2.1 Shifted Quantum Affine Algebra

In this subsection, we recall the definition of shifted quantum affine algebras. Our main reference is [Reference Finkelberg and Tsymbaliuk22, Section 5].

First, we fix some notations. Let $\mathfrak {g}$ be a simple Lie algebra, $\mathfrak {h} \subset \mathfrak {g}$ be a Cartan subalgebra, and $\big ( \cdot , \cdot \big )$ be a non-degenerated invariant symmetric bilinear form on $\mathfrak {g}$ . Let $\{\alpha _{i}\}_{i \in I} \subset \mathfrak {h}^*$ be the simple roots of $\mathfrak {g}$ relative to $\mathfrak {h}$ and $\{\alpha ^{\vee }_{i}\}_{i \in I} \subset \mathfrak {h}$ be the simple coroots. Let $c_{ij}:=2\frac {( \alpha _{i},\alpha _{j})}{( \alpha _{i},\alpha _{i} )}$ be the entries of the Cartan matrix and $d_{i}:=\frac { ( \alpha _{i},\alpha _{i} )}{2}$ such that $d_{i}c_{ij}=d_{j}c_{ji}$ for any $i,j \in I$ . We also fix the notations $q_{i}:=q^{d_{i}}$ , $[m]_{q}:=\frac {q^{m}-q^{-m}}{q-q^{-1}}$ and ${a \brack b}_{q}=\frac {[a-b+1]_{q}...[a]_{q}}{[1]_{q}...[b]_{q}}$ .

Definition 2.1 [Reference Finkelberg and Tsymbaliuk22, Subsection 5.1]. Given two coweights $\mu ^{+}, \mu ^{-}$ , set $b^{\pm }_{i}:=\alpha _{i}(\mu ^{\pm })$ . Then the shifted quantum affine algebra (simply-connected version), denoted by $\mathcal {U}_{\mu ^{+},\mu ^{-}}$ , is an associated $\mathbb {C}(q)$ algebra generated by

$$\begin{align*}\{e_{i,r},\ f_{i,r},\ (\psi^{\pm}_{i,\pm s^{\pm}_{i}}),\ (\psi^{\pm}_{i,\mp b^{\pm}_{i}})^{-1} \}^{r \in \mathbb{Z}, \ s_{i}^{\pm} \geq - b_{i}^{\pm}}_{i \in I} \end{align*}$$

subject to the following relations (for all $i, j \in I$ and $\epsilon , \epsilon ' \in \{\pm \}$ )

(U1) $$ \begin{align} [\psi^{\epsilon}_{i}(z),\psi^{\epsilon'}_{j}(w)]=0, \ \psi^{\pm}_{i,\mp b^{\pm}_{i}}(\psi^{\pm}_{i,\mp b^{\pm}_{i}})^{-1}=(\psi^{\pm}_{i,\mp b^{\pm}_{i}})^{-1}\psi^{\pm}_{i,\mp b^{\pm}_{i}}=1, \end{align} $$

(U2) $$ \begin{align} (z-q^{c_{ij}}_{i}w)e_{i}(z)e_{j}(w)=(q^{c_{ij}}_{i}z-w)e_{j}(w)e_{i}(z), \end{align} $$

(U3) $$ \begin{align} (q^{c_{ij}}_{i}z-w)f_{i}(z)f_{j}(w)=(z-q^{c_{ij}}_{i}w)f_{j}(w)f_{i}(z), \end{align} $$

(U4) $$ \begin{align} (z-q^{c_{ij}}_{i}w)\psi^{\epsilon}_{i}(z)e_{j}(w)=(q^{c_{ij}}_{i}z-w)e_{j}(w)\psi^{\epsilon}_{i}(z), \end{align} $$

(U5) $$ \begin{align} (q^{c_{ij}}_{i}z-w)\psi^{\epsilon}_{i}(z)f_{j}(w)=(z-q^{c_{ij}}_{i}w)f_{j}(w)\psi^{\epsilon}_{i}(z), \end{align} $$

(U6) $$ \begin{align} [e_{i}(z),f_{j}(w)]=\frac{\delta_{ij}}{q_{i}-q^{-1}_{i}}\delta(\frac{z}{w})(\psi_{i}^{+}(z)-\psi_{i}^{-}(z)), \end{align} $$

(U7) $$ \begin{align} & \mathrm{Sym}_{z_{1},...,z_{1-c_{ij}}} \sum^{1-c_{ij}}_{r=0} (-1)^{r} {1-c_{ij} \brack r}_{q_{i}} e_{i}(z_{1})....e_{i}(z_{r})e_{j}(w)e_{i}(z_{r+1})...e_{i}(z_{1-c_{ij}})=0, \end{align} $$

(U8) $$ \begin{align} &\mathrm{Sym}_{z_{1},...,z_{1-c_{ij}}} \sum^{1-c_{ij}}_{r=0} (-1)^{r} {1-c_{ij} \brack r}_{q_{i}} f_{i}(z_{1})....f_{i}(z_{r})f_{j}(w)f_{i}(z_{r+1})...f_{i}(z_{1-c_{ij}})=0, \end{align} $$

where $\mathrm {Sym}_{z_{1},...,z_{s}}$ stands for the symmetrization in $z_{1},...,z_{s}$ and the generating series are defined as follows

$$ \begin{align*} e_{i}(z):=\sum_{r\in \mathbb{Z}}e_{i,r}z^{-r}, f_{i}(z):=\sum_{r\in \mathbb{Z}}f_{i,r}z^{-r}, \psi^{\pm}_{i}(z):=\sum_{r \geq -b^{\pm}_{i}}\psi^{\pm}_{i, \pm r}z^{\mp r}, \delta(z):=\sum_{r \in \mathbb{Z}} z^{r}. \end{align*} $$

Let us introduce another set of Cartan generators $\{h_{i,r}\}^{r>0}_{i \in I}$ via

$$ \begin{align*} (\psi^{\pm}_{i, \mp b^{\pm}_{i}}z^{\pm b_{i}^{\pm}})^{-1}\psi^{\pm}_{i}(z)=\exp\Big(\pm (q_{i}-q^{-1}_{i}) \sum_{r>0} h_{i,\pm r}z^{\mp r}\Big). \end{align*} $$

With this, the relations (U4), (U5) are equivalent to the following:

$$ \begin{align*} &\psi^{\pm}_{i, \mp b^{\pm}_{i}}e_{j,s}=q^{\pm c_{ij}}_{i}e_{j,s}\psi^{\pm}_{i, \mp b^{\pm}_{i}}, \ [h_{i,r},e_{j,s}]=\frac{[rc_{ij}]_{q_{i}}}{r}e_{j,r+s}, \\ &\psi^{\pm}_{i, \mp b^{\pm}_{i}}f_{j,s}=q^{\mp c_{ij}}_{i}f_{j,s}\psi^{\pm}_{i, \mp b^{\pm}_{i}}, \ [h_{i,r},f_{j,s}]=-\frac{[rc_{ij}]_{q_{i}}}{r}f_{j,r+s}. \end{align*} $$

We mention some remarks about the properties of $\mathcal {U}_{\mu ^{+},\mu ^{-}}$ that have been addressed in [Reference Finkelberg and Tsymbaliuk22].

When $\mathcal {U}_{\mu ^{+},\mu ^{-}}$ is antidominantly shifted (i.e., $\mu =\mu ^{+}+\mu ^{-}$ is antidominant), then it admits another presentation, which is the so-called Levendorskii type presentation.

Definition 2.3 [Reference Finkelberg and Tsymbaliuk22, Subsection 5.5]. Given antidominant coweights $\mu _{1}, \mu _{2}$ , set $\mu =\mu _{1}+\mu _{2}$ . Define $b_{1,i}:=\alpha _{i}(\mu _{1})$ , $b_{2,i}:=\alpha _{i}(\mu _{2})$ , $b_{i}=b_{1,i}+b_{2,i}$ . Then we denote $\hat {\mathcal {U}}_{\mu _{1},\mu _{2}}$ to be the associated $\mathbb {C}(q)$ algebra generated by

$$\begin{align*}\{e_{i,r},\ f_{i,s},\ (\psi^{+}_{i,0})^{\pm 1},\ (\psi^{-}_{i,b_{i}})^{\pm 1},\ h_{i,\pm 1}\ | \ i\in I, \ b_{2,i}-1 \leq r \leq 0, \ b_{1,i} \leq s \leq 1 \} \end{align*}$$

subject to the following relations

(U1') $$ \begin{align} \{(\psi^{+}_{i,0})^{\pm 1},\ (\psi^{-}_{i,b_{i}})^{\pm 1},\ h_{i,\pm 1} \}_{i \in I} \ \mathrm{pairwise\ commute}, \end{align} $$

(U2') $$ \begin{align} (\psi^{+}_{i,0})^{\pm 1} \cdot (\psi^{+}_{i,0})^{\mp 1}= (\psi^{-}_{i,b_{i}})^{\pm 1} \cdot (\psi^{-}_{i,b_{i}})^{\mp 1}=1, \end{align} $$

(U3') $$ \begin{align} e_{i,r+1}e_{j,s}-q_{i}^{c_{ij}}e_{i,r}e_{j,s+1}=q_{i}^{c_{ij}}e_{j,s}e_{i,r+1}-e_{j,s+1}e_{i,r}, \end{align} $$

(U4') $$ \begin{align} q_{i}^{c_{ij}}f_{i,r+1}f_{j,s}-f_{i,r}f_{i,s+1}=f_{j,s}f_{i,r+1}-q_{i}^{c_{ij}}f_{j,s+1}f_{i,r}, \end{align} $$

(U5') $$ \begin{align} &\psi_{i,0}^{+}e_{j,r}=q_{i}^{c_{ij}}e_{j,r}\psi_{i,0}^{+}, \ \psi_{i,b_{i}}^{-}e_{j,r}=q_{i}^{-c_{ij}}e_{j,r}\psi_{i,b_{i}}^{-},\ [h_{i,\pm 1},e_{j,r}]=[c_{ij}]_{q_{i}} e_{j,r\pm 1}, \end{align} $$

(U6') $$ \begin{align} &\psi_{i,0}^{+}f_{j,s}=q_{i}^{-c_{ij}}f_{j,s}\psi_{i,0}^{+}, \ \psi_{i,b_{i}}^{-}f_{j,s}=q_{i}^{c_{ij}}f_{j,s}\psi_{i,b_{i}}^{-},\ [h_{i,\pm 1},f_{j,s}]=-[c_{ij}]_{q_{i}} f_{j,s\pm 1}, \end{align} $$

(U7') $$ \begin{align} [e_{i,r},f_{j,s}]=0 \ \mathrm{if} \ i \neq j\ \ \mathrm{and}\ \ [e_{i,r},f_{i,s}]= \begin{cases} \psi^{+}_{i,0}h_{i,1} & \text{if} \ r+s=1 \\ \frac{\psi^{+}_{i,0}-\delta_{b_{i},0}\psi_{i,b_{i}}^{-}}{q_{i}-q_{i}^{-1}} & \text{if} \ r+s=0 \\ 0 & \text{if} \ b_{i}+1 \leq r+s \leq-1 \\ \frac{-\psi^{-}_{i,b_{i}}+\delta_{b_{i},0}\psi_{i,0}^{-}}{q_{i}-q_{i}^{-1}} & \text{if} \ r+s=b_{i} \\ \psi^{-}_{i,b_{i}}h_{i,-1} & \text{if} \ r+s=b_{i}-1 \end{cases}, \end{align} $$

(U8') $$ \begin{align} \sum_{r=0}^{1-c_{ij}}(-1)^{r} { 1-c_{ij} \brack r}_{q_{i}} e^{r}_{i,0}e_{j,0}e_{i,0}^{1-c_{ij}-r}=0, \ \sum_{r=0}^{1-c_{ij}}(-1)^{r}{ 1-c_{ij} \brack r}_{q_{i}} f^{r}_{i,0}f_{j,0}f_{i,0}^{1-c_{ij}-r}=0, \end{align} $$

(U9') $$ \begin{align} [h_{i,1},[f_{i,1},[h_{i,1},e_{i,0}]]]=0,\ [h_{i,-1},[e_{i,b_{2,i}-1},[h_{i,-1},f_{i,b_{1,i}}]]]=0, \end{align} $$

for any $i,j \in I$ and $r,s$ such that the above relations make sense.

With those generators, then we define inductively

$$ \begin{align*} e_{i,r}&:=[2]^{-1}_{q_{i}}\begin{cases} [h_{i,1},e_{i,r-1}] & \text{if} \ r>0 \\ [h_{i,-1},e_{i,r+1}] & \text{if} \ r<b_{2,i}-1, \end{cases} \\ f_{i,r}&:=-[2]^{-1}_{q_{i}}\begin{cases} [h_{i,1},f_{i,r-1}] & \text{if} \ r>1 \\ [h_{i,-1},f_{i,r+1}] & \text{if} \ r<b_{1,i}, \end{cases} \\ \psi_{i,r}^{+}&:=(q_{i}-q^{-1}_{i})[e_{i,r-1},f_{i,1}] \ \text{for} \ r>0, \\ \psi_{i,r}^{-}&:=(q^{-1}_{i}-q_{i})[e_{i,r-b_{1,i}},f_{i,b_{1,i}}] \ \text{for} \ r<b_{i}. \end{align*} $$

Then we have the following theorem, which says that in the antidominantly shifted setting, the two presentations from Definition 2.1 and Definition 2.3 are equivalent.

Theorem 2.4 [Reference Finkelberg and Tsymbaliuk22, Theorem 5.5]

There is a $\mathbb {C}(q)$ -algebra isomorphism $\hat {\mathcal {U}}_{\mu _{1},\mu _{2}} \rightarrow \mathcal {U}_{0,\mu }$ such that

$$ \begin{align*} e_{i,r} \mapsto e_{i,r}, \ f_{i,r} \mapsto f_{i,r}, \ \psi^{\pm}_{i,\pm s^{\pm}_{i}} \mapsto \psi^{\pm}_{i,\pm s^{\pm}_{i}} \ \text{for} \ i \in I, \ r \in \mathbb{Z}, \ s^{+}_{i} \geq 0, \ s^{-}_{i} \geq -b_{i}. \end{align*} $$

Remark 2.5. We list some relations explicitly for the readers when $\mathfrak {g}=\mathfrak {sl}_{n}$ , which is the main type of Lie algebras that we will study for the shifted 0-affine algebra later. In this case, we have $c_{ij}=\begin {cases} 2 & \text {if} \ i=j \\ -1 & \text {if} \ |i-j|=1, \\ 0 & \text {if} \ |i-j| \geq 2 \end {cases}$ and $d_{i}=1$ for all i. The Cartan matrix is given by

$$\begin{align*}(c_{ij})= \begin{pmatrix} 2 & -1 & 0 & \dots & 0 & 0 \\ -1 & 2 & -1 & \dots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \dots & -1 & 2 \end{pmatrix}. \end{align*}$$

Then $q_{i}=q$ for all i and the numbers $c_{ij}$ in the relations of the algebra $\hat {\mathcal {U}}_{\mu _{1},\mu _{2}}$ in Definition 2.3 for $\mathfrak {g}=\mathfrak {sl}_{n}$ are known. For example, some of the relations in (U3') are $e_{i,r+1}e_{i,s}-q^{2}e_{i,r}e_{i,s+1}=q^{2}e_{i,s}e_{i,r+1}-e_{i,s+1}e_{i,r}$ , similarly for (U4'). The relations in (U5') are $\psi _{i,0}^{+}e_{i,r}=q^{2}e_{i,r}\psi _{i,0}^{+}$ , $\psi _{i,b_{i}}^{-}e_{i,r}=q^{-2}e_{i,r}\psi _{i,b_{i}}^{-}$ , and $[h_{i,\pm 1},e_{i,r}]=[2]_{q} e_{i,r\pm 1}$ , similarly for (U6'). The relation (U8') is just the (quantum) Serre relations, for example, $e_{i,0}e_{j,0}e_{i,0}=\frac {1}{[2]_{q}}(e^{2}_{i,0}e_{j,0}+e_{j,0}e^{2}_{i,0})$ .

2.2 Definition of the Shifted 0-Affine Algebras

In this section, we define the shifted 0-affine algebras. We define it by imitating Definition 2.3, which is by finite generators and relations. The main reason we use such a presentation is due to its simplicity and because we can easily define a categorical action for it (see next section).

On the other hand, we define another algebra in the appendix A by using the usual loop presentation (see Definition A.1). Similarly to Theorem 2.4, we conjecture that the two presentations, i.e., Definition 2.3 and Definition A.1, are equivalent (see Conjecture A.4).

In [Reference Beilinson, Lusztig and MacPherson8], they introduce the dot version (or idempotent modification) $\dot {\mathrm {\mathbf {U}}}_{q}(\mathfrak {sl}_2)$ of $\mathrm {\mathbf {U}}_{q}(\mathfrak {sl}_2)$ , since our motivation comes from their geometric construction, the shifted 0-affine algebras we introduce below is also an idempotent version. This means that we replace the identity by the direct sum of a system of projectors, one for each element of the weight lattices. They are orthogonal idempotents for approximating the unit element. We refer to [Reference Lusztig36, Chapter 23] for details of such modification.

Throughout the rest of this article, we fix a positive integer $N \geq 2$ . Let

$$\begin{align*}C(n,N):=\{\underline{k}=(k_1,...,k_{n}) \in (\mathbb{N} \cup \{0\})^n \ | \ k_{1}+...+k_{n}=N \}. \end{align*}$$

We regard each $\underline {k}$ as a weight for $\mathfrak {sl}_{n}$ via the identification of the weight lattice of $\mathfrak {sl}_n$ with the quotient $\mathbb {Z}^n/(1,1,...,1)$ . We choose the simple root $\alpha _{i}$ to be $(0...0,-1,1,0...0)$ where the $-1$ is in the i-th position for $1 \leq i \leq n-1$ . Then we introduce the shifted 0-affine algebra for $\mathfrak {sl}_n$ , which is defined by using finite generators and relations.

