Q}}^B]\,{\otimes}_{{\mathbb{C}}[{\mathcal{H}}]}\ {\operatorname{pr}}_{1'2}^*{\mathcal{M}}_{1'2})^{{{\operatorname{GL}}}(V'_1)}\\ &\quad\in {\mathsf{Coh}}^{{{\operatorname{GL}}}(V_1)\times{{\operatorname{GL}}}(V_2)}\big({{\operatorname{Hom}}}(V_1,V_2) \times{{\operatorname{Hom}}}(V_2,V_1)\big).\end{align*}

We will actually need the following modifications of these functors:

\[ {\stackrel {B}*}\colon D^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{G}}_{0,2}^\bullet)\times D^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{G}}_{0,2}^\bullet)\to D^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{G}}_{0,2}^\bullet),\quad {\stackrel {A}*}\colon D^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{G}}_{2,0}^\bullet)\times D^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{G}}_{2,0}^\bullet)\to D^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{G}}_{2,0}^\bullet), \]

obtained using the dg-algebras with trivial differentials ${\mathbb {C}}[{\mathcal {H}}]_{0,2}^\bullet$ and ${\mathbb {C}}[{\mathcal {H}}]_{2,0}^\bullet$ respectively, where

\begin{align*} {\mathbb{C}}[{\mathcal{H}}]_{0,2}^\bullet&={\operatorname{Sym}}({{\operatorname{Hom}}}(V_1,V_2))\,{\otimes}\,{\operatorname{Sym}}({{\operatorname{Hom}}}(V_2,V_1)[{-}2]) \,{\otimes}\,{\operatorname{Sym}}({{\operatorname{Hom}}}(V'_1,V_2))\\ &\quad \,{\otimes}\,{\operatorname{Sym}}({{\operatorname{Hom}}}(V_2,V'_1)[{-}2])\,{\otimes}\,{\operatorname{Sym}}({{\operatorname{Hom}}}(V_1,V'_1))\,{\otimes}\,{\operatorname{Sym}}({{\operatorname{Hom}}}(V'_1,V_1)[{-}2]), \end{align*}

\begin{align*} {\mathbb{C}}[{\mathcal{H}}]_{2,0}^\bullet&={\operatorname{Sym}}({{\operatorname{Hom}}}(V_1,V_2)[{-}2])\,{\otimes}\,{\operatorname{Sym}}({{\operatorname{Hom}}}(V_2,V_1)) \,{\otimes}\,{\operatorname{Sym}}({{\operatorname{Hom}}}(V'_1,V_2)[{-}2])\\ &\quad \,{\otimes}\,{\operatorname{Sym}}({{\operatorname{Hom}}}(V_2,V'_1))\,{\otimes}\,{\operatorname{Sym}}({{\operatorname{Hom}}}(V_1,V'_1)[{-}2])\,{\otimes}\,{\operatorname{Sym}}({{\operatorname{Hom}}}(V'_1,V_1)). \end{align*}

(we identify ${\mathbb {C}}[{\operatorname{Hom}}(U,W)]$ with ${\operatorname{Sym}}{\operatorname{Hom}}(W,U)$).

3.5 Localization, coherent

We identify

\[ {{\operatorname{Hom}}}(V_1,V_2)={{\operatorname{Hom}}}(V_2,V_1)^*,\quad {{\operatorname{Hom}}}(V_2,V_1)={{\operatorname{Hom}}}(V_1,V_2)^*, \]

so that

\[ {\operatorname{Sym}}({{\operatorname{Hom}}}(V_1,V_2))={\mathbb{C}}[{{\operatorname{Hom}}}(V_2,V_1)],\quad {\operatorname{Sym}}({{\operatorname{Hom}}}(V_2,V_1))={\mathbb{C}}[{{\operatorname{Hom}}}(V_1,V_2)]. \]

We have an open subvariety ${\operatorname{Isom}}(V_2,V_1)\subset {\operatorname{Hom}}(V_2,V_1)$, so that ${\mathbb {C}}[{\operatorname{Hom}}(V_2,V_1)]\subset {\mathbb {C}}[{\operatorname{Isom}}(V_2,V_1)]$. We set

\[ {\mathfrak{B}}^\bullet:={\mathbb{C}}[{{\operatorname{Isom}}}(V_2,V_1)]\,{\otimes}\,{\operatorname{Sym}}({{\operatorname{Hom}}}(V_2,V_1)[{-}2]) \]

(a dg-algebra with trivial differential). Similarly, we define

\[ {\mathfrak{A}}^\bullet:={\operatorname{Sym}}({{\operatorname{Hom}}}(V_1,V_2)[{-}2])\,{\otimes}\,{\mathbb{C}}[{{\operatorname{Isom}}}(V_1,V_2)]. \]

An equivalent formulation of [Reference Bezrukavnikov and FinkelbergBF08, Theorem 5] is an existence of a monoidal equivalence $D^{G_{\bar 0}}_{\operatorname{perf}}({\mathfrak {A}}^\bullet )\cong D_{{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}})$ (and, changing the roles of $V_1,V_2$, a monoidal equivalence $D^{G_{\bar 0}}_{\operatorname{perf}}({\mathfrak {B}}^\bullet )\cong D_{{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}})$).

Since ${\mathfrak {A}}^\bullet$ (respectively, ${\mathfrak {B}}^\bullet$) is a localization of ${\mathfrak {G}}_{2,0}^\bullet$ (respectively, of ${\mathfrak {G}}_{0,2}^\bullet$), we have the restriction of scalars functors

\[ \operatorname{Res}_A\colon D^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{A}}^\bullet)\to\hat{D}{}^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{G}}_{2,0}^\bullet),\quad \operatorname{Res}_B\colon D^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{B}}^\bullet)\to\hat{D}{}^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{G}}_{0,2}^\bullet), \]

where $\hat {D}{}^{G_{\bar 0}}_{\operatorname{perf}}({\mathfrak {G}}_{?,?}^\bullet )$ stands for the Ind-completion of $D^{G_{\bar 0}}_{\operatorname{perf}}({\mathfrak {G}}_{?,?}^\bullet )$.

However, one can check that, for ${\mathcal {N}}\in D^{G_{\bar 0}}_{\operatorname{perf}}({\mathfrak {A}}^\bullet )$ and ${\mathcal {M}}\in D^{G_{\bar 0}}_{\operatorname{perf}}({\mathfrak {G}}_{2,0}^\bullet )$, both convolutions $\operatorname {Res}_A({\mathcal {N}}){\stackrel {A}*}{\mathcal {M}}$ and ${\mathcal {M}}{\stackrel {A}*}\operatorname {Res}_A({\mathcal {N}})$ lie in $D^{G_{\bar 0}}_{\operatorname{perf}}({\mathfrak {G}}_{2,0}^\bullet )\subset \hat {D}{}^{G_{\bar 0}}_{\operatorname{perf}}({\mathfrak {G}}_{2,0}^\bullet )$. Thus we have the left and right convolution actions

\[ {\stackrel {A}*}\colon D^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{A}}^\bullet)\times D^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{G}}_{2,0}^\bullet)\to D^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{G}}_{2,0}^\bullet),\quad D^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{G}}_{2,0}^\bullet)\times D^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{A}}^\bullet) \to D^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{G}}_{2,0}^\bullet), \]

and similarly,

\[ {\stackrel {B}*}\colon D^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{B}}^\bullet)\times D^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{G}}_{0,2}^\bullet)\to D^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{G}}_{0,2}^\bullet),\quad D^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{G}}_{0,2}^\bullet)\times D^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{B}}^\bullet) \to D^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{G}}_{0,2}^\bullet). \]

3.6 Renormalizations

The action of the center $Z(\operatorname{GL}(V_1))\cong {\mathbb {G}}_m$ on an object ${\mathcal {M}}\in D^{G_{\bar 0}}_{\operatorname{perf}}({\mathfrak {G}}_{1,1}^\bullet )$ defines a grading, and the corresponding degrees will be denoted by $\deg _1$. Similarly, the action of $Z(\operatorname{GL}(V_2))\cong {\mathbb {G}}_m$ gives rise to another grading with degrees denoted by $\deg _2$. The cohomological degrees will be denoted simply by $\deg$. Clearly, the degrees of the generators are as follows:

\begin{align*} \deg_2({{\operatorname{Hom}}}(V_1,V_2))&=\deg_1({{\operatorname{Hom}}}(V_2,V_1))=1,\\ \deg_1({{\operatorname{Hom}}}(V_1,V_2))&=\deg_2({{\operatorname{Hom}}}(V_1,V_2))={-}1. \end{align*}

Hence changing cohomological degrees by the formula $\deg \leadsto \deg +\deg _1$ (respectively, $\deg \leadsto \deg -\deg _2$) yields equivalences

\begin{gather*} D^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{G}}_{2,0}^\bullet)\xrightarrow{\varrho_{\operatorname{right}}}D^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{G}}_{1,1}^\bullet) \xrightarrow{\varrho_{\operatorname{right}}}D^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{G}}_{0,2}^\bullet),\\ D^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{G}}_{2,0}^\bullet)\xrightarrow{\varrho_{\operatorname{left}}}D^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{G}}_{1,1}^\bullet) \xrightarrow{\varrho_{\operatorname{left}}}D^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{G}}_{0,2}^\bullet), \end{gather*}

respectively. The notation is due to the fact that $\varrho _{\operatorname {left}}$ commutes with the left action of ${\operatorname {Rep}}(\operatorname{GL}(V_1))$ on our categories, while $\varrho _{\operatorname {right}}$ commutes with the right action of ${\operatorname {Rep}}(\operatorname{GL}(V_2))$ on our categories.

