2.1 Notation

Let $G$ be a simply connected semisimple group. Let $T$ be a maximal torus of $G$ , let $\unicode[STIX]{x1D6EC}$ denote its weight lattice and let $W$ denote the Weyl group. Let $\unicode[STIX]{x1D6EC}_{+}$ denote the set of dominant weights and $\unicode[STIX]{x1D70C}^{\vee }$ half the sum of the fundamental coroots. For $\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}_{+}$ , let $V(\unicode[STIX]{x1D706})$ denote the irreducible representation of highest weight $\unicode[STIX]{x1D706}$ .

Recall that we have an isomorphism ${\mathcal{O}}(G)^{G}\cong {\mathcal{O}}(T)^{W}$ where $G$ acts on $G$ by the adjoint action. Let $E={\mathcal{O}}(T)^{W}\cong \mathbb{C}[\unicode[STIX]{x1D6EC}]^{W}$ . Recall that $E$ is a polynomial ring in $r$ variables, where $r$ is the rank of $G$ . We also have an isomorphism $R(G)\cong E$ , where $R(G)$ denotes the (complexified) representation ring of $G$ .

Let $G^{\vee }$ denote the Langlands dual group to $G$ and let $\widetilde{G^{\vee }}$ be its simply connected cover. Let $\unicode[STIX]{x1D6EC}^{\vee }$ be the weight lattice of $\widetilde{G^{\vee }}$ and let $E^{\vee }=R(\widetilde{G^{\vee }})=\mathbb{C}[\unicode[STIX]{x1D6EC}^{\vee }]^{W}$ .

Let $(\,,)$ be the $W$ -invariant bilinear form on $\mathfrak{t}^{\ast }$ (the dual of the Lie algebra of $T$ ), such that $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FC})=2$ for all short roots $\unicode[STIX]{x1D6FC}$ . This bilinear form gives an isomorphism $\unicode[STIX]{x1D704}:\mathfrak{t}^{\ast }\rightarrow \mathfrak{t}$ . From the $W$ -invariance, it follows that $\unicode[STIX]{x1D704}(\unicode[STIX]{x1D6EC})\subset \unicode[STIX]{x1D6EC}^{\vee }$ (with equality if $G$ is simply laced). Thus $\unicode[STIX]{x1D704}$ gives us an inclusion $E{\hookrightarrow}E^{\vee }$ (which is an isomorphism when $G$ is simply laced).

2.2 Classical representation categories

Let ${\mathcal{R}}\text{ep}(G)$ denote the usual category of finite-dimensional representations of $G$ . We will be interested in various enhancements/modifications of ${\mathcal{R}}\text{ep}(G)$ .

We have the adjoint actions of $G$ on $\mathfrak{g}$ and $G$ . This makes ${\mathcal{O}}(\mathfrak{g})$ and ${\mathcal{O}}(G)$ into (infinite-dimensional) $G$ representations. Let ${\mathcal{O}}(\frac{\mathfrak{g}}{G})\operatorname{ - mod}$ denote the full subcategory of $G$ -equivariant ${\mathcal{O}}(\mathfrak{g})$ -modules which are of the form ${\mathcal{O}}(\mathfrak{g})\otimes V$ , for $V$ in ${\mathcal{R}}\text{ep}(G)$ . Here $G$ acts diagonally on ${\mathcal{O}}(\mathfrak{g})\otimes V$ and ${\mathcal{O}}(\mathfrak{g})$ acts on the left tensor factor. Similarly we define ${\mathcal{O}}({\textstyle \frac{G}{G}})\operatorname{ - mod}$ to be the full subcategory of $G$ -equivariant ${\mathcal{O}}(G)$ -modules which are of the form ${\mathcal{O}}(G)\otimes V$ . Note that we can think of ${\mathcal{O}}(\frac{\mathfrak{g}}{G})\operatorname{ - mod}$ as the full subcategory of $G$ -equivariant coherent sheaves of $\mathfrak{g}$ consisting of trivial vector bundles with $G$ -action (and similarly for ${\mathcal{O}}({\textstyle \frac{G}{G}})\operatorname{ - mod}$ ).

2.3 Quantum representation categories

2.4 Horizontal trace and ${\mathcal{O}}_{q}({\textstyle \frac{G}{G}})\operatorname{ - mod}$

One reason why these categories ${\mathcal{O}}_{q}({\textstyle \frac{G}{G}})\operatorname{ - mod}$ are important for us is because they arise as horizontal traces. We now explain this construction. The results in this section seem to be known to experts, but we were not able to find an adequate reference.

2.5 Action of annular braids

As mentioned above, the procedure of horizontal trace is closely related to the circle or annulus. In particular, if ${\mathcal{C}}$ is braided monoidal, then the annular braid group acts on tensor products in ${\mathcal{C}}(S^{1})$ . Let us formulate this fact in a precise fashion in the case of ${\mathcal{R}}\text{ep}_{q}(G)$ .

2.6 Kostant slice functor

In [Reference Bezrukavnikov and FinkelbergBF08], an important role is played by the Kostant slice functor. In their setting of ${\mathcal{O}}(\frac{\mathfrak{g}}{G})\operatorname{ - mod}$ , they considered the Kostant slice $\mathfrak{t}/W{\hookrightarrow}\mathfrak{g}$ . Pulling back along the Kostant slice gives rise to a fully faithful functor $S:{\mathcal{O}}(\frac{\mathfrak{g}}{G})\operatorname{ - mod}\rightarrow Z_{\mathfrak{g}}\operatorname{ - mod}$ where $Z_{\mathfrak{g}}$ is the group scheme of regular centralizers regarded as a group scheme over $\mathfrak{t}/W$ .