Definition 2.6. We consider formal symbols of the form $1_{\lambda }x1_{\mu }$ ( $\lambda , \mu \in (\mathbb {N} \cup \{0\})^n$ ) and abbreviating $(1_{\lambda _1}x_{1}1_{\mu _1})...(1_{\lambda _i}x_{i}1_{\mu _i})=1_{\lambda _1}x_{1}...x_{i}1_{\mu _i}$ if the product is nonzero. Then we define the shifted 0-affine algebra, denoted by $\mathcal {U}=\dot {\mathrm {\mathbf {U}}}_{0,N}(L\mathfrak {sl}_n)$ , to be the associative $\mathbb {C}$ -algebra generated by

$$ \begin{align*} {\bigcup_{\underline{k} \in C(n,N)}\{1_{\underline{k}}, 1_{\underline{k}+\alpha_{i}}e_{i,r}1_{\underline{k}}, \ 1_{\underline{k}-\alpha_{i}}f_{i,s}1_{\underline{k}},\ 1_{\underline{k}}(\psi^{+}_{i})^{\pm 1}1_{\underline{k}}, \ 1_{\underline{k}}(\psi^{-}_{i})^{\pm 1}1_{\underline{k}} \}_{1 \leq i \leq n-1}^{-k_{i} \leq r \leq 0, \ 0 \leq s \leq k_{i+1}}} \end{align*} $$

subject to the following relations

(U01) $$ \begin{align} 1_{\underline{k}}1_{\underline{l}}=\delta_{\underline{k},\underline{l}}1_{\underline{k}}, \end{align} $$

(U02) $$ \begin{align} {(\psi^{+}_{i})^{\pm 1}(\psi^{+}_{j})^{\pm 1}1_{\underline{k}}=(\psi^{+}_{j})^{\pm 1}(\psi^{+}_{i})^{\pm 1}1_{\underline{k}} \ \text{for all} \ i,j,} \end{align} $$

(U03) $$ \begin{align} (\psi^{+}_{i})^{\pm 1} \cdot (\psi^{+}_{i})^{\mp 1} 1_{\underline{k}} = 1_{\underline{k}}=(\psi^{-}_{i})^{\pm 1} \cdot (\psi^{-}_{i})^{\mp 1}1_{\underline{k}}, \end{align} $$

(U04) $$ \begin{align} &e_{i,r}e_{j,s}1_{\underline{k}}=\begin{cases} -e_{i,s+1}e_{i,r-1}1_{\underline{k}} & \text{if} \ j=i \\ e_{i+1,s}e_{i,r}1_{\underline{k}}-e_{i+1,s-1}e_{i,r+1}1_{\underline{k}} & \text{if} \ j=i+1 \\ e_{i,r+1}e_{i-1,s-1}1_{\underline{k}}-e_{i-1,s-1}e_{i,r+1}1_{\underline{k}} &\text{if} \ j=i-1 \\ e_{j,s}e_{i,r}1_{\underline{k}} &\text{if} \ |i-j| \geq 2 \end{cases}, \end{align} $$

(U05) $$ \begin{align} &f_{i,r}f_{j,s}1_{\underline{k}}=\begin{cases} -f_{i,s-1}f_{i,r+1}1_{\underline{k}} & \text{if} \ j=i \\ f_{i,r-1}f_{i+1,s+1}1_{\underline{k}}-f_{i+1,s+1}f_{i,r-1}1_{\underline{k}} & \text{if} \ j=i+1 \\ f_{i-1,s}f_{i,r}1_{\underline{k}}-f_{i-1,s+1}f_{i,r-1}1_{\underline{k}} &\text{if} \ j=i-1 \\ f_{j,s}f_{i,r}1_{\underline{k}} &\text{if} \ |i-j| \geq 2 \end{cases}, \end{align} $$

(U06) $$ \begin{align} &\psi^{+}_{i}e_{j,r}1_{\underline{k}}= \begin{cases} -e_{i,r+1}\psi^{+}_{i}1_{\underline{k}} & \text{if} \ j=i \\ -e_{i+1,r-1}\psi^{+}_{i}1_{\underline{k}} & \text{if} \ j=i+1 \\ e_{i-1,r}\psi^{+}_{i}1_{\underline{k}} & \text{if} \ j=i-1 \\ e_{j,r}\psi^{+}_{i}1_{\underline{k}} & \text{if} \ |i-j| \geq 2 \\ \end{cases}, &\psi^{-}_{i}e_{j,r}1_{\underline{k}}= \begin{cases} -e_{i,r+1}\psi^{-}_{i}1_{\underline{k}} & \text{if} \ j=i \\ e_{i+1,r}\psi^{-}_{i}1_{\underline{k}} & \text{if} \ j=i+1 \\ -e_{i-1,r-1}\psi^{-}_{i}1_{\underline{k}} & \text{if} \ j=i-1 \\ e_{j,r}\psi^{-}_{i}1_{\underline{k}} & \text{if} \ |i-j| \geq 2 \\ \end{cases}, \end{align} $$

(U07) $$ \begin{align} &\psi^{+}_{i}f_{j,r}1_{\underline{k}}= \begin{cases} -f_{i,r-1}\psi^{+}_{i}1_{\underline{k}} & \text{if} \ j=i \\ -f_{i+1,r+1}\psi^{+}_{i}1_{\underline{k}} & \text{if} \ j=i+1 \\ f_{i-1,r}\psi^{+}_{i}1_{\underline{k}} & \text{if} \ j=i-1 \\ f_{j,r}\psi^{+}_{i}1_{\underline{k}} & \text{if} \ |i-j| \geq 2 \\ \end{cases}, &\psi^{-}_{i}f_{j,r}1_{\underline{k}}= \begin{cases} -f_{i,r-1}\psi^{-}_{i}1_{\underline{k}} & \text{if} \ j=i \\ f_{i+1,r}\psi^{-}_{i}1_{\underline{k}} & \text{if} \ j=i+1 \\ -f_{i-1,r+1}\psi^{-}_{i}1_{\underline{k}} & \text{if} \ j=i-1 \\ f_{j,r}\psi^{-}_{i}1_{\underline{k}} & \text{if} \ |i-j| \geq 2 \\ \end{cases}, \end{align} $$

(U08) $$ \begin{align} [e_{i,r},f_{j,s}]1_{\underline{k}}=0 \ \mathrm{if} \ i \neq j\ \ \mathrm{and}\ \ {[e_{i,r},f_{i,s}]1_{\underline{k}}= \begin{cases} \psi^{+}_{i}1_{\underline{k}} & \text{if} \ (r,s)=(0,k_{i+1}) \\ 0 & \text{if} \ -k_{i}+1 \leq r+s \leq k_{i+1}-1 \\ -\psi^{-}_{i}1_{\underline{k}} & \text{if} \ (r,s)=(-k_i,0) \end{cases},} \end{align} $$

for any $1 \leq i,j \leq n-1$ and $r,s$ such that the above relations make sense.

We discuss a bit about the use of the notation $\dot {\mathrm {\mathbf {U}}}_{0,N}(L\mathfrak {sl}_n)$ , and why we call it the name shifted 0-affine algebra. We present the discussion as a list of remarks.

Thus, although the precise relation between our algebras and the shifted quantum affine algebras is still unclear, the above evidence suggests we call this algebra the name shifted 0-affine algebra.

Finally, we give an example of shifted 0-affine algebra.

Example 2.10. The shifted 0-affine algebra $\dot {\mathrm {\mathbf {U}}}_{0,2}(L\mathfrak {sl}_2)$ is the path algebra of the following quiver

subject to the relations from (U01) to (U08). For example, we have $[e_{0},f_{1}]1_{(1,1)}=\psi ^{+}1_{(1,1)}$ .

5.1 An Overview

The utilization of FM transforms necessitates the introduction of additional geometric spaces (varieties) in conjunction with the spaces employed to define the categories $\mathcal {K}(\underline {k})$ . These additional spaces play the role of correspondences, enabling us to define kernels for the respective 1-morphisms. Here, we give the readers an overview of the main ideas behind the definitions and proofs first.

It suffices to consider the $\mathfrak {sl}_{2}$ case, in which the categories are the bounded derived category of coherent sheaves on Grassmannians $\mathrm {D}^b(\mathrm {Gr}(k,N))$ . The correspondence we use to define the $1$ -morphisms $\mathsf {E}_{r}\boldsymbol {1}_{(k,N-k)}$ is the $3$ -step partial flag variety $\mathrm {Fl}(k-1,k)$ in diagram (1.3), similar for $\mathsf {F}_{s}\boldsymbol {1}_{(k,N-k)}$ .

To verify those conditions in Definition 3.1, we have to calculate many convolutions of FM kernels, and most of them are done by using standard tools like base change, projection formula, and Proposition 4.4.

The most technical and interesting ones are the categorical commutator relations between $\mathsf {E}_{r}$ and $\mathsf {F}_s$ , i.e., condition (10). The final step of convolution is pushforward induced from the projection $\pi _{13}$ to the product of the first and third components. In the constructible/coherent setting, to calculate the convolutions of kernels for $\mathsf {E}\mathsf {F}$ / $\mathsf {E}_{r}\mathsf {F}_{s}$ and $\mathsf {F}\mathsf {E}$ / $\mathsf {F}_{s}\mathsf {E}_{r}$ , we need to know what are the geometric spaces that the projection $\pi _{13}$ really restrict to.

They are the following varieties

$$ \begin{align*} Z&=\{(V,V^{\prime},V^{\prime\prime}) \in \mathrm{Gr}(k,N) \times \mathrm{Gr}(k+1,N) \times \mathrm{Gr}(k,N) \ | \ V, \ V^{\prime\prime} \overset{1}{\subset} V^{\prime} \}, \\ Z^{\prime}&=\{(V,V^{\prime\prime\prime},V^{\prime\prime}) \in \mathrm{Gr}(k,N) \times \mathrm{Gr}(k-1,N) \times \mathrm{Gr}(k,N) \ | \ V^{\prime\prime\prime} \overset{1}{\subset} V, \ V^{\prime\prime}\} \end{align*} $$

that naturally arise when calculating convolution of kernels for $\mathsf {E}\mathsf {F}$ / $\mathsf {E}_{r}\mathsf {F}_{s}$ and $\mathsf {F}\mathsf {E}$ / $\mathsf {F}_{s}\mathsf {E}_{r}$ respectively (see Lemma 5.9 for details). Restricting the projections to Z and $Z'$ , in order to distinguish them we use $\pi ^{\prime }_{13}$ for $Z'$ . Then we obtain the following diagram

(5.1)

where Y is given by

(5.2) $$ \begin{align} Y=\{(V,V") \in \mathrm{Gr}(k,N) \times \mathrm{Gr}(k,N) \ | \ \dim(V \cap V") \geq k-1 \}. \end{align} $$

In the constructible setting, assume that $\lambda =N-2k \geq 0$ , then in the diagram (5.1) the map $\pi ^{\prime }_{13}$ is a small resolution and $\pi _{13}$ is not a small resolution. Using the theory about perverse sheaves (or IC sheaves), we obtain the isomorphism $\mathsf {{E}\mathsf {F}}|_{\mathcal {C}(\lambda )} \cong \mathsf {{F}\mathsf {E}}|_{\mathcal {C}(\lambda )} \bigoplus Id_{\mathcal {C}(\lambda )}^{\oplus \lambda }$ Footnote ² . The extra term $Id_{\mathcal {C}(\lambda )}^{\oplus \lambda }$ is contributed from $\pi _{13}$ since it is not small. Moreover, the extra term has a cohomological interpretation, which can be thought of as $Id_{\mathcal {C}(\lambda )} \otimes H_{sing}^{*}(\mathbb {P}^{\lambda -1},\mathbb {C})$ , see [Reference Cautis, Kamnitzer and Licata16].

However, in the coherent setting, we do not have powerful tools like the decomposition theorem. We use the fibered product $X:=Z' \times _{Y} Z$ which is defined as follows

(5.3)

and denote $p:X \rightarrow Y$ to be the natural projection. Moreover, there is a divisor $D \subset X$ which is the locus where $V=V"$ and cut out by the natural sections $\mathcal {V}"/\mathcal {V}"' \rightarrow \mathcal {V}'/\mathcal {V}$ where $\mathcal {V}^{\prime \prime \prime }, \ \mathcal {V}, \ \mathcal {V}^{\prime \prime }, \ \mathcal {V}'$ are the natural tautological bundles on X. Thus we have the following short exact sequence

(5.4) $$ \begin{align} 0 \rightarrow \mathcal{V}^{\prime\prime}/\mathcal{V}^{\prime\prime\prime} \rightarrow \mathcal{V}'/\mathcal{V} \rightarrow \mathcal{O}_{D} \otimes \mathcal{V}'/\mathcal{V} \rightarrow 0. \end{align} $$

To compare the kernels for $\mathsf {E}_{r}\mathsf {F}_{s}$ and $\mathsf {F}_{s}\mathsf {E}_{r}$ , instead of directly pushing forward to Y, we pull them back to X and use the short exact sequence (5.4) together with Proposition 4.4 to help us prove the result (see the proof of Proposition 5.10).

The kernels for $\mathsf {\Psi }^{\pm }\boldsymbol {1}_{(k,N-k)}$ are produced under the above comparison, and they are reflected by the non-vanishing of coherent sheaf cohomology $H^{*}(\mathbb {P}^{N-1}, \mathcal {O}_{\mathbb {P}^{N-1}}(-r-s-k)) \neq 0$ . Also, the kernels for $\mathsf {\Psi }^{\pm }\boldsymbol {1}_{(k,N-k)}$ are just line bundles up to homological shifts.

5.2 Correspondences and the Main Result

First, we define those geometric spaces (varieties) that will be used to define the categories $\mathcal {K}(\underline {k})$ and FM kernels for the 1-morphisms $\mathsf {E}_{i,r}\boldsymbol {1}_{\underline {k}}$ , $\mathsf {F}_{i,s}\boldsymbol {1}_{\underline {k}}$ , and $(\mathsf {\Psi }_{i}^{\pm })^{\pm 1}\boldsymbol {1}_{\underline {k}}$ . They can be viewed as a $\mathfrak {sl}_{n}$ generalization of the varieties mentioned in Subsection 5.1.

For each $\underline {k} \in C(n,N)$ , we define the n-step partial flag variety

(5.5) $$ \begin{align} \mathrm{Fl}_{\underline{k}}(\mathbb{C}^N):=\{V_{\bullet}=(0=V_{0} \subset V_1 \subset ... \subset V_{n}=\mathbb{C}^N) \ | \ \mathrm{dim} V_{i}/V_{i-1}=k_{i} \ \text{for} \ \text{all} \ i\}. \end{align} $$

We denote $Y(\underline {k})=\mathrm {Fl}_{\underline {k}}(\mathbb {C}^N)$ and $\mathrm {D}^b(Y(\underline {k}))$ to be the bounded derived categories of coherent sheaves on $Y(\underline {k})$ . On $Y(\underline {k})$ we denote $\mathcal {V}_{i}$ to be the tautological bundle whose fiber over a point $(0=V_{0} \subset V_1 \subset ... \subset V_{n}=\mathbb {C}^N)$ is $V_{i}$ . We define the following correspondence, which can be thought of as $\mathfrak {sl}_{n}$ generalization of $\mathrm {Fl}(k-1,k)$ .

Definition 5.2. We define $W^{1}_{i}(\underline {k})$ to be the subvariety of $Y(\underline {k}) \times Y(\underline {k}+\alpha _{i})$

$$ \begin{align*} W^{1}_{i}(\underline{k}):=\{(V_{\bullet},V_{\bullet}') \ | \ V_{j}=V^{\prime}_{j} \ \mathrm{for} \ j \neq i, \ V^{\prime}_{i} \subset V_{i}\} \subset Y(\underline{k}) \times Y(\underline{k}+\alpha_{i}) \end{align*} $$

and similarly its transpose correspondence

$$ \begin{align*} ^{\mathsf{{T}}}W^{1}_{i}(\underline{k}):=\{(V_{\bullet},V_{\bullet}') \ | \ V_{j}=V^{\prime}_{j} \ \mathrm{for} \ j \neq i, \ V_{i} \subset V^{\prime}_{i}\} \subset Y(\underline{k}+\alpha_{i}) \times Y(\underline{k}) \end{align*} $$

for all $\underline {k} \in C(n,N)$ and i.

We denote $\iota (\underline {k}):W^{1}_{i}(\underline {k}) \hookrightarrow Y(\underline {k}) \times Y(\underline {k}+\alpha _{i})$ and $^{\mathsf {{T}}}\iota (\underline {k}): {^{\mathsf {{T}}}W^{1}_{i}(\underline {k})} \hookrightarrow Y(\underline {k}+\alpha _{i}) \times Y(\underline {k})$ to be the natural inclusions. We also have the natural line bundle $\mathcal {V}_{i}/\mathcal {V}^{\prime }_{i}$ on $W^{1}_{i}(\underline {k})$ and similarly $\mathcal {V}^{\prime }_{i}/\mathcal {V}_{i}$ on $^{\mathsf {{T}}}W^{1}_{i}(\underline {k})$ , where $1 \leq i \leq n$ .

Next, we define the following varieties that can be thought as $\mathfrak {sl}_{n}$ generalization of X in (5.3).

Definition 5.3. We define $X_{i}(\underline {k})$ to be the subvariety of $Y(\underline {k}+\alpha _{i}) \times Y(\underline {k}) \times Y(\underline {k}) \times Y(\underline {k}-\alpha _{i})$

and the divisor $D_{i}(\underline {k}) \subset X_{i}(\underline {k})$ that is defined by

$$ \begin{align*} D_{i}(\underline{k}):=\{ (V_{\bullet}^{\prime\prime\prime},V_{\bullet},V_{\bullet}^{\prime\prime},V_{\bullet}^{\prime}) \ | \ V^{\prime\prime\prime}_{i}\subset V_{i}=V^{\prime\prime}_{i} \subset V^{\prime}_{i}, \ V^{\prime\prime\prime}_{j}=V_{j}=V^{\prime\prime}_{j}=V^{\prime}_{j} \ \forall \ j \neq i \}. \end{align*} $$

Note $D_{i}(\underline {k})$ is cut out by the natural section of the line bundles $\mathcal {H} om(\mathcal {V}_{i}"/\mathcal {V}_{i}"',\mathcal {V}_{i}'/\mathcal {V}_{i})$ . More precisely, we have $\mathcal {O}_{X_{i}(\underline {k})}(-D_{i}(\underline {k})) \cong \mathcal {V}_{i}"/\mathcal {V}_{i}"' \otimes (\mathcal {V}_{i}'/\mathcal {V}_{i})^{-1}$ and the following short exact sequences

$$ \begin{align*} 0 \rightarrow \mathcal{V}_{i}"/\mathcal{V}_{i}"' \rightarrow \mathcal{V}_{i}'/\mathcal{V}_{i} \rightarrow \mathcal{O}_{D_{i}(\underline{k})} \otimes \mathcal{V}_{i}'/\mathcal{V}_{i} \rightarrow 0. \end{align*} $$

We obtain similar results if we view $D_{i}(\underline {k})$ as the divisor that is cut out by the natural section of the line bundles $\mathcal {H} om(\mathcal {V}_{i}/\mathcal {V}_{i}"',\mathcal {V}_{i}'/\mathcal {V}^{\prime \prime }_{i})$ .

Finally, we define the $\mathfrak {sl}_{n}$ generalization of Y in diagram (5.1).

Definition 5.4.

$$ \begin{align*} Y_{i}(\underline{k})=\{(V_{\bullet},V_{\bullet}") \in Y(\underline{k}) \times Y(\underline{k}) \ | \ \dim V_{i} \cap V^{\prime\prime}_{i} \geq (\sum_{l=1}^{i}k_{l})-1, \ V_{j}=V^{\prime\prime}_{j} \ \forall \ j \neq i \}. \end{align*} $$

Let $\pi _{i}:Y(\underline {k}) \times Y(\underline {k}) \rightarrow Y(\underline {k})$ be the natural projection to the ith component for $i=1,2$ . Let $p_{i}(\underline {k}):X_{i}(\underline {k}) \rightarrow Y_{i}(\underline {k})$ be the natural projection defined by forgetting $V_{\bullet }^{\prime \prime \prime }$ and $V_{\bullet }'$ , $t_{i}(\underline {k}):Y_{i}(\underline {k}) \rightarrow Y(\underline {k}) \times Y(\underline {k})$ be the inclusion and $\Delta (\underline {k}):Y(\underline {k}) \rightarrow Y(\underline {k}) \times Y(\underline {k})$ to be the diagonal map.

