Exceptional Collection

Now we want to take $X=G/P$. for $P$ some parabolic group of $G$.

For any $\mathscr{F}\in \mathrm{Coh}^{G}(G/P)$, let $\mathscr{F}^{\vee}$ be its image under Grothendieck-Serre duality functor $\mathrm{RHom}(-,\mathscr{O}_ {G/P})$, so for any $\mathscr{G}\in \mathrm{Coh}^{G}(G/P)$, we have that $$\mathrm{RHom}(\mathscr{F},\mathscr{G})\cong R\Gamma(G/P, \mathscr{F}^{\vee}\otimes^{\mathbb{L}}\mathscr{G}).$$

Proposition 1. Suppose that $C=\{\mathscr{F}_ {w}\}_ {w}$ is an exceptional collection generating $\mathrm{Coh}(G/P)$, and $\{\mathscr{F}^{w}\}_ {w\in W}$ is the dual collection. We define a virtual $\mathbb{k}$-representation of $G$ by $$V_ {c} = \sum_ {w\in W} \mathrm{RHom}(\mathscr{F},\operatorname{Fr}^{\ast}(\mathscr{F}^{w}))$$ Then $V_ {c}|_ {G(\mathbb{F}_ {q})} = \mathbb{k}[G/P(\mathbb{F}_ {q})]$.

Proof.

First we claim that $$[\mathscr{O}_ {\Delta}] = \sum_ {w\in W}[\mathscr{F}^{\vee}_ {w}\boxtimes \mathscr{F}^{w}]$$ in $K_ {0}(\mathrm{Coh}_ {G}(G/P\times G/P))$.

There is a pairing $([\mathscr{F}],[\mathscr{G}])=\chi (R\Gamma(\mathscr{F}\otimes \mathscr{G}))$. To show the claim, we only need to check that the paring of each element with some elements of the form $[\mathscr{F}_ {y_ {1}}^{\vee}\boxtimes\mathscr{F}^{y_ {2}}]$ are equal.

Exercise: both sides pair with $[\mathscr{F}_ {y_ {1}}^{\vee}\boxtimes\mathscr{F}^{y_ {2}}]$ to give $\delta_ {y_ {1},\delta_ {y_ {2}}}$. Then by lemma in last time, we are done.

Fact 2. In type $A$, all irreducible unipotent representations live inside $\mathbb{k}[G/B(\mathbb{F}_ {q})]$. Furthermore, $\mathbb{k}[G/P(\mathbb{F}_ {q})]$ span the space of virtual characters as $P$ varies.

[Samokhin-van der Kallen] constructed an exceptional collection on each $G/P$. Originally we thought that via $V_ {C}$ construction for $C=$this collection, we get the same lift. BUt this is only true in Types $A_ {2},B_ {2},G_ {2}.A_ {3}$.

Admissible subcategories

For $\mathscr{C}\subseteq \mathscr{D}$ is a sub triangulated category. We write $\mathscr{C}^{\perp}$ (resp. ${}^{\perp}\mathscr{C}$) for the strictly full subcategory in $\mathscr{D}$ consists of objects $X$ such that $$\mathrm{Hom}(A,X) = 0\qquad \text{(resp.$\mathrm{Hom}(X,A) = 0$)}$$ for any $A\in \mathscr{C}$.

Definition 3. The subcategory $\mathscr{C}\subseteq \mathscr{D}$ is called right admissible, if

  1. $\mathscr{C}\hookrightarrow \mathscr{D}$ has a right adjoint.
  2. $\mathscr{D}=\langle \mathscr{C}, \mathscr{C}^{\perp} \rangle$.
  3. $[\mathscr{D}]=[\mathscr{C}]\ast [\mathscr{C}^{\perp}]$, where $[\mathscr{D}]$ denote the set of isomorphism classes, and the right hand side denotes the set of $[\mathscr{D}]$ consisting of all $Z$, such that there is a distinguished triangle $X\rightarrow Z\rightarrow Y\rightarrow $ with $X\in \mathscr{C}$ and $Y\in \mathscr{C}^{\perp}$.