In our situation, we have a Steinberg slice $T/W{\hookrightarrow}G$ . As in the Lie algebra case, the image of this Steinberg slice meets every regular $G$ -orbit once. Moreover, the set $G^{\text{reg}}$ of regular elements of $G$ is dense and its complement has codimension 2. Thus we see that the resulting functor $S:{\mathcal{O}}({\textstyle \frac{G}{G}})\operatorname{ - mod}\rightarrow Z_{G}\operatorname{ - mod}$ given by $M\mapsto M\otimes _{{\mathcal{O}}(G)}{\mathcal{O}}(T/W)$ is an equivalence, where $Z_{G}$ is the group scheme of group regular centralizers regarded as a group scheme over $T/W$ . This functor (as in the work of [Reference Bezrukavnikov and FinkelbergBF08]) can and will be used to give a more algebraic description of the category ${\mathcal{O}}({\textstyle \frac{G}{G}})\operatorname{ - mod}$ .

Unfortunately, in the quantum group situation we do not seem to have an analogous map ${\mathcal{O}}_{q}(G)\rightarrow E(q)$ (nor an analog of the group scheme of regular centralizers). However, it should be possible to construct a version of the Kostant slice functor by relating ${\mathcal{O}}_{q}({\textstyle \frac{G}{G}})\operatorname{ - mod}$ to a category of Harish-Chandra modules and using the work of Sevostyanov [Reference SevostyanovSev00].

2.7 Minuscule versions

A non-trivial irreducible representation $V$ of $G$ is called minuscule, if all weights of $V$ lie in single Weyl orbit. Equivalently $V(\unicode[STIX]{x1D706})$ is minuscule if and only if $\unicode[STIX]{x1D706}$ is a non-zero minimal element of $\unicode[STIX]{x1D6EC}_{+}$ (with respect to the usual partial order on $\unicode[STIX]{x1D6EC}_{+}$ ).

We consider the full subcategory ${\mathcal{R}}\text{ep}_{q}^{\min }(G)$ of ${\mathcal{R}}\text{ep}_{q}(G)$ consisting of tensor products of minuscule representations. Similarly, we consider ${\mathcal{O}}_{q}^{\min }({\textstyle \frac{G}{G}})\operatorname{ - mod}$ to be the subcategory of ${\mathcal{O}}_{q}({\textstyle \frac{G}{G}})\operatorname{ - mod}$ consisting of objects of the form ${\mathcal{O}}_{q}(G)\otimes V$ for $V\in {\mathcal{R}}\text{ep}_{q}^{\min }(G)$ .

When $G=\operatorname{SL}_{n}$ , the minuscule representations coincide with the fundamental representations $\bigwedge ^{k}\mathbb{C}^{n}$ for $k=1,\ldots ,n-1$ . In this case, the idempotent completion of ${\mathcal{R}}\text{ep}_{q}^{\min }(\operatorname{SL}_{n})$ is ${\mathcal{R}}\text{ep}_{q}(\operatorname{SL}_{n})$ . From Propositions 2.6 and 2.7, we deduce the following.

3.1 The affine Grassmannian

Let ${\mathcal{K}}=\mathbb{C}((t)),{\mathcal{O}}=\mathbb{C}[[t]]$ and let $Gr=G^{\vee }({\mathcal{K}})/G^{\vee }({\mathcal{O}})$ denote the affine Grassmannian of the Langlands dual group. See Zhu [Reference ZhuZhu16] for a thorough discussion about affine Grassmannians.

For every weight $\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}$ , there is a point $t^{\unicode[STIX]{x1D706}}\in Gr$ . For each $\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}_{+}$ , let $Gr^{\unicode[STIX]{x1D706}}=G^{\vee }({\mathcal{O}})t^{\unicode[STIX]{x1D706}}$ . We have $\dim Gr^{\unicode[STIX]{x1D706}}=2\unicode[STIX]{x1D70C}^{\vee }(\unicode[STIX]{x1D706})$ . We have the stratification $Gr=\bigcup _{\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}_{+}}Gr^{\unicode[STIX]{x1D706}}$ . Moreover, given any two points $L_{1},L_{2}\in Gr$ , there exists $g\in G^{\vee }({\mathcal{K}})$ such that $(gL_{1},gL_{2})=(t^{0},t^{\unicode[STIX]{x1D706}})$ for a unique dominant weight $\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}_{+}$ . In this case, we say that $L_{2}$ is distance $\unicode[STIX]{x1D706}$ from $L_{1}$ and we write $d(L_{1},L_{2})=\unicode[STIX]{x1D706}$ . With this definition, $Gr^{\unicode[STIX]{x1D706}}$ is the set of points of distance $\unicode[STIX]{x1D706}$ from $t^{0}$ .

The following well-known result explains the geometric significance of minuscule weights.

Given a sequence $\text{}\underline{\unicode[STIX]{x1D706}}=(\unicode[STIX]{x1D706}_{1},\ldots ,\unicode[STIX]{x1D706}_{m})$ of minuscule dominant weights, we can consider the variety

which comes with a morphism $p_{\text{}\underline{\unicode[STIX]{x1D706}}}:Gr^{\text{}\underline{\unicode[STIX]{x1D706}}}\rightarrow Gr$ given by $(L_{1},\ldots ,L_{m})\mapsto L_{m}$ . Note that $Gr^{\text{}\underline{\unicode[STIX]{x1D706}}}$ is an iterated bundle of $Gr^{\unicode[STIX]{x1D706}_{i}}$ ; there is a obvious map $Gr^{\text{}\underline{\unicode[STIX]{x1D706}}}\rightarrow Gr^{(\unicode[STIX]{x1D706}_{1},\ldots ,\unicode[STIX]{x1D706}_{m-1})}$ which is a $Gr^{\unicode[STIX]{x1D706}_{m}}$ -bundle.

There is an action of $G^{\vee }$ on $Gr$ by left multiplication. Also there is an action of $\mathbb{C}^{\times }$ on $Gr$ by ‘loop rotation’; this comes from the action of $\mathbb{C}^{\times }$ on ${\mathcal{O}}$ given by $s\cdot p(t)=p(st)$ .

3.2 Line bundles on the affine Grassmannian

Recall that the connected components of $Gr$ are labelled by the finite set $\unicode[STIX]{x1D6EC}/\mathbb{Z}\unicode[STIX]{x1D6E5}$ , where $\unicode[STIX]{x1D6E5}$ is the set of roots of $G$ . Each connected component has Picard group $\mathbb{Z}$ and we define a line bundle ${\mathcal{L}}$ on $Gr$ to be the positive generator of the Picard group on each connected component (see for example [Reference ZhuZhu16, § 2.4]).