Then, we define the 1-morphisms by using FM transforms with kernels involving the geometric spaces we introduced above.

Definition 5.5. We define $\boldsymbol {1}_{\underline {k}}$ , $\mathsf {E}_{i,r}\boldsymbol {1}_{\underline {k}}$ , $\boldsymbol {1}_{\underline {k}}\mathsf {F}_{i,s}$ , and $(\mathsf {\Psi }^{\pm }_{i})^{\pm 1}\boldsymbol {1}_{\underline {k}}$ to be FM transforms with the corresponding kernels

respectively.

Now, we can state the main result of this article, which is the following theorem.

We devote the rest of this section to proof of this theorem.

5.3 The $\mathfrak {sl}_2$ Case

In this subsection, we prove there is a categorical $\mathcal {U}$ -action on the derived category of coherent sheaves on Grassmannians first, which is the following theorem.

Now $n=2$ , to simplify the notations further, we drop i from all the functors with i in their notation, and thus the 1-morphisms are $\boldsymbol {1}_{(k,N-k)}$ , $\mathsf {E}_{r}\boldsymbol {1}_{(k,N-k)}$ , $\mathsf {F}_{s}\boldsymbol {1}_{(k,N-k)}$ , and $(\mathsf {\Psi }^{\pm })^{\pm 1}\boldsymbol {1}_{(k,N-k)}$ . Furthermore, for the weight $\underline {k}$ , we will just write $(k,N-k)$ with $0 \leq k \leq N$ .

To prove this, we need to prove the conditions (1), (3), (4), (5a), (6a), (7a), (8a), and (10) in Definition 3.1.

Note that condition (1) is obvious since the varieties are finite Grassmannians $\mathrm {Gr}(k,N)$ , which are smooth and proper, and the Hom spaces are finite-dimensional. Before we check the rest, let us remark that since all functors above are defined by using kernels, we check the conditions in terms of kernels by Proposition 4.2.

Due to the repetition of arguments, the proofs we give below for most of the conditions are only for the functors $\mathsf {E}_{r}\boldsymbol {1}_{(k,N-k)}$ and $\Psi ^+\boldsymbol {1}_{(k,N-k)}$ . For example, we will only prove $\mathsf {\Psi }^{+}\mathsf {E}_{r}\boldsymbol {1}_{(k,N-k)} \cong \mathsf {E}_{r+1}\mathsf {\Psi }^{+}\boldsymbol {1}_{(k,N-k)}[-1]$ for condition (7)(a). Finally, in order to make calculations simple, we will omit $(k,N-k)$ for all the maps in Definition 5.5, i.e. we will just write $\iota $ , $^{\mathsf {{T}}}\iota $ , $\Delta $ , t, p if there is no confusion.

It is helpful to keep the following picture.

(5.6)

where $\iota : \mathrm {Fl}(k-1,k) \rightarrow \mathrm {Gr}(k,N) \times \mathrm {Gr}(k-1,N)$ is the natural inclusion and $\pi _{1}$ , $\pi _{2}$ , $p_{1}$ , $p_{2}$ are the natural projections.

The first is the condition (3).

Lemma 5.8 (Condition (3)).

$\mathsf {E}_{r}\boldsymbol {1}_{(k,N-k)}$ and $\mathsf {F}_{s}\boldsymbol {1}_{(k,N-k)}$ are left and right adjoints up to homological shifts and twists of $\mathsf {\Psi }^{\pm }\boldsymbol {1}_{(k,N-k)}$ :

$$ \begin{align*} {({\mathcal{E}}_{r}\boldsymbol{1}_{(k,N-k)})_{R} \cong \boldsymbol{1}_{(k,N-k)}{({\Psi}^{+})^{r}} \ast {\mathcal{F}}_{N-k+1} \ast ({\Psi}^{+})^{-r-1}[-r].} \end{align*} $$

Proof. By Proposition 4.1, the right adjoint of $\mathcal {E}_{r}\boldsymbol {1}_{(k,N-k)}$ is given by

$$\begin{align*}\{\iota_{*} (\mathcal{V}/\mathcal{V}')^{\otimes r}\}^{\vee} \otimes \pi_{1}^*\omega_{\mathrm{Gr}(k,N)}[\mathrm{dim}\mathrm{Gr}(k,N)], \end{align*}$$

where

$$\begin{align*}\{\iota_{*} (\mathcal{V}/\mathcal{V}')^{\otimes r}\}^{\vee} \cong \iota_{*} ((\mathcal{V}/\mathcal{V}')^{\otimes -r} \otimes \omega_{\mathrm{Fl}(k-1,k)}) \otimes \omega_{\mathrm{Gr}(k,N)\times \mathrm{Gr}(k-1,N)}^{-1}[-\mathrm{codim} \mathrm{Fl}(k-1,k)]. \end{align*}$$

To calculate $\omega _{\mathrm {Fl}(k-1,k)}$ , we have $\omega _{\mathrm {Fl}(k-1,k)} \cong \omega _{p_{2}} \otimes p^*_{2}\omega _{\mathrm {Gr}(k-1,N)}$ where $\omega _{p_{2}}$ is the relative canonical bundle. Since the relative cotangent bundle is $\mathcal {T}^{\vee }_{p_{2}}\cong \mathcal {V}/\mathcal {V}' \otimes (\mathbb {C}^N/\mathcal {V})^{\vee }$ , we obtain $\omega _{p_{2}} \cong (\mathcal {V}/\mathcal {V}')^{\otimes (N-k)} \otimes \det (\mathbb {C}^N/\mathcal {V})^{-1}$ . A calculation gives $\mathrm {dim} \mathrm {Gr}(k,N)-\mathrm {codim} \mathrm {Fl}(k-1,k)=N-k$ .

So summarizing above and use $\mathcal {V}/\mathcal {V}' \cong \det (\mathbb {C}^N/\mathcal {V}') \otimes \det (\mathbb {C}^N/\mathcal {V})^{-1}$ we have

$$ \begin{align*} &\{\iota_{*} (\mathcal{V}/\mathcal{V}')^{\otimes r}\}^{\vee} \otimes \pi_{1}^*(\omega_{\mathrm{Gr}(k,N)})[\mathrm{dim}\mathrm{Gr}(k,N)] \cong \iota_{*}(\omega_{p_2}\otimes (\mathcal{V}/\mathcal{V}')^{\otimes -r} ) [N-k] \ \text{(using projection formula)} \\& \quad \cong \iota_{*} ((\mathcal{V}/\mathcal{V}')^{\otimes(-r+N-k+1)} \otimes \det(\mathbb{C}^N/\mathcal{V}')^{-1})[N-k]. \\& \quad \cong \iota_{*} ((\mathcal{V}/\mathcal{V}')^{\otimes {(N-k+1)}} \otimes \det(\mathbb{C}^N/\mathcal{V})^{\otimes {r}} \otimes \det(\mathbb{C}^N/\mathcal{V}')^{\otimes {(-r-1)}})[N-k] \\& \quad \cong \iota_{*} ((\mathcal{V}/\mathcal{V}')^{\otimes {(N-k+1)}} \otimes \det(\mathbb{C}^N/\mathcal{V})^{\otimes {r}}[{r}(1+k-N)] \otimes \det(\mathbb{C}^N/\mathcal{V}')^{\otimes {(-r-1)}}[{(r+1)}(N-k)])[{-r}] \end{align*} $$

Note that the kernel $(\Psi ^{+})^{-1}\boldsymbol {1}_{(k-1,N-k+1)}$ is defined by $\Delta _{*}\det (\mathbb {C}^N/\mathcal {V}')^{-1}[N-k]$ , while $\Psi ^{+}\boldsymbol {1}_{(k,N-k)}$ is defined by $\Delta _{*}\det (\mathbb {C}^N/\mathcal {V})[1+k-N]$ . We conclude that it is isomorphic to $\boldsymbol {1}_{(k,N-k)}{({\Psi }^{+})^{r}}\ast {\mathcal {F}}_{N-k+1}\ast ({\Psi }^{+})^{-r-1}[-r]$ .

Next, we consider conditions (5a), (6a), (7a), (8a), and (10) with $r=s=0$ . We summarize them in the following lemma.

Proof. Since the argument is fairly standard by using base change, projection formula, and Proposition 4.4. We decide to prove (3) only and leave (1) and (2) to the readers.

By definition

(5.7) $$ \begin{align} (\mathcal{E}_{0} \ast \mathcal{F}_{0})\boldsymbol{1}_{(k,N-k)} \cong\pi_{13*}(\pi^*_{12} {^{\mathsf{{T}}}}\iota_{*}\mathcal{O}_{\mathrm{Fl}(k,k+1)} \otimes \pi_{23}^*\iota_{*}\mathcal{O}_{\mathrm{Fl}(k,k+1)}). \end{align} $$

We will keep using the fibered product diagram, base change, and projection formula to calculate (5.7). Since the argument is pretty standard, we decided to omit the details and only mention the key steps in order to make the article short.

We have the following fibered product diagrams

where $a_{1}$ , $a_{2}$ are the natural projections. Then we obtain

(5.8) $$ \begin{align} (5.7) \cong \pi_{13*}((^{\mathsf{{T}}}\iota \times id)_{*}\mathcal{O}_{\mathrm{Fl}(k,k+1) \times \mathrm{Gr}(k,N)} \otimes (id\times \iota)_{*}\mathcal{O}_{\mathrm{Gr}(k,N) \times \mathrm{Fl}(k,k+1)}). \end{align} $$

Next, the following fibered product diagram

where Z is the following variety

gives us that

(5.9) $$ \begin{align} (5.8) \cong \pi_{13*}(id\times \iota)_{*}b_{2*}(\mathcal{O}_{Z}). \end{align} $$

Finally, we have the following commutative diagram

where $Y=\pi _{13}(Z)=\{(V,V")\ |\ \mathrm {dim} V\cap V" \geq k-1 \}$ , and $j_1=(id \times \iota ) \circ b_2$ , t are the inclusions.

Note that $Y \subset \mathrm {Gr}(k,N) \times \mathrm {Gr}(k,N)$ is a Schubert variety, it has rational singularities. So $(\pi _{13}|_{Z*})(\mathcal {O}_{Z}) \cong \mathcal {O}_{Y}$ and thus

$$ \begin{align*} (\mathcal{E}_{0} \ast \mathcal{F}_{0})\boldsymbol{1}_{(k,N-k)} \cong (5.9) \cong t_{*}(\pi_{13}|_{Z*})(\mathcal{O}_{Z}) \cong t_{*}\mathcal{O}_{Y}. \end{align*} $$

Using the same method for calculating $(\mathcal {F}_{0} \ast \mathcal {E}_{0})\boldsymbol {1}_{(k,N-k)}$ , we end up with the following commutative diagram

where

and $j_{2}$ is the inclusion.

Again, we have $\pi _{13}(Z')=\{(V,V") \ | \ \dim V\cap V" \geq k-1\}=Y$ , which is the same as we get when we calculate $(\mathcal {E}_{0} \ast \mathcal {F}_{0})\boldsymbol {1}_{(k,N-k)}$ . Thus

$$ \begin{align*} (\mathcal{F}_{0} \ast \mathcal{E}_{0})\boldsymbol{1}_{(k,N-k)} \cong \pi_{13'*}j_{2*}(\mathcal{O}_{Z'}) \cong t_{*}(\pi_{13'}|_{Z'*})(\mathcal{O}_{Z'}) \cong t_{*}\mathcal{O}_{Y} \end{align*} $$

which prove the lemma.

Then, we prove condition (10). Observe that from Lemma 5.9 (2), we have $(\Psi ^{+} \ast \mathcal {E}_{r})\boldsymbol {1}_{(k,N-k)} \cong (\mathcal {E}_{r+1}\ast \Psi ^{+})\boldsymbol {1}_{(k,N-k)}[-1]$ . Since $\Psi ^{+}\boldsymbol {1}_{(k,N-k)}$ is invertible, we get

$$ \begin{align*} [\Psi^{+}\ast \mathcal{E}_{r}\ast (\Psi^{+})^{-1} ]\boldsymbol{1}_{(k,N-k)} \cong \mathcal{E}_{r+1}\boldsymbol{1}_{(k,N-k)}[-1], \end{align*} $$

and apply this inductively, we obtain

(5.10) $$ \begin{align} [(\Psi^{+})^r \ast \mathcal{E}_{0} \ast (\Psi^{+})^{-r}] \boldsymbol{1}_{(k,N-k)} \cong \mathcal{E}_{r}\boldsymbol{1}_{(k,N-k)}[-r] \end{align} $$

where $(\Psi ^{+})^r$ means $(\Psi ^{+})$ convolution with itself r times.

Similarly, for $\mathcal {F}_{s}\boldsymbol {1}_{(k,N-k)}$ , we have

(5.11) $$ \begin{align} [(\Psi^{+})^{-s} \ast \mathcal{F}_{0} \ast (\Psi^{+})^{s}] \boldsymbol{1}_{(k,N-k)} \cong \mathcal{F}_{s}\boldsymbol{1}_{(k,N-k)}[-s]. \end{align} $$

By (5.10) and (5.11), we obtain that

$$ \begin{align*} & (\mathcal{E}_{r}\ast \mathcal{F}_{s})\boldsymbol{1}_{(k,N-k)} \cong [(\Psi^{+})^r \ast \mathcal{E}_{0} \ast \mathcal{F}_{s+r} \ast (\Psi^{+})^{-r} ]\boldsymbol{1}_{(k,N-k)}, \\ & (\mathcal{F}_{s}\ast \mathcal{E}_{r})\boldsymbol{1}_{(k,N-k)} \cong [(\Psi^{+})^r \ast \mathcal{F}_{r+s} \ast \mathcal{E}_{0} \ast (\Psi^{+})^{-r} ]\boldsymbol{1}_{(k,N-k)}. \end{align*} $$

Since $\Psi ^{+}\boldsymbol {1}_{(k,N-k)}$ is invertible, in order to compare $( \mathcal {F}_{s}\ast \mathcal {E}_{r})\boldsymbol {1}_{(k,N-k)}$ and $(\mathcal {E}_{r}\ast \mathcal {F}_{s})\boldsymbol {1}_{(k,N-k)}$ , it suffices to compare $(\mathcal {F}_{r+s} \ast \mathcal {E}_{0})\boldsymbol {1}_{(k,N-k)}$ and $ (\mathcal {E}_{0} \ast \mathcal {F}_{r+s})\boldsymbol {1}_{(k,N-k)}$ . Hence, we prove the following proposition.

Proposition 5.10 (Condition (10)(a)(c)).

We have the following (non-split) exact triangles in $\mathrm {D}^b(\mathrm {Gr}(k,N) \times \mathrm {Gr}(k,N))$ .

(5.12) $$ \begin{align} &{(\mathcal{F}_{N-k}\ast \mathcal{E}_{0})\boldsymbol{1}_{(k,N-k)} \rightarrow (\mathcal{E}_{0}\ast \mathcal{F}_{N-k})\boldsymbol{1}_{(k,N-k)} \rightarrow \Psi^{+}\boldsymbol{1}_{(k,N-k)},} \end{align} $$

and the following isomorphisms

(5.13) $$ \begin{align} (\mathcal{F}_{s}\ast \mathcal{E}_{0})\boldsymbol{1}_{(k,N-k)} \cong (\mathcal{E}_{0} \ast \mathcal{F}_{s})\boldsymbol{1}_{(k,N-k)}, \ \text{if} \ 0 \leq s \leq N-k-1. \end{align} $$

To compare $(\mathcal {E}_{0} \ast \mathcal {F}_{s})\boldsymbol {1}_{(k,N-k)}$ and $(\mathcal {F}_{s} \ast \mathcal {E}_{0})\boldsymbol {1}_{(k,N-k)}$ , by using the above fibered product diagrams, we have to compare the following two objects

$$\begin{align*}\pi_{13*}(j_{1*} (\mathcal{V}'/\mathcal{V})^{\otimes s} ) \cong t_{*}(\pi_{13}|_{Z*})(\mathcal{V}'/\mathcal{V})^{\otimes s} \ \text{and} \ \pi_{13'*}(j_{2*} (\mathcal{V}"/\mathcal{V}"')^{\otimes s} ) \cong t_{*}(\pi_{13'}|_{Z'*})(\mathcal{V}"/\mathcal{V}"')^{\otimes s} \end{align*}$$

in the derived category $\mathrm {D}^b(\mathrm {Gr}(k,N)\times \mathrm {Gr}(k,N))$ .

Note that both are pushforwards to Y. In order to handle the case where we tensor non-trivial line bundles, instead of directly pushing forward to Y, we lift the line bundles to a much larger space, i.e., their fibered product. The fibered product space $X:=Z \times _{Y} Z'$ , is given by

$$ \begin{align*} X=\{(V"',V,V",V')\ | \ V"' \subset V \subset V', \ V"' \subset V"\subset V' \}. \end{align*} $$

We have the following fibered product diagram

(5.14)

where $g_1$ and $g_2$ are the natural projections. We denote $p:X \rightarrow Y$ to be the natural projection.

On X, we have the divisor $D =\mathrm {Fl}(k-1,k,k+1)=\{(V"',V,V') \ | \ V"' \subset V \subset V'\}$ which is the locus where $V=V"$ and it is the vanishing locus of the natural sections $\mathcal {V}"/\mathcal {V}"' \rightarrow \mathcal {V}'/\mathcal {V}$ . Thus, we have the following short exact sequence

$$ \begin{align*} 0 \rightarrow \mathcal{V}"/\mathcal{V}"' \rightarrow \mathcal{V}'/\mathcal{V} \rightarrow \mathcal{O}_{D} \otimes \mathcal{V}'/\mathcal{V} \rightarrow 0. \end{align*} $$

We can relate it to Proposition 4.4. More precisely, it is easy to see that the restriction of the line bundle $\mathcal {V}'/\mathcal {V}$ to D is the pullback of the relative tautological bundle $\mathcal {O}_{\mathbb {P}(\mathbb {C}^N/\mathcal {V})}(-1)$ on $\mathbb {P}(\mathbb {C}^N/\mathcal {V})=\mathrm {Fl}(k,k+1) \subset Z$ . Thus we obtain

(5.15) $$ \begin{align} 0 \rightarrow \mathcal{V}"/\mathcal{V}"' \rightarrow \mathcal{V}'/\mathcal{V} \rightarrow \mathcal{O}_{D} \otimes \mathcal{O}_{\mathbb{P}(\mathbb{C}^N/\mathcal{V})}(-1) \rightarrow 0, \end{align} $$

and we will use the above short exact sequence to help us prove Proposition 5.10.

Proof of Proposition 5.10.

Note that $(\pi _{13}|_{Z*})(\mathcal {O}_{Z}) \cong \mathcal {O}_{Y}$ . This implies that $(\pi _{13}|_{Z*})(\pi _{13}|_{Z})^{*} \cong \text {Id}_{{\mathrm {D}^b(Y)}}$ . Using the fibered product diagram (5.14), since $g_2$ is the base change of $\pi _{13}|_{Z}$ , we have $g_{2*}g_{2}^{*} \cong \text {Id}_{{\mathrm {D}^b(Z')}}$ . Similarly, $(\pi _{13'}|_{Z'*})(\mathcal {O}_{Z'}) \cong \mathcal {O}_{Y}$ implies that $g_{1*}g_{1}^{*} \cong \text {Id}_{{\mathrm {D}^b(Z)}}$ .

