On the Frobenius direct image of the structure sheaf of a homogeneous projective variety

TL;DR

This paper provides an example of a homogeneous projective variety where the Frobenius direct image of its structure sheaf exhibits nonvanishing self-extensions, highlighting a unique property in algebraic geometry.

Contribution

It introduces a specific example demonstrating nonvanishing self-extensions of the Frobenius direct image, a novel phenomenon in the study of homogeneous projective varieties.

Findings

Frobenius direct image can have nonvanishing self-extensions.

Example of a homogeneous projective variety with this property.

Insights into the structure of sheaves under Frobenius morphism.

Abstract

We present an example of a homogeneous projective variety the Frobenius direct image of the structure sheaf of which has nonvanishing self extension.

Equations

$$\begin{gathered} \mathcal{E}(e)\otimes L(e)\oplus \mathcal{E}(s_2)\otimes \{L(s_2)\oplus L(s_1{s}_2{s}_1{s}_2)\}\oplus \mathcal{E}(s_1{s}_2)\otimes L(s_1{s}_2)\oplus {\mathcal{L}}_{G/P}(\mathfrak{p})(-1)\otimes L(e)\oplus \\ \mathcal{E}(s_2{s}_1{s}_2)\otimes L(s_2{s}_1{s}_2)\oplus \mathcal{E}(s_1{s}_2{s}_1{s}_2)\otimes L(s_1{s}_2{s}_1{s}_2)\oplus \mathcal{E}(s_2{s}_1{s}_2{s}_1{s}_2)\otimes \{L(s_2{s}_1{s}_2{s}_1{s}_2)\oplus L(s_1{s}_2)\}. \end{gathered}$$

$${\mathrm{soc}}_1{\hat{\nabla }}_P(\epsilon )$$

$${\mathrm{soc}}_2{\hat{\nabla }}_P(\epsilon )$$

$${\mathrm{soc}}_3{\hat{\nabla }}_P(\epsilon )$$

$$\oplus L(s_1{s}_2{s}_1{s}_2)\otimes (-{\varpi }_2{)}^{[1]}\oplus L(e)\otimes ({\mathrm{soc}}_{3,e}^1{)}^{[1]},$$

$${\mathrm{soc}}_4{\hat{\nabla }}_P(\epsilon )$$

$${\mathrm{soc}}_5{\hat{\nabla }}_P(\epsilon )$$

$$\oplus L(s_1{s}_2)\otimes (-2{\varpi }_2{)}^{[1]},$$

$${\mathrm{soc}}_6{\hat{\nabla }}_P(\epsilon )$$

$${\mathrm{ind}}_{G_{1}B}^G(M\otimes 3p{\varpi }_1)\to {\mathrm{ind}}_{G_{1}B}^G(\epsilon )\to {\mathrm{R}}^1{\mathrm{ind}}_{G_{1}B}^G({\mathrm{soc}}^2\otimes 3p{\varpi }_1)$$

$${\mathrm{R}}^1{\mathrm{ind}}_{G_{1}B}^G$$

$$\displaystyle\simeq\mathrm{R}^{1}\mathrm{ind}_{G_{1}B}^{G}(\begin{tabular}[]{|c|}\hline\cr$L(s_{2})\otimes(3\varpi_{1}-\varpi_{2})^{[1]}$\\ \hline\cr$3p\varpi_{1}$\\ \hline\cr\end{tabular}).$$

$${\mathrm{R}}^1{\mathrm{ind}}_{G_{1}B}^G(3p{\varpi }_1)$$

$$=0\text{by Kempf’s vanishing cite[cite][@@bibrefJ, II.B.4]},$$

$${\mathrm{R}}^1{\mathrm{ind}}_{G_{1}B}^G$$

$$\simeq L(s_2)\otimes {\mathrm{R}}^1{\mathrm{ind}}_{G_{1}B}^G(3{\varpi }_1-{\varpi }_2{)}^{[1]}=0\text{by Bott’s theorem cite[cite][@@bibrefJ, II.5.5]}.$$

$${\mathrm{ind}}_{G_{1}B}^G(M\otimes 3p{\varpi }_1)$$

$$\simeq {\mathrm{ind}}_{G_{1}B}^G(\hat{\nabla }(3p{\varpi }_1))\text{by the tensor identity again}$$

$$\simeq \nabla (3p{\varpi }_1)\text{by the transitivity of inductions cite[cite][@@bibrefJ, I.3.5]}.$$