Unfortunately, the line bundle ${\mathcal{L}}$ is not always $G^{\vee }$ -equivariant. It is however $\widetilde{G^{\vee }}$ -equivariant (since each connected component is isomorphic to the quotient of $\widehat{\widetilde{G^{\vee }}}$ (the affine Kac–Moody group) by a hyperspecial parahoric subgroup). In fact ${\mathcal{L}}$ carries a $\widetilde{G^{\vee }}\times \mathbb{C}^{\times }$ -equivariant structure where the action of $\widetilde{T^{\vee }}$ on the fibre over $t^{\unicode[STIX]{x1D707}}$ is given by the weight $\unicode[STIX]{x1D704}(\unicode[STIX]{x1D707})$ and the action of $\mathbb{C}^{\times }$ on the same fibre over $t^{\unicode[STIX]{x1D707}}$ is given by the pairing $(\unicode[STIX]{x1D707},\unicode[STIX]{x1D707})$ .

3.3 The equivariant Satake category

Consider the Satake category $P(Gr)$ of perverse sheaves on $Gr$ constructible with respect to the stratification by $G^{\vee }({\mathcal{O}})$ orbits.

We will need an equivariant version of this category. Note that we have an action of $G^{\vee }$ on $Gr$ which preserves the $G^{\vee }({\mathcal{O}})$ -orbits. Using the $W$ -invariant bilinear form, we may identify $H_{G^{\vee }}(\operatorname{pt})={\mathcal{O}}(\mathfrak{t}^{\vee })^{W}\cong {\mathcal{O}}(\mathfrak{t})^{W}$ .

Let $D_{G^{\vee }}(Gr)$ denote the equivariant derived category of $Gr$ for this action. We consider $D_{G^{\vee }}(Gr)$ with morphisms given by total $\text{Ext}$ . Thus the category $D_{G^{\vee }}(Gr)$ is enriched over graded ${\mathcal{O}}(\mathfrak{t})^{W}$ -modules.

We have a global section functor

By a theorem of Ginzburg (see [Reference Bezrukavnikov and FinkelbergBF08, Lemma 13]), this functor is fully faithful.

Since $Gr^{\text{}\underline{\unicode[STIX]{x1D706}}}$ is smooth and $p_{\text{}\underline{\unicode[STIX]{x1D706}}}$ is semismall, we can consider the perverse sheaf ${p_{\text{}\underline{\unicode[STIX]{x1D706}}}}_{\ast }\mathbb{C}_{Gr^{\text{}\underline{\unicode[STIX]{x1D706}}}}[2\unicode[STIX]{x1D70C}^{\vee }(\sum _{k}\unicode[STIX]{x1D706}_{k})]\in P(Gr)$ . It is the tensor product of the objects $\mathbb{C}_{Gr^{\unicode[STIX]{x1D706}_{k}}}[2\unicode[STIX]{x1D70C}^{\vee }(\unicode[STIX]{x1D706}_{k})]$ with respect to the usual monoidal structure on $P(Gr)$ . We let $D_{G^{\vee }}^{\min }(Gr)$ denote the full subcategory of $D_{G^{\vee }}(Gr)$ consisting of these objects.

3.4 Homology convolution categories

Recall convolution in homology following Chriss and Ginzburg [Reference Chriss and GinzburgCG97]. Let $I$ be an index set and let $\{Y_{i}\}_{i\in I}$ be a collection of smooth varieties along with proper maps $p_{i}:Y_{i}\rightarrow Y$ to some fixed variety $Y$ . Form the fibre products $Z_{ij}=Y_{i}\times _{Y}Y_{j}$ .

We can use this data to define a category $h\operatorname{Conv}(Y)$ (this category depends on more than just $Y$ , but we suppress the rest of the data in the notation). The objects in $h\operatorname{Conv}(Y)$ consist of the elements of the set $I$ . The morphisms are defined as

where $H_{\ast }(Z_{ij})$ denotes the total Borel–Moore homology of $Z_{ij}$ (with coefficients in $\mathbb{C}$ ). The composition operation in $h\operatorname{Conv}(Y)$ is defined by the convolution product

which is given by the formula

where ‘ $\cap$ ’ denotes the intersection product (with support), relative to the ambient manifold $Y_{i}\times Y_{j}\times Y_{k}$ and $\unicode[STIX]{x1D70B}_{12}:Y_{i}\times Y_{j}\times Y_{k}\rightarrow Y_{i}\times Y_{j}$ , etc. For more details about this construction see [Reference Chriss and GinzburgCG97, § 2.7].

The following result of Chriss and Ginzburg shows the importance of these categories. Consider the derived category of constructible sheaves on $Y$ , $D_{c}(Y)$ , which we regard as a category enriched over graded vector spaces by taking total $Ext$ .

If the varieties $Y_{i},Y$ all carry compatible actions of some group $H$ , we may consider the corresponding equivariant category $h\operatorname{Conv}^{H}(Y)$ where the morphisms are defined by equivariant homology.

We now apply this framework to obtain the category $h\operatorname{Conv}(Gr)$ where the base variety is $Gr$ and the mapping varieties are given by $p_{\text{}\underline{\unicode[STIX]{x1D706}}}:Gr^{\text{}\underline{\unicode[STIX]{x1D706}}}\rightarrow Gr$ , where $\text{}\underline{\unicode[STIX]{x1D706}}$ ranges over all sequence of minuscule dominant weights. The category $h\operatorname{Conv}(Gr)$ was first considered by the second author with Fontaine and Kuperberg in [Reference Fontaine, Kamnitzer and KuperbergFKK13]. The fibre products $Z(\text{}\underline{\unicode[STIX]{x1D706}},\text{}\underline{\unicode[STIX]{x1D707}})$ appearing in $h\operatorname{Conv}(Gr)$ are the ‘Steinberg-like’ varieties

Using the action of $G^{\vee }$ we may also define $h\operatorname{Conv}^{G^{\vee }}(Gr)$ . Applying Theorem 3.2 we deduce the following.

