Proof. Since \({\mathcal{O}}_{P^{\oplus \alpha_{s_k}^P}_{s_k}}\) is an open dense subvariety of \(\mathbf{E}_{\mathbf{V}_{s_k}}\) and \({\mathcal{O}}_{I^{\oplus \alpha_{s'_k}^I}_{s'_k}}\) is an open dense subvariety of \(\mathbf{E}_{\mathbf{V}_{s'_k}}\), Lemma 2 implies that any \[x\in \mathbf{E}_{\mathbf{V}_{s_k}}\setminus {\mathcal{O}}_{P^{\oplus \alpha_{s_k}^P}_{s_k}}, \quad M_x\;\text{has a direct summand }P_{\tilde{s_k}}\text{ with }\tilde{s_k}<s_k,\] and any \[x\in \mathbf{E}_{\mathbf{V}_{s'_k}}\setminus {\mathcal{O}}_{I^{\oplus \alpha_{s'_k}^I}_{s'_k}}, \quad M_x\;\text{has a direct summand }I_{\tilde{s'_k}}\text{ with }\tilde{s'_k}<s'_k.\]

By Lemma 1 and induction, we obtain \[\begin{align} &j_{\alpha,m}^*\Bigl( \overline{{\mathbb{Q}}_l}|_{\mathbf{E}_{\mathbf{V}_{s_1}}}\diamond \cdots \diamond \overline{{\mathbb{Q}}_l}|_{\mathbf{E}_{\mathbf{V}_{s_r}}} \diamond \overline{{\mathbb{Q}}_l}|_{\mathbf{E}_{\mathbf{V}_{\lambda_1}}}\diamond \overline{{\mathbb{Q}}_l}|_{\mathbf{E}_{\mathbf{V}_{\lambda_2}}}\cdots \diamond \overline{{\mathbb{Q}}_l}|_{\mathbf{E}_{\mathbf{V}_{\lambda_k}}} \diamond \overline{{\mathbb{Q}}_l}|_{\mathbf{E}_{\mathbf{V}'_{s'_1}}}\diamond \cdots \diamond \overline{{\mathbb{Q}}_l}|_{\mathbf{E}_{\mathbf{V}'_{s'_t}}} \Bigr)\\ ={}&j_{\alpha,m}^*\Bigl( \overline{{\mathbb{Q}}_l}|_{{\mathcal{O}}_{P^{\oplus \alpha_{s_1}^P}_{s_1}}}\diamond \cdots \diamond \overline{{\mathbb{Q}}_l}|_{{\mathcal{O}}_{P^{\oplus \alpha_{s_r}^P}_{s_r}}} \diamond \overline{{\mathbb{Q}}_l}|_{\mathbf{E}_{\mathbf{V}_{\lambda_1}}}\diamond \overline{{\mathbb{Q}}_l}|_{\mathbf{E}_{\mathbf{V}_{\lambda_2}}}\cdots \diamond \overline{{\mathbb{Q}}_l}|_{\mathbf{E}_{\mathbf{V}_{\lambda_k}}} \diamond \overline{{\mathbb{Q}}_l}|_{{\mathcal{O}}_{I^{\oplus \alpha_{{s'}_1}^I}_{{s'}_1}}}\diamond \cdots \diamond \overline{{\mathbb{Q}}_l}|_{{\mathcal{O}}_{I^{\oplus \alpha_{{s'}_t}^I}_{{s'}_t}}} \Bigr). \end{align}\]

Moreover, since \(\overline{X(0,m)}=\mathbf{E}_{\mathbf{V}_m}\), for any point \(x\in \mathbf{E}_{\mathbf{V}_m}\setminus X(0,m),\) \(M_x\) has a preprojective and a preinjective direct summand. By a direct dimension calculation and Lemma 1, if \(M\) is a regular representation and \(V\) is a subrepresentation (or a quotient) of \(M\) with \(\rm dimV=(u,u)\), then \(V\) is regular.

Therefore, \[\begin{align} &j_{\alpha,m}^*\Bigl( \overline{{\mathbb{Q}}_l}|_{{\mathcal{O}}_{P^{\oplus \alpha_{s_1}^P}_{s_1}}}\diamond \cdots \diamond \overline{{\mathbb{Q}}_l}|_{{\mathcal{O}}_{P^{\oplus \alpha_{s_r}^P}_{s_r}}} \diamond \overline{{\mathbb{Q}}_l}|_{\mathbf{E}_{\mathbf{V}_{\lambda_1}}}\diamond \overline{{\mathbb{Q}}_l}|_{\mathbf{E}_{\mathbf{V}_{\lambda_2}}}\cdots \diamond \overline{{\mathbb{Q}}_l}|_{\mathbf{E}_{\mathbf{V}_{\lambda_k}}} \diamond \overline{{\mathbb{Q}}_l}|_{{\mathcal{O}}_{I^{\oplus \alpha_{{s'}_1}^I}_{{s'}_1}}}\diamond \cdots \diamond \overline{{\mathbb{Q}}_l}|_{{\mathcal{O}}_{I^{\oplus \alpha_{{s'}_t}^I}_{{s'}_t}}} \Bigr)\\ ={}&\Bigl( \overline{{\mathbb{Q}}_l}|_{{\mathcal{O}}_{P^{\oplus \alpha_{s_1}^P}_{s_1}}}\diamond \cdots \diamond \overline{{\mathbb{Q}}_l}|_{{\mathcal{O}}_{P^{\oplus \alpha_{s_r}^P}_{s_r}}} \diamond \overline{{\mathbb{Q}}_l}|_{X(0,\lambda_1)}\diamond \overline{{\mathbb{Q}}_l}|_{X(0,\lambda_2)}\cdots \diamond \overline{{\mathbb{Q}}_l}|_{X(0,\lambda_k)} \diamond \overline{{\mathbb{Q}}_l}|_{{\mathcal{O}}_{I^{\oplus \alpha_{{s'}_1}^I}_{{s'}_1}}}\diamond \cdots \diamond \overline{{\mathbb{Q}}_l}|_{{\mathcal{O}}_{I^{\oplus \alpha_{{s'}_t}^I}_{{s'}_t}}} \Bigr). \end{align}\] ◻

Proof. First, let \(P=(l_1,\cdots,l_t),\) \(\sum l_i=m,\) and \(\underline{\mu_i}=(\mu_i^1,\cdots,\mu_i^{u_i}),\) such that \(\sum\mu^j_i=l_i,\) \(\mu_i^1\geqslant\cdots\geqslant\mu_i^{u_i}.\) Denote \(\mathbf{V}_{l_i}\) as the graded space of dimension \((l_i,l_i).\) Define the subset \[X_{P,(\underline{\mu_i}_{i=1,\cdots,t})}=\left\{x\in X(\alpha,m)\middle|\begin{align} &M_x\cong P_{\alpha}\oplus I_{\alpha}\oplus H_r,H_r\cong H_{r_1}\oplus\cdots\oplus H_{r_t}, H_{r_i}\cong \oplus_j H_{r_i^j},\\& H_{r_i^j}\text{ is indecomposable of dimension }(\mu_i^j,\mu_i^j),\\ & H_{r_i}\text{ and }H_{r_j}\text{ are in different tubes for }i\not=j. \end{align}\right\}.\] It is a locally closed subvariety of \(X(\alpha,m)\) [17]. In this way, we denote \(X(\alpha)\) as \(X_{P_0,(\underline{{\mu_0}_i})}.\)

It is straightforward to see that \(X(\alpha,m)=\bigsqcup_{(P,(\underline{\mu_i}_{i=1,\cdots,t}))\in \Theta} X_{P,(\underline{\mu_i}_{i=1,\cdots,t})},\) which is a finite union with index set \(\Theta\). Moreover, \(\rm dimX_{P,(\underline{\mu_i}_{i=1,\cdots,t})}=\rm dim{\mathcal{O}}_{x}+t,\) \(x\in X_{P,(\underline{\mu_i}_{i=1,\cdots,t})}.\) We also have that \(\rm dimW_{\alpha,\underline{\lambda}}=\rm dimX(\alpha)=m+\rm dim{\mathcal{O}}_{x_0},\) for any \(x_0\in X(\alpha).\)

For \(x\in X_{P,(\underline{\mu_i}_{i=1,\cdots,t})},\;x_0\in X(\alpha)\) \[\rm dim{\mathcal{O}}_{x_0}-\rm dim{\mathcal{O}}_{x}=\rm dim\rm Hom_{kQ}(M_x,M_x)-\rm dim\rm Hom_{kQ}(M_{x_0},M_{x_0})=\sum_{i=1}^t\rm dim\rm Hom_{kQ}(H_{r_i},H_{r_i})-m.\]

We consider the case \(\underline{\lambda}=(1,\cdots,1).\) For \(x\in X_{P,(\underline{\mu_i}_{i=1,\cdots,t})},\) \(\rm dimp_{\alpha,\underline{\lambda}}^{-1}(x)=\sum_{i=1}^t d_i,\) where \(d_i\) is the dimension of Springer fiber of \(x_i,\) where \(M_{x_i}\cong \oplus_{j=1}^{u_i} H_{{r'}_i^j},\) and \(H_{{r'}_i^j}\) is an indecomposable nilpotent representation of dimension \(\mu_i^j\) of Jordan quiver \(Q_J\).

By [18], Springer resolution is semismall, thus \(d_i\leqslant\frac{1}{2}(l_i(l_i-1)-\rm dim{\mathcal{O}}_{x_i}).\)

Then \[\begin{align} \rm dimp_{\alpha,\underline{\lambda}}^{-1}(x)=\sum_{i=1}^t d_i&\leqslant\sum_{i=1}^t \frac{1}{2}(l_i(l_i-1)-\rm dim{\mathcal{O}}_{x_i})\\ &=\sum_{i=1}^t \frac{1}{2}(l_i(l_i-1)-\rm dim\mathbf{G}l_{l_i}+\rm dim\rm Hom_{kQ_J}(M_{x_i},M_{x_i}))\\ &=\sum_{i=1}^t \frac{1}{2}(l_i(l_i-1)-\rm dim\mathbf{G}l_{l_i}+\rm dim\rm Hom_{kQ}(M_{r_i},M_{r_i}))\\ &=\frac{1}{2}(\rm dimW_{\alpha,\underline{\lambda}}-\rm dimX_{P,(\underline{\mu_i}_{i=1,\cdots,t})}+t)+\sum_{i=1}^t \frac{1}{2}(l_i(l_i-1)-\rm dim\mathbf{G}l_{l_i})\\ &=\frac{1}{2}(\rm dimW_{\alpha,\underline{\lambda}}-\rm dimX_{P,(\underline{\mu_i}_{i=1,\cdots,t})}+t-m). \end{align}\]

Thus when \(t\not=m,\) for \(x\in X_{P,(\underline{\mu_i}_{i=1,\cdots,t})},\) \(\rm dimp_{\alpha,\underline{\lambda}}^{-1}(x)<\frac{1}{2}(\rm dimW_{\alpha,\underline{\lambda}}-\rm dimX_{P,(\underline{\mu_i}_{i=1,\cdots,t})})\) and when \(t=m,\) \(X_{P,(\underline{\mu_i}_{i=1,\cdots,t})}=X(\alpha),\) for \(x\in X(\alpha),\) \(p_{\alpha,\underline{\lambda}}^{-1}(x)\) is a finite set.

For other \(\underline{\lambda},\) we have that \(p^{-1}_{\alpha,(1,\cdots,1)}(x)\rightarrow p^{-1}_{\alpha,\underline{\lambda}}(x)\) is a surjective map. Thus when \(t\not=m,\) for \(x\in X_{P,(\underline{\mu_i}_{i=1,\cdots,t})},\) \(\rm dimp_{\alpha,\underline{\lambda}}^{-1}(x)<\frac{1}{2}(\rm dimW_{\alpha,\underline{\lambda}}-\rm dimX_{P,(\underline{\mu_i}_{i=1,\cdots,t})})\) and when \(t=m,\) \(X_{P,(\underline{\mu_i}_{i=1,\cdots,t})})=X(\alpha),\) for \(x\in X(\alpha),\) \(p_{\alpha,\underline{\lambda}}^{-1}(x)\) is a finite set.

Since \(W_{\alpha,\underline{\lambda}}\) is open in the flag variety, it is smooth and irreducible. Denote \(L=\overline{{\mathbb{Q}}_l}[\rm dimX(\alpha)](\frac{\rm dimX(\alpha)}{2}).\)

We now consider \(Z=X(\alpha,m)-X(\alpha)=\bigsqcup_{(P,(\underline{\mu_i}_{i=1,\cdots,t}))\in \Theta-(P_0,(\underline{{\mu_0}_i}))} X_{P,(\underline{\mu_i}_{i=1,\cdots,t})},\) and \(i:Z\rightarrow X(\alpha,m).\) We will prove that \(i^*(p_{\alpha,\underline{\lambda}})!L\in {}^p{\mathcal{D}}^b_{\mathbf{G}_{\mathbf{V}},c}(Z,\overline{{\mathbb{Q}}_l})^{\leqslant-1}\) and \(i^!(p_{\alpha,\underline{\lambda}})!L\in {}^p{\mathcal{D}}^b_{\mathbf{G}_{\mathbf{V}},c}(Z,\overline{{\mathbb{Q}}_l})^{\geqslant 1}.\)

As in the proof in [16], \[\rm dim\operatorname{supp}L=\rm dimX(\alpha).\] Since for any \(y\in X_{P,(\underline{\mu_i}_{i=1,\cdots,t})},\) \[\operatorname{H}^j({p_{\alpha,\underline{\lambda}}}_!L)\cong \mathbf{H}^j_c({p_{\alpha,\underline{\lambda}}}^{-1}(y),L|_{{p_{\alpha,\underline{\lambda}}}^{-1}(y)}),\] and \(\rm dim{p_{\alpha,\underline{\lambda}}}^{-1}(y)<\frac{1}{2}(\rm dimX(\alpha)-\rm dimX_{P,(\underline{\mu_i}_{i=1,\cdots,t})}).\) By [16], \(\operatorname{H}^j({p_{\alpha,\underline{\lambda}}}_!L)=0\) unless \(j< -\rm dimX_{P,(\underline{\mu_i}_{i=1,\cdots,t})}.\) Thus \(\operatorname{supp} \operatorname{H}^j(i^*{p_{\alpha,\underline{\lambda}}}_!L)\subset\cup X_{P,(\underline{\mu_i}_{i=1,\cdots,t})},\) where \((P,(\underline{\mu_i}_{i=1,\cdots,t}))\in \Theta-(P_0,(\underline{{\mu_0}_i}))\) and \(j<- \rm dimX_{P,(\underline{\mu_i}_{i=1,\cdots,t})}.\) Then \(\rm dim\operatorname{supp}\operatorname{H}^j(i^*{p_{\alpha,\underline{\lambda}}}_!L)<-j,\) hence \[i^*{p_{\alpha,\underline{\lambda}}}_!L\in {}^p{\mathcal{D}}^b_{\mathbf{G}_{\mathbf{V}},c}(Z,\overline{{\mathbb{Q}}_l})^{\leqslant-1}.\]