So, from the above discussion, we obtain

$$ \begin{align*} &(\mathcal{E}_{0} \ast \mathcal{F}_{s})\boldsymbol{1}_{(k,N-k)} \cong t_{*}(\pi_{13}|_{Z*})(\mathcal{V}'/\mathcal{V})^{\otimes s} \cong t_{*}(\pi_{13}|_{Z*})g_{1*}g_{1}^{*}(\mathcal{V}'/\mathcal{V})^{\otimes s} \cong t_{*}p_{*}(\mathcal{V}'/\mathcal{V})^{\otimes s}, \\ &(\mathcal{F}_{s} \ast \mathcal{E}_{0})\boldsymbol{1}_{(k,N-k)} \cong t_{*}(\pi_{13'}|_{Z'*})(\mathcal{V}"/\mathcal{V}"')^{\otimes s} \cong t_{*}(\pi_{13'}|_{Z'*})g_{2*}g_{2}^{*} (\mathcal{V}"/\mathcal{V}"')^{\otimes s} \cong t_{*}p_{*}(\mathcal{V}"/\mathcal{V}"')^{\otimes s}. \end{align*} $$

Thus, at first, we have to compare $p_{*}(\mathcal {V}'/\mathcal {V})^{\otimes s}$ and $p_{*} (\mathcal {V}"/\mathcal {V}"')^{\otimes s}$ in $\mathrm {D}^b(Y)$ .

Note that for each $n \geq 1$ , we have the following short exact sequence on X

(5.16) $$ \begin{align} 0 \rightarrow (\mathcal{V}"/\mathcal{V}"')^{\otimes n} \rightarrow (\mathcal{V}'/\mathcal{V})^{\otimes n} \rightarrow \mathcal{O}_{nD} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes n} \rightarrow 0. \end{align} $$

Tensoring (5.15) by $(\mathcal {V}"/\mathcal {V}"')^{\otimes (n-1)}$ , we get

$$\begin{align*}0 \rightarrow (\mathcal{V}"/\mathcal{V}"')^{\otimes n} \rightarrow \mathcal{V}'/\mathcal{V} \otimes (\mathcal{V}"/\mathcal{V}"')^{\otimes (n-1)} \rightarrow \mathcal{O}_{D} \otimes \mathcal{V}'/\mathcal{V} \otimes (\mathcal{V}/\mathcal{V}"')^{\otimes (n-1)} \rightarrow 0. \end{align*}$$

Both of them are exact triangles in $\mathrm {D}^b(X)$ , and they can be completed together to form the following diagram of morphisms between exact triangles

So, we obtain the exact triangle

(5.17) $$ \begin{align} \mathcal{O}_{D} \otimes \mathcal{V}'/\mathcal{V} \otimes (\mathcal{V}/\mathcal{V}"')^{\otimes (n-1)} \rightarrow \mathcal{O}_{nD} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes n} \rightarrow \mathcal{O}_{(n-1)D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes n} \end{align} $$

for all $n \geq 1$ (here we take $\mathcal {O}_{(n-1)D}$ to be $0$ when $n=1$ ).

Note that the case $s=0$ has already been proved in Lemma 5.9 (3). Now, we move to the case where s is nonzero. The first case is $1 \leq s \leq N-k-1$ . For (5.16) with $n=s$ we have

$$\begin{align*}0 \rightarrow (\mathcal{V}"/\mathcal{V}"')^{\otimes s} \rightarrow (\mathcal{V}'/\mathcal{V})^{\otimes s} \rightarrow \mathcal{O}_{sD} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes s} \rightarrow 0. \end{align*}$$

Applying the derived pushforward $p_{*}$ , we obtain the following exact triangle in $\mathrm {D}^b(Y)$

$$ \begin{align*} p_{*}(\mathcal{V}"/\mathcal{V}"')^{\otimes s} \rightarrow p_{*}(\mathcal{V}'/\mathcal{V})^{\otimes s} \rightarrow p_{*}(\mathcal{O}_{sD} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes s}) \end{align*} $$

and it suffices to prove that $p_{*}(\mathcal {O}_{sD} \otimes (\mathcal {V}'/\mathcal {V})^{\otimes s}) \cong 0$ so that we can obtain $(\mathcal {E}_{0} \ast \mathcal {F}_{s})\boldsymbol {1}_{(k,N-k)} \cong (\mathcal {F}_{s} \ast \mathcal {E}_{0})\boldsymbol {1}_{(k,N-k)}$ after applying $t_{*}$ .

Using (5.17) with $n=s$ , we have

$$ \begin{align*} \mathcal{O}_{D} \otimes \mathcal{V}'/\mathcal{V} \otimes (\mathcal{V}/\mathcal{V}"')^{\otimes (s-1)} \rightarrow \mathcal{O}_{sD} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes s} \rightarrow \mathcal{O}_{(s-1)D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes s}. \end{align*} $$

Applying the derived pushforward $p_{*}$ to calculate $p_{*}(\mathcal {O}_{D} \otimes \mathcal {V}'/\mathcal {V} \otimes (\mathcal {V}/\mathcal {V}"')^{\otimes (s-1)} )$ , we use the projection formula.

$$ \begin{align*} p_{*}(\mathcal{O}_{D} \otimes \mathcal{V}'/\mathcal{V} \otimes (\mathcal{V}/\mathcal{V}"')^{\otimes (s-1)}) & \cong p_{*}(\mathcal{O}_{D} \otimes \mathcal{O}_{\mathbb{P}(\mathbb{C}^N/\mathcal{V})}(-1) \otimes \mathcal{O}_{\mathbb{P}(\mathcal{V}^{\vee})}(s-1)) \\ &\cong \pi_{13*}g_{1*}(\mathcal{O}_{D} \otimes g_{1}^*(\mathcal{O}_{\mathbb{P}(\mathbb{C}^N/\mathcal{V})}(-1)) \otimes \mathcal{O}_{\mathbb{P}(\mathcal{V}^{\vee})}(s-1)) \\ &\cong \pi_{13*}(\mathcal{O}_{\mathrm{Fl}(k,k+1)} \otimes \mathcal{O}_{\mathbb{P}(\mathbb{C}^N/\mathcal{V})}(-1) \otimes \mathrm{Sym}^{s-1}(\mathcal{V})) \\ &\cong \pi_{13*}(\mathcal{O}_{\mathrm{Fl}(k,k+1)} \otimes \mathcal{O}_{\mathbb{P}(\mathbb{C}^N/\mathcal{V})}(-1) \otimes \pi_{13}^*(\mathrm{Sym}^{s-1}(\mathcal{V}))) \\ &\cong \mathrm{Sym}^{s-1}(\mathcal{V}) \otimes \pi_{13*}(\mathcal{O}_{\mathbb{P}(\mathbb{C}^N/\mathcal{V})}(-1)) \cong 0. \end{align*} $$

Thus we get $p_{*}(\mathcal {O}_{sD} \otimes (\mathcal {V}'/\mathcal {V})^{\otimes s}) \cong p_{*}(\mathcal {O}_{(s-1)D} \otimes (\mathcal {V}'/\mathcal {V})^{\otimes s})$ . Next, consider the exact triangle (5.17) with $n=s-1$ , we have

$$ \begin{align*} \mathcal{O}_{D} \otimes \mathcal{V}'/\mathcal{V} \otimes (\mathcal{V}/\mathcal{V}"')^{\otimes (s-2)} \rightarrow \mathcal{O}_{(s-1)D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (s-1)} \rightarrow \mathcal{O}_{(s-2)D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (s-1)}. \end{align*} $$

Tensoring by $\mathcal {V}'/\mathcal {V}$ and then applying $p_{*}$ , using the same argument as above, we obtain $p_{*}(\mathcal {O}_{(s-1)D} \otimes (\mathcal {V}'/\mathcal {V})^{\otimes s}) \cong p_{*}(\mathcal {O}_{(s-2)D} \otimes (\mathcal {V}'/\mathcal {V})^{\otimes s})$ . Continuing this procedure, we will end up with

(5.18) $$ \begin{align} p_{*}(\mathcal{O}_{sD} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes s}) \cong ... \cong p_{*}(\mathcal{O}_{2D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes s}). \end{align} $$

For the exact triangle (5.17) with $n=2$ , tensoring by $(\mathcal {V}'/\mathcal {V})^{\otimes (s-2)}$ , we get

$$\begin{align*}\mathcal{O}_{D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (s-1)} \otimes \mathcal{V}/\mathcal{V}"' \rightarrow \mathcal{O}_{2D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes s} \rightarrow \mathcal{O}_{D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes s}. \end{align*}$$

Applying $p_{*}$ , then we get $p_{*}(\mathcal {O}_{D} \otimes (\mathcal {V}'/\mathcal {V})^{\otimes (s-1)} \otimes \mathcal {V}/\mathcal {V}"') \cong 0$ and $p_{*}(\mathcal {O}_{D} \otimes (\mathcal {V}'/\mathcal {V})^{\otimes s}) \cong 0$ . The first isomorphism is via projection formula and Proposition 4.4 with $1 \leq s \leq N-k-1$ , while the second one is via Proposition 4.4 with $1 \leq s \leq N-k-1$ .

Hence we get $p_{*}(\mathcal {O}_{2D} \otimes (\mathcal {V}'/\mathcal {V})^{\otimes s}) \cong 0$ and (5.18) implies that $p_{*}(\mathcal {O}_{sD} \otimes (\mathcal {V}'/\mathcal {V})^{\otimes s}) \cong 0$ , so we prove (5.13).

The remaining case is $s=N-k$ . Applying $p_{*}$ to (5.16) with $n=N-k$ , we have

$$\begin{align*}p_{*} (\mathcal{V}"/\mathcal{V}"')^{N-k} \rightarrow p_{*}(\mathcal{V}'/\mathcal{V})^{\otimes (N-k)} \rightarrow p_{*}(\mathcal{O}_{(N-k)D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k)}). \end{align*}$$

It suffices to prove $p_{*}(\mathcal {O}_{(N-k)D} \otimes (\mathcal {V}'/\mathcal {V})^{\otimes (N-k)}) \cong j_{*}\det (\mathbb {C}^N/\mathcal {V})[1+k-N]$ , where $j:\Delta =\mathrm {Gr}(k,N) \rightarrow Y$ is the natural inclusion. Note that we have the inclusion $t:Y \rightarrow \mathrm {Gr}(k,N) \times \mathrm {Gr}(k,N)$ , and so $\Delta =t \circ j$ .

Again, using the similar argument as in the proof of the case $1 \leq s \leq N-k-1$ , we also have

$$\begin{align*}p_{*}(\mathcal{O}_{(N-k)D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k)}) \cong ... \cong p_{*}(\mathcal{O}_{2D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k)}). \end{align*}$$

Considering (5.17) with $n=2$ and tensoring it with $(\mathcal {V}'/\mathcal {V})^{\otimes (N-k-2)}$ , we get

$$\begin{align*}\mathcal{O}_{D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k-1)} \otimes \mathcal{V}/\mathcal{V}"' \rightarrow \mathcal{O}_{2D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k)} \rightarrow \mathcal{O}_{D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k)}, \end{align*}$$

applying $p_{*}$ , we obtain $p_{*}(\mathcal {O}_{D} \otimes (\mathcal {V}'/\mathcal {V})^{\otimes (N-k-1)} \otimes \mathcal {V}/\mathcal {V}"') \cong 0$ via projection formula, while $p_{*}(\mathcal {O}_{D} \otimes (\mathcal {V}'/\mathcal {V})^{\otimes (N-k)}) \cong j_{*}\det (\mathbb {C}^N/\mathcal {V})[1+k-N]$ is by Proposition 4.4.

Hence we get

$$ \begin{align*} p_{*}(\mathcal{O}_{(N-k)D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k)}) &\cong p_{*}(\mathcal{O}_{2D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k)}) \\ &\cong p_{*}(\mathcal{O}_{D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k)}) \cong j_{*}\det(\mathbb{C}^N/\mathcal{V})[1+k-N] \end{align*} $$

and $t_{*}p_{*}(\mathcal {O}_{(N-k)D} \otimes (\mathcal {V}'/\mathcal {V})^{\otimes (N-k)}) \cong \Delta _{*}\det (\mathbb {C}^N/\mathcal {V})[1+k-N]$ , which proves (5.12).

Remark 5.11. In this remark, we show that the exact triangle (5.12) in Proposition 5.10 is non-split. We prove this by showing that the Hom spaces of morphisms between objects are zero. Indeed, using adjunction (condition (3)) we have

$$ \begin{align*} &\mathrm{Hom}( (\mathcal{E}_{0} \ast \mathcal{F}_{N-k})\boldsymbol{1}_{(k,N-k)}, (\mathcal{F}_{N-k} \ast \mathcal{E}_{0})\boldsymbol{1}_{(k,N-k)}) \\&\cong \mathrm{Hom}( (\mathcal{F}_{N-k})_{L} \ast \mathcal{E}_{0}\boldsymbol{1}_{(k-1,N-k+1)}, \mathcal{E}_{0} \ast (\mathcal{F}_{N-k})_{L}\boldsymbol{1}_{(k-1,N-k+1)}) \\& \cong \mathrm{Hom}( \mathcal{E}_{0} \ast (\Psi^{+})^{-1} \ast \mathcal{E}_{0}\boldsymbol{1}_{(k-1,N-k+1)}[1], \mathcal{E}_{0} \ast (\Psi^{+})^{-1} \ast \mathcal{E}_{0} \boldsymbol{1}_{(k-1,N-k+1)}) \\& \cong \mathrm{Hom}( \mathcal{E}_{0} \ast \mathcal{E}_{-1} \ast (\Psi^{+})^{-1} \boldsymbol{1}_{(k-1,N-k+1)}[2], \mathcal{E}_{0} \ast \mathcal{E}_{-1} \ast (\Psi^{+})^{-1} \boldsymbol{1}_{(k-1,N-k+1)}[1]) \\& \cong \mathrm{Hom}(0,0) \cong 0. \ (\text{using condition (5)(a)}) \end{align*} $$

Finally, since ${\Psi }^{\pm }\boldsymbol {1}_{(k,N-k)}$ are given by line bundles, condition (4) is standard to check. Thus, combining Lemma 5.8, Lemma 5.9, and Proposition 5.10, we prove Theorem 5.7.

5.4 The Remaining Relations

Since we have already proved the $\mathfrak {sl}_2$ case, many conditions in Definition 3.1 are direct generalization of the $\mathfrak {sl}_2$ version. To prove Theorem 5.6, we prove the remaining conditions for the $\mathfrak {sl}_n$ case in this section; this means that we prove those not addressed in the $\mathfrak {sl}_2$ case.

First, we note that condition (2) is apparent. Next, it is standard to check conditions (5c), (6c), (7c), and (8c) where $|i-j| \geq 2$ , and condition (9) where $i \neq j$ . So it remains to check the relations (4), (5b), (6b), (7b), and (8b). Again, condition (4) is standard to show since ${\Psi }^{\pm }_{i}\boldsymbol {1}_{\underline {k}}$ are just line bundles.

The next conditions we prove are conditions (5b) and (6b).

Lemma 5.12 (Conditions (5b) and (6b)).

We have the following exact triangle

$$ \begin{align*} (\mathcal{E}_{i+1,s}\ast \mathcal{E}_{i,r+1})\boldsymbol{1}_{\underline{k}} \rightarrow (\mathcal{E}_{i+1,s+1}\ast \mathcal{E}_{i,r})\boldsymbol{1}_{\underline{k}} \rightarrow (\mathcal{E}_{i,r}\ast \mathcal{E}_{i+1,s+1})\boldsymbol{1}_{\underline{k}}. \end{align*} $$

Proof. A simple calculation gives that $(\mathcal {E}_{i+1,s+1}\ast \mathcal {E}_{i,r})\boldsymbol {1}_{\underline {k}}$ is given by

$$\begin{align*}i_{1*}((\mathcal{V}_{i}/\mathcal{V}^{\prime\prime}_{i})^{\otimes r} \otimes (\mathcal{V}_{i+1}/\mathcal{V}^{\prime\prime}_{i+1})^{\otimes (s+1)}) \end{align*}$$

and $(\mathcal {E}_{i,r}\ast \mathcal {E}_{i+1,s+1})\boldsymbol {1}_{\underline {k}}$ is given by

$$\begin{align*}i_{2*}((\mathcal{V}_{i}/\mathcal{V}^{\prime\prime}_{i})^{\otimes r} \otimes (\mathcal{V}_{i+1}/\mathcal{V}^{\prime\prime}_{i+1})^{\otimes (s+1)}). \end{align*}$$

Here $i_{1}:W_{i+1,i}^{1,1}(\underline {k}) \rightarrow Y(\underline {k}) \times Y(\underline {k}+\alpha _{i}+\alpha _{i+1})$ , $i_{2}:W_{i,i+1}^{1,1}(\underline {k}) \rightarrow Y(\underline {k}) \times Y(\underline {k}+\alpha _{i}+\alpha _{i+1})$ are the natural inclusions with the subvarieties $W_{i+1,i}^{1,1}(\underline {k}) \subset Y(\underline {k}) \times Y(\underline {k}+\alpha _{i}+\alpha _{i+1})$ and $W_{i,i+1}^{1,1}(\underline {k}) \subset Y(\underline {k}) \times Y(\underline {k}+\alpha _{i}+\alpha _{i+1})$ given by

$$ \begin{align*} &W_{i+1,i}^{1,1}(\underline{k})=\{(V_{\bullet},V_{\bullet}") \ | \ V^{\prime\prime}_{i} \subset V_{i}, \ V^{\prime\prime}_{i+1} \subset V_{i+1}, \ V_{j}=V^{\prime\prime}_{j} \ \text{for} \ j \neq i,i+1 \}, \\ &W_{i,i+1}^{1,1}(\underline{k})=\{(V_{\bullet},V_{\bullet}") \ | \ V^{\prime\prime}_{i} \subset V_{i} \subset V^{\prime\prime}_{i+1} \subset V_{i+1}, \ V_{j}=V^{\prime\prime}_{j} \ \text{for} \ j \neq i,i+1 \}. \end{align*} $$

Note that $W_{i,i+1}^{1,1}(\underline {k}) \subset W_{i+1,i}^{1,1}(\underline {k})$ is a divisor that is cut out by the natural section of the line bundle $\mathcal {H}om(\mathcal {V}_{i}/\mathcal {V}^{\prime \prime }_{i},\mathcal {V}_{i+1}/\mathcal {V}^{\prime \prime }_{i+1})$ . This implies that

$$ \begin{align*} \mathcal{O}_{W_{i+1,i}^{1,1}(\underline{k})}(W_{i,i+1}^{1,1}(\underline{k})) \cong (\mathcal{V}_{i}/\mathcal{V}^{\prime\prime}_{i})^{\vee} \otimes \mathcal{V}_{i+1}/\mathcal{V}^{\prime\prime}_{i+1}. \end{align*} $$

From the divisor short exact sequence

$$\begin{align*}0 \rightarrow \mathcal{O}_{W_{i+1,i}^{1,1}(\underline{k})}(-W_{i,i+1}^{1,1}(\underline{k})) \cong \mathcal{V}_{i}/\mathcal{V}^{\prime\prime}_{i}\otimes (\mathcal{V}_{i+1}/\mathcal{V}^{\prime\prime}_{i+1})^{-1} \rightarrow \mathcal{O}_{W_{i+1,i}^{1,1}(\underline{k})} \rightarrow \mathcal{O}_{W_{i,i+1}^{1,1}(\underline{k})} \rightarrow 0. \end{align*}$$

Tensoring it with $(\mathcal {V}_{i}/\mathcal {V}^{\prime \prime }_{i})^{\otimes r} \otimes (\mathcal {V}_{i+1}/\mathcal {V}^{\prime \prime }_{i+1})^{\otimes (s+1)}$ we get the following exact triangle

$$ \begin{align*} (\mathcal{V}_{i}/\mathcal{V}^{\prime\prime}_{i})^{\otimes (r+1)} \otimes (\mathcal{V}_{i+1}/\mathcal{V}^{\prime\prime}_{i+1})^{\otimes s} \rightarrow (\mathcal{V}_{i}/\mathcal{V}^{\prime\prime}_{i})^{\otimes r} \otimes (\mathcal{V}_{i+1}/\mathcal{V}^{\prime\prime}_{i+1})^{\otimes (s+1)} \\ \rightarrow \mathcal{O}_{W_{i,i+1}^{1,1}(\underline{k})}\otimes (\mathcal{V}_{i}/\mathcal{V}^{\prime\prime}_{i})^{\otimes r} \otimes (\mathcal{V}_{i+1}/\mathcal{V}^{\prime\prime}_{i+1})^{\otimes (s+1)}. \end{align*} $$

Applying $i_{1*}$ and comparing the kernels, we get

$$ \begin{align*} (\mathcal{E}_{i+1,s}\ast \mathcal{E}_{i,r+1})\boldsymbol{1}_{\underline{k}} \rightarrow (\mathcal{E}_{i+1,s+1}\ast \mathcal{E}_{i,r})\boldsymbol{1}_{\underline{k}} \rightarrow (\mathcal{E}_{i,r}\ast \mathcal{E}_{i+1,s+1})\boldsymbol{1}_{\underline{k}}.\\[-35pt] \end{align*} $$

Next, we verify conditions (7b), (8b).