In other words, we can model $D_{G^{\vee }}^{\min }(Gr)$ using this homology convolution category. This motivates us to introduce a K-theory convolution category.

3.5 The category $K\operatorname{Conv}(Gr)$

The convolution formalism in homology from § 3.4 can be repeated in K-theory. More precisely, given a collection $Y_{i}\rightarrow Y$ as above, the objects in $K\operatorname{Conv}(Y)$ are still indexed by the set $I$ but the morphisms are now defined as

where $K(Z_{ij})$ denotes the complexified Grothendieck group of coherent sheaves on $Z_{ij}$ . The composition operation in $K\operatorname{Conv}(Y)$ is defined by a similar formula, namely

For more details about this construction see [Reference Chriss and GinzburgCG97, § 5.2]. As before, given a compatible action of a group $H$ we can work with equivariant K-theory $K^{H}(\cdot )$ to define the category $K\operatorname{Conv}^{H}(Y)$ .

Applying this construction to the collection $Gr^{\text{}\underline{\unicode[STIX]{x1D706}}}\rightarrow Gr$ , we obtain the categories $K\operatorname{Conv}^{\widetilde{G^{\vee }}}(Gr)$ and $K\operatorname{Conv}^{\widetilde{G^{\vee }}\times \mathbb{C}^{\times }}(Gr)$ where $\widetilde{G^{\vee }}$ acts as before and $\mathbb{C}^{\times }$ acts by loop rotation. (Here we are using $\widetilde{G^{\vee }}$ rather than $G^{\vee }$ , since our line bundles are only $\widetilde{G^{\vee }}$ -equivariant.) In these categories, the objects are sequences $(\unicode[STIX]{x1D706}_{1},\ldots ,\unicode[STIX]{x1D706}_{m})$ (we will say that this object has length $m$ ).

On the variety $Gr^{\text{}\underline{\unicode[STIX]{x1D706}}}$ we have $m$ line bundles ${\mathcal{L}}_{1},\ldots ,{\mathcal{L}}_{m}$ , where ${\mathcal{L}}_{k}$ is defined as $p_{k}^{\ast }{\mathcal{L}}\otimes p_{k-1}^{\ast }{\mathcal{L}}^{\vee }$ , where $p_{k}:Gr^{\text{}\underline{\unicode[STIX]{x1D706}}}\rightarrow Gr$ is given by $p_{k}(L_{1},\ldots ,L_{m})=L_{k}$ . For $i=1,\ldots ,m$ we define

as $X_{i}:=\unicode[STIX]{x1D6E5}_{\ast }[{\mathcal{L}}_{i}]$ where $\unicode[STIX]{x1D6E5}:Gr^{\text{}\underline{\unicode[STIX]{x1D706}}}\rightarrow Z(\text{}\underline{\unicode[STIX]{x1D706}},\text{}\underline{\unicode[STIX]{x1D706}})$ is the diagonal inclusion. These $X_{i}$ are invertible and so they define a map $\mathbb{Z}^{m}\rightarrow \operatorname{End}_{K\operatorname{Conv}^{\widetilde{G^{\vee }}\times \mathbb{C}^{\times }}}(\text{}\underline{\unicode[STIX]{x1D706}})$ . Thus, we see that $\mathbb{Z}^{m}$ acts on objects of length $m$ in $K\operatorname{Conv}^{\widetilde{G^{\vee }}}(Gr)$ and $K\operatorname{Conv}^{\widetilde{G^{\vee }}\times \mathbb{C}^{\times }}(Gr)$ .

3.6 Main conjecture

Motivated by Theorem 3.3 and the discussion in the introduction, we now formulate the following main conjecture.

By an equivalence of $E^{\vee }\operatorname{- mod}$ enriched categories, we mean that the map on Hom spaces should be an $E^{\vee }$ -module morphism map. On the left-hand side the $E^{\vee }$ -module structure comes from tensoring (and the $E$ -module structure was explained in § 2.6), while on the right-hand side it comes from $K^{\widetilde{G^{\vee }}}(pt)=E^{\vee }$ .

The compatibility between the actions of $\mathbb{Z}^{m}$ should be viewed as a partial substitute for the commutative diagram appearing in [Reference Bezrukavnikov and FinkelbergBF08] which involves the Kostant slice functor. The action of $\mathbb{Z}^{m}$ on the left-hand side is defined in § 2.5 and the action of $\mathbb{Z}^{m}$ on the right-hand side is defined in § 3.5.

4.1 Some $\operatorname{SL}_{n}$ notation

The remainder of the paper will be devoted to the proof of Conjecture 3.4 for the case $G=\operatorname{SL}_{n}$ .

The weight lattice of $\operatorname{SL}_{n}$ is $\unicode[STIX]{x1D6EC}=\mathbb{Z}^{n}/\mathbb{Z}(1,\ldots ,1)$ and $\unicode[STIX]{x1D6EC}_{+}$ is the subset consisting of those $\unicode[STIX]{x1D706}=(a_{1},\ldots ,a_{n})$ such that $a_{1}\geqslant \ldots \geqslant a_{n}$ . The minuscule dominant weights are $\unicode[STIX]{x1D714}_{k}=(1,\ldots ,1,0,\ldots ,0)$ , for $k=1,\ldots ,n-1$ . The corresponding representation of $U_{q}\mathfrak{s}\mathfrak{l}_{n}$ is $V(\unicode[STIX]{x1D714}_{k})=\bigwedge _{q}^{k}\mathbb{C}_{q}^{n}$ .

A sequence of minuscule dominant weights $\text{}\underline{\unicode[STIX]{x1D706}}=(\unicode[STIX]{x1D706}_{1},\ldots ,\unicode[STIX]{x1D706}_{m})$ will be encoded by a sequence $\text{}\underline{k}=(k_{1},\ldots ,k_{m})$ , where $\unicode[STIX]{x1D706}_{j}=\unicode[STIX]{x1D714}_{k_{j}}$ . Note that $G^{\vee }=PGL_{n}$ and $\widetilde{G^{\vee }}=\operatorname{SL}_{n}$ . We write

where $e_{k}$ is the usual $k$ th elementary symmetric function.