$${\mathrm{Ext}}_P^1$$

$$\simeq {\mathrm{Ext}}_P^1(\epsilon ,{\nabla }^P(-s_1(3{\varpi }_1-3{\varpi }_2))\otimes (-2{\varpi }_2))\text{as ∇P(3ϖ1−3ϖ2) is P-simple}$$

$$={\mathrm{Ext}}_P^1(\epsilon ,{\nabla }^P(3{\varpi }_1)\otimes (-2{\varpi }_2))$$

$$\simeq {\mathrm{Ext}}_P^1(\epsilon ,{\nabla }^P(3{\varpi }_1-2{\varpi }_2))\text{by the tensor identity}$$

$$\simeq {\mathrm{Ext}}_B^1(\epsilon ,3{\varpi }_1-2{\varpi }_2)\text{by the Frobenius reciprocity cite[cite][@@bibrefJ, I.3.4] and by Kempf}$$

$$={\mathrm{Ext}}_B^1(L(e),s_2\bullet 0)$$

$$\simeq {\mathbf{Mod}}_G(L(e),{\mathrm{R}}^1{\mathrm{ind}}_B^G(s_2\bullet 0))\text{by the Frobenius reciprocity and by Bott}$$

$$\simeq {\mathbf{Mod}}_G(L(e),L(e))\simeq \mathbb{k}.$$

$${\mathrm{ind}}_{G_{1}P}^G(M^{\prime }\otimes p{\varpi }_2)\to {\mathrm{ind}}_{G_{1}P}^G(\epsilon )\to {\mathrm{R}}^1{\mathrm{ind}}_{G_{1}P}^G({\mathrm{soc}}^2\otimes p{\varpi }_2)$$

$${\mathrm{R}}^1{\mathrm{ind}}_{G_{1}P}^G$$

$$=0\text{by the tensor identity and by Kempf}.$$

$$\cdots \to {\mathrm{Ext}}_P^1(-{\varpi }_2,-2{\varpi }_2)\to {\mathrm{Ext}}_P^1(-{\varpi }_2,M^2)\to {\mathrm{Ext}}_P^1(-{\varpi }_2,{\nabla }^P(3{\varpi }_1-3{\varpi }_2))\to \dots$$

$$\begin{gathered} 0\to {\mathrm{ind}}_{G_{1}B}^B((2p{\varpi }_1)\otimes {\mathrm{soc}}^2{\hat{\nabla }}_P(\epsilon ))\to {\mathrm{ind}}_{G_{1}B}^G((2p{\varpi }_1)\otimes M^{\prime \prime })\to {\mathrm{ind}}_{G_{1}B}^G(\epsilon ) \\ \to {\mathrm{R}}^1{\mathrm{ind}}_{G_{1}B}^B((2p{\varpi }_1)\otimes {\mathrm{soc}}^2{\hat{\nabla }}_P(\epsilon )). \end{gathered}$$

$${\mathrm{ind}}_{G_{1}B}^G$$

$$\displaystyle\simeq\mathrm{ind}_{G_{1}B}^{G}(\begin{tabular}[]{|c|}\hline\cr$L(s_{2})\otimes(2\varpi_{1}-\varpi_{2})^{[1]}$\\ \hline\cr$(2\varpi_{1})^{[1]}$\\ \hline\cr\end{tabular})$$

$$\simeq \nabla (2{\varpi }_1{)}^{[1]}\text{by Bott}$$

$$=L(2{\varpi }_1{)}^{[1]}\text{by the linkage principle cite[cite][@@bibrefJ, II.6.17] under the assumption p≥11}$$

$$\simeq L(2p{\varpi }_1).$$

$${\mathrm{R}}^1{\mathrm{ind}}_{G_{1}B}^G((2p{\varpi }_1)\otimes {\mathrm{soc}}^2{\hat{\nabla }}_P(\epsilon ))$$

$$\displaystyle\simeq\mathrm{R}^{1}\mathrm{ind}_{G_{1}B}^{G}(\begin{tabular}[]{|c|}\hline\cr$L(s_{2})\otimes(2\varpi_{1}-\varpi_{2})^{[1]}$\\ \hline\cr$(2\varpi_{1})^{[1]}$\\ \hline\cr\end{tabular})=0.$$

$$0\to L(2p{\varpi }_1)\to {\mathrm{ind}}_{G_{1}B}^G((2p{\varpi }_1)\otimes M^{\prime \prime })\to L(e)\to 0.$$