To prove \(i^!(p_{\alpha,\underline{\lambda}})!L\in {}^p{\mathcal{D}}^b_{\mathbf{G}_{\mathbf{V}},c}(Z,\overline{{\mathbb{Q}}_l})^{\geqslant 1}\) is equivalent to prove \(\mathbf{D}(i^!(p_{\alpha,\underline{\lambda}})!L)\in \mathbf{D}({}^p{\mathcal{D}}^b_{\mathbf{G}_{\mathbf{V}},c}(Z,\overline{{\mathbb{Q}}_l})^{\geqslant 1}).\) Since \(\overline{{\mathbb{Q}}_l}\) is a field, by [16], \(\mathbf{D}({}^p{\mathcal{D}}^b_{\mathbf{G}_{\mathbf{V}},c}(Z,\overline{{\mathbb{Q}}_l})^{\geqslant 1})={}^p{\mathcal{D}}^b_{\mathbf{G}_{\mathbf{V}},c}(Z,\overline{{\mathbb{Q}}_l})^{\leqslant-1}.\) And \(\mathbf{D}(i^!(p_{\alpha,\underline{\lambda}})!L)=i^*(p_{\alpha,\underline{\lambda}})!L\) because \(p_{\alpha,\underline{\lambda}}\) is proper and \(\mathbf{D}L=L.\) Then we prove that \[i^!(p_{\alpha,\underline{\lambda}})!L\in {}^p{\mathcal{D}}^b_{\mathbf{G}_{\mathbf{V}},c}(Z,\overline{{\mathbb{Q}}_l})^{\geqslant 1}.\]

By the same argument, we also have that \({p_{\alpha,\underline{\lambda}}}_!L\in \operatorname{Perv}(X(\alpha,m)).\)

We calculate that \(j_0^*{p_{\alpha,\underline{\lambda}}}_!L={p_{\alpha,\underline{\lambda}}^{rss}}_!(\overline{{\mathbb{Q}}_l}[\rm dimX(\alpha)](\frac{\rm dimX(\alpha)}{2}))=\oplus_{\chi\in X(S_m)}L_{\chi}[\rm dimX(\alpha)](\frac{\rm dimX(\alpha)}{2})^{\oplus K_{\underline{\lambda},\chi}}.\) Then by [[16]][19][Lemma 6.1], \[{p_{\alpha,\underline{\lambda}}}_!L=\oplus_{\chi\in X(S_m)} {j_{\alpha}^0}_{!*}L_{\chi}[\rm dimX(\alpha)](\frac{\rm dimX(\alpha)}{2})^{\oplus K_{\underline{\lambda},\chi}}.\]

Again by [16][Lemma 3.3.4][19][Lemma 6.1], it is easy to see that \[j_{\alpha,m}^*IC(X(\alpha),L_{\chi})\cong {j_{\alpha}^0}_{!*}L_{\chi}[\rm dimX(\alpha)](\frac{\rm dimX(\alpha)}{2}).\] ◻

Proof. By induction, it suffices to prove that \[\begin{align} &\Bigl( {j_{s_1}}_!j_{s_1}^*\overline{{\mathbb{Q}}_l}|_{\mathbf{E}_{\mathbf{V}_{s_1}}}\diamond {j_{\alpha-\alpha_{s_1}^PP_{s_1},m}}_!j_{\alpha-\alpha_{s_1}^PP_{s_1},m}^*\Bigl(\overline{{\mathbb{Q}}_l}|_{\mathbf{E}_{\mathbf{V}_{s_2}}}\diamond \overline{{\mathbb{Q}}_l}|_{\mathbf{E}_{\mathbf{V}_{s_r}}} \diamond IC(X(0,m),L_{\chi}) \diamond \overline{{\mathbb{Q}}_l}|_{\mathbf{E}_{\mathbf{V}'_{s'_1}}}\diamond \cdots \diamond \overline{{\mathbb{Q}}_l}|_{\mathbf{E}_{\mathbf{V}'_{s'_t}}}\Bigr) \Bigr)\\ ={}&{j_{\alpha,m}}_!j_{\alpha,m}^*\Bigl( \overline{{\mathbb{Q}}_l}|_{{\mathcal{O}}_{P^{\oplus \alpha_{s_1}^P}_{s_1}}}\diamond \cdots \diamond \overline{{\mathbb{Q}}_l}|_{{\mathcal{O}}_{P^{\oplus \alpha_{s_r}^P}_{s_r}}} \diamond IC(X(0,m),L_{\chi}) \diamond \overline{{\mathbb{Q}}_l}|_{{\mathcal{O}}_{I^{\oplus \alpha_{{s'}_1}^I}_{{s'}_1}}}\diamond \cdots \diamond \overline{{\mathbb{Q}}_l}|_{{\mathcal{O}}_{I^{\oplus \alpha_{{s'}_t}^I}_{{s'}_t}}} \Bigr). \end{align}\] Since \[\left\{(x,W)|x\in X(\alpha,m),x(W)\subset W,\rm dimW=(m,m)+\sum_{s\ge 2}\alpha_s^P(s-1,s)+\sum_{s\ge 1}\alpha_s^I(s,s-1),\right\}\] is the same as \[\left\{(x,W)|x\in X(\alpha,m),x|_{W}\in X(\alpha-\alpha^P_{s_1}P_{s_1},m),x|_{\mathbf{V}/W}\cong \alpha^P_{s_1}P_{s_1}\right\}.\]

By \[\begin{tikzcd} \mathbf{E}_{\mathbf{V}_{s_1}}\times \mathbf{E}_{\mathbf{V}'} & \mathbf{E}'_{\mathbf{V}} \arrow[r, "p_2"] \arrow[l, "p_1"'] & \mathbf{E}_{\mathbf{V}}'' \arrow[r, "p_3"] & \mathbf{E}_{\mathbf{V}} \\ {X(\alpha^P_{s_1}P_{s_1})\times X(\alpha-\alpha^P_{s_1}P_{s_1},m)} \arrow[u, "{j_{s_1}\times j_{\alpha-\alpha^P_{s_1}P_{s_1},m}}"] & {p_2^{-1}p_3^{-1}(X(\alpha,m))} \arrow[r] \arrow[l] \arrow[u, "j'"] & {p_3^{-1}(X(\alpha,m))} \arrow[r] \arrow[u, "j''"] & {X(\alpha,m)} \arrow[u, "{j_{\alpha,m}}"] \end{tikzcd}\] Since all squares in this diagram are Cartesian, the claim follows. The cases in which the first factor is \(IC(X(0,m),L_{\chi})\) or preinjective part, the proof is the same. ◻

Proof. Considering \(\mathcal{L}_{\alpha,\underline{\lambda}}\) in equation 3 , choosing \(\underline{\lambda}=(m),\) we know \(\mathbf{H}^i(j_{\alpha'}^*\mathcal{L}_{\alpha,\underline{\lambda}})\) is generated by \(L_{\chi'}\) and \(\operatorname{supp }\mathcal{L}_{\alpha,\underline{\lambda}}=\overline{X(\alpha,m)}.\) Thus if \(j_{\alpha'}^*\mathcal{L}_{\alpha,\underline{\lambda}}\not=0,\) \(X(\alpha')\subset \overline{X(\alpha,m)}.\) Moreover, \(X(\alpha',m')\subset \overline{X(\alpha')}.\) Then \(j_{\alpha'}^*\mathcal{L}_{\alpha,\underline{\lambda}}\not=0\) gives \(X(\alpha',m')\subset \overline{X(\alpha,m)}.\) If \(j_{\alpha'}^*\mathcal{L}_{\alpha,\underline{\lambda}}=0,\) by Theorem 5, it follows that \(j_{\alpha',m'}^*\mathcal{L}_{\underline{\lambda}}=0.\) For any geometric point \(x\in \overline{X(\alpha,m)}\cap X(\alpha',m'),\) we obtain that \(i_x^*\mathcal{L}_{\alpha,\underline{\lambda}}=0,\) contradicting \(\chi_n(\mathcal{L}_{\alpha,\underline{\lambda}})(x)\not=0,\) where \(\chi_n\) is the trace map of \(\operatorname{Fr_{q^n}}\) for \(n\gg0.\) Thus \[\overline{X(\alpha)}=\cup_{(\alpha',m'),j_{\alpha'}^*\mathcal{L}_{\alpha,\underline{\lambda}}\not=0} X(\alpha',m').\] ◻

Proof. We study the flag sheaf complex \(\mathcal{L}\) in Proposition 4, with the Weil structure induced by \(\mathcal{L}_0.\)

By Lemma 4 and equation 5 , after applying the trace map there exists a collection of polynomials \(h_{\alpha',\underline{\lambda'}}(v)\) such that \[\chi(j_{\alpha',m'}^*\mathcal{L}) = j_{\alpha',m'}^*\chi(\mathcal{L}) = j_{\alpha',m'}^*\!\left(\sum_{\alpha'',\underline{\lambda''}} h_{\alpha'',\underline{\lambda''}}(q)\, f^r_{\alpha'',\underline{\lambda''}}\right) = \sum_{\alpha'',\underline{\lambda''}} h_{\alpha'',\underline{\lambda''}}(q)\, j_{\alpha',m'}^* f^r_{\alpha'',\underline{\lambda''}}.\]

If \(\alpha''\not=\alpha',\) \(j_{\alpha',m'}^*f^r_{\alpha'',\underline{\lambda''}}=0.\) Since the elements \(f^r_{\alpha'',\underline{\lambda''}}\) form a basis, fix \((\alpha,m),\) \(\{f^r_{\alpha,\underline{\lambda}}|\underline{\lambda}=(\lambda_1,\cdots,\lambda_k),k\in {\mathbb{N}}_{\geqslant 1}, \sum_{i=1}^k \lambda_i=m, \lambda_j\geqslant\lambda_{j+1},j=1,\cdots,k-1\}\) are \({\mathbb{C}}\)-linearly independent.

There also exist Kostka numbers \(K_{\underline{\lambda'},\chi'}\in {\mathbb{N}}\) such that, for \(\underline{\lambda'}=(\lambda'_1,\ldots,\lambda'_k)\) with \(\sum \lambda'_j=m,\) \(\lambda_i\geqslant\lambda_{i+1},\;i=1\cdots k-1\) and \(\chi\in S_m\), we have \[j_{\alpha',m'}^*f^r_{\alpha',\underline{\lambda'}} = \chi({p_{\alpha,\underline{\lambda}}}_!\overline{{\mathbb{Q}}_l}[\rm dimX(\alpha)](\frac{\rm dimX(\alpha)}{2})) = \sum_{\chi} K_{\underline{\lambda'},\chi'}\, \chi\!\left(j_{\alpha',m'}^*(IC(X(\alpha'),L_{\chi'}),f_{\alpha',\chi'})\right),\] which follows from Remark 3. Since \(j_{\alpha',m'}^*f^r_{\alpha',\underline{\lambda'}}\) are \({\mathbb{C}}\)-linearly independent, by comparing dimensions, \(\chi\!\left(j_{\alpha',m'}^*(IC(X(\alpha'),L_{\chi'}),f_{\alpha',\chi'})\right)\) are \({\mathbb{C}}\)-linearly independent.

Thus \(\chi(j_{\alpha',m'}^*\mathcal{L})\in {\mathbb{Z}}[\sqrt{q},\sqrt{q}^{-1}](\chi(j_{\alpha',m'}^*(IC(X(\alpha'),L_{\chi'}),f_{\alpha',\chi'})),\chi'\in X(S_{m'})),\) and is independent of the choice of \(q\). Choosing \({\mathbb{F}}_{q^n}\)-points for \(n>0\) yields the desired conclusion. ◻

Proof. For a closed embedding \(Z\xrightarrow{i} X\) with complement \(U\xrightarrow{j} X,\) there is \[j_!j^*\mathcal{F}\rightarrow\mathcal{F}\rightarrow i_!i^*\mathcal{F}\xrightarrow{+1}.\] If there is a closed embedding \(Z'\xrightarrow{i'} Z\) with complement \(U'\xrightarrow{j'} Z,\) \[i_!{j'}_!{j'}^*i^*\mathcal{F}\rightarrow i_!i^*\mathcal{F}\rightarrow i_!{i'}_!{i'}^*i^{*}\mathcal{F}\xrightarrow{+1}.\] Therefore, for any stratification \(\sqcup U_i=X,\) if we denote the embedding \(U_i\xrightarrow{j_i} X,\) then in the modified Grothendieck group, we have \[[(\mathcal{F},f)]=\sum [(j_i)_!j_i^*(\mathcal{F},f)].\] Since \(\{X(\alpha',m')\mid \alpha'\in R,\;m'\in {\mathbb{N}}\}\) is a stratification of \(\mathbf{E}_{\mathbf{V}}\) and \(\overline{X(\alpha,m)}\subset \bigsqcup_{(\alpha',m')\leqslant(\alpha,m)}X(\alpha',m')\), we have in \({\mathcal{K}}^r\) that \[[IC(X(\alpha),L_{\chi}),f_{\alpha,\chi}] = \sum_{(\alpha',m')\leqslant(\alpha,m)} \bigl[{j_{\alpha',m'}}_!\, j_{\alpha',m'}^*\bigl(IC(X(\alpha),L_{\chi}),f_{\alpha,\chi}\bigr)\bigr].\]

By Proposition 6 and 8, we also have \[\bigl[j_{\alpha',m'}^*\bigl(IC(X(\alpha),L_{\chi}),f_{\alpha,\chi}\bigr)\bigr] = \sum_{\chi'\in X(S_{m'})} p_{(\alpha,\chi),(\alpha',\chi')}\!\left(-\sqrt{q}\right)\, \bigl[j_{\alpha',m'}^*\bigl(IC(X(\alpha'),L_{\chi'}),f_{\alpha',\chi'}\bigr)\bigr].\] Combining these equalities gives the desired formula. ◻

Proof. As in [1], we consider three bases. We denote the PBW basis \(\chi({j_{\alpha,m}}_!\, j_{\alpha,m}^*\bigl(IC(X(\alpha),L_{\chi}),f_{\alpha,\chi}\bigr))=Q_{\alpha,\chi},\) canonical basis \(\chi(IC(X(\alpha),L_{\chi}),f_{\alpha,\chi})=B_{\alpha,\chi}\) and polynomial basis \(P_{\alpha,\chi}=\sum_{\underline{\lambda}}c_{\chi,\underline{\lambda}}r_{\alpha,\underline{\lambda}}^r,\) where \((c_{\chi,\underline{\lambda}})_{\chi,\underline{\lambda}}\) is inverse matrix of \((K_{\underline{\lambda}},\chi)_{\underline{\lambda},\chi},\) thus fixed under bar involution.