Now recall the notation in the definition of categories $D_{?{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ (where $?={!},\ *,\ !*$) of § 3.1. Given ${\mathcal {F}}=({\mathcal {F}}_n)_{n>m}\in D_{!{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$, we define $\varrho _{\operatorname {right}}{\mathcal {F}}:=({\mathcal {F}}_n[-Nn])_{n>m}\in D_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$. Similarly, given ${\mathcal {F}}=({\mathcal {F}}_n)_{n>m}\in D_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ we define $\varrho _{\operatorname {right}}{\mathcal {F}}:=({\mathcal {F}}_n[-Nn])_{n>m}\in D_{*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$. The functors $\varrho _{\operatorname {right}}$ commute with the action of the monoidal category $D_{{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}})$ by right convolutions. Recall also that the affine Grassmannian is a union of connected components ${\mathbf {Gr}}={\bigsqcup} _{k\in {\mathbb {Z}}}{\mathbf {Gr}}^{(k)}$, where ${\mathbf {Gr}}^{(k)}$ parametrizes the lattices of virtual dimension $k$ (e.g. $\dim (t{\mathbf {V}}_0)=-N$). For ${\mathcal {F}}$ supported on ${\mathbf {Gr}}^{(k)}\times {\mathbf {V}}$ we set $\varrho _{\operatorname {left}}({\mathcal {F}}):=\varrho _{\operatorname {right}}({\mathcal {F}})[-k]$. Then the functors

\[ \varrho_{\operatorname{left}}\colon D_{!{\mathbf{G}}_{\mathbf{O}}}({\mathbf{Gr}}\times{\mathbf{V}})\to D_{!*{\mathbf{G}}_{\mathbf{O}}}({\mathbf{Gr}}\times{\mathbf{V}}),\ D_{!*{\mathbf{G}}_{\mathbf{O}}}({\mathbf{Gr}}\times{\mathbf{V}})\to D_{*{\mathbf{G}}_{\mathbf{O}}}({\mathbf{Gr}}\times{\mathbf{V}}) \]

commute with the action of the monoidal category $D_{{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}})$ by the left convolutions.

Our goal is the following theorem.

Theorem 3.6.1 There exist monoidalFootnote ⁷ equivalences of triangulated categories

(3.6.1)

(the vertical equivalences are not monoidal). The squares are commutative. The horizontal equivalences commute with the actions of the monoidal spherical Hecke category ${\operatorname {Perv}}_{{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}})\cong {\operatorname {Rep}}(\operatorname{GL}_N)$ by the left and right convolutions.

The proof will be given in § 3.16 after some necessary preparation.

3.7 Super

Strictly speaking, in accordance with the footnote at the beginning of § 3.4, we should consider the category $SD^{G_{\bar 0}}_{\operatorname{perf}}({\mathfrak {G}}_{1,1}^\bullet )$ of super dg-modules over the superalgebra ${\mathfrak {G}}_{1,1}^\bullet ={\operatorname{Sym}}({\mathfrak {g}}_{\bar 1}[-1])$. The latter superalgebra is supercommutative, hence we have a symmetric monoidal structure $\,{\otimes} _{{\mathfrak {G}}_{1,1}^\bullet }$ on the category $SD^{G_{\bar 0}}_{\operatorname{perf}}({\mathfrak {G}}_{1,1}^\bullet )$. The equivalence $\Phi ^{1,1}$ of Theorem 3.6.1 can be upgraded to a monoidal equivalence $S\Phi ^{1,1}\colon \big (SD^{G_{\bar 0}}_{\operatorname{perf}}({\mathfrak {G}}_{1,1}^\bullet ),\ \,{\otimes} _{{\mathfrak {G}}_{1,1}^\bullet }\big ){\buildrel \sim \over \longrightarrow} \big (SD_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}}),\ \star \big )$ to the derived category of sheaves with coefficients in super vector spaces.

However, the action of the central element $({\operatorname{Id}}_{V_1},-{\operatorname{Id}}_{V_2})\in G_{\bar 0}$ on an object of $D^{G_{\bar 0}}_{\operatorname{perf}}({\mathfrak {G}}_{1,1}^\bullet )$ equips this object with an extra ${\mathbb {Z}}/2{\mathbb {Z}}$-grading, and thus defines a fully faithful functor $D^{G_{\bar 0}}_{\operatorname{perf}}({\mathfrak {G}}_{1,1}^\bullet )\to SD^{G_{\bar 0}}_{\operatorname{perf}}({\mathfrak {G}}_{1,1}^\bullet )$ of a ‘superization’, such that its essential image is closed under the monoidal structure $\,{\otimes} _{{\mathfrak {G}}_{1,1}^\bullet }$. This defines the desired monoidal structure $\,{\otimes} _{{\mathfrak {G}}_{1,1}^\bullet }$ on the category $D^{G_{\bar 0}}_{\operatorname{perf}}({\mathfrak {G}}_{1,1}^\bullet )$.

3.8 Koszul equivalence

We consider the following complex $H^\bullet$ of odd vector spaces living in degrees $0,1\colon {\mathfrak {g}}_{\bar 1}\stackrel {{\operatorname{Id}}}{\longrightarrow }{\mathfrak {g}}_{\bar 1}$. We define the Koszul complex $K^\bullet$ as the symmetric algebra ${\operatorname{Sym}}(H^\bullet )$. The degree-$0$ part

\[ K^0=\Lambda\big({{\operatorname{Hom}}}(V_1,V_2)\oplus{{\operatorname{Hom}}}(V_2,V_1)\big)=:\Lambda \]

(as a vector space, with a superstructure disregarded). We turn $K^\bullet$ into a dg-${\mathfrak {G}}^\bullet _{1,1}-\Lambda$-bimodule by letting ${\mathfrak {G}}^\bullet _{1,1}$ act by multiplication, and $\Lambda$ by differentiation. Note that $K^{\bullet}$ is quasi-isomorphic to ${\mathbb {C}}$ in degree 0 as a complex of vector spaces, but not as a dg-${\mathfrak {G}}^\bullet _{1,1}-\Lambda$-bimodule. We consider the derived category $D_{\operatorname {fd}}^{G_{\bar 0}}(\Lambda )$ of finite-dimensional complexes of $G_{\bar 0}\ltimes \Lambda$-modules. If we remember the superstructure of $\Lambda$, we obtain the corresponding category of super dg-modules $SD_{\operatorname {fd}}^{G_{\bar 0}}(\Lambda )$. We have the Koszul equivalence functors

\[ \varkappa\colon D_{\operatorname{fd}}^{G_{\bar0}}(\Lambda){\buildrel \sim \over \longrightarrow} D^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{G}}_{1,1}^\bullet),\quad SD_{\operatorname{fd}}^{G_{\bar0}}(\Lambda){\buildrel \sim \over \longrightarrow} SD^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{G}}_{1,1}^\bullet),\quad {\mathcal{N}}\mapsto K^\bullet\,{\otimes}_\Lambda\ {\mathcal{N}}. \]

Here is an equivalent definition of the category $SD_{\operatorname {fd}}^{G_{\bar 0}}(\Lambda )$. We consider the following degeneration $\underline {\mathfrak {gl}}(N|N)$ of the Lie superalgebra ${\mathfrak {gl}}({N|N})$: the supercommutator of the even elements (with even or odd elements) remains intact, but the supercommutator of any two odd elements is set to zero. Let $SD_{\operatorname {int}}(\underline {\mathfrak {gl}}(N|N))$ denote the derived category of bounded complexes of integrable $\underline {\mathfrak {gl}}(N|N)$-modules (note that the even part of $\underline {\mathfrak {gl}}(N|N)$ is just ${\mathfrak {g}}_{\bar 0}$, and the integrability is nothing but ${\mathfrak {g}}_{\bar 0}$-integrability, i.e. $G_{\bar 0}$-equivariance). Then $SD_{\operatorname {int}}(\underline {\mathfrak {gl}}(N|N))\cong SD_{\operatorname {fd}}^{G_{\bar 0}}(\Lambda )$ tautologically. The resulting Koszul equivalence $\varkappa \colon SD_{\operatorname {int}}(\underline {\mathfrak {gl}}(N|N)){\buildrel \sim \over \longrightarrow} SD^{G_{\bar 0}}_{\operatorname{perf}}({\mathfrak {G}}_{1,1}^\bullet )$ is monoidal with respect to the usual tensor structure on the left-hand side and $\,{\otimes} _{{\mathfrak {G}}^\bullet _{1,1}}$ on the right-hand side.

As in § 3.7, the action of $({\operatorname{Id}}_{V_1},-{\operatorname{Id}}_{V_2})\in G_{\bar 0}$ gives rise to a ‘superization’ fully faithful functor $D_{\operatorname {fd}}^{G_{\bar 0}}(\Lambda )\to SD_{\operatorname {fd}}^{G_{\bar 0}}(\Lambda )\cong SD_{\operatorname {int}}(\underline {\mathfrak {gl}}(N|N))$ with the essential image closed under the tensor structure. This defines the tensor structure on $D_{\operatorname {fd}}^{G_{\bar 0}}(\Lambda )$ such that the Koszul equivalence $\varkappa \colon D_{\operatorname {fd}}^{G_{\bar 0}}(\Lambda ){\buildrel \sim \over \longrightarrow} D^{G_{\bar 0}}_{\operatorname{perf}}({\mathfrak {G}}_{1,1}^\bullet )$ is monoidal.

Corollary 3.8.1 (of Theorem 3.6.1)

(a) The composed equivalence

\[ \Phi^{1,1}\circ\varkappa\colon D_{\operatorname{fd}}^{G_{\bar0}}(\Lambda){\buildrel \sim \over \longrightarrow} D_{!*{\mathbf{G}}_{\mathbf{O}}}({\mathbf{Gr}}\times{\mathbf{V}}) \]

is exact with respect to the tautological $t$-structure on $D_{\operatorname {fd}}^{G_{\bar 0}}(\Lambda )$ and the perverse $t$-structure on $D_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$.

(b) This equivalence is monoidal with respect to the tensor structure on $D_{\operatorname {fd}}^{G_{\bar 0}}(\Lambda )$ and the fusion $\star$ on $D_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$.

(c) The equivariant derived category $D_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ is equivalent to the bounded derived category of the abelian category ${\operatorname {Perv}}_{{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$.