$${\mathrm{ind}}_{G_{1}B}^G((2p{\varpi }_1)\otimes M^{\prime \prime })$$

$$\le {\mathrm{ind}}_{G_{1}B}^G(\hat{\nabla }(2p{\varpi }_1))\simeq \nabla (2p{\varpi }_1).$$

$${\mathrm{Ext}}_P^i$$

Full text

On some homogeneous projective spaces $G/P$ in large positive characteristic we have found, for the projective spaces [K09], the quadrics [K14], Grassmannians $\mathrm{Gr}(2,n)$ [K17], and when $G$ is the special linear group of degree 3 [HKR], the symplectic group of degree 4 [AK00], [KY07], or when $G$ is in type $\mathrm{G}_{2}$ and $P$ a maximal parabolic subgroup [KY], a Karoubian complete strongly exceptional collection of coherent modules ${\mathcal{E}}(w)$ over $G/P$, parametrized by the minimal length representatives $w$ of the cosets of the Weyl group of $G$ by that of $P$, as subquotients of the Frobenius direct image of the structure sheaf of $G/P$, which are all defined over $\mathbb{Z}$. Except for the cases of $\mathrm{Gr}(2,n)$, $n\geq 4$, and when $G$ is in type $G_{2}$ with $P$ associated to a short simple root, we also know that the Frobenius direct image of the structure sheaf is a direct sum of copies of those ${\mathcal{E}}(w)$’s; the case for the quadrics is due to Langer [La]. In this paper we determine an extra summand in that exceptional case in type $\mathrm{G}_{2}$, and find that the summand causes nontrivial self extension of the Frobenius direct image of the structure sheaf. There follows nonvanishing of the 1st cohomology of the sheaf of rings of small differential operators on $G/P$ in this setting. The sheaf of rings of small differential operators is the first term of the $p$-filtration [Haa] of the sheaf $\mathcal{D}if\!f_{G/P}$ of rings of differential operators [EGA], and is a central reducion of the sheaf ${\mathcal{D}}_{G/P}^{(0)}$ of rings of arithmetic differential operators of level 0 [Ber] which is called the sheaf of rings of cristaline differential operators in [BMR]. One may recall that Kashiwara and Lauritzen [KaLa] found the nonvanishing of higher cohomolgy of $\mathcal{D}if\!f_{G/P}$ for $G/P=\mathrm{Gr}(2,5)$, while that the vanishing of the higher cohomology of ${\mathcal{D}}_{G/P}^{(0)}$ in general holds thanks to Bezrukavnikov, Mirkovic, and Rumynin [BMR].

In more details, let $G$ be a simple algebraic group over an algebraically closed field of characteristic at least 11, and $P$ the standard parabolic subgroup of $G$ associated to a short simple root $\alpha_{1}$. Let $W$ be the Weyl group of $G$ with simple reflections $s_{1}$ and $s_{2}$, and let $W^{P}=\{w\in W|w\alpha_{1}>0)=\{e,s_{2},s_{1}s_{2},s_{2}s_{1}s_{2},s_{1}s_{2}s_{1}s_{2},s_{2}s_{1}s_{2}s_{1}s_{2}\}$. For each $w\in W^{P}$ let $L(w)$ be the simple $G_{1}$-module of highest weight $w\rho-\rho$, $\rho$ a half sum of the positive roots and $G_{1}$ the Frobenius kernel of $G$. Let $\mathfrak{p}$ be the Lie algebra of $P$ under the adjoint represention, and let ${\mathcal{L}}_{G/P}(\mathfrak{p})(-1)$ be the sheaf over $G/P$ associated to $\mathfrak{p}$ with Serre-twist ${\mathcal{O}}_{G/P}(-1)$.

[TABLE]

The ${\mathcal{E}}(w)$, $w\in W^{P}$, are all locally free sheaves of finite rank, defined over $\mathbb{Z}$, and form a Karoubian complete strongly exceptional collection in the bounded derived category of coherent sheaves on $G/P$ such that $\mathbf{Mod}_{\mathcal{P}}({\mathcal{E}}(x),{\mathcal{E}}(y))\neq 0$ iff $x\geq y$ in the Chevalley-Bruhat order. However, $\mathrm{Ext}^{1}_{\mathcal{P}}(F_{*}{\mathcal{O}}_{G/P},F_{*}{\mathcal{O}}_{G/P})\neq 0$.