Lemma 5.13. (Conditions (7b), (8b))

$$ \begin{align*} (\Psi^{+}_{i} \ast\mathcal{E}_{i+1,r})\boldsymbol{1}_{\underline{k}} \cong ( \mathcal{E}_{i+1,r-1} \ast \Psi^{+}_{i})\boldsymbol{1}_{\underline{k}}[1]. \end{align*} $$

Proof. A direct calculation shows that $(\Psi ^{+}_{i} \ast \mathcal {E}_{i+1,r})\boldsymbol {1}_{\underline {k}}$ is given by

$$ \begin{align*} \iota_{*}((\mathcal{V}_{i+1}/\mathcal{V}^{\prime}_{i+1})^{\otimes r} \otimes \det(\mathcal{V}^{\prime}_{i+1}/\mathcal{V}_{i}))[2-k_{i+1}]. \end{align*} $$

while $(\mathcal {E}_{i+1,r-1}\ast \Psi ^{+}_{i})\boldsymbol {1}_{\underline {k}}$ is given by

$$ \begin{align*} \iota_{*} (\mathcal{V}_{i+1}/\mathcal{V}^{\prime}_{i+1})^{\otimes (r-1)} \otimes \det(\mathcal{V}_{i+1}/\mathcal{V}_{i})[1-k_{i+1}]. \end{align*} $$

Here $\iota : W^{1}_{i+1}(\underline {k}) \rightarrow Y(\underline {k}) \times Y(\underline {k}+\alpha _{i+1})$ is the inclusion. Note that the following short exact sequence

$$ \begin{align*} 0 \rightarrow \mathcal{V}^{\prime}_{i+1}/\mathcal{V}_{i} \rightarrow \mathcal{V}_{i+1}/\mathcal{V}_{i} \rightarrow \mathcal{V}_{i+1}/\mathcal{V}^{\prime}_{i+1} \rightarrow 0 \end{align*} $$

gives that $\det (\mathcal {V}^{\prime }_{i+1}/\mathcal {V}_{i}) \otimes \mathcal {V}_{i+1}/\mathcal {V}^{\prime }_{i+1} \cong \det (\mathcal {V}_{i+1}/\mathcal {V}_{i})$ .

Combining the above, we get

$$\begin{align*}\iota_{*} (\mathcal{V}_{i+1}/\mathcal{V}^{\prime}_{i+1})^{\otimes r} \otimes \det(\mathcal{V}^{\prime}_{i+1}/\mathcal{V}_{i})[2-k_{i+1}] \cong \iota_{*} (\mathcal{V}_{i+1}/\mathcal{V}^{\prime}_{i+1})^{\otimes (r-1)} \otimes \det(\mathcal{V}_{i+1}/\mathcal{V}_{i})[2-k_{i+1}] \end{align*}$$

which implies that $(\Psi ^{+}_{i} \ast \mathcal {E}_{i+1,r})\boldsymbol {1}_{\underline {k}} \cong (\mathcal {E}_{i+1,r-1}\ast \Psi ^{+}_{i})\boldsymbol {1}_{\underline {k}}[1]$ .

Combining the above results in this section and Theorem 5.7, we prove Theorem 5.6.

Finally, the following corollary is a direct consequence by Lemma 3.5 and Theorem 5.6.

6.1 The New Kernels $\mathcal {H}_{\pm 1}\boldsymbol {1}_{(k,N-k)}$

When we pass to the Grothendieck groups, the two exact triangles (6.2) and (6.3) should give the commutator relations $[e_0,f_{N-k+1}]1_{(k,N-k)}$ and $[e_{-k-1},f_0]1_{(k,N-k)}$ respectively.

Note that from Definition 2.6, we do not have $e_{-k-1}1_{(k,N-k)}$ and $f_{N-k+1}1_{(k,N-k)}$ in the generators of $\mathcal {U}$ . However, we have the following isomorphisms that can be deduced from (5.10) and (5.11),

(6.4) $$ \begin{align} \mathcal{E}_{-k-1}\boldsymbol{1}_{(k,N-k)} &\cong ({\Psi}^+)^{-1} \ast \mathcal{E}_{-k} \ast {\Psi}^+\boldsymbol{1}_{(k,N-k)}[-1], \nonumber\\ \mathcal{F}_{N-k+1}\boldsymbol{1}_{(k,N-k)} &\cong ({\Psi}^+)^{-1} \ast \mathcal{F}_{N-k} \ast {\Psi}^+\boldsymbol{1}_{(k,N-k)}[1]. \end{align} $$

Thus, it suggests that we can define $e_{-k-1}1_{(k,N-k)}$ and $f_{N-k+1}1_{(k,N-k)}$ in $\mathcal {U}$ by using the conjugation property, i.e.

(6.5)

By comparing to the shifted quantum affine algebra in Definition 2.3, it is reasonable to define the following two new generators in $\mathcal {U}$

(6.6)

such that (6.2) and (6.3) categorify the relations $[e_0,f_{N-k+1}]1_{(k,N-k)}=\psi ^+h_{1}1_{(k,N-k)}$ and $[e_{-k-1},f_{0}]1_{(k,N-k)}=-\psi ^-h_{-1}1_{(k,N-k)}$ respectively.

As a consequence, we define the following two FM kernels in $\mathrm {D}^b(\mathrm {Gr}(k,N) \times \mathrm {Gr}(k,N))$

(6.7)

(6.8)

such that their associated FM transforms categorify $h_{\pm 1}1_{(k,N-k)}$ respectively. Finally, we should mention that the above FM kernels $\mathcal {H}_{\pm 1}\boldsymbol {1}_{(k,N-k)}$ can be generalized to $\mathfrak {sl}_n$ case.

6.2 New Conditions and Properties of ${\mathcal {H}}_{\pm 1}\boldsymbol {1}_{(k,N-k)}$

With the introduction of the new elements $e_{-k-1}1_{(k,N-k)}, f_{N-k+1}1_{(k,N-k)}, h_{\pm 1}1_{(k,N-k)} \in \mathcal {U}$ from (6.5) and (6.6), we define the corresponding new 1-morphisms $\mathsf {E}_{-k-1}\boldsymbol {1}_{(k,N-k)}$ , $\mathsf {F}_{N-k+1}\boldsymbol {1}_{(k,N-k)}$ , and $\mathsf {{H}}_{\pm 1}\boldsymbol {1}_{(k,N-k)}$ in a categorical $\mathcal {U}$ -action.

Then, from (6.4) we should include the following as conditions in a categorical $\mathcal {U}$ -action

Similarly, from (6.2), (6.3), (6.7) and (6.8), we should include the following two exact triangles as conditions in a categorical $\mathcal {U}$ -action

(6.9) $$ \begin{align} &\mathsf{F}_{N-k+1}\mathsf{E}_{0}\boldsymbol{1}_{(k,N-k)} \rightarrow \mathsf{E}_{0}\mathsf{F}_{N-k+1}\boldsymbol{1}_{(k,N-k)} \rightarrow \mathsf{\Psi}^{+}\mathsf{{H}}_{1}\boldsymbol{1}_{(k,N-k)} \end{align} $$

(6.10) $$ \begin{align} &\mathsf{E}_{-k-1}\mathsf{F}_{0}\boldsymbol{1}_{(k,N-k)} \rightarrow \mathsf{F}_{0}\mathsf{E}_{-k-1} \boldsymbol{1}_{(k,N-k)} \rightarrow \mathsf{\Psi}^{-}\mathsf{{H}}_{-1}\boldsymbol{1}_{(k,N-k)}. \end{align} $$

In this subsection, we will provide an adjunction property about the 1-morphisms $\mathsf {{H}}_{\pm 1}\boldsymbol {1}_{(k,N-k)}$ . Moreover, we will also study the FM kernels $\mathcal {H}_{\pm 1}\boldsymbol {1}_{(k,N-k)}$ from (6.7) and (6.8).

6.2.1 Adjunctions

The first property is about the adjunctions between $\mathsf {{H}}_{\pm 1}\boldsymbol {1}_{(k,N-k)}$ .

Lemma 6.1. Assume we have the two exact triangles (6.9) and (6.10) in a categorical $\mathcal {U}$ -action. Then, the 1-morphisms $\mathsf {{H}}_{1}\boldsymbol {1}_{(k,N-k)}$ and $\mathsf {{H}}_{-1}\boldsymbol {1}_{(k,N-k)}$ are biadjoint up to conjugations of $\mathsf {\Psi }^{\pm }\boldsymbol {1}_{(k,N-k)}$ . More precisely,

$$ \begin{align*} (\mathsf{{H}}_{1}\boldsymbol{1}_{(k,N-k)})^{L} &\cong (\mathsf{\Psi}^{-})^{k} \mathsf{{H}}_{-1} (\mathsf{\Psi}^{-})^{-k} \boldsymbol{1}_{(k,N-k)} \cong (\mathsf{\Psi}^{+})^{k} \mathsf{{H}}_{-1}(\mathsf{\Psi}^{+})^{-k} \boldsymbol{1}_{(k,N-k)}, \\ (\mathsf{{H}}_{1}\boldsymbol{1}_{(k,N-k)})^{R} &\cong (\mathsf{\Psi}^{-})^{-N+k} \mathsf{{H}}_{-1} (\mathsf{\Psi}^{-})^{N-k}\boldsymbol{1}_{(k,N-k)} \cong (\mathsf{\Psi}^{+})^{-N+k} \mathsf{{H}}_{-1} (\mathsf{\Psi}^{+})^{N-k}\boldsymbol{1}_{(k,N-k)}. \end{align*} $$

Proof. We prove the case for the left adjoint, and the proof for the right adjoint is similar.

Taking the left adjoint of the following exact triangle

$$ \begin{align*} \mathsf{F}_{N-k+1}\mathsf{E}_{0}\boldsymbol{1}_{(k,N-k)} \rightarrow \mathsf{E}_{0}\mathsf{F}_{N-k+1}\boldsymbol{1}_{(k,N-k)} \rightarrow \mathsf{\Psi}^{+}\mathsf{{H}}_{1}\boldsymbol{1}_{(k,N-k)} \end{align*} $$

we obtain

(6.11) $$ \begin{align} \mathsf{{H}}_{1}^{L}(\mathsf{\Psi}^{+})^{L}\boldsymbol{1}_{(k,N-k)} \rightarrow \mathsf{F}_{N-k+1}^{L}\mathsf{E}_{0}^{L} \boldsymbol{1}_{(k,N-k)} \rightarrow \mathsf{E}_{0}^{L}\mathsf{F}_{N-k+1}^{L}\boldsymbol{1}_{(k,N-k)}. \end{align} $$

Using $\mathsf {F}_{N-k+1}\boldsymbol {1}_{(k,N-k)} = (\mathsf {\Psi }^+)^{-1}\mathsf {F}_{N-k}\mathsf {\Psi }^+\boldsymbol {1}_{(k,N-k)}[1]$ , we get

$$ \begin{align*} (\mathsf{F}_{N-k+1}\boldsymbol{1}_{(k,N-k)})^{L} &\cong (\mathsf{\Psi}^+)^{-1}(\mathsf{F}_{N-k}\boldsymbol{1}_{(k,N-k)})^{L}\mathsf{\Psi}^+\boldsymbol{1}_{(k,N-k)}[-1] \\ &\cong (\mathsf{\Psi}^+)^{-2}\mathsf{E}_{0}\mathsf{\Psi}^+\boldsymbol{1}_{(k,N-k)}[-1] \ (\text{by condition (3)(d)}). \end{align*} $$

Thus

$$ \begin{align*} \mathsf{F}_{N-k+1}^{L}\mathsf{E}_{0}^{L}\boldsymbol{1}_{(k,N-k)} &\cong (\mathsf{\Psi}^{+})^{-2}\mathsf{E}_{0}(\mathsf{\Psi}^{+})(\mathsf{\Psi}^{-})^{k}\mathsf{F}_{0} (\mathsf{\Psi}^{-})^{-k-1}\boldsymbol{1}_{(k,N-k)} [k] \ (\text{by conditions (3)(b)(d)}) \\ & \cong \mathsf{E}_{-2} (\mathsf{\Psi}^{+})^{-1} (\mathsf{\Psi}^{-})^{k} \mathsf{F}_{0} (\mathsf{\Psi}^{-})^{-k-1}\boldsymbol{1}_{(k,N-k)} [k+2] \ (\text{by conditions (7)(a)}) \\ & \cong \mathsf{E}_{-2} (\mathsf{\Psi}^{-})^{k} \mathsf{F}_{1} (\mathsf{\Psi}^{+})^{-1} (\mathsf{\Psi}^{-})^{-k-1}\boldsymbol{1}_{(k,N-k)} [k+1] \ (\text{by conditions (8)(a)}) \\ & \cong \mathsf{E}_{-2} (\mathsf{\Psi}^{-})^{k-1} \mathsf{F}_{0} (\mathsf{\Psi}^{+})^{-1} (\mathsf{\Psi}^{-})^{-k}\boldsymbol{1}_{(k,N-k)} [k] \ (\text{by conditions (8)(a)}) \\ & \cong (\mathsf{\Psi}^{-})^{k-1} \mathsf{E}_{-k-1} \mathsf{F}_{0} (\mathsf{\Psi}^{-})^{-k} (\mathsf{\Psi}^{+})^{-1} \boldsymbol{1}_{(k,N-k)} [1] \ (\text{by conditions (7)(a)}). \end{align*} $$

A similar calculation shows that

$$ \begin{align*} \mathsf{E}_{0}^{L}\mathsf{F}_{N-k+1}^{L}\boldsymbol{1}_{(k,N-k)} \cong (\mathsf{\Psi}^{-})^{k-1} \mathsf{F}_{0} \mathsf{E}_{-k-1} (\mathsf{\Psi}^{-})^{-k}(\mathsf{\Psi}^{+})^{-1} \boldsymbol{1}_{(k,N-k)} [1]. \end{align*} $$

Since $\mathsf {\Psi }^{+}\boldsymbol {1}_{(k,N-k)}$ is invertible, $(\mathsf {\Psi }^{+})^{L}\boldsymbol {1}_{(k,N-k)} \cong (\mathsf {\Psi }^{+})^{-1}\boldsymbol {1}_{(k,N-k)}$ . Thus (6.11) becomes

(6.12) $$ \begin{align} \mathsf{{H}}_{1}^{L}(\mathsf{\Psi}^{+})^{-1}\boldsymbol{1}_{(k,N-k)} \rightarrow (\mathsf{\Psi}^{-})^{k-1} \mathsf{E}_{-k-1} \mathsf{F}_{0} (\mathsf{\Psi}^{-})^{-k} (\mathsf{\Psi}^{+})^{-1} \boldsymbol{1}_{(k,N-k)} [1] \rightarrow (\mathsf{\Psi}^{-})^{k-1} \mathsf{F}_{0} \mathsf{E}_{-k-1} (\mathsf{\Psi}^{-})^{-k} (\mathsf{\Psi}^{+})^{-1} \boldsymbol{1}_{(k,N-k)} [1]. \end{align} $$

Applying $(\mathsf {\Psi }^{-})^{-k+1} - $ , $- (\mathsf {\Psi }^{-})^{k} (\mathsf {\Psi }^{+})$ and homological shift by one to (6.12), we get

$$ \begin{align*} \mathsf{E}_{-k-1}\mathsf{F}_{0}\boldsymbol{1}_{(k,N-k)} \rightarrow \mathsf{F}_{0}\mathsf{E}_{-k-1} \boldsymbol{1}_{(k,N-k)} \rightarrow (\mathsf{\Psi}^{-})^{-k+1}\mathsf{{H}}_{1}^{L}(\mathsf{\Psi}^{-})^{k} \boldsymbol{1}_{(k,N-k)}. \end{align*} $$

Comparing to the exact triangle (6.10), there is an isomorphism

$$ \begin{align*} (\mathsf{\Psi}^{-})^{-k+1} \mathsf{{H}}_{1}^{L} (\mathsf{\Psi}^{-})^{k} \boldsymbol{1}_{(k,N-k)} \cong \mathsf{\Psi}^{-}\mathsf{{H}}_{-1}\boldsymbol{1}_{(k,N-k)} \end{align*} $$

and hence we conclude that $\mathsf {{H}}_{1}^{L} \boldsymbol {1}_{(k,N-k)} \cong (\mathsf {\Psi }^{-})^{k} \mathsf {{H}}_{-1} (\mathsf {\Psi }^{-})^{-k} \boldsymbol {1}_{(k,N-k)}$ .

To obtain the second isomorphism, note that from conditions (7)(a) and (8)(a)

$$ \begin{align*} \mathsf{\Psi}^{+}(\mathsf{\Psi}^{-})^{-1}\mathsf{E}_{r}\mathsf{F}_{s}\boldsymbol{1}_{(k,N-k)} \cong \mathsf{E}_{r}\mathsf{F}_{s}\mathsf{\Psi}^{+}(\mathsf{\Psi}^{-})^{-1} \boldsymbol{1}_{(k,N-k)}, \end{align*} $$

for all r and s. This implies that $\mathsf {\Psi }^{+} (\mathsf {\Psi }^{-})^{-1} \mathsf {{H}}_{1} \boldsymbol {1}_{(k,N-k)} \cong \mathsf {{H}}_{1} \mathsf {\Psi }^{+} (\mathsf {\Psi }^{-})^{-1} \boldsymbol {1}_{(k,N-k)}$ . Thus $(\mathsf {\Psi }^{-})^{-1} \mathsf {{H}}_{1} \mathsf {\Psi }^{-} \boldsymbol {1}_{(k,N-k)} \cong (\mathsf {\Psi }^{+})^{-1} \mathsf {{H}}_{1} \mathsf {\Psi }^{+} \boldsymbol {1}_{(k,N-k)}$ and repeating this process we get the second isomorphism.