3.9 De-equivariantized Ext algebra

Recall from [Reference Finkelberg, Ginzburg and TravkinFGT09, Proposition 8] that the ${\mathbf {G}}_{\mathbf {O}}$-orbits in ${\mathbf {Gr}}\times {\vphantom {j^{X^2}}\smash {\overset {\circ }{\vphantom {\rule {0pt}{0.55em}}\smash {\mathbf {V}}}}}$ are numbered by bisignatures $({\boldsymbol {\lambda }},{\boldsymbol {\mu }})$ where both ${\boldsymbol {\lambda }}$ and ${\boldsymbol {\mu }}$ have length $N$. The IC extension of the constant one-dimensional local system on such an orbit is denoted by ${\operatorname {IC}}_{({\boldsymbol {\lambda }},{\boldsymbol {\mu }})}$. In particular, ${\operatorname {IC}}_{(0^N,0^N)}$ is the constant sheaf on ${\mathbf {Gr}}^0\times {\mathbf {V}}_0$, to be denoted by $E_0$ for short. Also recall from [Reference Finkelberg, Ginzburg and TravkinFGT09, § 3] that the left and right actions of the monoidal Satake category ${\operatorname {Perv}}_{{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}})\cong {\operatorname {Rep}}(\operatorname{GL}_N)$ on $D_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ by convolutions respect the perverse $t$-structure with the heart ${\operatorname {Perv}}_{{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})\subset D_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$. As has been mentioned in § 3.6, the right actions of $D_{{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}})$ on $D_{?{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ commute with the equivalences $\varrho _{\operatorname {right}}$, but the left actions only commute with $\varrho _{\operatorname {right}}$ up to cohomological shifts depending on the connected components of ${\mathbf {Gr}}$.

We restrict the left and right actions of $D_{{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}})$ on $D_{?{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ to the left and right actions of ${\operatorname {Perv}}_{{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}})\cong {\operatorname {Rep}}(\operatorname{GL}_N)$. Thus we obtain the action of ${\operatorname {Rep}}({\mathsf {G}})$ for ${\mathsf {G}}=\operatorname{GL}_N\times \operatorname{GL}_N$. Let $D^{{\operatorname{deeq}}}_{?{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ denote the corresponding de-equivariantized category (see [Reference Arkhipov and GaitsgoryAG03] in the setting of abelian categories and [Reference GaitsgoryGai15] in the setting of dg-categories). Recall that to construct $D^{{\operatorname{deeq}}}_{?{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$, we first consider objects of the ind-completion of $D_{?{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ endowed with an action of $R_{\mathsf {G}}$ where $R_{\mathsf {G}}$ is the regular representation of ${\mathsf {G}}$ considered as a ring object in ${\operatorname {Rep}}({\mathsf {G}})$. Then we take compact objects of the resulting category. More explicitly, an object of $D^{{\operatorname{deeq}}}_{?{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ is an object ${\mathcal {F}}$ of $D_{?{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ together with a system of isomorphisms ${\operatorname {IC}}_{\boldsymbol {\lambda }}*{\mathcal {F}}*{\operatorname {IC}}_{\boldsymbol {\mu }}{\buildrel \sim \over \longrightarrow} V^*_{\boldsymbol {\lambda }}\,{\otimes}\, V^*_{\boldsymbol {\mu }}\,{\otimes}\, {\mathcal {F}}$ (recall that the geometric Satake equivalence takes ${\operatorname {IC}}_{\boldsymbol {\lambda }}$ to $V^*_{\boldsymbol {\lambda }}$) for any signatures ${\boldsymbol {\lambda }},{\boldsymbol {\mu }}$, satisfying some natural compatibilities with respect to direct sums and tensor products. By definition we have a natural forgetful functor $D^{{\operatorname{deeq}}}_{?{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})\to D_{?{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$. This functor admits a left adjoint that sends an object ${\mathcal {F}}$ to $R_{\mathsf {G}}*{\mathcal {F}}=\bigoplus _{{\boldsymbol {\lambda }},{\boldsymbol {\mu }}}V_{\boldsymbol {\lambda }}\,{\otimes}\, V_{\boldsymbol {\mu }}\,{\otimes}\, {\operatorname {IC}}_{\boldsymbol {\lambda }}*{\mathcal {F}}*{\operatorname {IC}}_{\boldsymbol {\mu }}$.

Thus, given ${\mathcal {F}}_1,{\mathcal {F}}_2\in D_{?{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$, we denote the corresponding objects of the de-equivariantized category by the same symbols, and we have

(3.9.1)\begin{equation} \operatorname{RHom}_{D^{{{\operatorname{deeq}}}}_{?{\mathbf{G}}_{\mathbf{O}}}({\mathbf{Gr}}\times{\mathbf{V}})}({\mathcal{F}}_1,{\mathcal{F}}_2)= \bigoplus_{{\boldsymbol{\lambda}},{\boldsymbol{\mu}}}\operatorname{RHom}_{D_{?{\mathbf{G}}_{\mathbf{O}}}({\mathbf{Gr}}\times{\mathbf{V}})}({\mathcal{F}}_1,{\operatorname{IC}}_{\boldsymbol{\lambda}}*{\mathcal{F}}_2*{\operatorname{IC}}_{\boldsymbol{\mu}})\,{\otimes}\, V_{\boldsymbol{\lambda}}\,{\otimes}\, V_{\boldsymbol{\mu}}. \end{equation}

We denote the dg-algebra ${\operatorname{Ext}}^\bullet _{D^{{\operatorname{deeq}}}_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})}(E_0,E_0)$ (with trivial differential) by ${\mathfrak {E}}^\bullet$. Since it is an Ext-algebra in the de-equivariantized category between objects induced from the original category, it is automatically equipped with an action of $\operatorname{GL}_N\times \operatorname{GL}_N=\operatorname{GL}(V_1)\times \operatorname{GL}(V_2)=G_{\bar 0}$, and we can consider the corresponding triangulated category $D^{G_{\bar 0}}_{\operatorname{perf}}({\mathfrak {E}}^\bullet )$.

We also consider the left and right actions of the monoidal Satake category ${\operatorname {Perv}}_{{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}})\cong {\operatorname {Rep}}(\operatorname{GL}_N)$ on $D_{{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}})$ by convolutions. Let $D^{{\operatorname{deeq}}}_{{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}})$ denote the corresponding de-equivariantized category. Then the dg-algebra $\operatorname {RHom}_{D^{{\operatorname{deeq}}}_{{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}})}({\operatorname {IC}}_0,{\operatorname {IC}}_0)$ is formal, that is, it is quasi-isomorphic to the graded algebra ${\operatorname{Ext}}^\bullet _{D^{{\operatorname{deeq}}}_{{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}})}({\operatorname {IC}}_0,{\operatorname {IC}}_0)$ with trivial differential. Furthermore, it follows from [Reference Bezrukavnikov and FinkelbergBF08, Theorem 5] that there is a natural isomorphism ${\operatorname{Ext}}^\bullet _{D^{{\operatorname{deeq}}}_{{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}})}({\operatorname {IC}}_0,{\operatorname {IC}}_0)\cong {\mathfrak {A}}^\bullet$ (in the notation of § 3.5).

3.10 Localization, constructible

We have an automorphism $\alpha \colon {\mathbf {Gr}}\times {\mathbf {V}}{\buildrel \sim \over \longrightarrow} {\mathbf {Gr}}\times {\mathbf {V}}$, $(L,v)\mapsto (L,tv)$. We have a morphism of endofunctors $\alpha ^*\to {\operatorname{Id}}\colon D_{*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})\to D_{*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ constructed as follows. We consider a family of automorphisms $\boldsymbol {\alpha }\colon {\mathbb {A}}^1\times {\mathbf {Gr}}\times {\mathbf {V}}\hookrightarrow {\mathbf {Gr}}\times {\mathbf {V}}$, $(c,L,v)\mapsto (L,(c+t)v)$, so that $\boldsymbol {\alpha }_0=\alpha$. Note that $\boldsymbol {\alpha }_c^*\cong {\operatorname{Id}}$ on $D_{*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ for $c\ne 0$. Now the desired morphism $\alpha ^*\to {\operatorname{Id}}$ is just the cospecializationFootnote ⁸ morphism from the stalk at $0\in {\mathbb {A}}^1$ to the nearby stalk. For ${\mathcal {F}}_1,{\mathcal {F}}_2\in D_{*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$, we have an inductive system

\[ \cdots\to\operatorname{RHom}_{D_{*{\mathbf{G}}_{\mathbf{O}}}({\mathbf{Gr}}\times{\mathbf{V}})}((\alpha^{n-1})^*{\mathcal{F}}_1,{\mathcal{F}}_2)\to \operatorname{RHom}_{D_{*{\mathbf{G}}_{\mathbf{O}}}({\mathbf{Gr}}\times{\mathbf{V}})}((\alpha^n)^*{\mathcal{F}}_1,{\mathcal{F}}_2)\to\cdots. \]

Note that it stabilizes since, by definition of $D_{*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$, the restriction of $(\alpha ^n)^*{\mathcal {F}}_1$ to the support of ${\mathcal {F}}_2$ becomes the pullback of an appropriate sheaf in $D_{{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}})$ for $n\gg 0$.

We define the localized category $D^{\operatorname {loc}}_{*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ as the category with the same objects as $D_{*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$, and with morphisms

\[ \operatorname{RHom}_{D^{\operatorname{loc}}_{*{\mathbf{G}}_{\mathbf{O}}}({\mathbf{Gr}}\times{\mathbf{V}})}({\mathcal{F}}_1,{\mathcal{F}}_2):= \varinjlim\operatorname{RHom}_{D_{*{\mathbf{G}}_{\mathbf{O}}}({\mathbf{Gr}}\times{\mathbf{V}})}((\alpha^n)^*{\mathcal{F}}_1,{\mathcal{F}}_2). \]

The tautological functor $D_{*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})\to D^{\operatorname {loc}}_{*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ is denoted by $u_0^*$ (‘restriction to $v\thickapprox 0$’).