$1^{\circ}$ **Structure of the $G_{1}P$-Verma module **

(1.1) Let $\Bbbk$ be an algebraically closed field of positive characteristic $p$, $G$ a simple algebraic group over $\Bbbk$ in type $\mathrm{G}_{2}$, $B$ a Borel subgroup of $G$, $T$ a maximal torus of $B$, $R$ the root system of $G$ relative to $T$, $R^{+}$ the positive system of $R$ such that the roots of $B$ are $-R^{+}$, and $R^{s}=\{\alpha_{1},\alpha_{2}\}$ the set of simple roots with $\alpha_{1}$ short. Let $\Lambda$ be the character group of $T$, $\Lambda^{+}$ the set of dominant weights with the fundamental weights $\varpi_{1}$ and $\varpi_{2}$; $\langle\varpi_{i},\alpha_{j}^{\vee}\rangle=\delta_{ij}$ $\forall i,j\in[1,2]$ with simple coroots $\alpha_{j}^{\vee}$. We partially order $\Lambda$ by $R^{+}$ such that $\lambda\geq\mu$ iff $\lambda-\mu\in\sum_{\alpha\in R^{+}}\mathbb{N}\alpha$. Let $W$ be the Weyl group of $G$ with the simple reflections $s_{i}$ associated to the simple root $\alpha_{i}$, $i\in[1,2]$. Let $P$ denote the standard parabolic subgroup of $G$ associated to the short simple root $\alpha_{1}$ with the Weyl group $W_{P}$. Let $W^{P}=\{w\in W|w\alpha_{1}>0\}$ the set of minimal length representatives of $W/W_{P}$. Let $G_{1}$ be the Frobenius kernel of $G$ and let $\hat{\nabla}_{P}=\mathrm{ind}_{P}^{G_{1}P}$ be the induction functor from the category of $P$-modules to the category of $G_{1}P$-modules. Let $\hat{\nabla}_{P}(\varepsilon)$ be the $G_{1}P$-Verma module of highest weight 0 induced from the trivial 1-dimensional $P$-module $\varepsilon$. For $\lambda\in\Lambda^{+}$ we let $L(\lambda)$ denote the simple $G$-module of highest weight $\lambda$. We write each $\mu\in\Lambda$ as a sum $\mu=\mu^{0}+p\mu^{1}$ with $\langle\mu^{0},\alpha_{i}^{\vee}\rangle\in[0,p[$, $i\in\{1,2\}$. For $w\in W$ and $\mu\in\Lambda$ we let $w\bullet\mu=w(\mu+\rho)-\rho$ with $\rho=\frac{1}{2}\sum_{\alpha\in R^{+}}\alpha=\varpi_{1}+\varpi_{2}$. Put, in particular, $L(w)=L((w\bullet 0)^{0})$, which remains simple as $G_{1}$-module. For a $P$-module $M$ we let $M^{[1]}$ denote the Frobenius twist of $M$ [J, II.3.16]. Unless otherwise specified, $\otimes$ will stand for the tensor product over $\Bbbk$.

We consider the geometric Frobenius morphism $F:G/P\to G/P$ using the $\mathbb{F}_{p}$-form of $G/P$. It factors through the natural morphism $q:G/P\to G/G_{1}P$ to induce an isomorphism $G_{1}P\to G/P$, so the Frobenius direct image $F_{*}{\mathcal{O}}_{G/P}$ of the structure sheaf ${\mathcal{O}}_{G/P}$ of $G/P$ may be identified with the sheaf ${\mathcal{L}}_{G/G_{1}P}(\hat{\nabla}_{P}(\varepsilon))$ over $G/G_{1}P$ associated to the $G_{1}P$-module $\hat{\nabla}_{P}(\varepsilon)$. Thus the structure of $G_{1}P$-module on $\hat{\nabla}_{P}(\varepsilon)$ controls $G$-equivariantly the structure of $F_{*}{\mathcal{O}}_{G/P}$. Throughout the rest of the paper we will assume $p\geq 11$ so that Lusztig’s conjecture for the irreducible characters for $G$ and $G_{1}T$ hold [J, D], which enables us to compute the $G_{1}T$-socle series of $\hat{\nabla}_{P}(\varepsilon)$ by the formula [AbK, 5.2] using the periodic Kazhdan-Lusztig polynomials [L80] and [Kat].