By Proposition 4.1 in [5], we can find this kind of polynomial basis with \(c_{\chi,\underline{\lambda}}\in {\mathbb{Q}}.\)

By Theorem 10, we have the following equations, \[\begin{align} B_{\alpha,\chi}&=Q_{\alpha,\chi}+\sum_{(\alpha',m')< (\alpha,m),\chi'\in X(S_{m'})}p_{(\alpha,\chi),(\alpha',\chi')}(-\sqrt{q})Q_{\alpha',\chi'},\\ P_{\alpha,\chi}&=Q_{\alpha,\chi}+\sum_{(\alpha',m')< (\alpha,m),\chi'\in X(S_{m'})}u_{(\alpha,\chi),(\alpha',\chi')}(-\sqrt{q})Q_{\alpha',\chi'},\\ \overline{Q_{\alpha,\chi}}&=Q_{\alpha,\chi}+\sum_{(\alpha',m')< (\alpha,m),\chi'\in X(S_{m'})}v_{(\alpha,\chi),(\alpha',\chi')}(-\sqrt{q})Q_{\alpha',\chi'}, \end{align}\] where \(p_{(\alpha,\chi),(\alpha',\chi')}(v)\in {\mathbb{Z}}[v,v^{-1}],\) \(u_{(\alpha,\chi),(\alpha',\chi')}\in {\mathbb{Q}}[v,v^{-1}]\) and \(v_{(\alpha,\chi),(\alpha',\chi')}\in {\mathbb{Q}}[v,v^{-1}].\) If \(\alpha=\alpha',\) \(p_{(\alpha,\chi),(\alpha',\chi')}(v)=u_{(\alpha,\chi),(\alpha',\chi')}(v)=v_{(\alpha,\chi),(\alpha',\chi')}(v)=\delta_{\chi,\chi'}.\) The third equation follows from the second one and that polynomial basis is fixed under the bar involution.

In the Hall algebra, we calculate that \(u'_{(\alpha,\chi),(\alpha',\chi')}(v):=v^{\rm dim(X(\alpha))-\rm dim(X(\alpha'))}u_{(\alpha,\chi),(\alpha',\chi')}(v)\in {\mathbb{Q}}[v^2,v^{-2}].\)

Now applying bar involution on \(B_{\alpha,\chi}\) and \(P_{\alpha,\chi},\) we have the following equation, \[\begin{align} p_{(\alpha,\chi),(\alpha',\chi')}(v)&=\sum_{(\alpha',m')\leqslant(\alpha'',m'')\leqslant(\alpha,m),\chi''\in X(S_{m''})}\overline{p_{(\alpha,\chi),(\alpha'',\chi'')}(v)}v_{(\alpha'',\chi''),(\alpha',\chi')(v)},\\ u_{(\alpha,\chi),(\alpha',\chi')}(v)&=\sum_{(\alpha',m')\leqslant(\alpha'',m'')\leqslant(\alpha,m),\chi''\in X(S_{m''})}\overline{u_{(\alpha,\chi),(\alpha'',\chi'')}(v)}v_{(\alpha'',\chi''),(\alpha',\chi')(v)}. \end{align}\] If we denote \(v'_{(\alpha'',\chi''),(\alpha',\chi')}(v)=v^{\rm dim(X(\alpha'))-\rm dim(X(\alpha''))}v_{(\alpha'',\chi''),(\alpha',\chi')}(v)\) Then \[u'_{(\alpha,\chi),(\alpha',\chi')}(v)=v^{2\rm dim(X(\alpha))-2\rm dim(X(\alpha'))}\sum_{(\alpha',m')\leqslant(\alpha'',m'')\leqslant(\alpha,m),\chi''\in X(S_{m''})}\overline{u'_{(\alpha,\chi),(\alpha'',\chi'')}(v)}v'_{(\alpha'',\chi''),(\alpha',\chi')}(v).\] Thus \(v'_{(\alpha'',\chi''),(\alpha',\chi')}(v)\in {\mathbb{Q}}[v^2,v^{-2}].\)

If we denote \(p'_{(\alpha'',\chi''),(\alpha',\chi')}(v)=v^{\rm dim(X(\alpha''))-\rm dim(X(\alpha'))}p_{(\alpha'',\chi''),(\alpha',\chi')}(v),\) then \[p'_{(\alpha,\chi),(\alpha',\chi')}(v)=v^{2\rm dim(X(\alpha))-2\rm dim(X(\alpha'))}\sum_{(\alpha',m')\leqslant(\alpha'',m'')\leqslant(\alpha,m),\chi''\in X(S_{m''})}\overline{p'_{(\alpha,\chi),(\alpha'',\chi'')}(v)}v'_{(\alpha'',\chi''),(\alpha',\chi')}(v).\] Since we already know that \(p'_{(\alpha,\chi),(\alpha',\chi')}(v)\in {\mathbb{Q}}[v^2,v^{-2}]\) when \(\alpha=\alpha',\) by descending induction, we have \(p'_{(\alpha,\chi),(\alpha',\chi')}(v)-v^{2\rm dim(X(\alpha))-2\rm dim(X(\alpha'))}\overline{p'_{(\alpha,\chi),(\alpha',\chi')}(v)}\in {\mathbb{Q}}[v^2,v^{-2}].\)

Since \({j_{\alpha'}}^*(IC(X(\alpha),L_{\chi}))\in {}^p{\mathcal{D}}^{\leqslant-1}(X(\alpha'))\) by [[16]][19] \(p'_{(\alpha,\chi),(\alpha',\chi')}(v)\in v^{\rm dim(X(\alpha))-\rm dim(X(\alpha'))-1}{\mathbb{Q}}[v^{-1}].\) We obtain \[v^{2\rm dim(X(\alpha))-2\rm dim(X(\alpha'))}\overline{p'_{(\alpha,\chi),(\alpha',\chi')}(v)}\in v^{\rm dim(X(\alpha))-\rm dim(X(\alpha'))+1}{\mathbb{Q}}[v].\] Then we prove that \[p'_{(\alpha,\chi),(\alpha',\chi')}(v)\in {\mathbb{Q}}[v^2,v^{-2}].\] ◻

Proof. This corollary follows from the definition of \(p_{(\alpha,\chi),(\alpha',\chi')}(v)\) and Proposition 12. ◻

Proof. We have \[K=\left\{(x,f)\;\middle|\; \begin{align} &x\in \mathbf{E}_{\mathbf{V}},\;f=(0=V_0\subset V_1\subset V_2=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_j)\subset V_j,\;M_{x|_{V_2/V_1}}\cong S_1,\;M_{x|_{V_1}}\text{ is a direct sum of indecomposable regular modules} \end{align}\right\},\] which is isomorphic to the complement in \[\left\{(x,f)\;\middle|\; \begin{align} &x\in \mathbf{E}_{\mathbf{V}},\;f=(0=V_0\subset V_1\subset V_2=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_j)\subset V_j,\;M_{x|_{V_2/V_1}}\cong S_1 \end{align}\right\}\] of the closed subvariety \[\left\{(x,f)\;\middle|\; \begin{align} &x\in \mathbf{E}_{\mathbf{V}},\;f=(0=V_0\subset V_1\subset V_2=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_j)\subset V_j,\;M_{x|_{V_2/V_1}}\cong S_1,\;M_{x|_{V_1}}\text{ is not a direct sum of indecomposable regular modules} \end{align}\right\}.\] For any short exact sequence of representations of \(Q\), \[0\rightarrow H_r\rightarrow M\rightarrow S_1\rightarrow 0,\] where \(H_r\) is a direct sum of indecomposable regular modules, we have that \(M\) has no preprojective direct summands.

Thus, \(K\) is isomorphic to the complement in \[\left\{x\;\middle|\;x\in \mathbf{E}_{\mathbf{V}},\;M_x\text{ has no preprojective direct summands}\right\}\times \mathbb{P}_{n-1}\] of the closed subvariety \[K'=\left\{(x,f)\;\middle|\; \begin{align} &x\in \mathbf{E}_{\mathbf{V}},\;M_x\text{ has no preprojective direct summands},\\ &f=(0=V_0\subset V_1\subset V_2=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_j)\subset V_j,\;M_{x|_{V_2/V_1}}\cong S_1,\\ &M_{x|_{V_1}}\cong P_{s}\oplus H_r\oplus I_{s'},\;H_r\text{ is a direct sum of indecomposable regular modules} \end{align}\right\}.\]

Moreover, with the topological structure given by Lemma 2, \(K'\) is isomorphic to \[\bigsqcup_{s'}\left\{(x,f)\;\middle|\; \begin{align} &x\in \mathbf{E}_{\mathbf{V}},\;M_x\text{ has no preprojective direct summands},\\&f=(0=V_0\subset V_1\subset V_2\subset V_3=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_j)\subset V_j,\;M_{x|_{V_3/V_2}}\cong S_1,\\ & M_{x|_{V_1}}\cong I_{s'} \end{align}\right\},\] which is, in turn, isomorphic to \[\bigsqcup_{s'}\left\{(x,f)\;\middle|\; \begin{align} &x\in \mathbf{E}_{\mathbf{V}},\;M_x\text{ has no preprojective direct summands},\\&f=(0=V_0\subset V_1\subset V_2=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_j)\subset V_j,\;M_{x|_{V_1}}\cong I_{s'} \end{align}\right\}\times {\mathbb{P}}_{n-s'-1}.\] Computing the dimension of \(V_2/V_1\), the variety above is isomorphic to \[\bigsqcup_{s'}\left\{(x,f)\;\middle|\; \begin{align} &x\in \mathbf{E}_{\mathbf{V}},\;M_x\text{ has no preprojective direct summands},\\&f=(0=V_0\subset V_1\subset V_2=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_j)\subset V_j,\;M_{x|_{V_1}}\cong I_{s'},\;M_{x|_{V_2/V_1}}\text{ is a direct sum of indecomposable regular modules} \end{align}\right\}\times {\mathbb{P}}_{n-s'-1}.\]

Therefore, \(K\) is isomorphic to the complement in \[\bigsqcup_{s'}\left\{(x,f)\;\middle|\; \begin{align} &x\in \mathbf{E}_{\mathbf{V}},\;f=(0=V_0\subset V_1\subset V_2=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &M_{x|_{V_1}}\cong I_{s'},\;M_{x|_{V_2/V_1}}\text{ is a direct sum of indecomposable regular modules} \end{align}\right\}\times \mathbb{P}_{n-1}\] of the closed subvariety \[\bigsqcup_{s'}\left\{(x,f)\;\middle|\; \begin{align} &x\in \mathbf{E}_{\mathbf{V}},\;f=(0=V_0\subset V_1\subset V_2=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_j)\subset V_j,\;M_{x|_{V_1}}\cong I_{s'},\;M_{x|_{V_2/V_1}}\text{ is a direct sum of indecomposable regular modules} \end{align}\right\}\times {\mathbb{P}}_{n-s'-1}.\]

Finally, \[\left\{(x,f)\;\middle|\; \begin{align} &x\in \mathbf{E}_{\mathbf{V}},\;f=(0=V_0\subset V_1\subset V_2=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &M_{x|_{V_1}}\cong I_{s'},\;M_{x|_{V_2/V_1}}\text{ is a direct sum of indecomposable regular modules} \end{align}\right\}\] can be identified with \[\left\{x\;\middle|\; \begin{align} &x\in \mathbf{E}_{\mathbf{V}},\\ &M_x\cong I_{s'}\oplus H_r,\;\text{where } H_r\text{ is a direct sum of indecomposable regular modules} \end{align}\right\}.\] ◻

Proof. The variety \(K_s\) is isomorphic to the complement in \[\left\{(x,f)\;\middle|\; \begin{align} &x\in \mathbf{E}_{\mathbf{V}},\;f=(0=V_0\subset V_1\subset V_2=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_j)\subset V_j,\;M_{x|_{V_2/V_1}}\cong I_{s} \end{align}\right\}\] of the closed subvariety \[\left\{(x,f)\;\middle|\; \begin{align} &x\in \mathbf{E}_{\mathbf{V}},\;f=(0=V_0\subset V_1\subset V_2=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_j)\subset V_j,\;M_{x|_{V_2/V_1}}\cong I_{s},\;M_{x|_{V_1}}\text{ is not a direct sum of indecomposable regular modules} \end{align}\right\}.\] By Lemma 1, any extension of \(I_{s}\) by a regular module must be of the form a direct sum of an indecomposable preinjective representation \(I_{t}\), where \(t\ge s\), and a regular representation.

Therefore, \(K_s\) is isomorphic to the complement in \[\tilde{V}=\bigsqcup_{t\ge s}\left\{(x,f)\;\middle|\; \begin{align} &M_x\cong I_{t}\oplus H_r,\;f=(0=V_0\subset V_1\subset V_2=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_j)\subset V_j,\;M_{x|_{V_2/V_1}}\cong I_{s},\;H_r\text{ is a direct sum of indecomposable regular modules} \end{align}\right\}\] of the closed subvariety \[V=\bigsqcup_{t\ge s}\left\{(x,f)\;\middle|\; \begin{align} &M_x\cong I_{t}\oplus H_r,\;f=(0=V_0\subset V_1\subset V_2=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_j)\subset V_j,\;M_{x|_{V_2/V_1}}\cong I_{s},\\ &M_{x|_{V_1}}\cong P_{u}\oplus H_r\oplus I_{t},\;1\le u\le n-s-t+1,\\&H_r\text{ is a direct sum of indecomposable regular modules} \end{align}\right\}.\] We know by Lemma 2, fixed \(t'\ge s,\) \[\bigsqcup_{t'\ge t\ge s}\left\{(x,f)\;\middle|\; \begin{align} &M_x\cong I_{t}\oplus H_r,\;f=(0=V_0\subset V_1\subset V_2=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_j)\subset V_j,\;M_{x|_{V_2/V_1}}\cong I_{s},\\ &M_{x|_{V_1}}\cong P_{u}\oplus H_r\oplus I_{t},\;1\le u\le n-s-t+1,\;H_r\text{ is a direct sum of indecomposable regular modules} \end{align}\right\}\] is a closed subvariety in \(V.\) And \[\bigsqcup_{t'\ge t\ge s}\left\{(x,f)\;\middle|\; \begin{align} &M_x\cong I_{t}\oplus H_r,\;f=(0=V_0\subset V_1\subset V_2=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_j)\subset V_j,\;M_{x|_{V_2/V_1}}\cong I_{s},\;H_r\text{ is a direct sum of indecomposable regular modules} \end{align}\right\}\] is a closed subvariety in \(\tilde{V}.\) Moreover, \[\left\{(x,f)\;\middle|\; \begin{align} &M_x\cong I_{t}\oplus H_r,\;f=(0=V_0\subset V_1\subset V_2=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_j)\subset V_j,\;M_{x|_{V_2/V_1}}\cong I_{s},\\ &M_{x|_{V_1}}\cong P_{u}\oplus H_r\oplus I_{t},\;1\le u\le n-s-t+1,\;H_r\text{ is a direct sum of indecomposable regular modules} \end{align}\right\}\] is isomorphic to \[\left\{(x,f)\;\middle|\; \begin{align} &M_x\cong I_{t}\oplus H_r,\;f=(0=V_0\subset V_1\subset V_2\subset V_3=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_j)\subset V_j,\;M_{x|_{V_3/V_2}}\cong I_{s},\;M_{x|_{V_1}}\cong I_{t},\\ & M_{x|_{V_2/V_1}}\cong P_{u}\oplus H_r,\;1\le u\le n-s-t+1,\;H_r\text{ is a direct sum of indecomposable regular modules} \end{align}\right\},\] which is also isomorphic to \[\left\{(x,f)\;\middle|\; \begin{align} &M_x\cong I_{t}\oplus H_r,\;f=(0=V_0\subset V_1\subset V_2\subset V_3=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_j)\subset V_j,\;M_{x|_{V_3/V_2}}\cong I_{s},\\ &M_{x|_{V_1}}\cong I_{t},\;M_{x|_{V_3/V_1}}\cong H_r,\;H_r\text{ is a direct sum of indecomposable regular modules} \end{align}\right\}.\]

Fix an \(I\)-graded subspace \(\mathbf{V}'\subset \mathbf{V}\) with dimension \((t-1,t)\), and let \(P\subset \mathbf{G}_{\mathbf{V}}\) be its stabilizer. Then the variety above is isomorphic to \[\mathbf{G}_V\times_{P}\left\{(x,f)\;\middle|\; \begin{align} &x\in \mathbf{E}_{\mathbf{V}},\;f=(0=V_0\subset V_1=\mathbf{V}'\subset V_2\subset V_3=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_j)\subset V_j,\;M_{x|_{V_3/V_2}}\cong I_{s},\;M_{x|_{V_1}}\cong I_{t},\;M_{x|_{V_3/V_1}}\cong H_r,\\ &H_r\text{ is a direct sum of indecomposable regular modules} \end{align}\right\}.\] Now fix a representation \(I=(x_I,\mathbf{V}_I)\cong I_{s}\).