6.2.2 Non-triviality

The second property is about the geometric property of the FM kernels $\mathcal {H}_{\pm 1}\boldsymbol {1}_{(k,N-k)}$ defined in (6.7) and (6.8).

It suffices to consider $\mathcal {H}_{1}\boldsymbol {1}_{(k,N-k)}$ since the argument for $\mathcal {H}_{-1}\boldsymbol {1}_{(k,N-k)}$ is similar. A simple calculation shows that the FM kernel $\mathcal {H}_1\boldsymbol {1}_{(k,N-k)}$ for $\mathsf {{H}}_1\boldsymbol {1}_{(k,N-k)}$ is

(6.13) $$ \begin{align} \mathcal{H}_1\boldsymbol{1}_{(k,N-k)} \cong t_*p_*(\mathcal{O}_{2D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k+1)}) \otimes \pi_2^*\det(\mathbb{C}^N/\mathcal{V})^{-1}[N-k-1] \in \mathrm{D}^b(\mathrm{Gr}(k,N) \times \mathrm{Gr}(k,N)) \end{align} $$

where $p:X \rightarrow Y$ is the projection and $t:Y \rightarrow \mathrm {Gr}(k,N) \times \mathrm {Gr}(k,N)$ is the inclusion with X, Y defined in (5.3), (5.2) respectively.

On the other hand, tensoring (5.17) with $n=2$ by $(\mathcal {V}'/\mathcal {V})^{\otimes (N-k-1)}$ , we get

(6.14) $$ \begin{align} \mathcal{O}_{D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k)} \otimes \mathcal{V}/\mathcal{V}"' \rightarrow \mathcal{O}_{2D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k+1)} \rightarrow \mathcal{O}_{D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k+1)}. \end{align} $$

By using the projection formula and Proposition 4.4, after applying $t_{*}p_{*}$ to (6.14), we get the following exact triangle

(6.15) $$ \begin{align} \Delta_{*} \mathcal{V} \otimes \det(\mathbb{C}^N/\mathcal{V})[1+k-N] \rightarrow t_{*}p_{*}(\mathcal{O}_{2D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k+1)} ) \rightarrow \Delta_{*} \mathbb{C}^N/\mathcal{V} \otimes \det(\mathbb{C}^N/\mathcal{V})[1+k-N]. \end{align} $$

Tensoring (6.15) with $\pi _2^*\det (\mathbb {C}^N/\mathcal {V})^{-1}$ and homologically shifted by $[N-k-1]$ , by the projection formula and (6.13), we obtain the following exact triangle

(6.16) $$ \begin{align} \Delta_{*} \mathcal{V} \rightarrow \mathcal{H}_{1}\boldsymbol{1}_{(k,N-k)} \rightarrow \Delta_{*} \mathbb{C}^N/\mathcal{V} \end{align} $$

since $\pi _2 \circ \Delta =id$ . This implies that $\mathcal {H}_{1}\boldsymbol {1}_{(k,N-k)}$ is determined by $\mathrm {Ext}^1_{\mathrm {Gr}(k,N)\times \mathrm {Gr}(k,N)}(\Delta _{*}\mathbb {C}^N/\mathcal {V} ,\Delta _{*}\mathcal {V})$ . The following result shows that, in the non-extreme cases, the exact triangle (6.16) is not isomorphic to the derived pushforward of the tautological exact triangles on $\mathrm {Gr}(k,N)$ under the diagonal map $\Delta :\mathrm {Gr}(k,N) \rightarrow \mathrm {Gr}(k,N) \times \mathrm {Gr}(k,N)$ .

Proposition 6.2. When $k \neq 0, \ N$ , the Fourier-Mukai kernel $\mathcal {H}_1\boldsymbol {1}_{(k,N-k)}$ is neither isomorphic to $\Delta _{*}\mathbb {C}^N$ nor to $\Delta _*(\mathcal {V} \oplus \mathbb {C}^N/\mathcal {V})$ .

Proof. From the definition of $\mathcal {H}_1\boldsymbol {1}_{(k,N-k)}$ , it is equivalent to show $t_*p_*(\mathcal {O}_{2D} \otimes (\mathcal {V}'/\mathcal {V})^{\otimes (N-k+1)})$ is neither isomorphic to $\Delta _{*}\Bigl (\mathbb {C}^N\otimes \det (\mathbb {C}^N/\mathcal {V})\Bigr )[1+k-N]$ nor to $\Delta _*\Bigl ((\mathcal {V} \oplus \mathbb {C}^N/\mathcal {V}) \otimes \det (\mathbb {C}^N/\mathcal {V})\Bigr )[1+k-N]$ .

We have the following diagram

where the left square is a fibered product and $t \circ j=\Delta $ is the diagonal map.

Since $t_*$ is exact and $\Delta _*=t_*j_*$ , it is equivalent to show $p_*(\mathcal {O}_{2D} \otimes (\mathcal {V}'/\mathcal {V})^{\otimes (N-k+1)})$ is neither isomorphic to $j_{*}\Bigl (\mathbb {C}^N\otimes \det (\mathbb {C}^N/\mathcal {V})\Bigr )[1+k-N]$ nor to $j_*\Bigl ((\mathcal {V} \oplus \mathbb {C}^N/\mathcal {V}) \otimes \det (\mathbb {C}^N/\mathcal {V})\Bigr )[1+k-N]$ .

Assume otherwise, then applying $j^*$ we have

(6.17) $$ \begin{align} j^*p_*(\mathcal{O}_{2D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k+1)}) &\cong j^*j_{*}\Bigl(\mathbb{C}^N\otimes \det(\mathbb{C}^N/\mathcal{V})\Bigr)[1+k-N] \nonumber \\ &\mathrm{or} \ j^*j_*\Bigl((\mathcal{V} \oplus \mathbb{C}^N/\mathcal{V}) \otimes \det(\mathbb{C}^N/\mathcal{V})\Bigr)[1+k-N]. \end{align} $$

Taking the sheaf cohomology $\mathcal {H}^l$ to both sides of (6.17), by Lemma 4.6, we obtain

(6.18) $$ \begin{align} \mathcal{H}^l(j^*p_*(\mathcal{O}_{2D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k+1)}) ) &\cong \bigoplus_{s-r=l} \bigwedge^{r} \mathcal{N}^{\vee}_{j} \otimes \mathcal{H}^s(\mathbb{C}^N\otimes \det(\mathbb{C}^N/\mathcal{V})[1+k-N])\end{align} $$

(6.19) $$ \begin{align} &\mathrm{or} \ \bigoplus_{s-r=l} \bigwedge^{r} \mathcal{N}^{\vee}_{j} \otimes \mathcal{H}^s((\mathcal{V} \oplus \mathbb{C}^N/\mathcal{V})\otimes \det(\mathbb{C}^N/\mathcal{V})[1+k-N]). \end{align} $$

Since $\mathbb {C}^N\otimes \det (\mathbb {C}^N/\mathcal {V})[1+k-N]$ and $(\mathcal {V} \oplus \mathbb {C}^N/\mathcal {V})\otimes \det (\mathbb {C}^N/\mathcal {V})[1+k-N]$ are complex concentrate at degree $N-k-1$ , the only value s such that $\mathcal {H}^s \neq 0$ in (6.18) and (6.19) is $s=N-k-1$ . Thus, if we choose $l=N-k-1$ , then $r=s-l=0$ and

(6.20) $$ \begin{align} \mathcal{H}^{N-k-1}(j^*p_*(\mathcal{O}_{2D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k+1)}) ) \cong \mathbb{C}^N \otimes \det(\mathbb{C}^N/\mathcal{V}) \ \mathrm{or} \ (\mathcal{V} \oplus \mathbb{C}^N/\mathcal{V})\otimes \det(\mathbb{C}^N/\mathcal{V}). \end{align} $$

On the other hand, using base change, the left-hand side of (6.17) becomes

$$ \begin{align*} j^*p_{*}(\mathcal{O}_{2D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k+1)}) \cong p|_{D*}i^*(\mathcal{O}_{2D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k+1)}). \end{align*} $$

We tensor (5.16) with $n=2$ by $(\mathcal {V}'/\mathcal {V})^{\otimes (N-k-1)}$ such that it becomes

$$ \begin{align*} (\mathcal{V}"/\mathcal{V}"')^{\otimes 2} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k-1)} \xrightarrow{c\otimes id} (\mathcal{V}'/\mathcal{V})^{\otimes (N-k+1)} \rightarrow \mathcal{O}_{2D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k+1)} \end{align*} $$

where we denote $c:(\mathcal {V}"/\mathcal {V}"')^{\otimes 2} \rightarrow (\mathcal {V}'/\mathcal {V})^{\otimes 2}$ to be the natural inclusion. Applying the pullback $i^*$ , since we have $V=V"$ on D, the map c becomes $0$ and thus

$$ \begin{align*} i^*((\mathcal{V}"/\mathcal{V}"')^{\otimes 2} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k-1)}) \xrightarrow{0} i^*((\mathcal{V}'/\mathcal{V})^{\otimes (N-k+1)}) \rightarrow i^*(\mathcal{O}_{2D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k+1)}), \end{align*} $$

which is again an exact triangle. Thus,

$$ \begin{align*} i^*(\mathcal{O}_{2D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k+1)}) \cong i^*((\mathcal{V}"/\mathcal{V}"')^{\otimes 2} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k-1)}) [1] \bigoplus i^*((\mathcal{V}'/\mathcal{V})^{\otimes (N-k+1)}). \end{align*} $$

Using projection formula and Proposition 4.4, we conclude that

$$ \begin{align*} p|_{D*}i^*(\mathcal{O}_{2D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k+1)}) \cong \mathbb{C}^N/\mathcal{V} \otimes \det(\mathbb{C}^N/V) [1+k-N]. \end{align*} $$

When $l=N-k-1$ ,

$$ \begin{align*} \mathcal{H}^{N-k-1}(j^*p_{*}(\mathcal{O}_{2D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k+1)})) &\cong \mathcal{H}^{N-k-1}(p|_{D*}i^*(\mathcal{O}_{2D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k+1)})) \\ & \cong \mathcal{H}^{N-k-1}(\mathbb{C}^N/\mathcal{V} \otimes \det(\mathbb{C}^N/\mathcal{V}) [1+k-N]) \\ & \cong \mathbb{C}^N/\mathcal{V} \otimes \det(\mathbb{C}^N/\mathcal{V}) \end{align*} $$

which is not isomorphic to (6.20). Hence, we obtain a contradiction.

Remark 6.3. A similar argument shows that $\mathcal {H}_{-1}\boldsymbol {1}_{(k,N-k)}$ is neither isomorphic to $\Delta _{*}(\mathbb {C}^N)^{\vee }$ nor to $\Delta _*(\mathcal {V}^{\vee } \oplus (\mathbb {C}^N/\mathcal {V})^{\vee })$ .

6.3 Calculation of Convolutions and a Conjecture

Although the application of $\mathsf {{H}}_{\pm 1}\boldsymbol {1}_{(k,N-k)}$ is unclear to us, it is still interesting to know the categorical commutator relations between $\mathsf {{H}}_{\pm 1}\boldsymbol {1}_{(k,N-k)}$ and other 1-morphisms.

We will only consider the relations between $\mathsf {{H}}_1\boldsymbol {1}_{(k,N-k)}$ and $\mathsf {\Psi }^{+}\boldsymbol {1}_{(k,N-k)}$ , $\mathsf {E}_r\boldsymbol {1}_{(k,N-k)}$ since the arguments for the others are similar. Then we have to calculate the convolutions of the FM kernels between $\mathcal {H}_{1}\boldsymbol {1}_{(k,N-k)}$ and $\Psi ^{+}\boldsymbol {1}_{(k,N-k)}$ , $\mathcal {E}_{r}\boldsymbol {1}_{(k,N-k)}$ .

By Proposition 6.2, the kernel $\mathcal {H}_1\boldsymbol {1}_{(k,N-k)}$ is complicated. Thus, the computation of convolutions involving $\mathcal {H}_1\boldsymbol {1}_{(k,N-k)}$ is generally difficult to carry out. In this subsection, we provide the calculation in the simplest case $N=2$ .

When $N=2$ , $\mathrm {Gr}(0,2)=\mathrm {Gr}(2,2)=pt$ and $\mathrm {Gr}(1,2)=\mathbb {P}^1$ . The most interesting case is when $k=1$ where the fibered product variety X in (5.3) is given by

$$ \begin{align*} X=\{(0,V,V",\mathbb{C}^2) \in \mathrm{Gr}(0,2) \times \mathbb{P}^1 \times \mathbb{P}^1 \times \mathrm{Gr}(2,2) \ | \ 0 \subset V \subset \mathbb{C}^2, \ 0 \subset V" \subset \mathbb{C}^2 \} \end{align*} $$

and similarly the variety Y from diagram (5.1) is given by

$$ \begin{align*} Y=\{(V,V") \in \mathbb{P}^1 \times \mathbb{P}^1 \ | \ \dim(V \cap V") \geq 0 \}. \end{align*} $$

Thus we obtain $X=Y=\mathbb {P}^1 \times \mathbb {P}^1$ . Both the projection map $p:X \rightarrow Y$ and the inclusion map $t: Y \rightarrow \mathbb {P}^1 \times \mathbb {P}^1$ are the identity map. Let $\pi _{i}:\mathbb {P}^1 \times \mathbb {P}^1 \rightarrow \mathbb {P}^1$ be the natural projection to the i-th copy where $i=1,2$ . We also denote $\mathcal {V}$ and $\mathcal {V}"$ as the two tautological line bundles on the first and second copies of $\mathbb {P}^1$ . Then we have the two tautological line bundles $\pi ^*_{1}\mathcal {V}$ , $\pi ^*_{2}\mathcal {V}"$ on $\mathbb {P}^1 \times \mathbb {P}^1$ . Let $\mathbb {C}^2$ be the trivial bundle of rank two on $\mathbb {P}^1$ and denote $\Delta : \mathbb {P}^1 \rightarrow \mathbb {P}^1 \times \mathbb {P}^1$ as the diagonal map. By abusing notations, we still denote its image as $\Delta $ . Then the associated line bundle is $\mathcal {O}_{\mathbb {P}^1 \times \mathbb {P}^1}(-\Delta ) = \pi ^*_{2}\mathcal {V}" \otimes \pi ^*_{1}(\mathbb {C}^2/\mathcal {V})^{-1}$ . Then, from (6.7), we have $\mathcal {H}_1\boldsymbol {1}_{(1,1)}=\mathcal {O}_{2\Delta } \otimes \pi ^*_1(\mathbb {C}^2/\mathcal {V})^{\otimes 2} \otimes \pi ^*_2(\mathbb {C}^2/\mathcal {V}")^{-1}$ .

First, we compute the convolutions between $\mathcal {H}_1\boldsymbol {1}_{(1,1)}$ and $\Psi ^+\boldsymbol {1}_{(1,1)}$ and show that they are non-isomorphic.

Proof. It is standard to check that $(\Psi ^{+} \ast \mathcal {H}_1) \boldsymbol {1}_{(1,1)}=\mathcal {O}_{2\Delta } \otimes \pi ^*_{1}(\mathbb {C}^2/\mathcal {V})^{\otimes 2}$ . Now, we calculate $(\mathcal {H}_1 \ast \Psi ^{+}) \boldsymbol {1}_{(1,1)}$ . Using base-change with respect to

we have

$$ \begin{align*} (\mathcal{H}_1 \ast \Psi^{+}) \boldsymbol{1}_{(1,1)} &= \pi_{13*}\biggl(\pi^*_{12}\Delta_{*}(\mathbb{C}^2/\mathcal{V}) \otimes \pi^*_{23}\Bigl(\mathcal{O}_{2\Delta} \otimes \pi^*_{1}(\mathbb{C}^2/\mathcal{V})^{\otimes 2} \otimes \pi^*_{2}(\mathbb{C}^2/\mathcal{V}")^{-1} \Bigr) \biggr) \\ &\cong \pi_{13*}\biggl((\Delta \times id)_{*}\pi^*_{1}(\mathbb{C}^2/\mathcal{V}) \otimes \pi^*_{23}\Bigl(\mathcal{O}_{2\Delta} \otimes \pi^*_{1}(\mathbb{C}^2/\mathcal{V})^{\otimes 2} \otimes \pi^*_{2}(\mathbb{C}^2/\mathcal{V}")^{-1} \Bigr)\biggr) \\ &\cong (\pi_{13} \circ (\Delta \times id))_{*} \biggl( \pi^*_{1}(\mathbb{C}^2/\mathcal{V}) \otimes (\pi_{23} \circ (\Delta \times id))^{*}\Bigl(\mathcal{O}_{2\Delta} \otimes \pi^*_{1}(\mathbb{C}^2/\mathcal{V})^{\otimes 2} \otimes \pi^*_{2}(\mathbb{C}^2/\mathcal{V}")^{-1} \Bigr ) \biggr) \\ & = \mathcal{O}_{2\Delta} \otimes \pi^*_{1}(\mathbb{C}^2/\mathcal{V})^{\otimes 3} \otimes \pi^*_{2}(\mathbb{C}^2/\mathcal{V}")^{-1} \end{align*} $$

since $\pi _{13} \circ (\Delta \times id)=id$ , $\pi _{23} \circ (\Delta \times id)=id$ .