4.2 The spider

Our main tool for proving the conjecture in the $\operatorname{SL}_{n}$ case will be a combinatorially defined category called the annular spider.

We begin by recalling the definition of the spider category ${\mathcal{S}}p_{n}[q^{\pm }]$ from [Reference Cautis, Kamnitzer and MorrisonCKM14]. ${\mathcal{S}}p_{n}[q^{\pm }]$ has as objects sequences $\text{}\underline{k}$ in $\{1^{\pm },\ldots ,(n-1)^{\pm }\}$ , and as morphisms $\mathbb{C}[q^{\pm }]$ -linear combinations of oriented planar graphs by the following four types of vertices:

with all labels drawn from the set $\{1,\ldots ,n-1\}$ . The third and fourth graphs depict bivalent vertices, called ‘tags’, which are not rotationally symmetric, meaning that the tag provides a distinguished side. The bottom boundary of any planar graph in $\operatorname{Hom}(\text{}\underline{k},\text{}\underline{l})$ is $\text{}\underline{k}$ with the strand oriented up for each positive entry, and the strand oriented down for each negative entry. Similarly, the top boundary is determined by $\text{}\underline{l}$ in the same way.

On these diagrams, we impose the following relations, together with the mirror reflections and the arrow reversals of these.

The category ${\mathcal{S}}p_{n}[q^{\pm }]$ is a monoidal category with tensor product given on objects by concatenation of sequences and on morphisms by concatenation in the vertical direction. It also has a natural pivotal structure by the usual graphical calculus. We will denote by ${\mathcal{S}}p_{n}(q)$ the same category but over $\mathbb{C}(q)$ rather than $\mathbb{C}[q^{\pm }]$ . We also denote by ${\mathcal{S}}p_{n}$ the category specialized to $q=1$ .

It will be useful for us to add crossings to the spider. As in [Reference Cautis, Kamnitzer and MorrisonCKM14], we define a crossing of a strand labelled $k$ with a strand labelled $l$ to be a sum of webs.

This defines a functor ${\mathcal{B}}_{m}(\{1,\ldots ,n\})\rightarrow {\mathcal{S}}p_{n}$ which we can regard as giving a braiding on the category ${\mathcal{S}}p_{n}$ .

Figure 2.

An example of an annular web with source $(3,2,5,-4)$ and target $(1,2,3,-4,3,1)$ .

4.3 The annular spider

We now define the annular spider ${\mathcal{A}}{\mathcal{S}}p_{n}[q^{\pm }]$ . The objects are the same as in ${\mathcal{S}}p_{n}[q^{\pm }]$ . For the morphisms, we consider webs with the same kinds of vertices, except that the webs are now embedded in an annulus $A=S^{1}\times [0,1]$ . We consider the same local generating relations as in ${\mathcal{S}}p_{n}[q^{\pm }]$ . The inner boundary of the annulus is regarded as the source of the morphism and the outer boundary is the target. Let $\ast \in S^{1}$ denote the bottom of the circle. We do not allow strands to start or end at $\ast$ . Given a web $w$ , its source (respectively target) is found by reading the inner (respectively outer) circle clockwise starting at the $\ast$ (see Figure 2 for an example).

As before, we denote by ${\mathcal{A}}{\mathcal{S}}p_{n}(q)$ the same category but with the space of morphisms defined over $\mathbb{C}(q)$ rather than $\mathbb{C}[q^{\pm }]$ . We can also consider the $q=1$ version of ${\mathcal{A}}{\mathcal{S}}p_{n}[q^{\pm }]$ , which we denote  ${\mathcal{A}}{\mathcal{S}}p_{n}$ .

4.4 The functor $\unicode[STIX]{x1D6E4}$

We begin by defining a functor

following [Reference Cautis, Kamnitzer and MorrisonCKM14] (this functor was called $\unicode[STIX]{x1D6E4}_{n}$ in [Reference Cautis, Kamnitzer and MorrisonCKM14]). At the level of objects we take

(where $\unicode[STIX]{x1D700}_{i}\in \{+,-\}$ ).

The morphisms are mapped as

where $M_{k,l}:\bigwedge _{q}^{k}(\mathbb{C}_{q}^{n})\otimes \bigwedge _{q}^{l}(\mathbb{C}_{q}^{n})\rightarrow \bigwedge _{q}^{k+l}(\mathbb{C}_{q}^{n})$ is defined by

while $M_{k,l}^{\prime }:\bigwedge _{q}^{k+l}(\mathbb{C}_{q}^{n})\rightarrow \bigwedge _{q}^{k}(\mathbb{C}_{q}^{n})\otimes \bigwedge _{q}^{l}(\mathbb{C}_{q}^{n})$ is defined by

Here, if $S=\{k_{1},\ldots ,k_{a}\}\subset \{1,\ldots ,n\}$ , with $k_{1}>\cdots >k_{a}$ , we write $x_{S}=x_{k_{1}}\wedge _{q}\cdots \wedge _{q}x_{k_{a}}\in \bigwedge _{q}^{a}(\mathbb{C}_{q}^{n})$ and likewise for $x_{T}$ . Moreover, if $S,T$ are disjoint subsets of $\{1,\ldots ,n\}$ we define

The maps in (15) then force upon us the definition for tags. The main result from [Reference Cautis, Kamnitzer and MorrisonCKM14] can be stated as follows.

4.5 Horizontal trace and the annular spider category

We will now identify the annular spider category as the horizontal trace of the usual spider category. We define a functor ${\mathcal{S}}p_{n}(q)(S^{1})\rightarrow {\mathcal{A}}{\mathcal{S}}p_{n}(q)$ as follows. On objects it is the identity.

For morphisms, let $\text{}\underline{k},\text{}\underline{l}$ be two objects in ${\mathcal{S}}p_{n}(q)$ . Suppose that we have a morphism $[\text{}\underline{r},\unicode[STIX]{x1D719}]$ in ${\mathcal{S}}p_{n}(q)(S^{1})$ , where $\unicode[STIX]{x1D719}$ is a web with bottom boundary $\text{}\underline{k}\sqcup \text{}\underline{r}$ and top boundary $\text{}\underline{r}\sqcup \text{}\underline{l}$ . From $(\text{}\underline{r},\unicode[STIX]{x1D719})$ we can get a morphism $\overline{(r,\unicode[STIX]{x1D719})}$ in ${\mathcal{A}}{\mathcal{S}}p_{n}(q)$ by sending $\text{}\underline{r}$ around the circle as follows.