(1.2) Recall the $G_{1}T$-socle series of $\hat{\nabla}_{P}(\varepsilon)$: $0=\mathrm{soc}^{0}\hat{\nabla}_{P}(\varepsilon)<\mathrm{soc}\hat{\nabla}_{P}(\varepsilon)=\mathrm{soc}^{1}\hat{\nabla}_{P}(\varepsilon)<\mathrm{soc}^{2}\hat{\nabla}_{P}(\varepsilon)<\dots<\mathrm{soc}^{6}\hat{\nabla}_{P}(\varepsilon)=\hat{\nabla}_{P}(\varepsilon)$. Put $\mathrm{soc}_{i}\hat{\nabla}_{P}(\varepsilon)=(\mathrm{soc}^{i}\hat{\nabla}_{P}(\varepsilon))/(\mathrm{soc}^{i-1}\hat{\nabla}_{P}(\varepsilon))$, $i\in[1,6]$. One has a direct sum decomposition $\mathrm{soc}_{i}\hat{\nabla}_{P}(\varepsilon)=\coprod_{w\in W}L(w)\otimes G_{1}\mathbf{Mod}(L(w),\mathrm{soc}_{i}\hat{\nabla}_{P}(\varepsilon))$. As $G_{1}$ is normal in $G$, the decomposition holds as $G_{1}P$-modules. Untwisting the Frobenius or by the Frobenius contraction, put $\mathrm{soc}^{1}_{i,w}=G_{1}\mathbf{Mod}(L(w),\mathrm{soc}_{i}\hat{\nabla}_{P}(\varepsilon))^{[-1]}$ [J, II.3.16]/[GK]. Let $\nabla=\mathrm{ind}_{B}^{G}$ (resp. $\nabla^{P}=\mathrm{ind}_{B}^{P}$) denote the induction functor from the category of $B$-modules to the category of $G$- (resp. $P$-) modules. One has from [KY, 4.7, 4.8], see (A.1) in the appendix,

[TABLE]

where $\nabla(\varpi_{1})\twoheadrightarrow\nabla^{P}(\varpi_{1})$ is a unique epi of $P$-modules and $w^{P}=s_{2}s_{1}s_{2}s_{1}s_{2}$ is the longest element of $W^{P}$. We will determine the $P$-module $\mathrm{soc}_{3,e}^{1}$ left open in [KY], which will play the main role of the paper.

(1.3) By the weight consideration $\mathrm{soc}^{1}_{3,e}$ admits a $P$-module filtration $0<-2\varpi_{2}=M^{1}<M^{2}<M^{3}<M^{4}=\mathrm{soc}^{1}_{3,e}$ such that $M^{2}/M^{1}\simeq\nabla^{P}(3\varpi_{1}-3\varpi_{2})$, $M^{3}/M^{2}\simeq-\varpi_{2}$, and $M^{4}/M^{3}\simeq\nabla^{P}(2\varpi_{1}-2\varpi_{2})$. We will denote a module with a filtration with subquotients $M_{r},\dots,M_{1}$ from the top by

$M_{r}$

⋮

$M_{1}$

. Thus, \mathrm{soc}^{1}_{3,e}=\begin{tabular}[]{|c|}\hline\cr\nabla^{P}(2\varpi_{1}-2\varpi_{2})\\ \hline\cr-\varpi_{2}\\ \hline\cr\nabla^{P}(3\varpi_{1}-3\varpi_{2})\\ \hline\cr-2\varpi_{2}\\ \hline\cr\end{tabular}. Put $\mathrm{soc}^{i}=\mathrm{soc}^{i}\hat{\nabla}_{P}(\varepsilon)$, $i\in[1,6]$. Let also $\hat{\nabla}=\mathrm{ind}_{B}^{G_{1}B}$ denote the induction functor from the category of $B$-modules to the category of $G_{1}B$-modules. We let $?^{*}$ denote the $\Bbbk$-linear dual.

Just suppose the extension $M^{2}$ of $\nabla^{P}(3\varpi_{1}-3\varpi_{2})$ by $-2\varpi_{2}$ splits. As $-3\varpi_{1}$ is the lowest weight of $\nabla^{P}(3\varpi_{1}-3\varpi_{2})$ there would be a $G_{1}B$-submodule $M$ of $\mathrm{soc}^{3}$ containing $\mathrm{soc}^{2}$ such that $M/\mathrm{soc}^{2}\simeq-3p\varpi_{1}$, and hence an exact sequence $0\to\mathrm{soc}^{2}\otimes 3p\varpi_{1}\to M\otimes 3p\varpi_{1}\to\varepsilon\to 0$. Applying the induction functor $\mathrm{ind}_{G_{1}B}^{G}$ to the sequence would then induce an exact sequence of $G$-modules

[TABLE]

with

[TABLE]

One has

[TABLE]

and also

[TABLE]

It follows that $\mathrm{R}^{1}\mathrm{ind}_{G_{1}B}^{G}(\mathrm{soc}^{2}\otimes 3p\varpi_{1})=0$. On the other hand,

[TABLE]

As $\mathrm{ind}_{G_{1}B}^{G}(\varepsilon)\simeq\mathrm{ind}_{B}^{G}(\varepsilon)^{[1]}\simeq L(e)$, $L(e)$ would by (1) be a composition factor of $\nabla(3p\varpi_{1})$, absurd [A86, p. 150]. Thus the extension is non-split. Moreover,

[TABLE]

It follows that the non-split extension $M^{2}$ is unique up to isomorphism.