We also have the Verdier dual (to the above $\alpha ^*\to {\operatorname{Id}}$) morphism of endofunctors ${\operatorname{Id}}\to \alpha ^! \colon D_{!{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})\to D_{!{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$. This gives rise to an inductive system for ${\mathcal {F}}_1,{\mathcal {F}}_2\in D_{!{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$,

\[ \cdots\to\operatorname{RHom}_{D_{!{\mathbf{G}}_{\mathbf{O}}}({\mathbf{Gr}}\times{\mathbf{V}})}({\mathcal{F}}_1,(\alpha^{n-1})^!{\mathcal{F}}_2) \to\operatorname{RHom}_{D_{!{\mathbf{G}}_{\mathbf{O}}}({\mathbf{Gr}}\times{\mathbf{V}})}({\mathcal{F}}_1,(\alpha^n)^!{\mathcal{F}}_2)\to\cdots, \]

stabilizing for reasons similar to those above. We define the localized category $D^{{\operatorname {loc}}}_{!{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ as the category with the same objects as $D_{!{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$, and with morphisms

\[ \operatorname{RHom}_{D^{{\operatorname{loc}}}_{!{\mathbf{G}}_{\mathbf{O}}}({\mathbf{Gr}}\times{\mathbf{V}})}({\mathcal{F}}_1,{\mathcal{F}}_2):= \varinjlim\operatorname{RHom}_{D_{!{\mathbf{G}}_{\mathbf{O}}}({\mathbf{Gr}}\times{\mathbf{V}})}({\mathcal{F}}_1,(\alpha^n)^!{\mathcal{F}}_2). \]

The tautological functor $D_{!{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})\to D^{{\operatorname {loc}}}_{!{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ is denoted by $u_0^!$ (‘corestriction to $v\thickapprox 0$’).

We also have the projection ${\operatorname {pr}}\colon {\mathbf {Gr}}\times {\mathbf {V}}_0\to {\mathbf {Gr}}$, and the corresponding pullbacks ${\operatorname {pr}}^*\colon D_{{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}})\to D_{*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ and ${\operatorname {pr}}^!\colon D_{{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}})\to D_{!{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ (with the essential images supported on ${\mathbf {Gr}}\times {\mathbf {V}}_0\subset {\mathbf {Gr}}\times {\mathbf {V}}$).

Lemma 3.10.1 The compositions

\[ u_0^*\circ{\operatorname{pr}}^*\colon D_{{\mathbf{G}}_{\mathbf{O}}}({\mathbf{Gr}})\to D^{\operatorname{loc}}_{*{\mathbf{G}}_{\mathbf{O}}}({\mathbf{Gr}}\times{\mathbf{V}}),\quad u_0^!\circ{\operatorname{pr}}^!\colon D_{{\mathbf{G}}_{\mathbf{O}}}({\mathbf{Gr}})\to D^{{\operatorname{loc}}}_{!{\mathbf{G}}_{\mathbf{O}}}({\mathbf{Gr}}\times{\mathbf{V}}) \]

are equivalences of categories sending ${\operatorname {IC}}_0$ to $\varrho _{\operatorname {right}} E_0$, $\varrho _{\operatorname {right}}^{-1}E_0$, respectively.

The localizations of the de-equivariantized categories $D^{{\operatorname{deeq}}}_{*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ and $D^{{\operatorname{deeq}}}_{!{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ will be denoted by $D^{{\operatorname {loc}},{\operatorname{deeq}}}_{*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ and $D^{{\operatorname {loc}},{\operatorname{deeq}}}_{!{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$, respectively.

Recall the dg-algebras ${\mathfrak {A}}^\bullet ,{\mathfrak {B}}^\bullet$ introduced in § 3.5.

Note that the dg-algebra $\operatorname {RHom}_{D^{{\operatorname{deeq}}}_{*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\!\times\! {\mathbf {V}})}(\varrho _{\operatorname {right}} E_0,\varrho _{\operatorname {right}} E_0)$ (respectively, $\operatorname {RHom}_{D^{{\operatorname{deeq}}}_{!{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\!\times\! {\mathbf {V}})}$ $(\varrho _{\operatorname {right}}^{-1}E_0,\varrho _{\operatorname {right}}^{-1}E_0)$) is also formal, that is, it is quasi-isomorphic to ${\operatorname{Ext}}^\bullet _{D^{{\operatorname{deeq}}}_{*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})}(\varrho _{\operatorname {right}} E_0,$ $\varrho _{\operatorname {right}} E_0)$ (respectively, ${\operatorname{Ext}}^\bullet _{D^{{\operatorname{deeq}}}_{!{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})}(\varrho _{\operatorname {right}}^{-1}E_0,\varrho _{\operatorname {right}}^{-1}E_0)$) with trivial differential. If we disregard their gradings, they are both isomorphic to ${\operatorname{Ext}}^\bullet _{D^{{\operatorname{deeq}}}_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})}(E_0,E_0)$. We will denote the algebra ${\operatorname{Ext}}^\bullet _{D^{{\operatorname{deeq}}}_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})}(E_0,E_0)$ with grading forgotten by ${\operatorname{Ext}}_{D^{{\operatorname{deeq}}}_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})}(E_0,E_0)$; the same applies to ${\mathfrak {A}}^\bullet ,{\mathfrak {B}}^\bullet ,{\mathfrak {E}}^\bullet ,{\mathfrak {G}}^\bullet$. Thus we have equalities ${\operatorname{Ext}}_{D^{{\operatorname{deeq}}}_{*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})}(\varrho _{\operatorname {right}} E_0,\varrho _{\operatorname {right}} E_0)= {\operatorname{Ext}}_{D^{{\operatorname{deeq}}}_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})}(E_0,E_0)= {\operatorname{Ext}}_{D^{{\operatorname{deeq}}}_{!{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})}(\varrho _{\operatorname {right}}^{-1}E_0,\varrho _{\operatorname {right}}^{-1}E_0)$.

Lemma 3.10.3 The natural morphism

\[ {{\operatorname{Ext}}}^\bullet_{D^{{{\operatorname{deeq}}}}_{!{\mathbf{G}}_{\mathbf{O}}}({\mathbf{Gr}}\times{\mathbf{V}})}(\varrho_{\operatorname{right}}^{{-}1}E_0,\varrho_{\operatorname{right}}^{{-}1}E_0) \to{{\operatorname{Ext}}}^\bullet_{D^{{\operatorname{loc}},{{\operatorname{deeq}}}}_{!{\mathbf{G}}_{\mathbf{O}}}({\mathbf{Gr}}\times{\mathbf{V}})}(\varrho_{\operatorname{right}}^{{-}1}E_0,\varrho_{\operatorname{right}}^{{-}1}E_0) \]

is injective.

Proof. We have $\alpha ^!\varrho _{\operatorname {right}}^{-1}{\operatorname {IC}}_{(-1^N,1^N)}[N]\simeq \varrho _{\operatorname {right}}^{-1}E_0$, and we need to check that, for any ${\boldsymbol {\lambda }},{\boldsymbol {\mu }}$, the natural morphism

(3.10.1)\begin{align} & \vartheta\colon\operatorname{RHom}_{D_{!{\mathbf{G}}_{\mathbf{O}}}({\mathbf{Gr}}\times{\mathbf{V}})}(\varrho_{\operatorname{right}}^{{-}1}E_0, \varrho_{\operatorname{right}}^{{-}1}{\operatorname{IC}}_{({\boldsymbol{\lambda}},{\boldsymbol{\mu}})})\nonumber\\ &\quad \to\operatorname{RHom}_{D_{!{\mathbf{G}}_{\mathbf{O}}}({\mathbf{Gr}}\times{\mathbf{V}})}(\varrho_{\operatorname{right}}^{{-}1}{\operatorname{IC}}_{({-}1^N,1^N)}[{-}N], \varrho_{\operatorname{right}}^{{-}1}{\operatorname{IC}}_{({\boldsymbol{\lambda}},{\boldsymbol{\mu}})}) \end{align}

is injective. Indeed, since $\alpha ^!$ is a Hecke transformation endofunctor (namely, it is given by convolution ${\operatorname {IC}}_{(1^N)}*?*{\operatorname {IC}}_{(-1)^N}$ with invertible objects of the Satake category), the morphism $\vartheta$ is an endomorphism of the identity endofunctor of the de-equivariantized category $D^{{\operatorname{deeq}}}_{!{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$. In other words, $\vartheta$ is an element of the center of this category. But the natural morphism from the center to its localization is injective if and only if the multiplication by $\vartheta$ is injective.

We change the setting to the base field ${\mathbb {F}}_q$ as in the proof of Lemma 3.9.1. Then all the IC sheaves in question carry a natural Weil structure, and it was proved in the proof of Lemma 3.9.1 that both the left- and right-hand side of (3.10.1) are pure; it is immediate to see that they are pure of the same weight $w$. The cone of $\vartheta$ is $H^\bullet _{{\mathbf {G}}_{\mathbf {O}}}(\Omega _{(0^N,0^N)},\imath _0^!\varrho _{\operatorname {right}}^{-1}{\operatorname {IC}}_{({\boldsymbol {\lambda }},{\boldsymbol {\mu }})})$, where $\imath _0$ stands for the locally closed embedding of the ${\mathbf {G}}_{\mathbf {O}}$-orbit $\Omega _{(0^N,0^N)}={\mathbf {Gr}}^0\times ({\mathbf {V}}_0\,{\smallsetminus}\, t{\mathbf {V}}_0)\hookrightarrow {\mathbf {Gr}}\times {\mathbf {V}}$. Due to the pointwise purity of ${\operatorname {IC}}_{({\boldsymbol {\lambda }},{\boldsymbol {\mu }})}$ [Reference Finkelberg, Ginzburg and TravkinFGT09, § 3], $\imath _0^!{\operatorname {IC}}_{({\boldsymbol {\lambda }},{\boldsymbol {\mu }})}$ is a pure local system on $\Omega _{(0^N,0^N)}$; hence $H^\bullet _{{\mathbf {G}}_{\mathbf {O}}}(\Omega _{(0^N,0^N)},\imath _0^!{\operatorname {IC}}_{({\boldsymbol {\lambda }},{\boldsymbol {\mu }})})$ is also pure of weight $w$. It follows that the kernel of $\vartheta$ vanishes, and $\vartheta$ is injective.