To see that $(\Psi ^{+} \ast \mathcal {H}_1) \boldsymbol {1}_{(1,1)} \ncong (\mathcal {H}_1 \ast \Psi ^{+}) \boldsymbol {1}_{(1,1)}$ , we apply the pushforward $\pi _{1*}$ to them and using the projection formula we obtain

$$ \begin{align*} \pi_{1*}(\Psi^{+} \ast \mathcal{H}_{1}) \boldsymbol{1}_{(1,1)} &= \pi_{1*}(\mathcal{O}_{2\Delta} \otimes \pi^*_{1}(\mathbb{C}^2/\mathcal{V})^{\otimes 2}) = \pi_{1*}(\mathcal{O}_{2\Delta}) \otimes (\mathbb{C}^2/\mathcal{V})^{\otimes 2}, \\ \pi_{1*}(\mathcal{H}_1 \ast \Psi^{+}) \boldsymbol{1}_{(1,1)} &= \pi_{1*}(\mathcal{O}_{2\Delta} \otimes \pi^*_{1}(\mathbb{C}^2/\mathcal{V})^{\otimes 3} \otimes \pi^*_{2}(\mathbb{C}^2/\mathcal{V}")^{-1}) \\ &=\pi_{1*}(\mathcal{O}_{2\Delta} \otimes \pi^*_{2}(\mathbb{C}^2/\mathcal{V}")^{-1}) \otimes (\mathbb{C}^2/\mathcal{V})^{\otimes 3}. \end{align*} $$

To calculate $\pi _{1*}(\mathcal {O}_{2\Delta })$ , we use the following exact triangle

(6.21) $$ \begin{align} \mathcal{O}_{\mathbb{P}^1 \times \mathbb{P}^1}(-2\Delta) \rightarrow \mathcal{O}_{\mathbb{P}^1 \times \mathbb{P}^1} \rightarrow \mathcal{O}_{2\Delta}. \end{align} $$

Note that $\mathcal {O}_{\mathbb {P}^1 \times \mathbb {P}^1}(-\Delta ) = \pi ^*_{2}\mathcal {V}" \otimes \pi ^*_{1}(\mathbb {C}^2/\mathcal {V})^{-1}$ , thus

$$ \begin{align*} \pi_{1*}\mathcal{O}_{\mathbb{P}^1 \times \mathbb{P}^1}(-2\Delta)=\pi_{1*}(\pi^*_{2}(\mathcal{V}")^{\otimes 2} \otimes \pi^*_{1}(\mathbb{C}^2/\mathcal{V})^{\otimes (-2)}) \cong \pi_{1*}\pi^*_{2}(\mathcal{V}")^{\otimes 2} \otimes (\mathbb{C}^2/\mathcal{V})^{\otimes (-2)} \cong (\mathbb{C}^2/\mathcal{V})^{\otimes (-2)}[-1] \end{align*} $$

since $H^*(\mathbb {P}^1,\mathcal {O}_{\mathbb {P}^1}(-2))=\mathbb {C}[-1]$ . Then, after applying $\pi _{1*}$ to (6.21), it becomes

(6.22) $$ \begin{align} \mathcal{O}_{\mathbb{P}^1} \rightarrow \pi_{1*}\mathcal{O}_{2\Delta} \rightarrow (\mathbb{C}^2/\mathcal{V})^{\otimes (-2)} \end{align} $$

and $\pi _{1*}(\mathcal {O}_{2\Delta })$ is determined by $\mathrm {Ext}^1((\mathbb {C}^2/\mathcal {V})^{\otimes (-2)}, \mathcal {O}_{\mathbb {P}^1})=\mathrm {Ext}^1(\mathcal {O}_{\mathbb {P}^1}(-2), \mathcal {O}_{\mathbb {P}^1})=H^1(\mathbb {P}^1,\mathcal {O}_{\mathbb {P}^1}(2))=0$ .

Thus

$$ \begin{align*} \pi_{1*}(\mathcal{O}_{2\Delta})=(\mathbb{C}^2/\mathcal{V})^{\otimes (-2)} \oplus \mathcal{O}_{\mathbb{P}^1} \end{align*} $$

and we obtain

(6.23) $$ \begin{align} \pi_{1*}(\Psi^{+} \ast \mathcal{H}_{1}) \boldsymbol{1}_{(1,1)} = \pi_{1*}(\mathcal{O}_{2\Delta}) \otimes (\mathbb{C}^2/\mathcal{V})^{\otimes 2} =(\mathbb{C}^2/\mathcal{V})^{\otimes 2} \oplus \mathcal{O}_{\mathbb{P}^1} = \mathcal{O}_{\mathbb{P}^1}(2) \oplus \mathcal{O}_{\mathbb{P}^1}. \end{align} $$

On the other hand, to calculate $\pi _{1*}(\mathcal {O}_{2\Delta } \otimes \pi ^*_{2}(\mathbb {C}^2/\mathcal {V}")^{-1})$ , we tensor (6.21) with $\pi ^*_{2}(\mathbb {C}^2/\mathcal {V}")^{-1}$ to get

$$ \begin{align*} \mathcal{O}_{\mathbb{P}^1 \times \mathbb{P}^1}(-2\Delta)\otimes \pi^*_{2}(\mathbb{C}^2/\mathcal{V}")^{-1} \rightarrow \pi^*_{2}(\mathbb{C}^2/\mathcal{V}")^{-1} \rightarrow \mathcal{O}_{2\Delta}\otimes \pi^*_{2}(\mathbb{C}^2/\mathcal{V}")^{-1}. \end{align*} $$

Then, applying $\pi _{1*}$ , we calculate

$$ \begin{align*} \pi_{1*}\pi^*_{2}(\mathbb{C}^2/\mathcal{V}")^{-1} &\cong 0 \\ \pi_{1*}(\pi^*_{2}(\mathcal{V}")^{\otimes 2} \otimes \pi^*_{2}(\mathbb{C}^2/\mathcal{V}")^{-1} \otimes \pi^*_{1}(\mathbb{C}^2/\mathcal{V})^{\otimes (-2)}) &\cong \pi_{1*}(\pi^*_{2}(\mathcal{V}")^{\otimes 2} \otimes \pi^*_{2}(\mathbb{C}^2/\mathcal{V}")^{-1}) \otimes (\mathbb{C}^2/\mathcal{V})^{\otimes (-2)} \\ &\cong ((\mathbb{C}^2/\mathcal{V})^{\otimes (-2)})^{\oplus 2}[-1] \end{align*} $$

since $H^*(\mathbb {P}^1,\mathcal {O}_{\mathbb {P}^1}(-1))=0$ and $H^*(\mathbb {P}^1,\mathcal {O}_{\mathbb {P}^1}(-3))=\mathbb {C}^2[-1]$ .

Thus,

(6.24) $$ \begin{align} \pi_{1*}(\mathcal{O}_{2\Delta}\otimes \pi^*_{2}(\mathbb{C}^2/\mathcal{V}")^{-1})= ((\mathbb{C}^2/\mathcal{V})^{\otimes (-2)})^{\oplus 2} \end{align} $$

which implies

(6.25) $$ \begin{align} \pi_{1*}(\mathcal{H}_1 \ast \Psi^{+}) \boldsymbol{1}_{(1,1)} =\pi_{1*}(\mathcal{O}_{2\Delta} \otimes \pi^*_{2}(\mathbb{C}^2/\mathcal{V}")^{-1}) \otimes (\mathbb{C}^2/\mathcal{V})^{\otimes 3} =(\mathbb{C}^2/\mathcal{V})^{\oplus 2}=\mathcal{O}_{\mathbb{P}^1}(1)^{\oplus 2} \end{align} $$

which is not isomorphic to (6.23).

Next, we compute the convolutions between $\mathcal {H}_1\boldsymbol {1}_{(1,1)}$ and $\mathcal {E}_{r}\boldsymbol {1}_{(1,1)}$ . Note that besides $r=-1,-0$ , we also have the FM kernel $\mathcal {E}_{-2}\boldsymbol {1}_{(1,1)}$ since we have defined the new 1-morphism $\mathsf {E}_{-2}\boldsymbol {1}_{(1,1)}$ .

Proposition 6.6. We have the following exact triangle in $\mathrm {D}^b(\mathrm {Gr}(1,2) \times \mathrm {Gr}(0,2))=\mathrm {D}^b(\mathbb {P}^1)$

$$ \begin{align*} (\mathcal{H}_{1} \ast \mathcal{E}_{-1})\boldsymbol{1}_{(1,1)} \rightarrow (\mathcal{E}_{-1} \ast \mathcal{H}_1)\boldsymbol{1}_{(1,1)} \rightarrow \mathcal{E}_{0}\boldsymbol{1}_{(1,1)} \oplus \mathcal{E}_{0}\boldsymbol{1}_{(1,1)}[1] \end{align*} $$

and $(\mathcal {H}_{1} \ast \mathcal {E}_{0})\boldsymbol {1}_{(1,1)} \cong (\mathcal {E}_{0} \ast \mathcal {H}_1)\boldsymbol {1}_{(1,1)}$ , $(\mathcal {H}_{1} \ast \mathcal {E}_{-2})\boldsymbol {1}_{(1,1)} \cong (\mathcal {E}_{-2} \ast \mathcal {H}_1)\boldsymbol {1}_{(1,1)}$ .

Proof. First, we calculate $(\mathcal {H}_1 \ast \mathcal {E}_{r}) \boldsymbol {1}_{(1,1)}$ . To know $\mathcal {H}_{1}\boldsymbol {1}_{(0,2)}$ , from Proposition 5.10, we have the following exact triangle

$$ \begin{align*} (\mathcal{F}_{3} \ast \mathcal{E}_{0}) \boldsymbol{1}_{(0,2)} \rightarrow (\mathcal{E}_{0} \ast \mathcal{F}_{3})\boldsymbol{1}_{(0,2)} \rightarrow (\Psi^{+} \ast \mathcal{H}_{1})\boldsymbol{1}_{(0,2)}. \end{align*} $$

Since $(\mathcal {F}_{3} \ast \mathcal {E}_{0}) \boldsymbol {1}_{(0,2)}=0$ and $\Psi ^{+}\boldsymbol {1}_{(0,2)}=\Delta _{*}\mathbb {C}[-1] \in \mathrm {D}^b(\mathrm {Gr}(0,2) \times \mathrm {Gr}(0,2))$ , a direct calculation shows that $(\mathcal {E}_{0} \ast \mathcal {F}_{3})\boldsymbol {1}_{(0,2)} \cong \mathbb {C}^2[-1] \in \mathrm {D}^b(\mathrm {Gr}(0,2) \times \mathrm {Gr}(0,2))$ .

So, $\mathcal {H}_{1}\boldsymbol {1}_{(0,2)}=((\Psi ^{+})^{-1} \ast \mathcal {E}_{0} \ast \mathcal {F}_{3})\boldsymbol {1}_{(0,2)} \cong \mathbb {C}^2$ , and $(\mathcal {H}_1 \ast \mathcal {E}_{r}) \boldsymbol {1}_{(1,1)} = \mathcal {O}_{\mathbb {P}^1}(-r)^{\oplus 2}$ since $\pi _{12}=\pi _{23}=\pi _{13}=id$ .

Next, we calculate $(\mathcal {E}_{r} \ast \mathcal {H}_{1}) \boldsymbol {1}_{(1,1)}$ , which is given by

$$ \begin{align*} (\mathcal{E}_{r} \ast \mathcal{H}_{1}) \boldsymbol{1}_{(1,1)} &= \pi_{13*}(\pi^*_{12}\mathcal{H}_{1}\boldsymbol{1}_{(1,1)} \otimes \pi^*_{23}\mathcal{E}_{r}\boldsymbol{1}_{(1,1)}) \\ &= \pi_{1*}(\mathcal{H}_{1}\boldsymbol{1}_{(1,1)} \otimes \pi^*_{2}\mathcal{E}_{r}\boldsymbol{1}_{(1,1)}) \\ &= \pi_{1*}\Bigl(\mathcal{O}_{2\Delta} \otimes \pi^*_{1}(\mathbb{C}^2/\mathcal{V})^{\otimes 2} \otimes \pi^*_{2}(\mathbb{C}^2/\mathcal{V}")^{-1} \otimes \pi^*_{2}(\mathcal{V}")^{\otimes r}\Bigr) \\ &= \pi_{1*}\Bigl(\mathcal{O}_{2\Delta} \otimes \pi^*_{2}(\mathbb{C}^2/\mathcal{V}")^{-1} \otimes \pi^*_{2}(\mathcal{V}")^{\otimes r} \Bigr ) \otimes (\mathbb{C}^2/\mathcal{V})^{\otimes 2}. \end{align*} $$

From (6.3) and (6.24), we know that

$$ \begin{align*} (\mathcal{E}_{0} \ast \mathcal{H}_{1}) \boldsymbol{1}_{(1,1)} &=\pi_{1*}(\mathcal{O}_{2\Delta} \otimes \pi^*_{2}(\mathbb{C}^2/\mathcal{V}")^{-1}) \otimes (\mathbb{C}^2/\mathcal{V})^{\otimes 2} = ((\mathbb{C}^2/\mathcal{V})^{\otimes (-2)})^{\oplus 2} \otimes (\mathbb{C}^2/\mathcal{V})^{\otimes 2} = \mathcal{O}_{\mathbb{P}^1}^{\oplus 2},\\(\mathcal{E}_{-1} \ast \mathcal{H}_{1}) \boldsymbol{1}_{(1,1)} &=\pi_{1*}(\mathcal{O}_{2\Delta} \otimes \pi^*_{2}(\mathbb{C}^2/\mathcal{V}")^{-1} \otimes \pi^*_{2}(\mathcal{V}")^{-1}) \otimes (\mathbb{C}^2/\mathcal{V})^{\otimes 2} \\&= \pi_{1*}(\mathcal{O}_{2\Delta}) \otimes (\mathbb{C}^2/\mathcal{V})^{\otimes 2} = ((\mathbb{C}^2/\mathcal{V})^{\otimes (-2)} \oplus \mathcal{O}_{\mathbb{P}^1}) \otimes (\mathbb{C}^2/\mathcal{V})^{\otimes 2} = \mathcal{O}_{\mathbb{P}^1}(2) \oplus \mathcal{O}_{\mathbb{P}^1}. \end{align*} $$

Finally, to calculate

$$ \begin{align*} (\mathcal{E}_{-2} \ast \mathcal{H}_{1}) \boldsymbol{1}_{(1,1)} &=\pi_{1*}(\mathcal{O}_{2\Delta} \otimes \pi^*_{2}(\mathbb{C}^2/\mathcal{V}")^{-1} \otimes \pi^*_{2}(\mathcal{V}")^{\otimes (-2)}) \otimes (\mathbb{C}^2/\mathcal{V})^{\otimes 2} \\ &= \pi_{1*}(\mathcal{O}_{2\Delta}\otimes \pi^*_{2}(\mathcal{V}")^{-1}) \otimes (\mathbb{C}^2/\mathcal{V})^{\otimes 2}, \end{align*} $$

we need to calculate $\pi _{1*}(\mathcal {O}_{2\Delta }\otimes \pi ^*_{2}(\mathcal {V}")^{-1})$ .

Tensoring (6.21) with $\pi ^*_{2}(\mathcal {V}")^{-1}$ and applying $\pi _{1*}$ , we get $\pi _{1*}(\pi ^*_{2}\mathcal {V}" \otimes \pi ^*_{1}(\mathbb {C}^2/\mathcal {V})^{\otimes (-2)}) \cong \pi _{1*}\pi ^*_{2}\mathcal {V}" \otimes (\mathbb {C}^2/\mathcal {V})^{\otimes (-2)}=0$ by projection formula and $\pi _{1*}\pi ^*_{2}(\mathcal {V}")^{-1} \cong \mathcal {O}_{\mathbb {P}^1}^{\oplus 2}$ . So we have

$$ \begin{align*} (\mathcal{E}_{-2} \ast \mathcal{H}_{1}) \boldsymbol{1}_{(1,1)} = \pi_{1*}(\mathcal{O}_{2\Delta}\otimes \pi^*_{2}(\mathcal{V}")^{-1}) \otimes (\mathbb{C}^2/\mathcal{V})^{\otimes 2} = \mathcal{O}_{\mathbb{P}^1}^{\oplus 2} \otimes (\mathbb{C}^2/\mathcal{V})^{\otimes 2} = \mathcal{O}_{\mathbb{P}^1}(2)^{\oplus 2}. \end{align*} $$

As a consequence, we have the following isomorphisms

$$ \begin{align*} & (\mathcal{H}_1 \ast \mathcal{E}_{0}) \boldsymbol{1}_{(1,1)} = \mathcal{O}_{\mathbb{P}^1}^{\oplus 2} \cong \mathcal{O}_{\mathbb{P}^1}^{\oplus 2} = (\mathcal{E}_{0} \ast \mathcal{H}_{1}) \boldsymbol{1}_{(1,1)}, \\ &(\mathcal{H}_1 \ast \mathcal{E}_{-2}) \boldsymbol{1}_{(1,1)} = \mathcal{O}_{\mathbb{P}^1}(2)^{\oplus 2} \cong \mathcal{O}_{\mathbb{P}^1}(2)^{\oplus 2} = (\mathcal{E}_{-2} \ast \mathcal{H}_{1}) \boldsymbol{1}_{(1,1)}. \end{align*} $$

On the other hand, for $r=-1$ , we have the following exact triangle

$$ \begin{align*} (\mathcal{H}_1 \ast \mathcal{E}_{-1}) \boldsymbol{1}_{(1,1)} = \mathcal{O}_{\mathbb{P}^1}(1)^{\oplus 2} \rightarrow (\mathcal{E}_{-1} \ast \mathcal{H}_1) \boldsymbol{1}_{(1,1)}= \mathcal{O}_{\mathbb{P}^1}(2) \oplus \mathcal{O}_{\mathbb{P}^1} \rightarrow \mathcal{O}_{\mathbb{P}^1}[1] \oplus \mathcal{O}_{\mathbb{P}^1} =\mathcal{E}_{0}\boldsymbol{1}_{(1,1)} \oplus \mathcal{E}_{0}\boldsymbol{1}_{(1,1)}[1]. \end{align*} $$

Remark 6.7. Like Proposition 6.4 and Proposition 6.6, similar results also hold for the convolutions of $\mathcal {H}_{-1}\boldsymbol {1}_{(1,1)}$ , which is given by

$$ \begin{align*} \mathcal{H}_{-1}\boldsymbol{1}_{(1,1)} =\mathcal{O}_{2\Delta} \otimes \pi^*_1\mathcal{V}^{\otimes (-2)} \otimes \pi^*_2\mathcal{V}". \end{align*} $$

Finally, based on the above observation, we formulate a conjecture, which is the $\mathfrak {sl}_n$ version of Proposition 6.6.

6.4 The Commutator Relation on the Grothendieck Groups for All Loop Generators

From Corollary 5.14, we know that there is an action of $\mathcal {U}$ on $\bigoplus _{\underline {k}}K(\mathrm {Fl}_{\underline {k}}(\mathbb {C}^N))$ which comes from decategorifying the categorical $\mathcal {U}$ -action in Theorem 5.6. Since the 1-morphisms act by FM transforms with kernels in Definition 5.5, the generators of $\mathcal {U}$ act on $\bigoplus _{\underline {k}}K(\mathrm {Fl}_{\underline {k}}(\mathbb {C}^N))$ by the K-theoretic FM transform. More precisely,

$$ \begin{align*} e_{i,r}1_{\underline{k}}:K(\mathrm{Fl}_{\underline{k}}(\mathbb{C}^N)) \rightarrow K(\mathrm{Fl}_{\underline{k}+\alpha_{i}}(\mathbb{C}^N)), \ x \mapsto \pi_{2*}(\pi^{*}_{1}(x) \otimes [\iota(\underline{k})_{*}(\mathcal{V}_{i}/\mathcal{V}^{\prime}_{i})^{\otimes r}]) \end{align*} $$

where we denote $[\iota (\underline {k})_{*}(\mathcal {V}_{i}/\mathcal {V}^{\prime }_{i})^{\otimes r}]$ to be the class of $\iota (\underline {k})_{*}(\mathcal {V}_{i}/\mathcal {V}^{\prime }_{i})^{\otimes r}$ in $K(\mathrm {Fl}_{\underline {k}}(\mathbb {C}^N) \times \mathrm {Fl}_{\underline {k}+\alpha _{i}}(\mathbb {C}^N))$ , and similarly for $f_{i,s}1_{\underline {k}}$ in the opposite direction. For other generators, it is direct to see that

$$ \begin{align*} &\psi_i^{+}1_{\underline{k}}:K(\mathrm{Fl}_{\underline{k}}(\mathbb{C}^N)) \rightarrow K(\mathrm{Fl}_{\underline{k}}(\mathbb{C}^N)), \ x \mapsto x \otimes (-1)^{k_{i+1}-1}[\det(\mathcal{V}_{i+1}/\mathcal{V}_i)] , \\ &\psi_i^{-}1_{\underline{k}}:K(\mathrm{Fl}_{\underline{k}}(\mathbb{C}^N)) \rightarrow K(\mathrm{Fl}_{\underline{k}}(\mathbb{C}^N)), \ x \mapsto x \otimes (-1)^{k_i-1}[\det(\mathcal{V}_i/\mathcal{V}_{i-1})^{-1}]. \end{align*} $$

As we mentioned at the beginning of this section, the FM kernels $\mathcal {E}_{i,r}\boldsymbol {1}_{\underline {k}}$ , $\mathcal {F}_{i,r}\boldsymbol {1}_{\underline {k}}$ are defined for all $r \in \mathbb {Z}$ . Thus, the action of $e_{i,r}1_{\underline {k}}$ , $f_{i,r}1_{\underline {k}}$ by K-theoretic FM transforms are also defined for all $r \in \mathbb {Z}$ . In this subsection, we want to understand the commutator relation $[e_{i,r},f_{i,s}]1_{\underline {k}}$ for all $r,s \in \mathbb {Z}$ .