Just suppose the extension $M^{3}$ of $-\varpi_{2}$ by $M^{2}$ is split. There would then be a $G_{1}P$-submodule $M^{\prime}$ of $\mathrm{soc}^{3}$ containing $\mathrm{soc}^{2}$ such that $M^{\prime}/\mathrm{soc}^{2}\simeq-\varpi_{2}$, and hence an exact sequence $0\to\mathrm{soc}^{2}\otimes p\varpi_{2}\to M^{\prime}\otimes p\varpi_{2}\to\varepsilon\to 0$. Applying the induction functor $\mathrm{ind}_{G_{1}P}^{G}$ to the sequence would induce an exact sequence of $G$-modules

[TABLE]

with

[TABLE]

Also, $\mathrm{ind}_{G_{1}P}^{G}(M^{\prime}\otimes p\varpi_{2})\leq\mathrm{ind}_{G_{1}P}^{G}(\hat{\nabla}_{P}(\varepsilon)\otimes p\varpi_{2})\simeq\mathrm{ind}_{G_{1}P}^{G}(\hat{\nabla}_{P}(p\varpi_{2}))\simeq\nabla(p\varpi_{2})$. Then $L(e)$ would be a composition factor of $\nabla(p\varpi_{2})$, absurd again [A86]. Thus the extension $M^{3}$ is non-split. Moreover, there is a long exact sequence

[TABLE]

with $\mathrm{Ext}^{i}_{P}(-\varpi_{2},-2\varpi_{2})\simeq\mathrm{Ext}^{i}_{P}(\varepsilon,-\varpi_{2})=0$ $\forall i\in\mathbb{N}$ by Bott. Then $\mathrm{Ext}^{1}_{P}(-\varpi_{2},M^{2})\simeq\mathrm{Ext}^{1}_{P}(-\varpi_{2},\nabla^{P}(3\varpi_{1}-3\varpi_{2}))\simeq\mathrm{Ext}^{1}_{P}(\varepsilon,\nabla^{P}(3\varpi_{1}-2\varpi_{2}))\simeq\Bbbk$ as in (2), and hence the extension $M^{3}$ is unique up to isomorphism.

We verify finally that the extension $M^{4}=\mathrm{soc}^{1}_{3,e}$ of $\nabla^{P}(2\varpi_{1}-2\varpi_{2})$ by $M^{3}$ is also non-split and uniquely. Just suppose it split. As $-2\varpi_{1}$ is the lowest weight of $\nabla^{P}(2\varpi_{1}-2\varpi_{2})$, there would be a $G_{1}B$-submodule $M^{\prime\prime}$ of $\mathrm{soc}^{3}\hat{\nabla}_{P}(\varepsilon)$ containing $\mathrm{soc}^{2}\hat{\nabla}_{P}(\varepsilon)$ to form an exact sequence $0\to\mathrm{soc}^{2}\hat{\nabla}_{P}(\varepsilon)\to M^{\prime\prime}\to-2p\varpi_{1}\to 0$, which would induce an exact sequence of $G$-modules

[TABLE]

There are isomorphisms of $G$-modules $\mathrm{ind}_{G_{1}B}^{G}(\varepsilon)\simeq\nabla(\varepsilon)^{[1]}=L(e)$,

[TABLE]

Likewise

[TABLE]

Thus the exact sequence (3) reads

[TABLE]

Also,

[TABLE]

As $L(2p\varpi_{1}))$ is the $G$-socle of $\nabla(2p\varpi_{1})$, we must have $\mathrm{Ext}_{G}^{1}(L(e),L(2p\varpi_{1}))\neq 0$. But the distance between the alcoves containing [math] and $2p\varpi_{1}$ is 12 even, contradicting [A86, 2.10]/[J, C.3]. Thus, $\mathrm{soc}_{3,e}^{1}$ is a nonsplit $P$-extension of $\nabla^{P}(2\varpi_{1}-2\varpi_{2})$ by $M^{3}$. We show next that the extension is unique up to isomorphism. One has for each $i\in\mathbb{N}$

[TABLE]

and

[TABLE]

Then

[TABLE]

as desired.