Finally, since \[\left\{(x,f)\;\middle|\; \begin{align} &M_x\cong I_{t}\oplus H_r,\;f=(0=V_0\subset V_1\subset V_2=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_j)\subset V_j,\;M_{x|_{V_2/V_1}}\cong I_{s},\;H_r\text{ is a direct sum of indecomposable regular modules} \end{align}\right\}\] is isomorphic to \[\left\{(x,g)\;\middle|\; \begin{align} &M_x\cong I_{t}\oplus H_r,\;g\in \rm Hom_{kQ}(M_x,I)\text{ is surjective}\\ &H_r\text{ is a direct sum of indecomposable regular modules} \end{align}\right\}/k^*,\] And for \[\mathbf{G}_V\times_{P}\left\{(x,f)\;\middle|\; \begin{align} &x\in \mathbf{E}_{\mathbf{V}},\;f=(0=V_0\subset V_1=\mathbf{V}'\subset V_2\subset V_3=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_j)\subset V_j,\;M_{x|_{V_3/V_2}}\cong I_{s},\;M_{x|_{V_1}}\cong I_{t},\;M_{x|_{V_3/V_1}}\cong H_r,\\ &H_r\text{ is a direct sum of indecomposable regular modules} \end{align}\right\},\] we consider \[\left\{(x,f)\;\middle|\; \begin{align} &x\in \mathbf{E}_{\mathbf{V}},\;f=(0=V_0\subset V_1=\mathbf{V}'\subset V_2\subset V_3=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_j)\subset V_j,\;M_{x|_{V_3/V_2}}\cong I_{s},\;M_{x|_{V_1}}\cong I_{t},\;M_{x|_{V_3/V_1}}\cong H_r,\\ &H_r\text{ is a direct sum of indecomposable regular modules} \end{align}\right\}\] is isomorphic to \[\left\{(x,g)\;\middle|\; \begin{align} &x\in \mathbf{E}_{\mathbf{V}},\;M_x\cong H_r\oplus I_t,M_{x|_{\mathbf{V}'}}\cong I_t,\;g\in \rm Hom_{kQ}(H_r,I)\text{ is surjective},\\ &H_r\text{ is a direct sum of indecomposable regular modules} \end{align}\right\}/k^*.\] Since the complement of \[\left\{(x,g)\;\middle|\; \begin{align} &x\in \mathbf{E}_{\mathbf{V}},\;M_x\cong H_r\oplus I_t,M_{x|_{\mathbf{V}'}}\cong I_t,\;g\in \rm Hom_{kQ}(H_r,I)\text{ is surjective}\\ &H_r\text{ is a direct sum of indecomposable regular modules} \end{align}\right\}/k^*.\] in \[\left\{(x,g)\;\middle|\; \begin{align} &x\in \mathbf{E}_{\mathbf{V}},\;M_x\cong I_{t}\oplus H_r,\;M_{x|_{\mathbf{V}'}}\cong I_t,\;g\in \rm Hom_{kQ}(M_x,I)\text{ is surjective}\\ &H_r\text{ is a direct sum of indecomposable regular modules} \end{align}\right\}/k^*,\] is the same as the complement of \[\left\{(x,g)\;\middle|\; \begin{align} &x\in \mathbf{E}_{\mathbf{V}},\;M_x\cong H_r\oplus I_t,M_{x|_{\mathbf{V}'}}\cong I_t,\;0\not=g\in \rm Hom_{kQ}(H_r,I),\\ &H_r\text{ is a direct sum of indecomposable regular modules} \end{align}\right\}/k^*\] in \[\left\{(x,g)\;\middle|\; \begin{align} &x\in \mathbf{E}_{\mathbf{V}},\;M_x\cong I_{t}\oplus H_r,M_{x|_{\mathbf{V}'}}\cong I_t,\;0\not=g\in \rm Hom_{kQ}(M_x,I),\\ &H_r\text{ is a direct sum of indecomposable regular modules} \end{align}\right\}/k^*.\] Since \[\left\{(x,g)\;\middle|\; \begin{align} &x\in \mathbf{E}_{\mathbf{V}},\;M_x\cong I_{t}\oplus H_r,M_{x|_{\mathbf{V}'}}\cong I_t, \;g\in \rm Hom_{kQ}(M_x,I),\\ &H_r\text{ is a direct sum of indecomposable regular modules} \end{align}\right\}\] is the kernel of a morphism of vector bundles on \(W_{I_{t},(n-t)}\) with constant rank for any \(x\in W_{I_{t},(n-t)}\), it follows that it admits a locally trivial fibration over \(W_{I_{t},(n-t)}\). Hence \[\left\{(x,g)\;\middle|\; \begin{align} &x\in \mathbf{E}_{\mathbf{V}},\;M_x\cong I_{t}\oplus H_r,\;0\not=g\in \rm Hom_{kQ}(M_x,I),\\ &H_r\text{ is a direct sum of indecomposable regular modules} \end{align}\right\}/k^*\] also admits a locally trivial fibration over \(W_{I_{t},(n-t)},\) whose fiber is a projective space. For \[\left\{(x,g)\;\middle|\; \begin{align} &x\in \mathbf{E}_{\mathbf{V}},\;M_x\cong H_r\oplus I_t,M_{x|_{\mathbf{V}'}}\cong I_t,\;0\not=g\in \rm Hom_{kQ}(H_r,I),\\ &H_r\text{ is a direct sum of indecomposable regular modules} \end{align}\right\}/k^*\] we have the same result, thus applying \(\mathbf{G}_{\mathbf{V}}\times_P\), we have that \[\mathbf{G}_V\times_{P}\left\{(x,f)\;\middle|\; \begin{align} &x\in \mathbf{E}_{\mathbf{V}},\;f=(0=V_0\subset V_1=\mathbf{V}'\subset V_2\subset V_3=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_j)\subset V_j,\;M_{x|_{V_3/V_2}}\cong I_{s},\;M_{x|_{V_1}}\cong I_{t},\;M_{x|_{V_3/V_1}}\cong H_r,\\ &H_r\text{ is a direct sum of indecomposable regular modules} \end{align}\right\}.\] admits a locally trivial fibration over \(W_{I_{t},(n-t)}.\) In fact, the fiber is a projective space. This completes the proof. ◻

Proof. If \(0\rightarrow P_{s}\rightarrow M\rightarrow I_{s'}\rightarrow 0\) is a short exact sequence of representations of \(Q\), then either \(M\cong P_{s}\oplus I_{s'}\), or \(M\) is a direct sum of regular indecomposable representations.

It is straightforward to see that \[\left\{(x,f)\;\middle|\; \begin{align} &M_x\cong P_{s}\oplus S_1,\;f=(0=V_0\subset V_1\subset V_2=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_i)\subset V_i,\;M_{x|_{V_1}}\cong P_{s},\;M_{x|_{V_2/V_1}}\cong S_1 \end{align}\right\}\] admits a locally trivial fibration over \(\{x\in \mathbf{E}_{\mathbf{V}}\mid M_x\cong P_{s}\oplus S_1\}\), which is a closed subvariety of \(L\). Thus it suffices to consider the case where \(M\) is a direct sum of regular indecomposable representations.

We now give a more explicit description. \[L=\left\{(x,f)\;\middle|\; \begin{align} &x\in \mathbf{E}_{\mathbf{V}},\;f=(0=V_0\subset V_1\subset V_2=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_i)\subset V_i,\;M_{x|_{V_1}}\cong P_{s},\;M_{x|_{V_2/V_1}}\cong S_1 \end{align}\right\},\] which is isomorphic to the complement in \[\left\{(x,f)\;\middle|\; \begin{align} &x\in \mathbf{E}_{\mathbf{V}},\;f=(0=V_0\subset V_1\subset V_2=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_i)\subset V_i,\;M_{x|_{V_2/V_1}}\cong S_1 \end{align}\right\}\] of the closed subvariety \[\left\{(x,f)\;\middle|\; \begin{align} &x\in \mathbf{E}_{\mathbf{V}},\;f=(0=V_0\subset V_1\subset V_2=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_i)\subset V_i,\;M_{x|_{V_1}}\not\cong P_{s},\;M_{x|_{V_2/V_1}}\cong S_1 \end{align}\right\}.\]

After restricting to \(x\in X(0,n)\), this is the complement of \[\left\{(x,f)\;\middle|\; \begin{align} &x\in X(0,n),\;f=(0=V_0\subset V_1\subset V_2=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_i)\subset V_i,\;M_{x|_{V_2/V_1}}\cong S_1 \end{align}\right\}\] of the closed subvariety \[\left\{(x,f)\;\middle|\; \begin{align} &x\in X(0,n),\;f=(0=V_0\subset V_1\subset V_2=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_i)\subset V_i,\;M_{x|_{V_1}}\not\cong P_{s},\;M_{x|_{V_2/V_1}}\cong S_1 \end{align}\right\}.\] The variety above is isomorphic to the complement in \(\{(x,f)\mid x\in X(0,n)\}\times {\mathbb{P}}_{n-1}\) of the closed subvariety \[L'=\left\{(x,f)\;\middle|\; \begin{align} &x\in X(0,n),\;f=(0=V_0\subset V_1\subset V_2=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\;x(V_i)\subset V_i,\\ &M_{x|_{V_1}}\cong P_{s''}\oplus H_{r_{n-s''}},\;s''<s,\;M_{x|_{V_2/V_1}}\cong S_1,\\ &H_{r_{n-s''}}\text{ is a direct sum of indecomposable regular modules} \end{align}\right\},\] which is isomorphic to \[\label{1} \bigsqcup_{s''<s}\left\{(x,f)\;\middle|\; \begin{align} &x\in X(0,n),\;f=(0=V_0\subset V_1\subset V_2\subset V_3=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\;x(V_i)\subset V_i,\\ &M_{x|_{V_1}}\cong H_{r_{n-s''}},\;M_{x|_{V_2/V_1}}\cong P_{s''},\;M_{x|_{V_3/V_2}}\cong S_1,\\ &H_{r_{n-s''}}\text{ is a direct sum of indecomposable regular modules} \end{align}\right\}.\tag{6}\] Fixed \(s'<s,\) \[\bigsqcup_{s''<s'}\left\{(x,f)\;\middle|\; \begin{align} &x\in X(0,n),\;f=(0=V_0\subset V_1\subset V_2\subset V_3=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\;x(V_i)\subset V_i,\\ &M_{x|_{V_1}}\cong H_{r_{n-s''}},\;M_{x|_{V_2/V_1}}\cong P_{s''},\;M_{x|_{V_3/V_2}}\cong S_1,\\ &H_{r_{n-s''}}\text{ is a direct sum of indecomposable regular modules} \end{align}\right\}\] is a closed subvariety of the variety above. Then \(L\) is isomorphic to \(\widetilde{L_{(n)}}\), which is an open subset of \(L_{(n)}\). ◻

Proof. If \(0\rightarrow P_{s}\rightarrow M\rightarrow I_{s'}\rightarrow 0\) is a short exact sequence of representations of \(Q\), then either \(M\cong P_{s}\oplus I_{s'}\), or \(M\) is a direct sum of regular indecomposable representations.

It is straightforward to see that \[\left\{(x,f)\;\middle|\; \begin{align} &M_x\cong P_{s}\oplus I_{s'},\;f=(0=V_0\subset V_1\subset V_2=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_i)\subset V_i,\;M_{x|_{V_1}}\cong P_{s},\;M_{x|_{V_2/V_1}}\cong I_{s'} \end{align}\right\}\] admits a locally trivial fibration over \(\{x\in \mathbf{E}_{\mathbf{V}}\mid M_x\cong P_{s}\oplus I_{s'}\}\), which is a closed subvariety of \(L\). Hence it suffices to consider the case where \(M\) is a direct sum of regular indecomposable representations.