Figure 3.

An annular web $\unicode[STIX]{x1D713}$ and its unfolding $\unicode[STIX]{x1D713}^{\prime }$ . In this example $\text{}\underline{k}=(3,2,5,4^{-})$ , $\text{}\underline{l}=(1,2,3,4^{-},3,1)$ and $\text{}\underline{r}=(3,1)$ .

Because ${\mathcal{S}}p_{n}(q)$ is a braided monoidal category, ${\mathcal{S}}p_{n}(q)(S^{1})$ carries a monoidal structure and thus ${\mathcal{A}}{\mathcal{S}}p_{n}(q)$ acquires a monoidal structure. Diagrammatically, the tensor product of two morphisms $\unicode[STIX]{x1D719}_{1}$ and $\unicode[STIX]{x1D719}_{2}$ is given by placing the source and target of $\unicode[STIX]{x1D719}_{1}$ to the left of those of $\unicode[STIX]{x1D719}_{2}$ and having the strands of $\unicode[STIX]{x1D719}_{1}$ cross over those of $\unicode[STIX]{x1D719}_{2}$ .

4.6 Action of annular braids and loops

Suppose we have an annular braid with strands labelled from $1,\ldots ,n-1$ . Since we have crossings in ${\mathcal{A}}{\mathcal{S}}p_{n}(q)$ , we can interpret this annular braid as a morphism in ${\mathcal{A}}{\mathcal{S}}p_{n}(q)$ . Thus we get a functor

4.7 The action of $E$

One can define a map $E[q^{\pm }]\rightarrow \operatorname{End}_{{\mathcal{A}}{\mathcal{S}}p_{n}[q^{\pm }]}(\text{}\underline{k})$ by taking $e_{k}$ to a counterclockwise loop labelled $k$ passing over all the strands.

Such maps commute with all morphisms in ${\mathcal{A}}{\mathcal{S}}p_{n}[q^{\pm }]$ and hence give a map $E[q^{\pm }]\rightarrow \operatorname{End}_{{\mathcal{A}}{\mathcal{S}}p_{n}[q^{\pm }]}(\operatorname{id})$ to the endomorphisms of the identity functor.

On the other hand, by the construction in § 2.3.2, we have a map

By a similar argument as in the proof of Proposition 4.7, we deduce the following.

5.1 The quantum loop algebra

We first recall the integral, idempotent form $\dot{U}_{[q^{\pm }]}L\mathfrak{g}\mathfrak{l}_{m}$ of the quantum loop algebra of $\mathfrak{g}\mathfrak{l}_{m}$ . This is a category whose objects are indexed by $\text{}\underline{k}=(k_{1},\ldots ,k_{m})\in \mathbb{Z}^{m}$ . For $i=1,\ldots ,m-1$ we denote $\unicode[STIX]{x1D6FC}_{i}=(0,\ldots ,1,-1,\ldots ,0)$ where the $1$ is in position $i$ . Moreover, we denote $\unicode[STIX]{x1D6FC}_{0}=(-1,0,\ldots ,0,1)$ .

As usual, we will write $1_{\text{}\underline{k}}$ for the identity morphism of $\text{}\underline{k}$ . The morphisms in $\dot{U}_{[q^{\pm }]}L\mathfrak{g}\mathfrak{l}_{m}$ are further generated over $\mathbb{C}[q^{\pm }]$ by $E_{i}^{(s)},F_{i}^{(s)}$ , for $i=0,\ldots ,m-1$ , $s\in \mathbb{N}$ and by $R$ . These satisfy the following relations:

1. (i) $1_{\text{}\underline{k}+s\unicode[STIX]{x1D6FC}_{i}}E_{i}^{(s)}=1_{\text{}\underline{k}+s\unicode[STIX]{x1D6FC}_{i}}E_{i}^{(s)}1_{\text{}\underline{k}}=E_{i}^{(s)}1_{\text{}\underline{k}}$ and $1_{\text{}\underline{k}}F_{i}^{(s)}=1_{\text{}\underline{k}}F_{i}^{(s)}1_{\text{}\underline{k}+s\unicode[STIX]{x1D6FC}_{i}}=F_{i}^{(s)}1_{\text{}\underline{k}+s\unicode[STIX]{x1D6FC}_{i}}$ ;

2. (ii) $R1_{\text{}\underline{k}}=1_{r\cdot \text{}\underline{k}}R$ where $r\cdot (k_{1},\ldots ,k_{m})=(k_{m},k_{1},\ldots ,k_{m-1})$ is ‘rotation’ of the weights;

3. (iii) $[E_{i}^{(s)},F_{i}^{(t)}]1_{\text{}\underline{k}}=\sum _{j\geqslant 0}\left[\begin{smallmatrix}\langle \text{}\underline{k},\unicode[STIX]{x1D6FC}_{i}\rangle -t+s\\ j\\ \end{smallmatrix}\right]F_{i}^{(t-j)}E_{i}^{(s-j)}1_{\text{}\underline{k}}$ where $\langle \cdot ,\cdot \rangle$ is the standard inner product (here by convention $F^{(r)}=E^{(r)}=0$ if $r<0$ , so this is actually a finite sum);

4. (iv) $[E_{i}^{(s)},F_{j}^{(t)}]=0$ if $i\neq j$ ;

5. (v) if $|i-j|=1$ (modulo $m$ ) then $E_{i}E_{j}E_{i}=E_{i}^{(2)}E_{j}+E_{j}E_{i}^{(2)}$ and similarly where the generators $E_{i}$ are replaced by the generators $F_{i}$ ;

6. (vi) if $|i-j|>1$ (modulo $m$ ) then $[E_{i}^{(r)},E_{j}^{(s)}]=0=[F_{i}^{(r)},F_{j}^{(s)}]$ ;

7. (vii) $RE_{i}^{(s)}=E_{i+1}^{(s)}R$ (modulo $m$ ) and similarly $RF_{i}^{(s)}=F_{i+1}^{(s)}R$ (modulo $m$ ).