3.11 Calculation of the Ext algebra

Recall that the first fundamental coweight of $\operatorname{GL}_N$ is $\omega _1=(1,0,\ldots ,0)$, and $\omega _1^*=(0,\ldots ,0,-1)$. We have $\operatorname {RHom}_{D_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})}({\operatorname {IC}}_{\omega _1}*E_0,E_0*{\operatorname {IC}}_{\omega _1})$ $= \operatorname {RHom}_{D_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})}(E_0,{\operatorname {IC}}_{\omega _1^*}*E_0*{\operatorname {IC}}_{\omega _1})$, $\operatorname {RHom}_{D_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})}(E_0*{\operatorname {IC}}_{\omega _1},{\operatorname {IC}}_{\omega _1}*E_0)$ $= \operatorname {RHom}_{D_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})}(E_0,{\operatorname {IC}}_{\omega _1}*E_0*{\operatorname {IC}}_{\omega _1^*})$. Now ${\operatorname {IC}}_{\omega _1}*E_0$ is the constant IC sheaf of the stratum closure formed by all the pairs $(L,v)$ such that the lattice $L$ contains ${\mathbf {V}}_0$ as a hyperplane, and $v\in L$. Furthermore, $E_0*{\operatorname {IC}}_{\omega _1}$ is the constant IC sheaf of the stratum closure formed by all the pairs $(L,v)$ such that the lattice $L$ contains ${\mathbf {V}}_0$ as a hyperplane, and $v\in {\mathbf {V}}_0$. In particular, the latter stratum closure is a smooth divisor in the former stratum closure, so we have canonical elements $h\in {\operatorname{Ext}}^1_{D_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})}({\operatorname {IC}}_{\omega _1}*E_0,E_0*{\operatorname {IC}}_{\omega _1})$ and $h^*\in {\operatorname{Ext}}^1_{D_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})}(E_0*{\operatorname {IC}}_{\omega _1},{\operatorname {IC}}_{\omega _1}*E_0)$. Hence we obtain the subspaces $h\,{\otimes}\, V_{\omega _1^*}\,{\otimes}\, V_{\omega _1}\subset {\mathfrak {E}}^1$ and $h^*\,{\otimes}\, V_{\omega _1}\,{\otimes}\, V_{\omega _1^*}\subset {\mathfrak {E}}^1$ (see § 3.9 for the definition of ${\mathfrak {E}}^\bullet$ and (3.9.1)). We identify the former subspace with ${\operatorname{Hom}}(V_1,V_2)$ and the latter one with ${\operatorname{Hom}}(V_2,V_1)$. Thus we obtain a homomorphism $\phi ^\bullet \colon {\operatorname{Sym}}\big (\Pi {\operatorname{Hom}}(V_1,V_2)[-1]\oplus \Pi {\operatorname{Hom}}(V_2,V_1)[-1]\big )$ $= {\operatorname{Sym}}({\mathfrak {g}}_{\bar 1}[-1])={\mathfrak {G}}_{1,1}^\bullet \to {\mathfrak {E}}^\bullet$ (due to commutativity of ${\mathfrak {E}}$).

Proof. We can and will disregard the grading. The morphism $\phi$ induces the morphism $\phi ^*\colon {\operatorname{Spec}}{\mathfrak {E}}\to {\mathfrak {g}}^*_{\bar 1}$ that is an isomorphism over the open subset ${\operatorname{Isom}}(V_2,V_1)\times {\operatorname{Hom}}(V_1,V_2)\subset {\mathfrak {g}}^*_{\bar 1}$ due to Corollary 3.10.2(a). Similarly, $\phi ^*$ is an isomorphism over the open subset ${\operatorname{Hom}}(V_2,V_1)\times {\operatorname{Isom}}(V_1,V_2)$ due to Corollary 3.10.2(b).

Since the complement to the union of these two open subsets has codimension 2 in ${\mathfrak {g}}_{\bar 1}^*$, we can apply Lemma 3.11.2 below. Note that the irreducibility of ${\operatorname{Spec}}{\mathfrak {E}}$ is guaranteed by Corollary 3.10.4. It remains only to check that the ratio of the above isomorphisms is the identity birational isomorphism between the varieties ${\operatorname{Isom}}(V_2,V_1)\times {\operatorname{Hom}}(V_1,V_2)$ and ${\operatorname{Hom}}(V_2,V_1)\times {\operatorname{Isom}}(V_1,V_2)$.

The composition $h\circ h^*\in {\operatorname{Ext}}^2_{D_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})}(E_0*{\operatorname {IC}}_{\omega _1},E_0*{\operatorname {IC}}_{\omega _1})$ is multiplication by the first Chern class of the normal line bundle ${\mathcal {N}}$ to the divisor $\operatorname {supp}(E_0*{\operatorname {IC}}_{\omega _1})$ in $\operatorname {supp}({\operatorname {IC}}_{\omega _1}*E_0)$. Recall that ${\mathbf {Gr}}^{\omega _1}\simeq {\mathbb {P}}^{N-1}$. The line bundle ${\mathcal {N}}$ is pulled back from the line bundle ${\mathcal {O}}(1)$ on ${\mathbb {P}}^{N-1}\simeq {\mathbf {Gr}}^{\omega _1}$. Recall also that the restriction of the determinant line bundle ${\mathcal {L}}$ from ${\mathbf {Gr}}$ to ${\mathbf {Gr}}^{\omega _1}$ is also isomorphic to ${\mathcal {O}}(1)$. We conclude that $h\circ h^*=c_1({\mathcal {L}})$.

On the other hand, in the equivariant Satake category

\[ D_{{\mathbf{G}}_{\mathbf{O}}}({\mathbf{Gr}})\cong D_{{\operatorname{perf}}}^{{{\operatorname{GL}}}_N}({\operatorname{Sym}}({\mathfrak{g}}{\mathfrak{l}}_N[{-}2]))\cong D_{{\operatorname{perf}}}^{G_{\bar0}}({\mathfrak{A}}^\bullet), \]

the first Chern class $c_1({\mathcal {L}})\in {\operatorname{Ext}}^2_{D_{{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}})}({\operatorname {IC}}_0*{\operatorname {IC}}_{\omega _1},{\operatorname {IC}}_0*{\operatorname {IC}}_{\omega _1})\subset {\mathfrak {A}}^2$ corresponds to the identity element (shifted by 2) ${\operatorname{Id}}\in {\operatorname{Hom}}(V_2,V_1)^*\,{\otimes}\, {\operatorname{Hom}}(V_2,V_1)$.

The lemma is proved.

Proof. Let $X_f=\pi ^{-1}(U_f)$ (respectively, $X_g=\pi ^{-1}(U_g)$), and write $j\colon U_f\cup U_g\hookrightarrow {\mathbb {A}}^n$ (respectively, $j_X\colon X_f\cup X_g\hookrightarrow X$) for the open embedding. We have the commutative diagram

The map $j^*$ in this diagram is an isomorphism by the assumption of codimension at least $2$. The map $j_X^*$ is injective since $X$ is irreducible. The assumptions imply also that the vertical map $(\pi |_{X_f})^*\oplus (\pi |_{X_g})^*$ on the right is an isomorphism. It follows that the vertical map $\pi ^*$ on the left must be an isomorphism, as required.

3.12 Restriction to ${\mathbf {Gr}}\times {\mathbf {V}}_0$

The existence of the desired equivalence $\Phi ^{1,1}$ of Theorem 3.6.1 follows from Lemmas 3.9.2 and 3.11.1. The equivalences $\Phi ^{0,2}$ and $\Phi ^{2,0}$ are obtained by conjugating with $\varrho _{\operatorname {right}}^{\pm 1}$. It remains to check their compatibility with monoidal structures.

We denote by $\bar \jmath _0$ (respectively, $\bar \jmath _{-1}$) the closed embedding ${\mathbf {Gr}}\times {\mathbf {V}}_0\hookrightarrow {\mathbf {Gr}}\times {\mathbf {V}}$ (respectively, ${\mathbf {Gr}}\times t{\mathbf {V}}_0\hookrightarrow {\mathbf {Gr}}\times {\mathbf {V}}$). We also denote by $\jmath _0$ the locally closed embedding ${\mathbf {Gr}}\times ({\mathbf {V}}_0\,{\smallsetminus}\, t{\mathbf {V}}_0)\hookrightarrow {\mathbf {Gr}}\times {\mathbf {V}}$. Our goal in this section is a description in terms of $\Phi ^{1,1}$ of the endofunctors $\bar \jmath _{0*}\bar \jmath {}_0^!$, $\jmath _{0*}\jmath _0^!\colon D_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})\to D_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$.

Recall the setup and notation of § 3.4. We have the natural morphisms

\begin{gather*} p\colon{\mathcal{Q}}^A\to{{\operatorname{Hom}}}(V_1,V_2)\times{{\operatorname{Hom}}}(V_2,V_1)=\Pi{\mathfrak{g}}^*_{\bar1},\\ q\colon{\mathcal{Q}}^A\to{{\operatorname{Hom}}}(V'_1,V_2)\times{{\operatorname{Hom}}}(V_2,V'_1)=\Pi{\mathfrak{g}}^*_{\bar1};\\ p(A,A',A'',B,B',B'')=(A,B),\quad q(A,A',A'',B,B',B'')=(A',B'). \end{gather*}

We also have the natural morphisms

\[ p,q\colon G_{\mathcal{Q}}\to G_{\bar0},\quad p(g_1,g'_1,g_2)=(g_1,g_2),\quad q(g_1,g'_1,g_2)=(g'_1,g_2). \]

Clearly, $p,q\colon {\mathcal {Q}}^A\to \Pi {\mathfrak {g}}^*_{\bar 1}$ are equivariant with respect to $p,q\colon G_{\mathcal {Q}}\to G_{\bar 0}$. Hence we have the convolution functor

\[ p_*q^*\colon{\mathsf{Coh}}^{G_{\bar0}}(\Pi{\mathfrak{g}}^*_{\bar1})={\mathsf{Coh}}(G_{\bar0}\backslash\Pi{\mathfrak{g}}^*_{\bar1})\xrightarrow{q^*} {\mathsf{Coh}}(G_{\mathcal{Q}}\backslash{\mathcal{Q}}^A)\xrightarrow{p_*}{\mathsf{Coh}}(G_{\bar0}\backslash\Pi{\mathfrak{g}}^*_{\bar1})= {\mathsf{Coh}}^{G_{\bar0}}(\Pi{\mathfrak{g}}^*_{\bar1}) \]

(in particular, $p_*$ involves taking $\operatorname{GL}(V'_1)$-invariants). We will actually need the same-named functor $p_*q^*\colon D_{\operatorname{perf}}^{G_{\bar 0}}({\mathfrak {G}}_{1,1}^\bullet )\to D_{\operatorname{perf}}^{G_{\bar 0}}({\mathfrak {G}}_{1,1}^\bullet )$ defined similarly using the dg-algebra with trivial differential ${\mathfrak {G}}_{1,1}^\bullet \,{\otimes}\, {\mathbb {C}}[{\operatorname{Hom}}(V_1,V'_1)]$ (the grading on ${\mathbb {C}}[{\operatorname{Hom}}(V_1,V'_1)]$ is trivial, and if we disregard the grading, then ${\mathfrak {G}}_{1,1}^\bullet \,{\otimes}\, {\mathbb {C}}[{\operatorname{Hom}}(V_1,V'_1)]\simeq {\mathbb {C}}[{\mathcal {Q}}^A]$).