In this case, \(U\) is isomorphic to the complement in \[\left\{(x,f)\;\middle|\; \begin{align} &x\in X(0,n),\;f=(0=V_0\subset V_1\subset V_2=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_i)\subset V_i,\;M_{x|_{V_2/V_1}}\cong I_{s'} \end{align}\right\}\] of the closed subvariety \[\left\{(x,f)\;\middle|\; \begin{align} &x\in X(0,n),\;f=(0=V_0\subset V_1\subset V_2=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_i)\subset V_i,\;M_{x|_{V_1}}\not\cong P_{s},\;M_{x|_{V_2/V_1}}\cong I_{s'} \end{align}\right\}.\] The closed subvariety above is isomorphic to \[\label{eqq} \bigsqcup_{s''<s}\left\{(x,f)\;\middle|\; \begin{align} &x\in X(0,n),\;f=(0=V_0\subset V_1\subset V_2\subset V_3=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\;x(V_i)\subset V_i,\\ &M_{x|_{V_1}}\cong H_r,\;M_{x|_{V_2/V_1}}\cong P_{s''},\;M_{x|_{V_3/V_2}}\cong I_{s'},\\ &H_r\text{ is a direct sum of indecomposable regular modules} \end{align}\right\}.\tag{7}\] Moreover, fixed \(s'<s,\) \[\bigsqcup_{s''<s'}\left\{(x,f)\;\middle|\; \begin{align} &x\in X(0,n),\;f=(0=V_0\subset V_1\subset V_2\subset V_3=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\;x(V_i)\subset V_i,\\ &M_{x|_{V_1}}\cong H_r,\;M_{x|_{V_2/V_1}}\cong P_{s''},\;M_{x|_{V_3/V_2}}\cong I_{s'},\\ &H_r\text{ is a direct sum of indecomposable regular modules} \end{align}\right\}\] is a closed subvariety of the variety above. For \(\underline{\lambda}=(\lambda_1,\ldots,\lambda_k)\), define the variety \({}^{pre}W'_{0,\underline{\lambda}}\) by \[\left\{(x,f,g)\;\middle|\; \begin{align} &x\in X(0,n),\;f=(0=V_0\subset V_1\subset \cdots \subset V_{k-1}\subset V_k=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_i)\subset V_i,\;M_{x|_{V_i/V_{i-1}}}\cong H_{r_i},\;0\not=g\in \rm Hom(M_{x|_{V_k/V_{k-1}}},I_{s'}),\\ &H_{r_i}\text{ is a direct sum of indecomposable regular modules, }\rm dim(H_{r_i})=(\lambda_i,\lambda_i),\;i=1,\ldots,k \end{align}\right\}.\] The variety \(W^{sur}_{0,\underline{\lambda}}\) is defined as \[\left\{(x,f,g)\;\middle|\; \begin{align} &x\in X(0,n),\;f=(0=V_0\subset V_1\subset \cdots \subset V_{k-1}\subset V_k=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_i)\subset V_i,\;M_{x|_{V_i/V_{i-1}}}\cong H_{r_i},\;0\not=g\in \rm Hom(M_{x|_{V_k/V_{k-1}}},I_{s'})\text{ is surjective},\\ &H_{r_i}\text{ is a direct sum of indecomposable regular modules, }\rm dim(M_{r_i})=(\lambda_i,\lambda_i),\;i=1,\ldots,k \end{align}\right\}.\] Inductively, set the open subset of \({}^{pre}W'_{0,\underline{\lambda}}\) \[\widetilde{{}^{pre}W'_{0,\underline{\lambda}}} = {}^{pre}W'_{0,\underline{\lambda}} - \bigsqcup_{\lambda_i=\lambda'_{i},\;i\le k-1,\;\lambda_k=\lambda'_{k+1}+\lambda'_{k}} \widetilde{{}^{pre}W'_{0,\underline{\lambda'}}},\] and the open subset of \(W^{sur}_{0,\underline{\lambda}}\) \[\widetilde{W^{sur}_{0,\underline{\lambda}}} = W^{sur}_{0,\underline{\lambda}} - \bigsqcup_{\lambda_i=\lambda'_{i},\;i\le k-1,\;\lambda_k=\lambda'_{k+1}+\lambda'_{k}} \widetilde{W^{sur}_{0,\underline{\lambda'}}},\] where the embeddings and the stratification come from the dimension of the kernel of map \(g\) and equation 7 with Lemma 2.

By induction, we obtain \[\begin{align} \widetilde{{}^{pre}W'_{0,\underline{\lambda}}}=& \left\{(x,f,g)\;\middle|\; \begin{aligned} &x\in \mathbf{E}_{\mathbf{V}},\;f=(0=V_0\subset V_1\subset \cdots \subset V_{k-1}\subset V_k=\mathbf{V})\text{ is a filtration of }{\mathbb{N}}I\text{-graded spaces},\\ &x(V_i)\subset V_i,\;x(V')\subset V',\;M_{x|_{V_i/V_{i-1}}}\cong H_{r_i},\;\\ &0\not=g\in \rm Hom(M_{x|_{V_k/V_{k-1}}},I_{s'})\text{ is injective not surjective},\\ &H_{r_i}\text{ is a direct sum of indecomposable regular modules of dimension }(\lambda_i,\lambda_i),\;i=1,\ldots,k \end{aligned}\right\}\\ \bigcup& \widetilde{W^{sur}_{0,\underline{\lambda}}}.\end{align}\] In fact, one of the two sets above is empty.

Indeed, when \(\underline{\lambda}=(n),\) there does not exist \(g\in \rm Hom(x,I_s)\) injective for \(x\in X(0,n),\) \(\widetilde{W^{sur}_{0,(n)}}=\widetilde{{}^{pre}W'_{0,(n)}}.\) And \(\widetilde{W^{sur}_{0,(n)}}/k^*=U.\)

By the same argument as in Lemma 6, \(W'_{0,\underline{\lambda}}:={}^{pre}W'_{0,\underline{\lambda}}/k^*\) admits a locally trivial fibration with projective fiber over \(W_{0,\underline{\lambda}}\). Moreover, \(W'_{0,\underline{\lambda}}\) is irreducible. ◻

Proof. First, \(\mathbf{H}^i(f_!\overline{{\mathbb{Q}}_l}|_{X})\) is a local system. Hence it suffices to prove that the structure group acts as \(\operatorname{Gal}(\overline{{\mathbb{F}}_q}/{\mathbb{F}}_q)\) on \(\mathbf{H}^j({\mathbb{P}}^n,\overline{{\mathbb{Q}}_l})\). Since we can choose an open covering \(\cup U=Y\) and \(f|_{U}\) as \(U\times {\mathbb{P}}^n\rightarrow U.\) We just need to consider the gluing on these \(U.\)

We know that \(\rm Aut({\mathbb{P}}^n)\cong \operatorname{PGL}_{n+1}\) by [24] and by [25] that \[\mathbf{H}^*({\mathbb{P}}^n,\overline{{\mathbb{Q}}_l})=\overline{{\mathbb{Q}}_l}[h]/(h^{n+1}).\] Thus it suffices to show that \(\operatorname{PGL}_{n+1}\) acts trivially on \(h\). Since \(h\) comes from the first Chern class of \({\mathcal{O}}_{{\mathbb{P}}^n}(1)\), it is enough to note that for any \(g\in \operatorname{PGL}_{n+1}\), the pullback preserves \({\mathcal{O}}_{{\mathbb{P}}^n}(1)\). This proves the claim. ◻

Proof. Fix \(x_0\) such that \(M_{x_0}\cong I_t\oplus I_t,\) and denote its automorphism group by \(H=\mathbf{G}l(2,k).\) Let \[A'_{n,m}=\left\{(x_0,f)\middle|\begin{align} & f=(0\subset V_1\subset V_2=\mathbf{V})\\ &x_0(V_i)\subset V_i, M_{x_0|_{V_1}}\cong I_m, M_{x_0|_{V_2/V_1}}\cong I_n. \end{align}\right\},\] Then \(A_{n,m}\cong A'_{n,m}\times_{H} \mathbf{G}_{\mathbf{V}}.\) Any \(H\)-stable stratification on \(A'_{n,m}\) induces \(A_{n,m}.\) Also fix \(\tilde{V}\subset \mathbf{V}\) such that \(M_{x_0|_{\tilde{V}}}\cong I_t.\)

For \(0\rightarrow I_m\xrightarrow{f} I_t\oplus I_t,\) we have \(\rm dim\operatorname{coker} f=\rm dimI_n,\) and \(\operatorname{coker} f\) is preinjective, thus \(\operatorname{coker} f\cong I_n.\) Thus for \(A'_{n,m},\) we only need to fix \(x_m\) such that \(M_{x_m}\cong I_m,\) then \(A'_{n,m}\cong \{f\in \rm Hom_{kQ}(M_{x_m},M_{x_0})/k^*|\ker f=0\},\) which is an open subset in \((\rm Hom_{kQ}(M_{x_m},M_{x_0})-{0})/k^*.\) Since \((\rm Hom_{kQ}(M_{x_m},M_{x_0})-{0})/k^*\cong (\rm Hom_{kQ}(I_m,I_t)\oplus \rm Hom_{kQ}(I_m,I_t)-{0})/k^*,\) if we denote \(C=\rm Hom_{kQ}(x_m,x_0),\) \(A'_{n,m}\) is an open subvariety in \(\{[a,b]\in C\oplus C-{0}/k^*|a,b\text{ are linear independent}\}\) with \(H=\mathbf{G}l(2,k)\) action. \(C\) is \(m-t+1\) dimensional \(k\)-space. Because \(C\oplus C\cong \rm Hom_k(k^2,C),\) in our case, we only consider the subvariety \(\rm Hom^0_k(k^2,C)=\{f\in \rm Hom_k(k^2,C)|rank(f)=2\}.\) \(\rm Hom^0_k(k^2,C)/k^*\) has a canonical map \(\pi\) to Grassmannian \(Gr(2,C).\) For the Grassmannian \(Gr(2,C),\) we give a natural stratification.

Actually, when we give \(C\) a \(k-\)basis \(e_1,\cdots,e_{m-t+1},\) the following locally closed subvariety, for \(0<a<b\leqslant m-t+1,\) \(U'_{a,b}=\{[x,y]\in Gr(2,C)|x=e_a+\sum_{j<a} \alpha_j e_j, y=e_b+\sum_{j'<b,\not=a}\beta_{j'}e_{j'}, \alpha_j,\beta_{j'}\in k\}\) which is a Schubert cell, gives the desired stratification. We use the stratification \(\tilde{U}_{a,b}=\pi^{-1}(U'_{a,b})\cap A'_{n,m}.\) It is clearly stable under \(H.\) For \(0<a<b\leqslant m-t+1,\) denote the subvariety \(\{[x,y]\in (C\oplus C-{0})/k^*|x=e_a+\sum_{j<a} \alpha_j e_j, y=e_b+\sum_{j'<b,\not=a}\beta_{j'}e_{j'}, \alpha_j,\beta_{j'}\in k\}\) as \(L_{a,b}\) and \(L'_{a,b}=A'_{n,m}\cap L_{a,b}.\) Then \(\tilde{U}_{a,b}\cong L'_{a,b}\times PGl(2,k).\)

The stratification \(\tilde{U}_{a,b}\) induces the stratification \(U_{a,b}\cong L'_{a,b}\times (G_{\mathbf{V}}/k^*)\) on \(A_{n,m}'.\) Then we define a morphism from \(U_{a,b}\) to \(A_{t,t}\) as follows. \[\begin{tikzcd} {L'_{a,b}\times (G_{\mathbf{V}}/k^*)} \arrow[r, "{p_{a,b}}"] & { A_{t,t}} \\ {((x_0,f),g)} \arrow[r, maps to] & {(g(x_0),g(\tilde{V}))} \end{tikzcd}\] It is straightforward to see that \(U_{a,b}\rightarrow \mathbf{E}_{\mathbf{V}}\) factors through \(A_{t,t}\) by \(U_{a,b}\xrightarrow{p_{a,b}}A_{t,t}\rightarrow \mathbf{E}_{\mathbf{V}}.\) This completes the proof. Indeed, we can choose \(x_m,\) \((x_0, \tilde{V}),\) and \(e_i\) above \(\operatorname{Fr^r}-\)fixed. ◻

Proof. We consider the sequence \(\hat{\underline{\nu}}=(\hat{\nu_1},\cdots,\hat{\nu_n}),\) where \(\sum \hat{\nu_i}=\nu,\) and for each \(\hat{\nu}_i=((\hat{\nu}_i)_1,(\hat{\nu}_i)_2),\) we assume \(|(\hat{\nu}_i)_1-(\hat{\nu}_i)_2)|\leqslant 1.\) We denote the set consisting of such sequences \(\mathcal{S}q_{\nu}.\) Then there is a fixed \(\mathbf{E}_{\hat{\nu}_i}^0\subset \mathbf{E}_{\mathbf{V}_i},\rm dim\mathbf{V}_i=\hat{\nu}_i\), where \(\mathbf{E}_{\hat{\nu}_i}^0\) is the orbit of indecomposable preprojective representation, indecomposable preinjective representation, or the subvariety consisting of regular representations. Consider \[S_{\underline{\hat{\nu}}}=\left\{(x\in \mathbf{E}_{\mathbf{V}},f)\middle|\begin{align} &f=(0=V_0\subset\cdots\subset V_{k-1}\subset V_k\subset V_{k+1}\subset\cdots\subset V_{n}=\mathbf{V}, \rm dim(V_i/V_{i-1})=\hat{\nu}_i,\\ &x|_{V_i/V_{i-1}}\in \mathbf{E}_{\hat{\nu}_i}^0. \end{align}\right\}.\] Claim: To study the simple perverse constituents, assume we start with \(\overline{{\mathbb{Q}}_l}|_{S_{\underline{\hat{\nu}}}}\boxtimes\overline{{\mathbb{Q}}_l}(\zeta)\) on some \(S_{\underline{\hat{\nu}}}\), where \(\overline{{\mathbb{Q}}_l}(\zeta)\) is pure on \(\operatorname{Spec}({\mathbb{F}}_q)\) and \(\operatorname{egf}(\overline{{\mathbb{Q}}_l}(\zeta))=\overline{{\mathbb{Q}}_l}.\) All the simple perverse constituents come from \({p_{\alpha',\underline{\lambda'}}}_!\overline{{\mathbb{Q}}_l}|_{W_{\alpha',\underline{\lambda'}}}\boxtimes\overline{{\mathbb{Q}}_l}(\zeta),\) for \(\overline{{\mathbb{Q}}_l}(\zeta)\) above. \(-\boxtimes \overline{{\mathbb{Q}}_l}(\zeta)\) is exact and perverse \(t\)-exact with \(f_!(\mathcal{F})\boxtimes \overline{{\mathbb{Q}}_l}(\zeta)\cong (f_!\mathcal{F})\boxtimes \overline{{\mathbb{Q}}_l}(\zeta)\) and \(g^*(\mathcal{F}\boxtimes \overline{{\mathbb{Q}}_l}(\zeta))\cong (g^*\mathcal{F})\boxtimes \overline{{\mathbb{Q}}_l}(\zeta).\) Thus we may start with the constant sheaf \(\overline{{\mathbb{Q}}_l}|_{S_{\underline{\hat{\nu}}}}.\)

If there exists \(\hat{\nu}_k=(s'-1,s')\) and \(\hat{\nu}_{k+1}=(s,s-1)\) in \(\underline{\hat{\nu}},\) we fix \(\mathbf{V}_{k-1}\subset \mathbf{V}_{k+1}\) with dimensions \(\sum_{j=1}^{k-1}\hat{\nu}_j,\sum_{j=1}^{k+1}\hat{\nu}_j\), and the stabilizer \(P\subset \mathbf{G}_{\mathbf{V}}\) of \(\mathbf{V}_{k+1}\) and \(\mathbf{V}_{k-1}.\) Also denote the unipotent radical of \(P\) as \(U.\)