Proposition 4. Let $\nabla\subseteq \mathrm{Ob}(\mathscr{D})$ be a finite exceptional set.

  1. The triangulated subcategory of $\mathscr{D}$ generated by $\nabla$ is both left and right admissible.
  2. The dual exceptional set $\Delta$ exists.

Let $\nabla = (\nabla^{i})_ {i=1}^{n}$ be a full exceptional set in a finite type triangulated $\mathbb{k}$-linear category $\mathscr{D}$.

Proposition 5. There exists a unique $t$-structure $(\mathscr{D}^{\geq 0},\mathscr{D}^{<0})$ on $\mathscr{D}$ such that $$\nabla^{i}\in \mathscr{D}^{\geq 0}, \Delta_ {i}\in \mathscr{D}^{\leq 0},$$ with

  1. $\mathscr{D}^{\geq 0}=\langle\nabla^{i}[d],i\in I,d\leq 0 \rangle$, and $\mathscr{D}^{< 0}=\langle \Delta^{i}[d],i\in I,d>0 \rangle$, where $I=\{1,2,\dots, n\}$, and $\langle \rangle$ here means generated under extensions.
  2. The $t$-structure is bounded.
  3. For any $X\in \mathscr{D}$, we have \begin{align} X\in \mathscr{D}^{\geq 0} &\Leftrightarrow \operatorname{Ext}^{<0}(\Delta_ {i},X)=0, \forall i \\ X\in \mathscr{D}^{<0} &\Leftrightarrow \operatorname{Ext}^{\leq 0}(X,\nabla^{i})=0, \forall i \end{align}
  4. Let $\mathscr{A}=\mathscr{D}^{\heartsuit}$ be the heart of this $t$-structure. Then every object of $\mathscr{A}$ has finite length. Further, the image $L_ {i}$ of $\tau_ {\geq 0}(\Delta_ {i})\rightarrow \tau_ {\leq 0}(\nabla^{i})$ is irreducible. These $L_ {i}$ are distinct and they are all the irreducible in $\mathscr{A}$.
  5. If $\mathscr{D}_ {i}=\langle \nabla_ {1},\dots,\nabla_ {i} \rangle$, then there is an induced $t$-structure, and $\mathscr{A}_ {i}=\mathscr{D}_ {i}^{\heartsuit}=$the Serre subcategory of $\mathscr{A}$ generated by $\{L_ {1},\dots,L_ {i}\}$. Moreover, $\tau_ {\geq 0}(\Delta_ {i})\rightarrow L_ {i}$ is a projective cover of $L_ {i}$ in $\mathscr{A}_ {i}$, and dually $L_ {i}\rightarrow \tau_ {\leq 0} (\nabla^{i})$ is an injective hull in $\mathscr{A}_ {i}$.

Mutation of an exceptional set

Suppose that $\widetilde {\leq} $ is another order on $I$. Then we can write \begin{align} \mathscr{D}_ {\widetilde {\leq}i }& = \langle \nabla^{j}| j \widetilde {\leq} i \rangle \\ \mathscr{D}_ {\widetilde { <} i } & = \langle \nabla^{j}| j \widetilde {<} i \rangle. \end{align}

Lemma 6.

  1. For $i\in I$, there is a unique up to unique isomorphism object $\nabla_ {\text{mut}}^{i}$ such that $$\nabla_ {\text{mut}}^{i}\in \mathscr{D}_ {\widetilde {\leq}i }\cap \mathscr{D}^{\perp}_ {\widetilde {<} i},$$ and $$\nabla_ {\text{mut}}^{i}\cong \nabla^{i} \text{ mod }\mathscr{D}_ {\widetilde {<}i}.$$
  2. The objects $\nabla_ {\text{mut}}^{i}$ form an exceptional set indexed by $(I,\widetilde {\leq} )$.
  3. We have that $\mathscr{D}_ {\widetilde {\leq} i} = \langle \nabla_ {\text{mut}}^{i}| j\widetilde {\leq}i \rangle$
Proof.