Comparing $\varpi_{2}\otimes\mathrm{soc}^{1}_{3,e}$ with the adjoint representation of $P$ on its Lie algebra $\mathfrak{p}$, we obtain

[TABLE]

(1.4) Corollary: All multiplicity spaces $\mathrm{soc}^{1}_{i,w}$, $i\in[1,6]$, $w\in W^{P}$, are indecomposable as $P$-modules.

$2^{\circ}$ Decomposition of $F_{*}{\mathcal{O}}_{G/P}$

(2.1) Put ${\mathcal{P}}=G/P$. For each $P$-module $M$ let ${\mathcal{L}}_{\mathcal{P}}(M)$ denote the $G$-equivariant sheaf over ${\mathcal{P}}$ associated to $M$. Sheafifying the socle series of $\hat{\nabla}_{P}(\varepsilon)$ one obtains a filtration of $F_{*}{\mathcal{O}}_{\mathcal{P}}$ with subquotients $\coprod_{w\in W^{P}}L(w)\otimes{\mathcal{L}}_{\mathcal{P}}(\mathrm{soc}_{i,w}^{1})$. We will show that the filtration splits, i.e., the $G_{1}T$-socle series of $\hat{\nabla}_{P}(\varepsilon)$ geometrically splits in the terminology of [DG], to give a decomposition of $F_{*}{\mathcal{O}}_{\mathcal{P}}$ into the direct sum $F_{*}{\mathcal{O}}_{\mathcal{P}}=\coprod_{i=1}^{6}\coprod_{w\in W^{P}}L(w)\otimes{\mathcal{L}}_{\mathcal{P}}(\mathrm{soc}_{i,w}^{1})$ and that ${\mathcal{L}}_{\mathcal{P}}(\mathrm{soc}_{3,e}^{1})$ causes an obstruction to the self extension of $F_{*}{\mathcal{O}}_{\mathcal{P}}$: $\mathrm{Ext}_{\mathcal{P}}^{1}(F_{*}{\mathcal{O}}_{\mathcal{P}},F_{*}{\mathcal{O}}_{\mathcal{P}})\neq 0$. Put ${\mathcal{M}}={\mathcal{L}}_{\mathcal{P}}(\mathrm{soc}_{3,e}^{1})$.

Let $\ell$ denote the length function on $W$ with respect to the simple reflections. For each $w\in W^{P}=\{e,s_{2},s_{1}s_{2},s_{2}s_{1}s_{2},s_{1}s_{2}s_{1}s_{2},w^{P}\}$ put ${\mathcal{E}}(w)={\mathcal{L}}_{\mathcal{P}}(\mathrm{soc}_{\ell(w)+1,w}^{1})$ the $G$-equivariant sheaf over ${\mathcal{P}}$ associated to the $P$-module $\mathrm{soc}_{\ell(w)+1,w}^{1}$. We know from [KY, 3.3], see Appendix, that the ${\mathcal{E}}(w)$, $w\in W^{P}$, form a Karoubian complete strongly exceptional collection of coherent modules over ${\mathcal{P}}$ such that $\forall x,y\in W^{P}$, $\mathbf{Mod}_{\mathcal{P}}({\mathcal{E}}(x),{\mathcal{E}}(y))\neq 0$ iff $x\geq y$ in the Chevalley-Bruhat order. Thus, in order to show that the socle series is geometrially split, it is enough to show that

[TABLE]

Moreover, the $G_{1}T$-socle series on $\hat{\nabla}_{P}(\varepsilon)$ coincides with its radical series [AbK], so that $\mathrm{soc}_{6}\hat{\nabla}_{P}(\varepsilon)$ coincides with the head $\hat{\nabla}_{P}(\varepsilon)/\mathrm{rad}(\hat{\nabla}_{P}(\varepsilon))$ of $\hat{\nabla}_{P}(\varepsilon)$. We know from [K17, §5] that the inclusion $\mathrm{soc}\hat{\nabla}_{P}(\varepsilon)\hookrightarrow\hat{\nabla}_{P}(\varepsilon)$ and the quotient $\hat{\nabla}_{P}(\varepsilon)\twoheadrightarrow\hat{\nabla}_{P}(\varepsilon)/\mathrm{rad}(\hat{\nabla}_{P}(\varepsilon))$ are both geometrically split to yield direct summands $L(e)\otimes{\mathcal{E}}(e)$ and $L(w^{P})\otimes{\mathcal{E}}(w^{P})$ of $F_{*}{\mathcal{O}}_{\mathcal{P}}$, and hence we have only to deal with $w\in\{s_{2}s_{1}s_{2},s_{1}s_{2}s_{1}s_{2}\}$ in (1) and $w=s_{2}$ in (2). We will actually show that all higher extension modules $\mathrm{Ext}^{i}_{\mathcal{P}}({\mathcal{E}}(s_{2}s_{1}s_{2}),{\mathcal{M}})$ $\mathrm{Ext}^{i}_{\mathcal{P}}({\mathcal{E}}(s_{1}s_{2}s_{1}s_{2}),{\mathcal{M}})$, and $\mathrm{Ext}^{i}_{\mathcal{P}}({\mathcal{M}},{\mathcal{E}}(s_{2}))$, $i>0$, vanish.