We also have a $G_{\mathcal {Q}}$-invariant subvariety ${\mathcal {Q}}^A_0\subset {\mathcal {Q}}^A$ given by the equation that $A''$ is non-invertible. The restriction of $p,q$ to ${\mathcal {Q}}^A_0$ will be denoted by $p_0,q_0$. As above, we obtain the functor $p_{0*}q_0^*\colon D_{\operatorname{perf}}^{G_{\bar 0}}({\mathfrak {G}}_{1,1}^\bullet )\to D_{\operatorname{perf}}^{G_{\bar 0}}({\mathfrak {G}}_{1,1}^\bullet )$.

Proposition 3.12.1 (a) There is an isomorphism of functors

\[ \bar\jmath_{0*}\bar\jmath{}_0^!\circ\Phi^{1,1}\simeq \Phi^{1,1}\circ p_*q^*\colon D^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{G}}_{1,1}^\bullet)\to D_{!*{\mathbf{G}}_{\mathbf{O}}}({\mathbf{Gr}}\times{\mathbf{V}}). \]

(b) The isomorphism in (a) can be extended to the following commutative diagram of morphisms:

(c) There is an isomorphism of functors

\[ \jmath_{0*}\jmath_0^!\circ\Phi^{1,1}\simeq \Phi^{1,1}\circ p_{0*}q_0^*\colon D^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{G}}_{1,1}^\bullet)\to D_{!*{\mathbf{G}}_{\mathbf{O}}}({\mathbf{Gr}}\times{\mathbf{V}}). \]

Proof. (a) The support of ${\operatorname {IC}}_{({\boldsymbol {\nu }},{\boldsymbol {\mu }})}$ lies in ${\mathbf {Gr}}\times {\mathbf {V}}_0$ if and only if ${\boldsymbol {\nu }}$ is a negative partition ($0\geq \nu _1\geq \cdots \geq \nu _N$) (see [Reference Finkelberg, Ginzburg and TravkinFGT09, Proof of Proposition 8]). So $D_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}}_0)$ is generated by the collection of objects ${\operatorname {IC}}_{({\boldsymbol {\nu }},{\boldsymbol {\mu }})}$ where ${\boldsymbol {\nu }}$ is a negative partition. Thus we see that $\bar \jmath _{0*}$ is a fully faithful functor whose image is generated by $\{{\operatorname {IC}}_{({\boldsymbol {\nu }},{\boldsymbol {\mu }})}\mid {\boldsymbol {\nu }}\le 0\}$ and $\bar \jmath _0^!$ is the right adjoint of $\bar \jmath _{0*}$. Recall that $V^*_{\boldsymbol {\nu }}$ denotes an irreducible representation of $\operatorname{GL}(V)$ obtained by applying the corresponding Schur functor to $V^*$. The result of application of the same Schur functor to $V^*_1$ (respectively, $V^{\prime *}_1,V^*_2$) will be denoted by $V^*_{1,{\boldsymbol {\nu }}}$ (respectively, $V^{\prime *}_{1,{\boldsymbol {\nu }}},V^*_{2,{\boldsymbol {\nu }}}$). Since

\[ {\operatorname{IC}}_{({\boldsymbol{\nu}},{\boldsymbol{\mu}})}=\Phi^{1,1}(V_{1,{\boldsymbol{\nu}}}^*\,{\otimes}\,{\mathfrak{G}}_{1,1}^\bullet{\,{\otimes}\,} V_{2,{\boldsymbol{\mu}}}^*)= \Phi^{1,1}(V_{1,{\boldsymbol{\nu}}^*}\,{\otimes}\,{\mathfrak{G}}_{1,1}^\bullet{\,{\otimes}\,} V_{2,{\boldsymbol{\mu}}^*}) \]

(where ${\boldsymbol {\nu }}^*=(-\nu _N,-\nu _{N-1},\ldots ,-\nu _2,-\nu _1)$ for ${\boldsymbol {\nu }}=(\nu _1,\ldots ,\nu _N)$), we have to show that $p_*q^*$ lands in the subcategory $D_{\operatorname{perf}}^{G_{\bar 0},\geq }({\mathfrak {G}}_{1,1}^\bullet )\subset D_{\operatorname{perf}}^{G_{\bar 0}}({\mathfrak {G}}_{1,1}^\bullet )$ generated by objects $\{V_{1,{\boldsymbol {\lambda }}}\,{\otimes}\, {\mathfrak {G}}_{1,1}^\bullet \,{\otimes}\, V_{2,{\boldsymbol {\mu }}}\mid {\boldsymbol \lambda} \geq 0\}$ and to construct the adjunction isomorphism

\begin{align*} & {{\operatorname{Hom}}}_{D_{{\operatorname{perf}}}^{G_{\bar0}}({\mathfrak{G}}_{1,1}^\bullet)}(V_{1,{\boldsymbol{\lambda}}}\,{\otimes}\,{\mathfrak{G}}_{1,1}^\bullet{\,{\otimes}\,} V_{2,{\boldsymbol{\mu}}}, V_{1,{\boldsymbol{\lambda}}'}\,{\otimes}\,{\mathfrak{G}}_{1,1}^\bullet{\,{\otimes}\,} V_{2,{\boldsymbol{\mu}}'})\\ &\quad {\buildrel \sim \over \longrightarrow} {{\operatorname{Hom}}}_{D_{{\operatorname{perf}}}^{G_{\bar0}}({\mathfrak{G}}_{1,1}^\bullet)}(V_{1,{\boldsymbol{\lambda}}}\,{\otimes}\,{\mathfrak{G}}_{1,1}^\bullet{\,{\otimes}\,} V_{2,{\boldsymbol{\mu}}}, p_*q^*(V'_{1,{\boldsymbol{\lambda}}'}\,{\otimes}\,{\mathfrak{G}}_{1,1}^\bullet{\,{\otimes}\,} V_{2,{\boldsymbol{\mu}}'})) \end{align*}

for a partition ${\boldsymbol {\lambda }}$. Equivalently, we have to construct an isomorphism

\begin{align*} & \big((V_{1,{\boldsymbol{\lambda}}}^{\prime*}\,{\otimes}\, V'_{1,{\boldsymbol{\lambda}}'})\,{\otimes}\,{\mathbb{C}}[{{\operatorname{Hom}}}(V'_1,V_2)\times{{\operatorname{Hom}}}(V_2,V'_1)] \,{\otimes}\,(V_{2,{\boldsymbol{\mu}}}^*\,{\otimes}\, V_{2,{\boldsymbol{\mu}}'})\big)^{{{\operatorname{GL}}}(V'_1)\times{{\operatorname{GL}}}(V_2)}\\ &\quad {\buildrel \sim \over \longrightarrow} \big(V_{1,{\boldsymbol{\lambda}}}^*\,{\otimes}\, p_*q^*(V'_{1,{\boldsymbol{\lambda}}'}\,{\otimes}\,{\mathbb{C}}[{{\operatorname{Hom}}}(V'_1,V_2)\times{{\operatorname{Hom}}}(V_2,V'_1)]\,{\otimes}\, V_{2,{\boldsymbol{\mu}}'}) \,{\otimes}\, V_{2,{\boldsymbol{\mu}}}^*\big)^{{{\operatorname{GL}}}(V_1)\times{{\operatorname{GL}}}(V_2)}\\ &\quad :=\big(V^*_{1,{\boldsymbol{\lambda}}}\,{\otimes}\, V'_{1,{\boldsymbol{\lambda}}'}\,{\otimes}\,{\mathbb{C}}[{\mathcal{Q}}^A]\,{\otimes}\,(V^*_{2,{\boldsymbol{\mu}}}\,{\otimes}\, V_{2,{\boldsymbol{\mu}}'})\big)^{G_{\mathcal{Q}}}. \end{align*}

Recall that ${\mathcal {Q}}^A={\operatorname{Hom}}(V_1,V'_1)\times {\operatorname{Hom}}(V'_1,V_2)\times {\operatorname{Hom}}(V_2,V_1)$. We apply Lemma 3.13.1(c) below to $U_1=V_2$, $U_2=V_1$, $U_3=V'_1$, ${\boldsymbol {\nu }}={\boldsymbol {\lambda }}$ (in the notation of § 3.13) to obtain an isomorphism

\[ V^{\prime*}_{1,{\boldsymbol{\lambda}}}\,{\otimes}\,{\mathbb{C}}[{{\operatorname{Hom}}}(V_2,V'_1)]{\buildrel \sim \over \longrightarrow}\big({\mathbb{C}}[{{\operatorname{Hom}}}(V_2,V_1)\,{\otimes}\, V^*_{1,{\boldsymbol{\lambda}}}\,{\otimes}\,{\mathbb{C}}[{{\operatorname{Hom}}}(V_1,V'_1)]\big)^{{{\operatorname{GL}}}(V_1)} \]

whose inverse induces the desired adjunction isomorphism.