Let $\Pi_ {i}: \mathscr{D}_ {\widetilde {\leq}i }\rightarrow \mathscr{D}_ {\widetilde {\leq}i }/\mathscr{D}_ {\widetilde {<} i}$ be the projection functor. Let $\Pi_ {i}^{r}$ be the right adjoint. We then set $$\nabla_ {\text{mut}}^{i}:=\Pi_ {i}^{r}\Pi_ {i}\langle \nabla^{i} \rangle$$ one can check this gives an exceptional collection. Part 3 follows by induction.

Exceptional collections on $G/B$ explicitly

Given any weight $\chi \in \mathbb{X}^{\ast}(T)$, let $\mathscr{L}(\chi)$ denote the corresponding $G$-equivariant bundle on $G/B$. Define the vector bundle $\Psi_ {i}^{\omega_ {i}}$ as the kernel of the canonical morphism $$\nabla_ {\omega_ {i}}\otimes \mathscr{L}(0)\rightarrow \mathscr{L}(\omega_ {i})$$ i.e. we have a short exact sequence $$0\rightarrow \Psi_ {i}^{\omega_ {i}}\rightarrow \nabla_ {\omega_ {i}}\otimes \mathscr{L}(0)\rightarrow \mathscr{L}(\omega_ {i})\rightarrow 0 .$$

For any simple root $\alpha_ {i}$, let $P_ {\alpha_ {i}}$ be the minimal parabolic generated by $B$ and $U_ {-\alpha_ {i}}$. Denote $Y_ {i} = G/P_ {\alpha_ {i}}$ and $\pi_ {\alpha_ {i}}: G/B\rightarrow G/P_ {\alpha_ {i}}.$

Given a reduced expression $w=s_ {i_ {1}}\cdots s_ {i_ {n}}$, we build an endofunctor of $\mathrm{Coh}(G/B)$ by $$D_ {w}=D_ {\alpha_ {i_ {1}}}\circ \cdots D_ {\alpha_ {i_ {n}}}$$ and $$D_ {\alpha_ {i}}:=\pi_ {\alpha_ {i}}^{\ast}\pi_ {\alpha_ {i},\ast}.$$

Here is a full exceptional collection in $\mathrm{Coh}(\operatorname{SL}_ {3}/B)$:

$\mathscr{A}_ {-3}$$\mathscr{A}_ {-2}$$\mathscr{A}_ {-1}$$\mathscr{A}_ {0}$
$\mathscr{L}(-\rho)$$D_ {\alpha_ {1}}(\Psi_ {1}^{\omega_ {1}})$$\Psi_ {1}^{\omega_ {1}}$$\mathscr{L}(0)$
$D_ {\alpha_ {2}}(\Psi_ {1}^{\omega_ {2}})$$\Psi_ {1}^{\omega_ {2}}$

Look at their classes in $K_ {0}(\mathrm{Coh}^{G}(G/B))$. It turns out that $D_ {\alpha_ {1}}(\Psi_ {1}^{\omega_ {1}}) = \mathrm{Ker}(\Psi^{\omega_ {1}}_ {1}\rightarrow \mathscr{L}(\omega_ {1}-\alpha_ {1}))$ and thus one can deduce that

\begin{align} D_ {\alpha_ {1}}(\Psi_ {1}^{\omega_ {1}}) & \cong \mathscr{L}(-\omega_ {2}),\\ D_ {\alpha_ {2}}(\Psi_ {1}^{\omega_ {2}}) & \cong \mathscr{L}(-\omega_ {1}). \end{align}

The weights of $\Psi_ {1}^{\omega_ {1}}$ are $\omega_ {1}-\alpha_ {1}$ and $-\omega_ {2}$ and similarly $\Psi_ {1}^{\omega_ {2}}$ gives weights $\omega_ {2}-\alpha_ {2}$ and $-\omega_ {1}$.

One can compare these explicit computations to $f_ {w}$ and $f^{w}$ (the Kazhdan-Lusztig-Steinberg basis).