For \[\left\{(x\in \mathbf{E}_{\mathbf{V}},f)\middle|\begin{align} &f=(0=V_0\subset \cdots\subset V_{k-1}\subset V_k\subset V_{k+1}\subset\cdots \subset V_{n}=\mathbf{V}), x(V_i)\subset V_i,\\ &\rm dim(V_i/V_{i-1})=\hat{\nu}_i,x|_{V_i/V_{i-1}}\in \mathbf{E}_{\hat{\nu}_i}^0,i\not=k,k+1,\\ &M_{x|_{V_{k}/V_{k-1}}}\cong P_{s'}, M_{x|_{V_{k+1}/V_k}}\cong I_{s}. \end{align}\right\}\] is isomorphic to \[\label{con} \mathbf{G}_{\mathbf{V}}\times_{P}\left\{(x\in \mathbf{E}_{\mathbf{V}},f)\middle|\begin{align} &f=(0=V_0\subset\cdots \subset\mathbf{V}_{k-1}=V_{k-1}\subset V_k\subset \mathbf{V}_{k+1}=V_{k+1}\subset\cdots\subset V_{n}=\mathbf{V}),x(V_i)\subset V_i,,\\ &\rm dim(V_i/V_{i-1})=\hat{\nu}_i,x|_{V_i/V_{i-1}}\in \mathbf{E}_{\hat{\nu}_i}^0,i\not=k,k+1,\\ &M_{x|_{V_{k}/\mathbf{V}_{k-1}}}\cong P_{s'}, M_{x|_{\mathbf{V}_{k+1}/V_k}}\cong I_{s},x(\mathbf{V}_{k+1})\subset \mathbf{V}_{k+1},x(\mathbf{V}_{k-1})\subset \mathbf{V}_{k-1}. \end{align}\right\}.\tag{8}\] We denote \[F=\left\{(x\in \mathbf{E}_{\mathbf{V}},f)\middle|\begin{align} &f=(0=V_0\subset\cdots \subset\mathbf{V}_{k-1}=V_{k-1}\subset V_k\subset \mathbf{V}_{k+1}=V_{k+1}\subset\cdots\subset V_{n}=\mathbf{V}),x(V_i)\subset V_i,\\ &\rm dim(V_i/V_{i-1})=\hat{\nu}_i,x|_{V_i/V_{i-1}}\in \mathbf{E}_{\hat{\nu}_i}^0,i\not=k,k+1,\\ &M_{x|_{V_{k}/\mathbf{V}_{k-1}}}\cong P_{s'}, M_{x|_{\mathbf{V}_{k+1}/V_k}}\cong I_{s},x(\mathbf{V}_{k+1})\subset \mathbf{V}_{k+1},x(\mathbf{V}_{k-1})\subset \mathbf{V}_{k-1},. \end{align}\right\}.\] If we denote

\[E_1=\left\{(x\in \mathbf{E}_{\mathbf{V}_{k-1}},f)\middle|\begin{align} &f=(0=V_0\subset V_1\cdots \subset\mathbf{V}_{k-1}),x(V_i)\subset V_i,\\ &x|_{V_i/V_{i-1}}\in \mathbf{E}_{\hat{\nu}_i}^0, \end{align}\right\},\] \[E_2=\left\{(x\in \mathbf{E}_{\mathbf{V}_{k+1}/\mathbf{V}_{k-1}},f)\middle|\begin{align} &f=(0\subset V_k/\mathbf{V}_{k-1}\subset \mathbf{V}_{k+1}/\mathbf{V}_{k-1}),x(V_{k}/\mathbf{V}_{k-1})\subset V_{k}/\mathbf{V}_{k-1},,\\ &M_{x|_{V_{k}/\mathbf{V}_{k-1}}}\cong P_{s'}, M_{x|_{\mathbf{V}_{k+1}/V_k}}\cong I_{s}. \end{align}\right\}\] and \[E_3=\left\{(x\in \mathbf{E}_{\mathbf{V}/\mathbf{V}_{k+1}},f)\middle|\begin{align} &f=(0\subset V_{k+2}/\mathbf{V}_{k+1}\cdots\subset V_{n}/\mathbf{V}_{k+1}=\mathbf{V}/\mathbf{V}_{k+1}),x(V_i/\mathbf{V}_{k+1})\subset V_i/\mathbf{V}_{k+1},\\ &\rm dim(V_i/V_{i-1})=\hat{\nu}_i,x|_{V_i/V_{i-1}}\in \mathbf{E}_{\hat{\nu}_i}^0,\\ \end{align}\right\}\]

Then for \[E_1\times E_2\times E_3\xleftarrow{p_1} F\times_{U}\mathbf{G}_{\mathbf{V}}\xrightarrow{p_2} F\times_{P}\mathbf{G}_{\mathbf{V}},\] \(p_2\) is principal bundle and \(p_1\) is smooth with connected fibers. The stratification of \(E_1\times E_2\times E_3\) factors through \(p_2\circ p_1^{-1}\) gives a stratification on \(F\times_{P}\mathbf{G}_{\mathbf{V}}.\)

By Lemma 8, \(E_2\) is isomorphic to \(K\sqcup U\) and by Lemma 9, the trivial fibration morphism \(f\) with projective fiber satisfying that \(\mathbf{H}^j(f_!\overline{{\mathbb{Q}}_l})\) is constant sheaf with Tate twist. We apply Lemma 8 to \(E_2\) and by using \(f_!\) for the corresponding projection map \(f:W'_{0,\underline{\lambda}}\rightarrow W_{0,\underline{\lambda}}\). For the closed subset \(K,\) any \(\mathbf{G}_{\mathbf{V}}\)-equivariant simple lisse sheaf over an orbit with integral weights has the form \(\overline{{\mathbb{Q}}_l}\boxtimes \overline{{\mathbb{Q}}_l}(\zeta),\) where \(\overline{{\mathbb{Q}}_l}(\zeta)\) is pure on \(\operatorname{Spec}({\mathbb{F}}_q).\) This is because \(\rm Aut_x\) is connected for any \(x\in \mathbf{E}_{\mathbf{V}}.\) Thus for \(K\xrightarrow{f}{\mathcal{O}},\) \(\mathbf{H}^i(f_!\overline{{\mathbb{Q}}_l})\) is generated by \(\overline{{\mathbb{Q}}_l}|_{{\mathcal{O}}}\boxtimes\overline{{\mathbb{Q}}_l}(\zeta),\) for some \(\zeta.\) Denote orbit \({\mathcal{O}}\) as the orbit of \(P_s'\oplus I_s\) in \(\mathbf{E}_{\mathbf{V}_{k+1}/\mathbf{V}_{k-1}},\) \[F_{\mathcal{O}}=\left\{(x\in \mathbf{E}_{\mathbf{V}},f)\middle|\begin{align} &f=(0=V_0\subset\cdots \subset\mathbf{V}_{k-1}=V_{k-1}\subset V_{k+1}=\mathbf{V}_{k+1}\subset\cdots\subset V_{n}=\mathbf{V}),x(V_i)\subset V_i,\\ &\rm dim(V_i/V_{i-1})=\hat{\nu}_i,x|_{V_i/V_{i-1}}\in \mathbf{E}_{\hat{\nu}_i}^0,i\not=k,k+1,\\ &M_{x|_{\mathbf{V}_{k+1}/\mathbf{V}_{k-1}}}\cong I_{s}\oplus P_{s'}. \end{align}\right\},\] \[F_K=\left\{(x\in \mathbf{E}_{\mathbf{V}},f)\middle|\begin{align} &f=(0=V_0\subset\cdots \subset\mathbf{V}_{k-1}=V_{k-1}\subset V_k\subset \mathbf{V}_{k+1}=V_{k+1}\subset\cdots\subset V_{n}=\mathbf{V}),x(V_i)\subset V_i,,\\ &\rm dim(V_i/V_{i-1})=\hat{\nu}_i,x|_{V_i/V_{i-1}}\in \mathbf{E}_{\hat{\nu}_i}^0,i\not=k,k+1,\\ &M_{x|_{\mathbf{V}_{k+1}/\mathbf{V}_{k-1}}}\cong I_{s}\oplus P_{s'},M_{x|_{V_{k}/\mathbf{V}_{k-1}}}\cong P_{s'}, M_{x|_{\mathbf{V}_{k+1}/V_k}}\cong I_{s}. \end{align}\right\},\] Consider \[\begin{tikzcd} E_1\times K\times E_3 \arrow[d, "\pi_1"] & F_K\times_U \mathbf{G}_{\mathbf{V}} \arrow[l, "p_1"'] \arrow[r, "p_2"] \arrow[d] & F_K\times_P \mathbf{G}_{\mathbf{V}} \arrow[d, "\pi_2"] \\ E_1\times {\mathcal{O}}\times E_3 & F_{{\mathcal{O}}}\times_U \mathbf{G}_{\mathbf{V}} \arrow[l, "p_1"'] \arrow[r, "p_2"] & F_{{\mathcal{O}}}\times_P \mathbf{G}_{\mathbf{V}} \end{tikzcd}\] we have that \({\pi_2}_!{p_2}_{\flat}p_1^*(\overline{{\mathbb{Q}}_l})\cong {p_2}_{\flat}p_1^*{\pi_1}_!(\overline{{\mathbb{Q}}_l}),\) hence \(\mathbf{H}^i({\pi_2}_!{p_2}_{\flat}p_1^*(\overline{{\mathbb{Q}}_l}))\cong \mathbf{H}^i({p_2}_{\flat}p_1^*{\pi_1}_!(\overline{{\mathbb{Q}}_l}))\cong {p_2}_{\flat}p_1^*\mathbf{H}^i({\pi_1}_!(\overline{{\mathbb{Q}}_l})).\) We already have that \(\mathbf{H}^i({\pi_1}_!(\overline{{\mathbb{Q}}_l}))\) is generated by \(\overline{{\mathbb{Q}}_l}|_{E_1\times {\mathcal{O}}\times E_3}\boxtimes\overline{{\mathbb{Q}}_l}(\zeta),\) for some \(\zeta.\) Furthermore, \(p_1^*\) and \({p_2}_{\flat}\) are exact functors, thus by Lemma 11 \(\mathbf{H}^i({p_2}_{\flat}p_1^*{\pi_1}_!(\overline{{\mathbb{Q}}_l}))\) is generated by \(\overline{{\mathbb{Q}}_l}|_{F_{{\mathcal{O}}}\times_P \mathbf{G}_{\mathbf{V}}}\boxtimes\overline{{\mathbb{Q}}_l}(\zeta),\) for some \(\zeta.\)

If we denote \[F_{W_{0,\underline{\lambda}}}=\left\{(x\in \mathbf{E}_{\mathbf{V}},f)\middle|\begin{align} &f=(0=V_0\subset\cdots \subset V_{k-1}=\mathbf{V}_{k-1}=V_k^0\subset\cdots\subset V_k^{t-1}\subset V_k^t=\mathbf{V}_{k+1}=V_{k+1}\cdots\subset V_{n}=\mathbf{V}),\\ &x(V_i)\subset V_i,\rm dim(V_i/V_{i-1})=\hat{\nu}_i,x|_{V_i/V_{i-1}}\in \mathbf{E}_{\hat{\nu}_i}^0,i\not=k,k+1, x(V_k^i)\subset V_k^i,\\ &M_{x|_{V_{k}^i/V_{k}^{i-1}}} \text{ is regular of dimension }(\lambda_i,\lambda_i). \end{align}\right\}\] and \[F_{W'_{0,\underline{\lambda}}}=\left\{(x\in \mathbf{E}_{\mathbf{V}},f,g)\middle|\begin{align} &f=(0=V_0\subset\cdots \subset V_{k-1}=\mathbf{V}_{k-1}=V_k^0\subset\cdots\subset V_k^{t-1}\subset V_k^t=\mathbf{V}_{k+1}=V_{k+1}\cdots\subset V_{n}=\mathbf{V}),\\ &x(V_i)\subset V_i,\rm dim(V_i/V_{i-1})=\hat{\nu}_i,x|_{V_i/V_{i-1}}\in \mathbf{E}_{\hat{\nu}_i}^0,i\not=k,k+1, x(V_k^i)\subset V_k^i,\\ &M_{x|_{V_{k}^i/V_{k}^{i-1}}} \text{ is regular of dimension }(\lambda_i,\lambda_i),\\ &0\not=g\in \rm Hom(M_{x|_{V_k^t/V_{k}^{t-1}}},I_{s'})/k^*. \end{align}\right\}\] \(r\in P\) acts on \(F_{W'_{0,\underline{\lambda}}}\) as \(r(x,f,g)=(rxr^{-1},rf,gr^{-1})\) where for \(g\in \rm Hom(M_{x|_{V_k^t/V_{k}^{t-1}}},I_{s'})/k^*,\) \(gr^{-1}\in \rm Hom(M_{x|_{r(V_k^t)/r(V_{k}^{t-1})}},I_{s'})/k^*.\)

By the same stratification in Lemma 8, we only need to consider \(\mathbf{G}\times_{P}F_{W'_{0,\underline{\lambda}}}\xrightarrow{\pi} \mathbf{E}_{\mathbf{V}},\) which factors through \(\mathbf{G}\times_{P}F_{W'_{0,\underline{\lambda}}}\xrightarrow{\pi_{\underline{\lambda}}}\mathbf{G}\times_{P}F_{W_{0,\underline{\lambda}}}\xrightarrow{\pi'} \mathbf{E}_{\mathbf{V}}.\) And \(\pi_{\underline{\lambda}}\) is a locally trivial fibration with fiber \({\mathbb{P}}_{\lambda_t-1},\) thus by Lemma 9, \(\mathbf{H}^i({\pi_{\underline{\lambda}}}_!(\overline{{\mathbb{Q}}_l}))\) has the form \(\overline{{\mathbb{Q}}_l}(-\frac{i}{2}).\)

Then, by Lemma 11, we know that we only need to consider sheaf \(\overline{{\mathbb{Q}}_l}\) on \(\mathbf{G}_{\mathbf{V}}\times_{P} F_{{\mathcal{O}}}\) and \(\mathbf{G}_{\mathbf{V}}\times_{P}F_{W_{0,\underline{\lambda}}}\) which both have the form \(S_{\hat{\underline{\nu}}'},\) for some \(\hat{\underline{\nu}}'\in \mathcal{S}q_{\nu}.\)