We will denote by $(\dot{U}_{[q^{\pm }]}L\mathfrak{g}\mathfrak{l}_{m})^{n}$ the quotient of $\dot{U}_{[q^{\pm }]}L\mathfrak{g}\mathfrak{l}_{m}$ obtained by killing any object not of the form $(k_{1},\ldots ,k_{m})$ with $0\leqslant k_{i}\leqslant n$ . We will also denote by $\dot{U}_{[q^{\pm }]}\mathfrak{g}\mathfrak{l}_{m}$ the subcategory generated by $E_{i}$ and $F_{i}$ for $i=1,\ldots ,m-1$ and by $(\dot{U}_{[q^{\pm }]}\mathfrak{g}\mathfrak{l}_{m})^{n}$ the corresponding quotient.

The category $(\dot{U}_{[q^{\pm }]}L\mathfrak{g}\mathfrak{l}_{m})^{n}$ (or more precisely the direct sum of all Hom spaces in this category) is a special case of an affine generalized $q$ -Schur algebra (or Beilinson–Lusztig–MacPherson (BLM) algebra), as defined in [Reference McGertyMcG07, § 6.1]. As explained by McGerty [Reference McGertyMcG07, Proposition 6.4], the following result is a consequence of the work of Beck and Nakajima [Reference Beck and NakajimaBN04].

5.2 The braiding functor

Like the braiding structure (14) in ${\mathcal{S}}p_{n}$ we can define a functor ${\mathcal{B}}_{m}(\{0,\ldots ,n\})\rightarrow (\dot{U}_{[q^{\pm }]}\mathfrak{g}\mathfrak{l}_{m})^{n}$ as follows (following [Reference LusztigLus93, 5.2.1]). On objects it takes $\text{}\underline{k}\mapsto \text{}\underline{k}$ . On morphisms it takes

5.3 Representations of $(\dot{U}_{[q^{\pm }]}L\mathfrak{g}\mathfrak{l}_{m})^{n}$

The description above gives $\dot{U}_{[q^{\pm }]}L\mathfrak{g}\mathfrak{l}_{m}$ in its Kac–Moody form rather than its more common loop form. In the latter form, one also considers generators $E_{i}\otimes t^{r},F_{i}\otimes t^{r}$ , for $r\in \mathbb{Z}$ , but no $E_{0}$ or $F_{0}$ .

The advantage of the Kac–Moody presentation of $\dot{U}_{[q^{\pm }]}L\mathfrak{g}\mathfrak{l}_{m}$ is that it is more closely related to the annular spider category ${\mathcal{A}}{\mathcal{S}}p_{n}$ . However, we will need to define representations of $(\dot{U}_{[q^{\pm }]}L\mathfrak{g}\mathfrak{l}_{m})^{n}$ which are more naturally defined in the loop presentation. The following technical lemma tells us how to deal with this issue. It shows that it suffices to define the functor on $(\dot{U}_{[q^{\pm }]}\mathfrak{g}\mathfrak{l}_{m})^{n}$ together with a compatible functor from the annular braid groupoid.

5.4 Embeddings

Observe that we have $2(m+1)$ faithful functors $(\dot{U}_{[q^{\pm }]}L\mathfrak{g}\mathfrak{l}_{m})^{n}\rightarrow (\dot{U}_{[q^{\pm }]}L\mathfrak{g}\mathfrak{l}_{m+1})^{n}$ , given by adding either a $0$ or an $n$ in some slot. For concreteness let us describe the functor of adding a $0$ at the end. On objects it is given by $(k_{1},\ldots ,k_{m})\mapsto (k_{1},\ldots ,k_{m},0)$ . On morphisms it is given by

5.5 From quantum loop algebras to annular spiders

In [Reference Cautis, Kamnitzer and MorrisonCKM14] we studied the spider category ${\mathcal{S}}p_{n}[q^{\pm }]$ by defining a functor $\unicode[STIX]{x1D6F9}_{m}:(\dot{U}_{[q^{\pm }]}\mathfrak{g}\mathfrak{l}_{m})^{n}\rightarrow {\mathcal{S}}p_{n}[q^{\pm }]$ for each $m$ . In exactly the same way one can define a functor

More precisely, on objects it takes $\text{}\underline{k}$ to $\overline{\text{}\underline{k}}$ where $\overline{\text{}\underline{k}}$ is the sequence obtained from $\text{}\underline{k}$ by deleting all $0,n$ . On morphisms we define $A\unicode[STIX]{x1D6F9}_{m}$ by forming ‘ladder-like’ webs, as in [Reference Cautis, Kamnitzer and MorrisonCKM14], the only difference being that $E_{0}^{(s)},F_{0}^{(s)}$ are sent to webs which ‘wrap-around’ the annulus crossing the cut line, as illustrated in (18) with $E_{0}^{(s)}$ . Note that this functor was also defined in [Reference QueffelecQue15, § 2.3].

6.1 The varieties

We work with the lattice description of $Gr=Gr_{P\operatorname{GL}_{n}}\!$ . This means that we identify $Gr$ with the space of ${\mathcal{O}}$ -lattices in the vector space ${\mathcal{K}}^{n}$ (where ${\mathcal{O}}=\mathbb{C}[[z]]$ and ${\mathcal{K}}=\mathbb{C}((z))$ ) modulo the equivalence relation of homothety, i.e. $L_{1}\sim L_{2}$ if and only if $z^{k}L_{1}=L_{2}$ for some $k\in \mathbb{Z}$ .

Note that under this identification, if $\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}_{+}\subset \mathbb{Z}^{n}/\mathbb{Z}(1,\ldots ,1)$ , then we have

where we regard $z|_{L/L_{0}}$ as a nilpotent linear operator on a finite-dimensional vector space and consider its Jordan type to be the length of blocks in its Jordan form. (Note that $\unicode[STIX]{x1D706}$ is only defined up to shift by a constant sequence; this matches the fact that $L$ is only defined up to homothety.)