It suffices to consider the $\mathfrak {sl}_2$ case where the action is on $\bigoplus _k K(\mathrm {Gr}(k,N))$ . Then, condition (10) gives the following commutator relation

(6.26) $$ \begin{align} [e_{0},f_{N-k}]1_{(k,N-k)}&=\psi^+1_{(k,N-k)}=\otimes (-1)^{N-k-1}[\det(\mathbb{C}^N/\mathcal{V})], \nonumber \\ [e_{r},f_{s}]1_{(k,N-k)}&=0, \ \text{if} \ -k+1 \leq r+s \leq N-k-1, \\ \nonumber [e_{-k},f_{0}]1_{(k,N-k)}&=\psi^-1_{(k,N-k)}= \otimes (-1)^{k}[\det(\mathcal{V})^{-1}]. \end{align} $$

Moreover, the two exact triangles (6.2) and (6.3) give the following commutator relations

(6.27) $$ \begin{align} [e_{0},f_{N-k+1}]1_{(k,N-k)}&=\psi^+h_{1}1_{(k,N-k)}, \nonumber \\ [e_{-k-1},f_{0}]1_{(k,N-k)}&=-\psi^-h_{-1}1_{(k,N-k)}. \end{align} $$

To know the action of $h_{\pm 1}1_{(k,N-k)}$ , although their FM kernels $\mathcal {H}_{\pm 1}\boldsymbol {1}_{(k,N-k)}$ are complicate from (6.7) and (6.8), from Subsection 6.2 we know that $\mathcal {H}_{1}\boldsymbol {1}_{(k,N-k)}$ fits in the exact triangle (6.16).

Passing to the Grothendieck group, we have the following equality in $K(\mathrm {Gr}(k,N) \times \mathrm {Gr}(k,N))$

$$ \begin{align*} [\mathcal{H}_{1}\boldsymbol{1}_{(k,N-k)}]=[\Delta_{*}\mathcal{V}]+[\Delta_{*}\mathbb{C}^N/\mathcal{V}]=[\Delta_{*}\mathbb{C}^N]. \end{align*} $$

Thus the action of $h_{1}1_{(k,N-k)}$ is given by

$$ \begin{align*} h_{1}1_{(k,N-k)}:K(\mathrm{Gr}(k,N)) \rightarrow K(\mathrm{Gr}(k,N)), \ x \mapsto x \otimes [\mathbb{C}^N]. \end{align*} $$

A similar argument also shows that the action of $h_{-1}1_{(k,N-k)}$ is given by

$$ \begin{align*} h_{-1}1_{(k,N-k)}:K(\mathrm{Gr}(k,N)) \rightarrow K(\mathrm{Gr}(k,N)), \ x \mapsto x \otimes [(\mathbb{C}^N)^{\vee}]. \end{align*} $$

Note that even though the convolutions between $\mathcal {H}_{1}\boldsymbol {1}_{(k,N-k)}$ and $\Psi ^{+}\boldsymbol {1}_{(k,N-k)}$ are non-isomorphic in general when we pass to the Grothendieck group, they are commutative, i.e., $\psi ^{+}h_{1}1_{(k,N-k)}=h_{1}\psi ^+1_{(k,N-k)}$ (similarly for $h_{-1}1_{(k,N-k)}$ and $\psi ^{-}1_{(k,N-k)}$ ).

Next, it is standard to check that we have the following conjugation property

$$ \begin{align*} e_r1_{(k,N-k)}&=(-1)^r(\psi^+)^re_0(\psi^+)^{-r}1_{(k,N-k)}=(-1)^r(\psi^-)^re_0(\psi^-)^{-r}1_{(k,N-k)}, \\ f_r1_{(k,N-k)}&=(-1)^r(\psi^+)^rf_0(\psi^+)^{-r}1_{(k,N-k)}=(-1)^r(\psi^-)^rf_0(\psi^-)^{-r}1_{(k,N-k)}, \end{align*} $$

for all $r \in \mathbb {Z}$ (see (3.2), (3.3), and (3.4) for their categorical version).

The above conjugation property can help us to know other commutator relations. For example, we can write

$$ \begin{align*} [e_{r},f_{s}]1_{(k,N-k)}&=e_rf_s1_{(k,N-k)}-f_se_r1_{(k,N-k)}=(\psi^{+})^re_0f_{r+s}(\psi^{+})^{-r}-(\psi^+)^rf_{r+s}e_0(\psi^+)^{-r}1_{(k,N-k)} \\ &=(\psi^+)^r[e_0,f_{r+s}](\psi^+)^{-r}1_{(k,N-k)}. \end{align*} $$

Thus, by applying suitable conjugation to (6.26) and (6.27), we obtain

$$ \begin{align*} [e_{r},f_{s}]1_{(k,N-k)}= \begin{cases} \otimes (-1)^{N-k-1}[\det(\mathbb{C}^N/\mathcal{V})][\mathbb{C}^N] & \text{if} \ r+s=N-k+1 \\ \otimes (-1)^{N-k-1}[\det(\mathbb{C}^N/\mathcal{V})] & \text{if} \ r+s=N-k \\ 0 & \text{if} \ -k+1 \leq r+s \leq N-k-1 \\ \otimes (-1)^{k}[\det(\mathcal{V})^{-1}] & \text{if} \ r+s=-k \\ \otimes (-1)^{k}[\det(\mathcal{V})^{-1}][(\mathbb{C}^N)^{\vee}]& \text{if} \ r+s=-k-1 \end{cases}. \end{align*} $$

For the rest of the cases, we have to understand the categorical commutator relation between $\mathsf {E}_{r}\mathsf {F}_{s}\boldsymbol {1}_{(k,N-k)}$ and $\mathsf {F}_{s}\mathsf {E}_{r}\boldsymbol {1}_{(k,N-k)}$ (or more precisely $(\mathcal {E}_{r} \ast \mathcal {F}_{s})\boldsymbol {1}_{(k,N-k)}$ and $(\mathcal {F}_{s} \ast \mathcal {E}_{r})\boldsymbol {1}_{(k,N-k)}$ ) for $r+s \geq N-k+2$ and $r+s \leq -k-2$ .

Consider first case $r+s=N-k+2$ , similarly like Proposition 5.10, it suffices to compare $(\mathcal {E}_{0} \ast \mathcal {F}_{s})\boldsymbol {1}_{(k,N-k)}$ and $(\mathcal {F}_{s} \ast \mathcal {E}_{0})\boldsymbol {1}_{(k,N-k)}$ for $s=N-k+2$ .

Then like the exact triangle (6.1) we have the following exact triangle

(6.28) $$ \begin{align} p_{*} (\mathcal{V}"/\mathcal{V}"')^{\otimes (N-k+2)} \rightarrow p_{*}(\mathcal{V}'/\mathcal{V})^{\otimes (N-k+2)} \rightarrow p_{*}(\mathcal{O}_{(N-k+2)D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k+2)}) \end{align} $$

by applying $p_{*}$ to (5.16) with $n= N-k+2$ . Again, a similar argument as in the proof of Proposition 5.10 tells us that

$$ \begin{align*} p_{*}(\mathcal{O}_{(N-k+2)D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k+2)}) \cong p_{*}(\mathcal{O}_{(N-k+1)D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k+2)}) \cong ... \cong p_{*}(\mathcal{O}_{3D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k+2)}), \end{align*} $$

and applying $t_{*}$ to (6.28) we obtain the following exact triangle

(6.29) $$ \begin{align} (\mathcal{F}_{N-k+2} \ast \mathcal{E}_{0})\boldsymbol{1}_{(k,N-k)} \rightarrow (\mathcal{E}_{0} \ast \mathcal{F}_{N-k+2})\boldsymbol{1}_{(k,N-k)} \rightarrow t_{*}p_{*}(\mathcal{O}_{3D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k+2)}). \end{align} $$

Thus the first question is we have to study $t_{*}p_{*}(\mathcal {O}_{3D} \otimes (\mathcal {V}'/\mathcal {V})^{\otimes (N-k+2)})$ . Tensoring (5.17) with $n=3$ by $(\mathcal {V}'/\mathcal {V})^{\otimes (N-k-1)}$ , we obtain

$$ \begin{align*} \mathcal{O}_{D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k)} \otimes (\mathcal{V}/\mathcal{V}"')^{2} \rightarrow \mathcal{O}_{3D}\otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k+2)} \rightarrow \mathcal{O}_{2D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k+2)} \end{align*} $$

applying $t_{*}p_{*}$ , using the projection formula and Proposition 4.4, we get the following exact triangle

(6.30) $$ \begin{align} \Delta_{*}\mathrm{Sym}^{2}(\mathcal{V}) \otimes \det(\mathbb{C}^N/\mathcal{V})[1+k-N]\rightarrow t_{*}p_{*}(\mathcal{O}_{3D}\otimes(\mathcal{V}'/\mathcal{V})^{\otimes N-k+2}) \rightarrow t_{*}p_{*}(\mathcal{O}_{2D}\otimes(\mathcal{V}'/\mathcal{V})^{\otimes N-k+2}). \end{align} $$

Next, tensoring (5.17) with $n=2$ by $(\mathcal {V}'/\mathcal {V})^{\otimes (N-k)}$ , we obtain

$$ \begin{align*} \mathcal{O}_{D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k+1)} \otimes (\mathcal{V}/\mathcal{V}"') \rightarrow \mathcal{O}_{2D}\otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k+2)} \rightarrow \mathcal{O}_{D}\otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k+2)} \end{align*} $$

again applying $t_{*}p_{*}$ , using the projection formula and Proposition 4.4, we get the following exact triangle

(6.31) $$ \begin{align} \begin{split} &\Delta_{*}\mathcal{V} \otimes \mathbb{C}^N/\mathcal{V} \otimes \det(\mathbb{C}^N/\mathcal{V})[1+k-N]\rightarrow t_{*}p_{*}(\mathcal{O}_{2D}\otimes(\mathcal{V}'/\mathcal{V})^{\otimes (N-k+2)}) \\& \quad \rightarrow \Delta_{*}\mathrm{Sym}^{2}(\mathbb{C}^N/\mathcal{V}) \otimes \det(\mathbb{C}^N/\mathcal{V})[1+k-N]. \end{split} \end{align} $$

Thus, understanding $t_{*}p_{*}(\mathcal {O}_{3D} \otimes (\mathcal {V}'/\mathcal {V})^{\otimes (N-k+2)})$ is equivalent to understanding the two exact triangles (6.30) and (6.31). Moreover, we define

(6.32) $$ \begin{align} \mathcal{H}_{2}\boldsymbol{1}_{(k,N-k)}&:=(\Psi^{+}\boldsymbol{1}_{(k,N-k)})^{-1} \ast [t_{*}p_{*}(\mathcal{O}_{3D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k+2)})] \nonumber \\ &=t_{*}p_{*}(\mathcal{O}_{3D} \otimes (\mathcal{V}'/\mathcal{V})^{\otimes (N-k+2)}) \otimes \pi^*_{2}\det(\mathbb{C}^N/\mathcal{V})^{-1}[N-k-1] \in \mathrm{D}^b(\mathrm{Gr}(k,N)\times \mathrm{Gr}(k,N)) \end{align} $$

which can be thought of as an FM kernel for a certain (undefined) 1-morphism $\mathsf {{H}}_{2}\boldsymbol {1}_{(k,N-k)}$ . Then $\mathcal {H}_{2}\boldsymbol {1}_{(k,N-k)}$ is build up from the following two exact triangles in $\mathrm {D}^b(\mathrm {Gr}(k,N) \times \mathrm {Gr}(k,N))$

(6.33) $$ \begin{align} &\Delta_{*}\mathrm{Sym}^{2}(\mathcal{V}) \rightarrow \mathcal{H}_{2}\boldsymbol{1}_{(k,N-k)} \rightarrow (\Psi^{+}\boldsymbol{1}_{(k,N-k)})^{-1} \ast [t_{*}p_{*}(\mathcal{O}_{2D}\otimes(\mathcal{V}'/\mathcal{V})^{\otimes (N-k+2)})], \end{align} $$

(6.34) $$ \begin{align} &\Delta_{*}\mathcal{V} \otimes \mathbb{C}^N/\mathcal{V} \rightarrow (\Psi^{+}\boldsymbol{1}_{(k,N-k)})^{-1} \ast [t_{*}p_{*}(\mathcal{O}_{2D}\otimes(\mathcal{V}'/\mathcal{V})^{\otimes (N-k+2)})] \rightarrow \Delta_{*}\mathrm{Sym}^{2}(\mathbb{C}^N/\mathcal{V}). \end{align} $$

However, the above discussions can help us to understand what exactly are the commutators for the loop generators $e_{r}1_{(k,N-k)}$ , $f_{s}1_{(k,N-k)}$ at the level of Grothendieck groups. From (6.33) and (6.34), we have the following equality in $K(\mathrm {Gr}(k,N) \times \mathrm {Gr}(k,N))$

$$ \begin{align*} [\mathcal{H}_{2}\boldsymbol{1}_{(k,N-k)}] &= [\Delta_{*}\mathrm{Sym}^{2}(\mathcal{V})] + [(\Psi^{+}\boldsymbol{1}_{(k,N-k)})^{-1} \ast t_{*}p_{*}(\mathcal{O}_{2D}\otimes(\mathcal{V}'/\mathcal{V})^{\otimes (N-k+2)}) ] \\ &=[\Delta_{*}\mathrm{Sym}^{2}(\mathcal{V})] + [\Delta_{*}\mathcal{V} \otimes \mathbb{C}^N/\mathcal{V}] +[\Delta_{*}\mathrm{Sym}^{2}(\mathbb{C}^N/\mathcal{V})] \\ & = [\mathrm{Sym}^2(\mathcal{V} \oplus \mathbb{C}^N/\mathcal{V})]=[\mathrm{Sym}^{2}(\mathbb{C}^N)]. \end{align*} $$

Thus, by the definition of $\mathcal {H}_{2}\boldsymbol {1}_{(k,N-k)}$ and the exact triangle (6.29), we get the following commutator relation after a suitable conjugation

$$ \begin{align*} [e_{r},f_{s}]1_{(k,N-k)}=\otimes (-1)^{N-k-1} [\mathrm{Sym}^2(\mathbb{C}^N)][\det(\mathbb{C}^N/\mathcal{V})], \ \text{if} \ r+s=N-k+2. \end{align*} $$

The above argument can be applied to other cases where $r+s \geq N-k+3$ and similarly $r+s \leq -k-2$ . Moreover, the result can be generalized to the $\mathfrak {sl}_n$ case directly. In conclusion, we have the following result.

Corollary 6.10. The commutator relations at the level of Grothendieck groups $K(\mathrm {Fl}_{\underline {k}}(\mathbb {C}^N))$ for all $e_{i,r}1_{\underline {k}}$ , $f_{i,s}1_{\underline {k}}$ with $r,s \in \mathbb {Z}$ is given by

$$ \begin{align*} [e_{i,r},f_{i,s}]1_{\underline{k}}= \begin{cases} \otimes (-1)^{k_{i+1}-1}[\det(\mathcal{V}_{i+1}/\mathcal{V}_{i})][\mathrm{Sym}^{r+s-k_{i+1}}(\mathcal{V}_{i+1}/\mathcal{V}_{i-1})] & \text{if} \ r+s \geq k_{i+1} \\ 0 & \text{if} \ -k_{i}+1 \leq r+s \leq k_{i+1}-1 \\ \otimes (-1)^{k_{i}}[\det(\mathcal{V}_{i}/\mathcal{V}_{i-1})^{-1}][\mathrm{Sym}^{-r-s-k_{i}}(\mathcal{V}_{i+1}/\mathcal{V}_{i-1})^{\vee}]& \text{if} \ r+s \leq -k_{i} \end{cases}. \end{align*} $$

Finally, we give a remark.

7.1 Affine 0-Hecke Algebras

We begin with the definition of the affine 0-Hecke algebras of type A.

Definition 7.1. The affine 0-Hecke algebra $\mathcal {H}_{N}(0)$ is defined to be the unital associative $\mathbb {C}$ -algebra generated by the elements $T_{1},...,T_{N-1}$ and the polynomial algebra $\mathbb {C}[X_{1}^{\pm 1},...,X_{N}^{\pm 1}]$ subject to the following relations

(H01) $$ \begin{align} T_{i}^2=T_{i}, \end{align} $$

(H02) $$ \begin{align} T_{i}T_{i+1}T_{i}=T_{i+1}T_{i}T_{i+1}, \ T_{i}T_{j}=T_{j}T_{i}, \ \text{if} \ |i-j| \geq 2, \end{align} $$

(H03) $$ \begin{align} T_{i}X_{j}=X_{j}T_{i} \ \text{if} \ j \neq i,i+1, \end{align} $$

(H04) $$ \begin{align} X_{i+1}T_{i}=T_{i}X_{i}+X_{i+1}, \end{align} $$

(H05) $$ \begin{align} X_{i}T_{i}=T_{i}X_{i+1}-X_{i+1}. \end{align} $$

We give a remark about the affine 0-Hecke algebra.

Next, we recall the action of $\mathcal {H}_{N}(0)$ on the Grothendieck group of the full flag variety. Let $G=\mathrm {SL}_{N}(\mathbb {C})$ and $B \subset G$ be the Borel of upper triangular matrices. Then, the full flag variety is defined by

(7.1) $$ \begin{align} G/B:=\{0 \subset V_{1} \subset V_{2} \subset ... \subset V_{N}=\mathbb{C}^N \ | \ \dim V_{k}=k \ \text{for} \ \text{all} \ k \}. \end{align} $$

The Weyl group W is isomorphic to the symmetric group, denoted by $S_N$ . Let $\{s_1,...,s_{N-1}\}$ be the simple reflections that generate the Weyl group $W=S_{N}$ . Then, for each $1 \leq i \leq N-1$ , there is the minimal parabolic subgroup $P_i=B \cup Bs_iB$ . Similarly, the quotient $G/P_i$ , which is called a partial flag variety, can be identified with the following space