If there exists \(\hat{\nu_k}=(r,r),\) \(\hat{\nu}_{k+1}=(s,s-1),\) or \(\hat{\nu_k}=(s-1,s),\) \(\hat{\nu}_{k+1}=(r,r),\) this is the case of Lemma 6 and its preprojective analogue, the proof is the same. If \(\hat{\nu_k}=(s',s'-1),\) \(\hat{\nu}_{k+1}=(s,s-1),\) with \(s'>s\) or \(\hat{\nu_k}=(s'-1,s'),\) \(\hat{\nu}_{k+1}=(s-1,s),\) with \(s'<s,\) by Lemma 10, we only need to consider \[F_{a,b}=\left\{(x\in \mathbf{E}_{\mathbf{V}},f)\middle|\begin{align} &f=(0=V_0\subset\cdots \subset\mathbf{V}_{k-1}=V_{k-1}\subset V_k\subset \mathbf{V}_{k+1}=V_{k+1}\subset\cdots\subset V_{n}=\mathbf{V}),x(V_i)\subset V_i,\\ &\rm dim(V_i/V_{i-1})=\hat{\nu}_i,x|_{V_i/V_{i-1}}\in \mathbf{E}_{\hat{\nu}_i}^0,i\not=k,k+1,\\ &M_{x|_{\mathbf{V}_{k+1}/\mathbf{V}_{k-1}}}\cong I_{t}\oplus I_{t},(x,f)|_{{\mathbf{V}_{k+1}/\mathbf{V}_{k-1}}}\in U_{a,b}. \end{align}\right\},\]

and \[F_t=\left\{(x\in \mathbf{E}_{\mathbf{V}},f)\middle|\begin{align} &f=(0=V_0\subset\cdots \subset\mathbf{V}_{k-1}=V_{k-1}\subset V_k\subset \mathbf{V}_{k+1}=V_{k+1}\subset\cdots\subset V_{n}=\mathbf{V}),x(V_i)\subset V_i,\\ &\rm dim(V_i/V_{i-1})=\hat{\nu}_i,x|_{V_i/V_{i-1}}\in \mathbf{E}_{\hat{\nu}_i}^0,i\not=k,k+1,\\ &M_{x|_{\mathbf{V}_{k+1}/\mathbf{V}_{k-1}}}\cong I_{t}\oplus I_{t},(x,f)|_{{\mathbf{V}_{k+1}/\mathbf{V}_{k-1}}}\in A_{t,t}. \end{align}\right\}.\]

Denote \(\operatorname{Pr}:\mathbf{V}_{k+1}\rightarrow \mathbf{V}_{k+1}/\mathbf{V}_{k-1}.\) And morphism from \(F_{a,b}\xrightarrow{P_{a,b}} F_t\) induced by \(p_{a,b}=(p^1_{a,b},p_{a,b}^2)\) in Lemma 10, where \(p^1_{a,b}\) gives part in \(\mathbf{E}_{\mathbf{V}_{k+1}/\mathbf{V}_{k-1}}\) and \(p^2_{a,b}\) gives the Grassmannian, is defined as \[\begin{align} &P_{a,b}(x,(0=V_0\subset\cdots \subset\mathbf{V}_{k-1}=V_{k-1}\subset V_k\subset \mathbf{V}_{k+1}=V_{k+1}\subset\cdots\subset V_{n}=\mathbf{V}))\\&=(x,(0=V_0\subset\cdots \subset\mathbf{V}_{k-1}=V_{k-1}\subset \operatorname{Pr}^{-1}(p^2_{a,b}((x,f)|_{\mathbf{V}_{k+1}/\mathbf{V}_{k-1}}))\subset \mathbf{V}_{k+1}=V_{k+1}\subset\cdots\subset V_{n}=\mathbf{V})). \end{align}\] It is \(P\)-equivariant morphism. Moreover, \(A_{t,t}\) is \(\mathbf{G}_{\mathbf{V}_{k+1}/\mathbf{V}_{k-1}}\)-orbit, then, using the same argument as for \({\mathcal{O}}\) and \(K,\) we only need to study \(F_t\times_P\mathbf{G}_{\mathbf{V}},\) which also has the form \(S_{\hat{\underline{\nu''}}},\) for some \(\hat{\underline{\nu''}}\in \mathcal{S}q_{\nu}.\) If the middle term has the form \(I_{t'}\oplus I_{t''}, t'\not=t'',\) the proof is the same as dealing with \({\mathcal{O}}\) and \(K.\)

Since no new \(I_{s'}\) or \(P_{s'}\) with smaller dimension appears in the above steps, after finitely many steps, we only need to consider the following case.

Fix \((s_i)_{i=1,\cdots,k}\) and \((s'_i)_{i=1,\cdots,l}\) such that \(i\leqslant j\), \(s_i\leqslant s_j\), and \(i\leqslant j\), \(s'_i\geqslant s'_j\), \[W'=\left\{(x,f)\middle|\begin{align} &x\in \mathbf{E}_{\mathbf{V}}, f=(0=V_0\subset \cdots\subset V_k=V_k^0\subset V_k^1\subset\cdots\subset V_k^t=V_{k+1}\subset\cdots V_{k+1+l}=\mathbf{V}),\\ &x(V_i)\subset V_i, x(V_k^j)\subset V_k^j, M_{x|_{V_i/V_{i-1}}}\cong I_{s_i}, i=1,\cdots,k, \\ & M_{x|_{V_{i+k+1}/V_{i+k}}}\cong P_{s'_i}, i=1,\cdots,l,\\ &M_{x|_{V_k^r/V_k^{r-1}}}\text{ is regular with dimension }(\lambda_r,\lambda_r). \end{align}\right\}\xrightarrow{\pi} \mathbf{E}_{\mathbf{V}}.\] We group together the identical preprojective and preinjective parts, and for \((s_i),(s_i')\) above, if \((s_i)=(s_1,\cdots,s_1, s_2,\cdots, s_2,\cdots),\) we denote \(t_1=s_1,\) \(t_2=s_2,\cdots\) with \(a_i\) is the number of occurrence of \(s_i\) in \((s_i)\) and if \((s'_i)=(s'_1,\cdots,s'_1, s'_2,\cdots, s'_2,\cdots),\) we denote \(t'_1=s'_1,\) \(t'_2=s'_2,\cdots\) with \(b_i\) is the number of occurrence of \(s'_i\) in \((s'_i).\) Now with new sequence \((t_i)_{i=1,\cdots,u}\) and \((t'_i)_{i=1,\cdots,v},\) there is \[W=\left\{(x,f)\middle|\begin{align} &x\in \mathbf{E}_{\mathbf{V}}, f=(0=V'_0\subset \cdots\subset V'_u=V_u^0\subset {V'_u}^1\subset\cdots\subset {V'_u}^t=V'_{u+1}\subset\cdots \subset V'_{u+v+1}=\mathbf{V}),\\ &x(V'_i)\subset V'_i, x({V'_k}^j)\subset {V'_k}^j, M_{x|_{V'_i/V_{i'-1}}}\cong I_{t_i}^{\oplus a_i}, i=1,\cdots,u, \\ & M_{x|_{V'_{i+u+1}/V'_{i+u}}}\cong P_{t'_i}^{\oplus b_{i}}, i=1,\cdots,v, \\ &M_{x|_{{V'_u}^r/{V'_u}^{r-1}}}\text{ is regular with dimension }(\lambda_r,\lambda_r). \end{align}\right\},\] \(\pi\) factors through \(W'\xrightarrow{\pi'}W\xrightarrow{\pi''} \mathbf{E}_{\mathbf{V}}.\) We only need to study \(\pi'_!(\overline{{\mathbb{Q}}_l}).\) As in the treatment of the orbit \({\mathcal{O}}\) and of \(K\) above, we only need to consider \(\overline{{\mathbb{Q}}_l}|_W\boxtimes \overline{{\mathbb{Q}}_l}(\zeta)\) on \(W.\) We work in the mixed derived category of sheaf complexes. Thus \(\overline{{\mathbb{Q}}_l}(\zeta)\boxtimes\overline{{\mathbb{Q}}}(\zeta')\) appeared above is isomorphic to \(\overline{{\mathbb{Q}}_l}(\zeta')\) for some \(\zeta'\) such that \(\overline{{\mathbb{Q}}_l}(\zeta'')\) is pure on \(\operatorname{Spec}({\mathbb{F}}_q)\) and \(\operatorname{egf}(\overline{{\mathbb{Q}}_l}(\zeta''))=\overline{{\mathbb{Q}}_l}.\)

After the preceding reductions, we finally obtain varieties of the form \(W_{\alpha'',\underline{\lambda}''}\) and consider the constant sheaf on them. It is straightforward to see that \[p_{\alpha,\underline{\lambda}}:W_{\alpha,\underline{\lambda}}\rightarrow \mathbf{E}_{\mathbf{V}}\] factors through \[p_{\alpha,\underline{\lambda}}:W_{\alpha,\underline{\lambda}}\rightarrow X(\alpha,m)\xrightarrow{j_{\alpha,m}}\mathbf{E}_{\mathbf{V}}.\] And by [16], \(j_{\alpha',m'}^*{j_{\alpha,m}}_!=0\) for \((\alpha',m')\not=(\alpha,m).\) By Lemma 11, simple constituents of \({}^p\mathbf{H}^j\!\left({j_{\alpha',m'}}^*{\pi_{\underline{\nu}}}_!\overline{{\mathbb{Q}}_l}\right)\) are among the simple constituents of \(^p\mathbf{H}^j({p_{\alpha',\underline{\lambda}'}}_!\overline{{\mathbb{Q}}_l})\boxtimes \overline{{\mathbb{Q}}_l}(\zeta).\) This completes the proof. ◻

Proof. We already have that \({\rm supp}(\mathcal{L}_0)=\overline{X(\alpha,m)_0}.\)

If \(x\in X(\alpha,m)_0({\mathbb{F}}_{q^l}), \operatorname{Spec}({\mathbb{F}}_{q^l})\xrightarrow{i_x} \mathbf{E}_\mathbf{V},\) since the property of \[\label{eq12} i_x^*\overline{{\mathbb{Q}}_l}|_{\mathbf{E}_{\mathbf{V}_{s_1}}}\diamond \cdots \diamond \overline{{\mathbb{Q}}_l}|_{\mathbf{E}_{\mathbf{V}_{s_r}}} \diamond \overline{{\mathbb{Q}}_l}|_{\mathbf{E}_{\mathbf{V}_{\lambda_1}}}\diamond \overline{{\mathbb{Q}}_l}|_{\mathbf{E}_{\mathbf{V}_{\lambda_2}}}\cdots \diamond \overline{{\mathbb{Q}}_l}|_{\mathbf{E}_{\mathbf{V}_{\lambda_k}}} \diamond \overline{{\mathbb{Q}}_l}|_{\mathbf{E}_{\mathbf{V}'_{s'_1}}}\diamond \cdots \diamond \overline{{\mathbb{Q}}_l}|_{\mathbf{E}_{\mathbf{V}'_{s'_t}}},\lambda_i=1,\tag{9}\] is from a map \(S\rightarrow \operatorname{Spec({\mathbb{F}}_{q^l})}\), where \(S\) admits an affine paving as in Jordan quiver [28]. Thus the proposition is proved for \(x\in X(\alpha,m)_0({\mathbb{F}}_{q^l})\). Furthermore, \[{j_{\alpha}}^*\mathcal{L}_0=\oplus_{\chi\in X(S_m)}L_{\chi}^{\oplus s_{\chi}}[\rm dimX(\alpha)](\frac{\rm dimX(\alpha)}{2})\] by the etale covering map, and \(\operatorname{egf}({}^p\mathbf{H}^{i}(j_{\alpha,m}^*\mathcal{L}_0))={}^p\mathbf{H}^{i}(j_{\alpha,m}^*\mathcal{L})=0\) for \(i\not=0.\) The functor \(\operatorname{egf}\) is faithful by [16]. Thus \({}^p\mathbf{H}^{i}(j_{\alpha,m}^*\mathcal{L}_0)=0\) for \(i\not=0,\) hence \(j_{\alpha,m}^*\mathcal{L}_0\) is perverse. If we denote the closed embedding \(i:X(\alpha,m)-X(\alpha)\rightarrow X(\alpha,m),\) As well as \(\operatorname{egf}({}^p{\mathbf{H}}^{-k} (i^!j_{\alpha,m}^*\mathcal{L}_0))=0,\) \(\operatorname{egf}({}^p{\mathbf{H}}^{k}( i^*j_{\alpha,m}^*\mathcal{L}_0))=0,\) for \(k\geqslant 0.\) Thus \({}^p\tau^{\leqslant 0} i^!j_{\alpha,m}^*\mathcal{L}_0=0,\) \({}^p\tau^{\geqslant 0} i^*j_{\alpha,m}^*\mathcal{L}_0=0.\) By [[16]][19], it follows that \(j_{\alpha,m}^*\mathcal{L}_0=\oplus_{\chi\in X(S_m)}j_{\alpha,m}^*IC(X(\alpha)_0,L_{\chi})^{\oplus s_{\chi}}.\) Thus it follows that \(j_{\alpha,m}^*IC(X(\alpha)_0,L_{\chi})\) is pointwise pure. By the same argument, \(j_{\alpha',m'}^*IC(X(\alpha')_0,L_{\chi'})\) are pointwise pure for any \(\alpha',m'.\)

We now prove that for \(X(\alpha')_0\subset \overline{X(\alpha)_0},\) \({\mathcal{L}}_0\) is pointwise pure on \(X(\alpha')_0.\) For \(x\in X(\alpha')_0({\mathbb{F}}_{q^n})\subset X(\alpha',m')_0({\mathbb{F}}_{q^n}),\) where \((\alpha',m')<(\alpha,m),\) denote the corresponding representation by \(M_x\). By Theorem 5, and since the category of lisse sheaves (isomorphic to \(\overline{{\mathbb{Q}}_l}[\pi_1(X(\alpha'))]-\operatorname{mod}^{fc}\)) is closed under kernels, cokernels, and extensions (by reducing to a finite extension \(E\) of \({\mathbb{Q}}_l\) and its integral ring \(o\), and considering \(E\otimes_{o}\rm Ext_{{\mathcal{D}}_c^b(X(\alpha'),o)}(L_1,L_2)\) for \(o\)-lisse sheaf \(L_1,L_2.\) For \(E\otimes_o L_{torsion}=0,\) we can choose \(L_1,L_2\) torsion free), we have \({j_{\alpha'}}^*\mathcal{L}\in {\mathcal{D}}^b_{\mathbf{G}_{\mathbf{V}},loc}(X(\alpha')).\) Using fpqc property as in Remark 14, \(j_{\alpha'}^*\mathcal{L}_0\) is lisse sheaf complex whose simple constituents of \(\mathbf{H}^i(j_{\alpha'}^*\mathcal{L}_0)\) have the form \(L_{\chi'}\boxtimes \overline{{\mathbb{Q}}_l}(\zeta)\), where \(\overline{{\mathbb{Q}}_l}(\zeta)\) is pure of weight \(l\) on \(\operatorname{Spec}{\mathbb{F}}_q\) for some \(l\in {\mathbb{Z}}\) and \(\operatorname{egf}(\overline{{\mathbb{Q}}_l}(\zeta))=\overline{{\mathbb{Q}}_l},\) \(\chi\in X(S_{m'})\). By [29], any lisse sheaf (local system) has a weight filtration, \(0=L_0\subset L_1\subset\cdots\subset L_n=L\), where \(L_i/L_{i-1}\) is a pure lisse sheaf and \(wt(L_i/L_{i-1})<wt(L_{i+1}/L_i)\) for \(i=1,\cdots,n-1.\) Moreover, any \({\mathbb{F}}_{q^l}\)-point gives the same weight on \(L_i/L_{i-1}.\) Thus, if for \({\mathbb{F}}_{q^n}\)-point \(x,\) \((L_{i}/L_{i-1})_x\) is of weight \(0,\) then \((L_{i}/L_{i-1})\) is pointwise pure of weight \(0.\) Indeed, \(X(\alpha')\) is irreducible.