As before, we encode of sequence of minuscule dominant weights $\text{}\underline{\unicode[STIX]{x1D706}}$ by a sequence $\text{}\underline{k}=(k_{1},\ldots ,k_{m})$ with $1\leqslant k_{i}\leqslant n-1$ . Let us introduce the notation $Y(\text{}\underline{k}):=Gr^{\text{}\underline{\unicode[STIX]{x1D706}}}$ . The varieties $Y(\text{}\underline{k})$ can be described explicitly as sequences of lattices (not considered up to homothety).

For convenience of notation, we also allow sequences $\text{}\underline{k}$ containing $0,n$ . We have a canonical identification $Y(\text{}\underline{k})=Y(\overline{\text{}\underline{k}})$ where $\overline{\text{}\underline{k}}$ is defined by removing all $0,n$ .

Assume that $\text{}\underline{k}=(k_{1},\ldots ,k_{m})$ and $\text{}\underline{k}^{\prime }=(k_{1}^{\prime },\ldots ,k_{m^{\prime }}^{\prime })$ are two sequences with the same sum, $k_{1}+\cdots +k_{m}=k_{1}^{\prime }+\cdots +k_{m^{\prime }}^{\prime }$ . From the lemma, we reach a simple description of the varieties

We consider the action of $\operatorname{SL}_{n}\times \mathbb{C}^{\times }$ on ${\mathcal{K}}^{n}=\mathbb{C}^{n}\otimes {\mathcal{K}}$ where $\operatorname{SL}_{n}$ acts on $\mathbb{C}^{n}$ and $\mathbb{C}^{\times }$ acts on ${\mathcal{K}}$ by $s\cdot z^{k}=s^{k}z^{k}$ . This gives us an action of $\operatorname{SL}_{n}\times \mathbb{C}^{\times }$ on $Gr$ and on each $Y(\text{}\underline{k})$ .

6.2 The action of $(\dot{U}_{[q^{\pm }]}\mathfrak{g}\mathfrak{l}_{m})^{n}$

We first define a functor

On objects $\unicode[STIX]{x1D719}_{m}$ takes $\text{}\underline{k}$ to $\overline{\text{}\underline{k}}$ where $\overline{\text{}\underline{k}}$ is obtained from $\text{}\underline{k}$ by removing all $0,n$ . To define it on morphisms, we must define maps

for every $\text{}\underline{k},\text{}\underline{k}^{\prime }$ . Since $(\dot{U}_{[q^{\pm }]}\mathfrak{g}\mathfrak{l}_{m})^{n}$ is defined by generators and relations, we will define $\unicode[STIX]{x1D719}_{m}$ on the generators $E_{i}^{(s)}$ and $F_{i}^{(s)}$ for $i=1,\ldots ,m-1$ . To do this we use the varieties

Note that these varieties are actually supported on $Z(\text{}\underline{k},\text{}\underline{k}-s\unicode[STIX]{x1D6FC}_{i})$ . Then we define

where the prime denotes the corresponding bundle pulled back from the second factor. (These kernels live in $K^{\operatorname{SL}_{n}\times \mathbb{C}^{\times }}(Z(\text{}\underline{k}-s\unicode[STIX]{x1D6FC}_{i},\text{}\underline{k}))$ and $K^{\operatorname{SL}_{n}\times \mathbb{C}^{\times }}(Z(\text{}\underline{k},\text{}\underline{k}-s\unicode[STIX]{x1D6FC}_{i}))$ respectively.) It follows from the calculations in [Reference Cautis, Kamnitzer and LicataCKL10, Reference CautisCau05] that these satisfy the relations of $\dot{U}_{[q^{\pm }]}\mathfrak{g}\mathfrak{l}_{m}$ (see for instance [Reference CautisCau05, Theorem 8.2]).

6.3 Extending to an action of $(\dot{U}_{[q^{\pm }]}L\mathfrak{g}\mathfrak{l}_{m})^{n}$

To extend the action above we first define an action of the annular braid group on $K\operatorname{Conv}^{\operatorname{SL}_{n}\times \mathbb{C}^{\times }}(Gr)$ and then use Lemma 5.5.

In order to apply Lemma 5.5 we need to check the following two conditions:

1. (i) $\unicode[STIX]{x1D6FD}(X_{1})$ commutes with $\unicode[STIX]{x1D719}_{m}(E_{i}^{(s)})$ and $\unicode[STIX]{x1D719}_{m}(F_{i}^{(s)})$ for $i=2,\ldots ,m-1$ ;

2. (ii) $\unicode[STIX]{x1D6FD}(X_{1}\ldots X_{m})$ commutes with everything in the image of $\unicode[STIX]{x1D719}_{m}$ .

The first condition follows since $\unicode[STIX]{x1D6FD}(X_{1})$ only involves the line bundle $\det (L_{1}/L_{0})$ while $\unicode[STIX]{x1D719}_{m}(E_{i}^{(s)})$ only involves changing the flag $L_{i}$ . The second condition follows since $\unicode[STIX]{x1D6FD}(X_{1}\ldots X_{m})$ is given by $\unicode[STIX]{x1D6E5}_{\ast }\det (L_{m}/L_{0})^{\vee }$ which clearly commutes with the image of every $E_{i}^{(s)}$ and $F_{i}^{(s)}$ . Thus, by Lemma 5.5, $\unicode[STIX]{x1D719}_{m}$ extends to a functor

6.4 The functor $\unicode[STIX]{x1D6F7}$

The remainder of the paper will be devoted to proving the following result which combined with Corollary 4.5 completes the proof of Conjecture 3.4 in the case $G=\operatorname{SL}_{n}$ .

6.5 Compatibility with $E$ -action

Recall that inside ${\mathcal{A}}{\mathcal{S}}p_{n}[q^{\pm }]$ we have the elements $e_{j}$ corresponding to counterclockwise loops. This was used to define a $E[q^{\pm }]$ -module structure to the Hom spaces in ${\mathcal{A}}{\mathcal{S}}p_{n}[q^{\pm }]$ . The following result determines the image of $e_{j}$ under $\unicode[STIX]{x1D6F7}$ .