We still have to check that the essential image of $p_*q^*$ lies in the subcategory $D_{\operatorname{perf}}^{G_{\bar 0},\geq }({\mathfrak {G}}_{1,1}^\bullet )\subset D_{\operatorname{perf}}^{G_{\bar 0}}({\mathfrak {G}}_{1,1}^\bullet )$ generated by $\{V_{1,{\boldsymbol {\lambda }}}\,{\otimes}\, {\mathfrak {G}}_{1,1}^\bullet \,{\otimes}\, V_{2,{\boldsymbol {\mu }}}\mid {\boldsymbol \lambda} \geq 0\}$. We consider the homomorphism ${\mathfrak {G}}_{1,1}^\bullet \to {\mathbb {C}}$ killing all the generators, and for ${\mathcal {M}}\in D^{G_{\bar 0}}_{\operatorname{perf}}({\mathfrak {G}}_{1,1}^\bullet )$ we set $z_0^*{\mathcal {M}}:={\mathbb {C}}\,{\otimes} _{{\mathfrak {G}}_{1,1}^\bullet }^L {\mathcal {M}}\in D^{G_{\bar 0}}({\mathbb {C}})$ (‘the fiber at $0\in {\operatorname{Hom}}(V_1,V_2)\times {\operatorname{Hom}}(V_2,V_1)$’). Note that $z_0^*p_*q^*$ lands in the category generated by $\{V_{1,{\boldsymbol {\lambda }}}\,{\otimes}\, V_{2,{\boldsymbol {\mu }}}\mid {\boldsymbol \lambda} \geq 0\}$, that is, the category of modules with polynomial action of $\operatorname{GL}(V_1)$. Indeed, recall that $G_{\mathcal {Q}}$ acts on ${\mathcal {Q}}^A$ via

\[ (g_1,g_1',g_2)(A',A'',B)=(g_2A'g_1^{\prime-1},g'_1A''g_1^{{-}1},g_1Bg_2^{{-}1}). \]

Since we impose the conditions $B=0=A:=A'A''$, the action of $\operatorname{GL}(V_1)$ on $z_0^*p_*q^*{\mathcal {N}}$ (for a free dg-${\mathfrak {G}}_{1,1}^\bullet$-module ${\mathcal {N}}$) comes from its action on functions of $A''$, and the latter action is polynomial.

Finally, we claim that if the action of $\operatorname{GL}(V_1)$ on $z_0^*{\mathcal {M}}$ is polynomial, then ${\mathcal {M}}$ lies in the subcategory generated by $\{V_{1,{\boldsymbol {\lambda }}}\,{\otimes}\, {\mathfrak {G}}_{1,1}^\bullet \,{\otimes}\, V_{2,{\boldsymbol {\mu }}}\mid {\boldsymbol \lambda} \geq 0\}$. To this end we apply the Koszul equivalence $\varkappa \colon D_{\operatorname {fd}}^{G_{\bar 0}}(\Lambda ){\buildrel \sim \over \longrightarrow} D^{G_{\bar 0}}_{\operatorname{perf}}({\mathfrak {G}}_{1,1}^\bullet )$ of § 3.8. It is easy to see that if the action of $\operatorname{GL}(V_1)$ on the total cohomology of ${\mathcal {K}}\in D_{\operatorname {fd}}^{G_{\bar 0}}(\Lambda )$ is polynomial, then ${\mathcal {K}}$ lies in the subcategory $\operatorname {Pol}\subset D_{\operatorname {fd}}^{G_{\bar 0}}(\Lambda )$ generated by the $G_{\bar 0}\ltimes \Lambda$-modules with polynomial action of $\operatorname{GL}(V_1)$ and trivial action of $\Lambda$. Now $\varkappa (\operatorname {Pol})$ is the subcategory $D_{\operatorname{perf}}^{G_{\bar 0},\geq }({\mathfrak {G}}_{1,1}^\bullet )\subset D_{\operatorname{perf}}^{G_{\bar 0}}({\mathfrak {G}}_{1,1}^\bullet )$. And if the action of $\operatorname{GL}(V_1)$ on $z_0^*{\mathcal {M}}$ is polynomial, then ${\mathcal {M}}\simeq \varkappa ({\mathcal {K}})$, where the action of $\operatorname{GL}(V_1)$ on the total cohomology of ${\mathcal {K}}$ is polynomial. This completes the proof of (a).

(b) As $\alpha ^*\bar \jmath _{-1*}\bar \jmath {}_{-1}^!\simeq \bar \jmath _{0*}\bar \jmath {}_0^!\alpha ^*$ (in the notation of § 3.10), we deduce an isomorphism of functors

\[ \bar\jmath_{{-}1*}\bar\jmath{}_{{-}1}^!\circ\Phi^{1,1}\simeq\Phi^{1,1}\circ\big(\det V_1\,{\otimes}\, (p_*q^*)\circ(\det\!{}^{{-}1}V'_1\,{\otimes}\,{-})\big)\colon D^{G_{\bar0}}_{{\operatorname{perf}}}({\mathfrak{G}}_{1,1}^\bullet)\to D_{!*{\mathbf{G}}_{\mathbf{O}}}({\mathbf{Gr}}\times{\mathbf{V}}). \]

Thus the upper and lower rectangles of the diagram in (b) are commutative by construction. We have to prove the commutativity of the big curved quadrangle. The endofunctors $\bar \jmath _{0*}\bar \jmath {}_0^!$, $\bar \jmath _{-1*}\bar \jmath {}_{-1}^!$ of $D_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ are equipped with the structure of idempotent comonads. The desired commutativity follows from the fact that, given two idempotent comonads $T_0,T_1\colon {\mathcal {C}}\to {\mathcal {C}}$, there is at most one morphism of functors $\chi \colon T_1\to T_0$ such that $\varepsilon _1=\varepsilon _0\circ \chi$ for the counits $\varepsilon _i\colon T_i\to {\operatorname{Id}}_{\mathcal {C}}$. Indeed, $\chi =T_0(\varepsilon _1)\circ \chi _1$, where $\chi _1$ is defined as the composition

\[ T_1\cong T_1\circ T_1\xrightarrow{\chi\circ T_1}T_0\circ T_1. \]

We claim that $\chi _1$ is an isomorphism uniquely defined as the inverse to the morphism $\varepsilon _0\circ T_1\colon T_0\circ T_1{\buildrel \sim \over \longrightarrow} T_1$. In effect, the composition

\[ T_1\cong T_1\circ T_1\xrightarrow{\chi\circ T_1}T_0\circ T_1 \xrightarrow{\varepsilon_0\circ T_1} T_1 \]

equals the composition

\[ T_1\cong T_1\circ T_1\xrightarrow{\varepsilon_1\circ T_1}{{\operatorname{Id}}}_{\mathcal{C}}\circ T_1=T_1 \]

which in turn equals ${\operatorname{Id}}_{T_1}$. Conversely, the composition

\[ T_0\circ T_1\xrightarrow{\varepsilon_0\circ T_1} T_1\cong T_1\circ T_1 \xrightarrow{\chi\circ T_1}T_0\circ T_1 \]

equals the composition

\[ T_0\circ T_1\cong T_0\circ T_1\circ T_1\xrightarrow{T_0\circ\chi\circ T_1} T_0\circ T_0\circ T_1\cong T_0\circ T_1 \]

which in turn equals ${\operatorname{Id}}_{T_0\circ T_1}$.

This completes the proof of (b), but we would like to give one more independent argument that will prove useful later on.

Recall that we have to prove the commutativity of the big curved quadrangle. To this end we change the setting to the base field ${\mathbb {F}}_q$ as in the proof of Lemma 3.9.1. That is, we replace $D_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ by the equivalent equivariant derived category of sheaves on $({\mathbf {Gr}}\times {\mathbf {V}})_{\bar {\mathbb {F}}_q}$ as in [Reference Bezrukavnikov and FinkelbergBF08, Proposition 5] (in particular, choosing an isomorphism ${\mathbb {C}}\simeq \bar {\mathbb {Q}}_\ell$). However, we preserve the notation $D_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ for this category in order not to overload our notation (anyway, it will only be used during the current proof). All the irreducible perverse sheaves ${\operatorname {IC}}_{({\boldsymbol {\lambda }},{\boldsymbol {\mu }})}$ carry a natural Tate Weil structure by [Reference Finkelberg, Ginzburg and TravkinFGT09, Proposition 11]. They (along with their Tate twists) generate a subcategoryFootnote ⁹ $\hat {D}_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ of the mixed version $D^{\operatorname {mix}}_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ of $D_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$. We will use a particular dg-model of $\hat {D}_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ viewed as a category enriched over complexes equipped with an action of the Frobenius automorphism $\operatorname {Fr}$. Note that the absolute values of the eigenvalues of $\operatorname {Fr}$ lie in $\sqrt {q}^{\mathbb {Z}}$, and hence our complexes carry an additional grading according to the absolute values of the eigenvalues of $\operatorname {Fr}$. If we forget the mixed structure and remember only this additional grading, we obtain a category $\tilde {\underline D}{}_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ enriched over complexes equipped with an additional grading. Its localization with respect to quasi-isomorphisms will be denoted $\tilde {D}_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$.

On the other hand, we consider the category $\underline {D}{}_{\operatorname{perf}}^{G_{\bar 0}\times {\mathbb {C}}^\times }({\mathfrak {G}}^\bullet _{1,1})$ of perfect $G_{\bar 0}\times {\mathbb {C}}^\times$-equivariant dg-${\mathfrak {G}}^\bullet _{1,1}$-modules and its localization (with respect to quasi-isomorphisms) $D_{\operatorname{perf}}^{G_{\bar 0}\times {\mathbb {C}}^\times }({\mathfrak {G}}^\bullet _{1,1})$. Here all the generators of ${\mathfrak {G}}^\bullet _{1,1}$ have weight 1 with respect to the action of ${\mathbb {C}}^\times$. Then the standard modification of our construction of the equivalence $\Phi ^{1,1}\colon D^{G_{\bar 0}}_{\operatorname{perf}}({\mathfrak {G}}_{1,1}^\bullet ){\buildrel \sim \over \longrightarrow} D_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ produces a functor $\tilde {\underline \Phi }{}^{1,1}\colon \underline {D}{}^{G_{\bar 0}\times {\mathbb {C}}^\times }_{\operatorname{perf}}({\mathfrak {G}}_{1,1}^\bullet )\to \tilde {\underline D}{}_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ and its localization $\tilde {\Phi }{}^{1,1}\colon D^{G_{\bar 0}\times {\mathbb {C}}^\times }_{\operatorname{perf}}({\mathfrak {G}}_{1,1}^\bullet ){\buildrel \sim \over \longrightarrow} \tilde {D}{}_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$.