By [2], \(X(\alpha')_0\) is smooth with dimension \(m'+\rm dim{\mathcal{O}}_x.\) We consider the following map, \[\phi:\oplus_{j\in I}\rm Hom_k(\mathbf{V}_i,\mathbf{V}_i)/\rm Hom_{kQ}(M_x,M_x)\rightarrow \mathbf{E}_{\mathbf{V}}\] defined by \(g\rightarrow gx-xg.\)

Write \(\mathbf{V}=\oplus \mathbf{V}_s^P\oplus \mathbf{V}_1\oplus \cdots \oplus \mathbf{V}_{m'}\oplus \oplus\mathbf{V}_r^I\) for the decomposition \(M_x=P_{\alpha'}\oplus r_1\oplus \cdots \oplus r_{m'}\oplus I_{\alpha'},\) with \(M_{x|_{\mathbf{V}_s^P}}=P_s^{{\alpha'_s}^P},\) \(M_{x|_{\mathbf{V}_r^I}}=I_r^{{\alpha'_r}^I}\) and \(M_x|_{\mathbf{V}_i}=r_i.\) After replacing \(n\) if necessary, \(x\) is decomposed as above over the field \({\mathbb{F}}_{q^n}.\) We choose a basis of \(\mathbf{V}_i\) such that \(x|_{\mathbf{V}_i}=(x_\alpha,x_\beta)=(1,t_i),\) where \(\alpha,\beta\) are arrows, and consider the space \(T_i:=k(0,1)\subset \mathbf{E}_{\mathbf{V}_i}.\) Denote \(\mathbf{E}_{\mathbf{V}.\mathbf{V}'}=\oplus_{h}\rm Hom_k(\mathbf{V}_{s(h)},\mathbf{V}'_{t(h)}).\)

Let \(\mathcal{T}=\rm Im\,\phi\times \Pi_{i=1}^{m'}T_i\subset \mathbf{E}_{\mathbf{V}},\) with the embedding \((x,x_1,\cdots,x_{m'})\rightarrow x+\sum_{i=1}^{m'} x_i\) and consider \(x+\mathcal{T}\). It is straightforward to see that \(x+\mathcal{T}\) is the tangent space of \(X(\alpha')\) at \(x.\) And \(\mathcal{T}\) contains \(\Pi_{s\leqslant s'}\mathbf{E}_{\mathbf{V}^P_{s},\mathbf{V}^P_{s'}}\times \Pi_{s,i}\mathbf{E}_{\mathbf{V}_s^P,\mathbf{V}_i}\times\Pi_{j,r}\mathbf{E}_{\mathbf{V}_j,\mathbf{V}^I_r}\times \Pi_{s,r}\mathbf{E}_{\mathbf{V}^P_s,\mathbf{V}^I_r}\times\Pi_{r\geqslant r'}\mathbf{E}_{\mathbf{V}^I_{r},\mathbf{V}^I_{r'}}\times \Pi_{i\leqslant j}\mathbf{E}_{\mathbf{V}_i,\mathbf{V}_j}.\)

We only need to show that \(\mathbf{E}_{\mathbf{V}_i,\mathbf{V}_i}\subset \mathcal{T},\) for other sets are contained in \(\rm Im\,\phi.\)

For \(T_i\) and \(\phi(\oplus_{j\in I}\rm Hom_k({\mathbf{V}_i}_j,{\mathbf{V}_i}_j)/\rm Hom_{kQ}({r_i},r_i))\) span \(\mathbf{E}_{\mathbf{V}_i,\mathbf{V}_i},\) this completes the proof.

Consider the map \(\lambda(t)\in \mathbf{G}_{\mathbf{V}}\) as \(\lambda(t)|_{\mathbf{V}_s^P}=t^{a_s},\) \(\lambda(t)|_{\mathbf{V}_r^I}=t^{b_r},\) \(\lambda(t)|_{\mathbf{V}_i}=t^{c_i},\) such that \(a_s<c_i<b_r\) and \(a_s<a_{s'}\) for \(s<s'\), \(b_r>b_{r'}\) for \(r<r'\), \(c_1<c_2<\cdots<c_{m'}.\) Then \(\lambda(t)\rm Im\,\phi\subset \rm Im\,\phi\) and \(\lambda(t)(T_i)\subset T_i.\) Thus \(\mathcal{T}\) is \(\lambda(t)\)-stable.

We then choose the \(\lambda(t)\)-stable complement of \(\mathcal{T}\) in \(\mathbf{E}_{\mathbf{V}}\) as \(\mathcal{T}'.\) \[\mathcal{T}'\subset \Pi_{s> s'}\mathbf{E}_{\mathbf{V}^P_{s},\mathbf{V}^P_{s'}}\times \Pi_{s,i}\mathbf{E}_{\mathbf{V}_i,\mathbf{V}_s^P}\times\Pi_{j,r}\mathbf{E}_{\mathbf{V}^I_r,\mathbf{V}_j}\times \Pi_{s,r}\mathbf{E}_{\mathbf{V}^I_r,\mathbf{V}^P_s}\times\Pi_{r< r'}\mathbf{E}_{\mathbf{V}^I_{r},\mathbf{V}^I_{r'}}\times \Pi_{i> j}\mathbf{E}_{\mathbf{V}_i,\mathbf{V}_j}.\]

We now consider \(x+(\mathcal{T'}\times\Pi_{i=1}^{m'}T_i),\) it is transverse to the orbit \({\mathcal{O}}_x\) at \(x.\)

The action of \(\lambda(t)\) preserves \(\mathcal{T}\) and \(\mathcal{T'}.\) And \(\lambda(t)\) acts trivially on \(x+T_i,\) \(i=1,\cdots,m'.\)

Furthermore, \(\lim_{t\rightarrow0}\lambda(t)u=0,\) for \(u\in \mathcal{T'}.\) Thus fixed-point set \((x+(\mathcal{T'}\times\Pi_{i=1}^{m'}T_i))^{G_m}\) is \(x+\Pi_{i=1}^{m'}T_i.\)

Following [16], when we consider \(\mathbf{G}_{\mathbf{V}}\times (x+(\mathcal{T'}\times\Pi_{i=1}^{m'}T_i))\xrightarrow{f} \mathbf{E}_{\mathbf{V}},\) with \(f(g,x')=g(x'),\) \(x\) is a smooth point of \(f.\) And the set of smooth points of \(f,\) denoted as \(\tilde{S},\) is open and stable under \(\mathbf{G}_{\mathbf{V}}\) and \(\operatorname{Fr}^n.\) Thus we can find an open subset \(S\) of \((x+(\mathcal{T'}\times\Pi_{i=1}^{m'}T_i))\) such that \(\tilde{S}\cong \mathbf{G}_{\mathbf{V}}\times S\xrightarrow{f} \mathbf{E}_{\mathbf{V}}\) is smooth with \(S\) stable under \(\operatorname{Fr}^n.\) Since \(G_m\) acts on \((x+(\mathcal{T'}\times\Pi_{i=1}^{m'}T_i))\) and \(G_m\subset \mathbf{G}_{\mathbf{V}},\) it follows that \(S\) is \(G_m\)-stable. We now consider \(U=X(\alpha')\cap S\cap (x+\Pi_{i=1}^{m'}T_i),\) one checks that \(U\) is an open subset of \((x+\Pi_{i=1}^{m'}T_i),\) thus \(U\) is smooth.

Let \(U'=S\cap (U\times\mathcal{T'}).\) It is an open subset of \((x+(\mathcal{T'}\times\Pi_{i=1}^{m'}T_i))\) and stable under \(G_m,\) with fixed-point set \(U.\) Then \(\mathbf{G}_{\mathbf{V}}\times U'\xrightarrow{f} \mathbf{E}_{\mathbf{V}}\) is smooth morphism with relative dimension \(\rm dim\mathbf{G}-\rm dim{\mathcal{O}}_x.\) Let \(U'\xrightarrow{i}\mathbf{G}_{\mathbf{V}}\times U'\) with \(i(u)=(1,u).\) As in [16], it follows that \(i^*[-\rm dim\mathbf{G}_{\mathbf{V}}](\frac{-\rm dim\mathbf{G}_{\mathbf{V}}}{2})=i^\) is equivalence of categories. Furthermore, \(i^*\) takes pure sheaves to pure sheaves. Thus \(i^*f^*[-\rm dim{\mathcal{O}}_x](\frac{-\rm dim{\mathcal{O}}_x}{2})\) takes pure sheaves to pure sheaves. We now consider inclusion map \(e:U\rightarrow U'\) and \(p:U'\rightarrow U\) with \(p(u)=\lim_{t\rightarrow0}\lambda(t)u\) as in [16].

If we denote the inclusion map \(i_{U'}:U'\rightarrow \mathbf{E}_{\mathbf{V}}\) and \(i_{U'}=f\circ i,\) then \(i_{U'}\) is \(G_m\)-equivariant. By [15], for proper map \(p,\) \(p_!\) preserves the weight. Thus \({\mathcal{L}}_0\) is pure of weight \(0\). Thus \(i^*f^*(\mathcal{L}_0)\) is \(G_m\)-equivariant and pure of weight \(0\). By [16], it follows that \(e^*i^*f^*(\mathcal{L}_0)\cong p_*i^*f^*(\mathcal{L}_0).\) Because \(e^*\) preserves weight \(\leqslant 0\) and \(p_*\) preserves weight \(\geqslant 0,\) \(e^*i^*f^*(\mathcal{L}_0)\) is pure. Since \(U\) is smooth, considering \(U\xrightarrow{i_{u}} X(\alpha)_0,\) it follows that \(e^*i^*f^*(\mathcal{L}_0)\in {\mathcal{D}}^b_{loc,m}(U).\) Thus by [29], it follows that \(e^*i^*f^*(\mathcal{L}_0)\) is pointwise pure of weight \(0\). Thus \(j_{\alpha}^*\mathcal{L}_0\) is pointwise pure of weight \(0\).

Now consider the subset \(X(\alpha',m')_0\subset \overline{X(\alpha)_0}.\) If we denote the open embedding \(j^0:X(\alpha')_0\rightarrow X(\alpha',m')_0,\) since \({j^0}^*\) preserves the perverse \(t\)-structure and weight and \({}^p\mathbf{H}^n(\mathcal{F})=\mathbf{H}^{n-\rm dim(X(\alpha'))}(\mathcal{F})[\rm dimX(\alpha')]\) for \(\mathcal{F}\in {\mathcal{D}}^b_{\mathbf{G}_0,loc}(X(\alpha')_0)\), by Theorem 5 and the faithfulness of the functor \(\operatorname{egf}\) on \(\operatorname{Perv}(X_0,\overline{{\mathbb{Q}}_l}),\) it follows that \({}^p\mathbf{H}^l(j_{\alpha',m'}^*\mathcal{L}_0)\) is pure of weight \(l.\) In the triangle \[{}^p\tau^{<l}(j_{\alpha',m'}^*\mathcal{L}_0)\rightarrow {}^p\tau^{\leqslant l}(j_{\alpha',m'}^*\mathcal{L}_0)\rightarrow {}^p\mathbf{H}^l(j_{\alpha',m'}^*\mathcal{L}_0)[-l]\xrightarrow{+1},\] if \({}^p\tau^{<l}(j_{\alpha',m'}^*\mathcal{L}_0)\) and \({}^p\mathbf{H}^l(j_{\alpha',m'}^*\mathcal{L}_0)[-l]\) are pure of weight \(0,\) then \({}^p\tau^{\leqslant l}(j_{\alpha',m'}^*\mathcal{L}_0)\) is pure of weight \(0.\) Thus by induction on \(l,\) \(j_{\alpha',m'}^*\mathcal{L}_0\) is pure of weight zero and \(j_{\alpha',m'}^*\mathcal{L}=\oplus_l {}^p\mathbf{H}^l(j_{\alpha',m'}^*\mathcal{L})[-l]\) has the form \(\oplus_{l,\chi'}j_{\alpha',m'}^*(IC(X(\alpha'),L_{\chi'}))^{\oplus s_{l,\chi'}}[l].\)

Let closed embedding \(i':X(\alpha',m')_0-X(\alpha')_0\rightarrow X(\alpha',m')_0\) and consider \({}^p\tau^{\leqslant 0}{i'}^!{}^p{\mathbf{H}}^l(j^*_{\alpha',m'}\mathcal{L}_0)\) and \({}^p\tau^{\geqslant 0}{i'}^*{}^p{\mathbf{H}}^l(j^*_{\alpha',m'}\mathcal{L}_0).\) After applying \(\operatorname{egf},\) \({}^p{\mathbf{H}}^{-k}({i'}^!{}^p{\mathbf{H}}^l(j^*_{\alpha',m'}\mathcal{L}_0))\) and \({}^p{\mathbf{H}}^{k}({i'}^*{}^p{\mathbf{H}}^l(j^*_{\alpha',m'}\mathcal{L}_0))\) are \(0\) for \(k\geqslant 0.\) By the faithfulness of \(\operatorname{egf},\) it follows that \({i'}^*{}^p{\mathbf{H}}^l(j^*_{\alpha',m'}\mathcal{L}_0)\in {}^p{\mathcal{D}}^{\leqslant-1}\) and \({i'}^!{}^p{\mathbf{H}}^l(j^*_{\alpha',m'}\mathcal{L}_0)\in {}^p{\mathcal{D}}^{\geqslant 1},\) by [16] and [19], \({}^p{\mathbf{H}}^l(j^*_{\alpha',m'}\mathcal{L}_0)=j^0_{!*}({}^p{\mathbf{H}}^l({j_{\alpha'}}^*\mathcal{L}_0)).\) By [16], and the faithfulness of \(\operatorname{egf},\) any simple perverse constituent of \({}^p\mathbf{H}^l(j_{\alpha',m'}^*\mathcal{L}_0)\) has the form \(j_{\alpha',m'}^*(IC(X(\alpha')_0,L_{\chi}))\boxtimes \overline{{\mathbb{Q}}_l}(\zeta),\) where \(\overline{{\mathbb{Q}}_l}(\zeta)\) is pure of weight \(l\) on \(\operatorname{Spec}{\mathbb{F}}_q\) and \(\operatorname{egf}(\overline{{\mathbb{Q}}_l}(\zeta))=\overline{{\mathbb{Q}}_l}.\) Moreover, \(j_{\alpha',m'}^*(IC(X(\alpha')_0,L_{\chi})\) is pointwise pure of weight \(0\) and \(\boxtimes \overline{{\mathbb{Q}}_l}(\zeta)\) preserves pointwise purity and purity. If \(A\rightarrow B\rightarrow C\xrightarrow{+1}\) is a triangle and \(A\) and \(C\) are pointwise pure of weight \(a\), then \(B\) is pointwise pure of weight \(a\). Thus \(j_{\alpha',m'}^*\mathcal{L}_0\) is pointwise pure.

This completes the proof that \(\mathcal{L}_0\) is pointwise pure. ◻