We have the similar diagram of morphisms of functors $D^{G_{\bar 0}\times {\mathbb {C}}^\times }_{\operatorname{perf}}({\mathfrak {G}}_{1,1}^\bullet )\to \tilde {D}{}_{!*{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ with commutative upper and lower rectangles

and we have to prove the commutativity of the big curved quadrangle. The endofunctors $\det V_1\,{\otimes}\, (p_*q^*)\circ (\det \!{}^{-1}V'_1\,{\otimes}\, {-})$, ${\operatorname{Id}}$, $p_*q^*$ of $D^{G_{\bar 0}\times {\mathbb {C}}^{\times}}_{\operatorname{perf}}({\mathfrak {G}}_{1,1}^\bullet )$ are given by their respective kernels $K_1,K_2,K_3$ in $\underline {D}{}^{G_{\bar 0}^2{\times} {\mathbb {C}}^{\times} }_{\operatorname{perf}}({\mathfrak {G}}_{1,1}^\bullet \,{\otimes}\, {\mathfrak {G}}_{1,1}^\bullet )$. Note that the equivariance with respect to ${\mathbb {C}}^{\times}$ (as opposed to $({\mathbb {C}}^{\times} )^2$) suffices since all the three functors under consideration commute with the shifts of the additional grading. Similarly, all the three functors on the constructible side commute with the Tate twists. All three kernels are pure of weight 0, that is, their additional gradings coincide with their cohomological gradings. The category of pure weight 0 objects in $D^{G_{\bar 0}^2\times {\mathbb {C}}^\times }_{\operatorname{perf}}({\mathfrak {G}}_{1,1}^\bullet \,{\otimes}\, {\mathfrak {G}}_{1,1}^\bullet )$ is equivalent to the abelian category of $G_{\bar 0}^2\times {\mathbb {C}}^\times$-equivariant ${\mathfrak {G}}_{1,1}\,{\otimes}\, {\mathfrak {G}}_{1,1}$-modules (the equivalence being obtained by taking cohomology). Therefore, the morphisms of functors $(\det V_1\,{\otimes}\, {-})\circ (p_*q^*)\circ (\det \!{}^{-1}V'_1\,{\otimes}\, {-})\to {\operatorname{Id}}$ and $p_*q^*\to {\operatorname{Id}}$ arise from the morphisms between the respective kernels that are injective as morphisms of ${\mathfrak {G}}_{1,1}\,{\otimes}\, {\mathfrak {G}}_{1,1}$-modules. We have to compare certain morphisms to $\tilde {\underline \Phi }{}^{1,1}K_3$, and we know that their compositions with the monomorphism $\tilde {\underline \Phi }{}^{1,1}K_3\to \tilde {\underline \Phi }{}^{1,1}K_2$ coincide, hence the desired equality of morphisms. This completes our second proof of (b).

(c) This follows from the comparison of the distinguished triangles

\[ (\bar\jmath_{{-}1*}\bar\jmath{}_{{-}1}^!\to\bar\jmath_{0*}\bar\jmath{}_0^!\to\jmath_{0*}\jmath_0^!) \circ\Phi^{1,1} \]

and

\[ \Phi^{1,1}\circ\big(\det V_1\,{\otimes}\, (p_*q^*)\circ(\det\!{}^{{-}1}V'_1\,{\otimes}\,{-}) \to p_*q^*\to p_{0*}q_0^*\big). \]

The proposition is proved.

3.13 Some invariant theory

Let $U_1,U_2,U_3$ be vector spaces of dimensions $n_1,n_2,n_3$. The irreducible polynomial (respectively, antipolynomial) representations of $\operatorname{GL}(U_i)$ are realized in the Schur spaces ${\mathbb {S}}_{\boldsymbol {\lambda }} U_i$ (respectively, ${\mathbb {S}}_{\boldsymbol {\lambda }} U_i^*$), where ${\boldsymbol {\lambda }}$ is a partition with $\ell ({\boldsymbol {\lambda }})\leq n_i$. We will also write ${\mathbb {S}}_{{\boldsymbol {\lambda }}^*}U_i$ for ${\mathbb {S}}_{\boldsymbol {\lambda }} U_i^*$, where

\[ {\boldsymbol{\lambda}}^*={-}w_0{\boldsymbol{\lambda}}=(-\lambda_{n_i},-\lambda_{n_i-1},\ldots,-\lambda_2,-\lambda_1) \]

for ${\boldsymbol {\lambda }}=(\lambda _1,\ldots ,\lambda _{n_i})$. We set ${\mathbb {S}}_{\boldsymbol {\lambda }} U_i=0={\mathbb {S}}_{{\boldsymbol {\lambda }}^*}U_i$ for $\ell ({\boldsymbol {\lambda }})>n_i$. We denote by ${\operatorname{Hom}}_{\leq n_2}(U_1,U_3)\subset {\operatorname{Hom}}(U_1,U_3)$ the subvariety formed by all the homomorphisms of rank at most $n_2$.

Proof. (a) We have ${\mathbb {C}}[{\operatorname{Hom}}(U_i,U_j)]=\bigoplus _{m\geq 0}{\operatorname{Sym}}^m(U_j^*\,{\otimes}\, U_i)= \bigoplus _{\boldsymbol {\lambda }}{\mathbb {S}}_{\boldsymbol {\lambda }} U_i\,{\otimes}\, {\mathbb {S}}_{{\boldsymbol {\lambda }}^*} U_j$ as a $\operatorname{GL}(U_i)\times \operatorname{GL}(U_j)$-module. Also, ${\mathbb {C}}[{\operatorname{Hom}}_{\leq r}(U_i,U_j)]= \bigoplus _{\ell ({\boldsymbol {\lambda }})\leq r}{\mathbb {S}}_{\boldsymbol {\lambda }} U_i\,{\otimes}\, {\mathbb {S}}_{{\boldsymbol {\lambda }}^*} U_j$ as a $\operatorname{GL}(U_i)\times \operatorname{GL}(U_j)$-module. Clearly, $({\mathbb {S}}_{{\boldsymbol {\lambda }}^*}U_2\,{\otimes}\, {\mathbb {S}}_{\boldsymbol {\mu }} U_2)^{\operatorname{GL}(U_2)}={\mathbb {C}}^{\delta _{{\boldsymbol {\lambda }}{\boldsymbol {\mu }}}}$. So the two sides of (a) are isomorphic as $\operatorname{GL}(U_1)\times \operatorname{GL}(U_3)$-modules. On the other hand, the morphism in question is injective since the composition morphism ${\operatorname{Hom}}(U_1,U_2)\times {\operatorname{Hom}}(U_2,U_3)\to {\operatorname{Hom}}_{\leq n_2}(U_1,U_3)$ is dominant. Hence the morphism in question is an isomorphism.

(b) We consider a copy $U'_2$ of $U_2$, we tensor both sides of (b) with ${\mathbb {S}}_{{\boldsymbol {\nu }}^*}U'_2$, and take the direct sum over all partitions ${\boldsymbol {\nu }}$ with $\ell ({\boldsymbol {\nu }})\leq n_2$. Then we have to prove that the morphism

\begin{align*} &\gamma\colon {\mathbb{C}}[{{\operatorname{Hom}}}(U_1,U_2')]\,{\otimes}\,{\mathbb{C}}[{{\operatorname{Hom}}}(U_1,U_3)]\\ &\quad\to \big({\mathbb{C}}[{{\operatorname{Hom}}}(U_1,U_2)]\,{\otimes}\, {\mathbb{C}}[{{\operatorname{Hom}}}(U_2,U_2')]\,{\otimes}\,{\mathbb{C}}[{{\operatorname{Hom}}}(U_2,U_3)]\big)^{{{\operatorname{GL}}}(U_2)}, \end{align*}

induced by the composition of arrows of the $D_4$-quiver in

(3.13.1)

is an isomorphism. Now the statement can be reduced to (a) using the substitution $U_3 \rightsquigarrow U_3\oplus U'_2$. Alternatively, the condition $n_1\leq n_2$ guarantees that the morphism from the representation space of the $D_4$-quiver to the representation space of the dashed $A_3$-quiver is dominant. Hence $\gamma$ is injective. The surjectivity of $\gamma$ follows, for example, from [Reference Le Bruyn and ProcesiLP90, Theorem 1].

(c) This is dual to (b).

3.14 The monoidal property of $\Phi ^{2,0}$

Recall the notation of § 3.4. The monoidal structure ${\stackrel {A}*}$ on $D^{G_{\bar 0}}_{\operatorname{perf}}({\mathfrak {G}}_{2,0}^\bullet )$ is defined via the kernel ${\mathbb {C}}[{\mathcal {Q}}^A]_{2,0}^\bullet$: a $G_{\mathcal {Q}}$-equivariant dg-${\mathbb {C}}[{\mathcal {H}}]_{2,0}^\bullet$-module. The monoidal structure ${\stackrel{*}*}$ on $D_{!{\mathbf {G}}_{\mathbf {O}}}({\mathbf {Gr}}\times {\mathbf {V}})$ transferred to $D^{G_{\bar 0}}_{\operatorname{perf}}({\mathfrak {G}}_{2,0}^\bullet )$ via the equivalence $\Phi ^{2,0}$ is also defined via a kernel ${\mathcal {K}}^\bullet$ (a $G_{\mathcal {Q}}$-equivariant dg-${\mathbb {C}}[{\mathcal {H}}]_{2,0}^\bullet$-module). We have to construct an isomorphism of $G_{\mathcal {Q}}$-equivariant dg-${\mathbb {C}}[{\mathcal {H}}]_{2,0}^\bullet$-modules ${\mathbb {C}}[{\mathcal {Q}}^A]_{2,0}^\bullet {\buildrel \sim \over \longrightarrow} {\mathcal {K}}^\bullet$.