(2.2) Let us start the computations. Put ${\mathcal{B}}=G/B$ and let ${\mathcal{L}}(M)$ denote the sheaf over ${\mathcal{B}}$ associated to a $B$-module $M$. Let $i\in\mathbb{N}$. One has isomorphisms of $G$-modules

[TABLE]

(2.3) Note that $\nabla(\varpi_{2})$ coincides with the adjoint representation of $G$ on its Lie algebra, and hence that \nabla(\varpi_{2})/\mathfrak{p}\simeq\begin{tabular}[]{|c|}\hline\cr\varpi_{2}\\ \hline\cr\nabla^{P}(3\varpi_{1}-\varpi_{2})\\ \hline\cr\end{tabular}. We will frequently make use of identifications

[TABLE]

For each $i\in\mathbb{N}$ one has isomorhisms of $G$-modules

[TABLE]

giving rise to a long exact sequence

[TABLE]

with isomorphisms of $G$-modules

[TABLE]

and

[TABLE]

Thus the sequence (2) reads as an exact sequence

[TABLE]

and $\mathrm{Ext}^{j}_{\mathcal{P}}({\mathcal{E}}(s_{2}s_{1}s_{2}),{\mathcal{M}})=0$ $\forall j\geq 2$. On the other hand,

[TABLE]

which induces another exact sequence

[TABLE]

Comparing with (3), we must have

[TABLE]

(2.4) For each $i\in\mathbb{N}$ one has isomorhisms of $G$-modules

[TABLE]

giving rise to a long exact sequence

[TABLE]

with $\mathrm{Ext}^{i}_{\mathcal{P}}({\mathcal{E}}(s_{2}s_{1}s_{2}),{\mathcal{M}})\simeq\delta_{i,0}\nabla(\varpi_{1})$ by (2.3). Thus the sequence (1) reads as

[TABLE]

and gives isomorphisms for each $j\geq 1$

[TABLE]

There is a long exact sequence

[TABLE]

with $\mathrm{H}^{i}({\mathcal{P}},{\mathcal{L}}_{\mathcal{P}}(\nabla^{P}(2\varpi_{1})\otimes\nabla(\varpi_{2})))\simeq\delta_{i,0}\nabla(2\varpi_{1})\otimes\nabla(\varpi_{2})$ and also

[TABLE]

Thus the sequence (4) reads as an exact sequence

[TABLE]

and gives $\mathrm{Ext}^{j}_{\mathcal{P}}(\nabla^{P}(2\varpi_{1}-3\varpi_{2})),{\mathcal{M}})=0$ $\forall j\geq 2$.

On the other hand,

[TABLE]

which induces an exact sequence

[TABLE]

Then, together with (5), we must have

[TABLE]

It now follows from (2) and (3) that

[TABLE]

(2.5) As $\nabla(\varpi_{1})$ is simple, $\nabla(\varpi_{1})$ is also a Weyl module of highest weight $\varpi_{1}$, and there is a closed imbedding $i:{\mathcal{P}}\to\mathbb{P}(\nabla(\varpi_{1}))$ such that $i^{*}({\mathcal{O}}_{\mathbb{P}(\nabla(\varpi_{1}))}(1))\simeq{\mathcal{L}}_{\mathcal{P}}(\varpi_{2})$ [J, II.8.5]. For an ${\mathcal{O}}_{\mathcal{P}}$-module ${\mathcal{F}}$ and $n\in\mathbb{Z}$ let us abbreviate ${\mathcal{F}}\otimes_{\mathcal{P}}{\mathcal{L}}_{\mathcal{P}}(n\varpi_{2})$ as ${\mathcal{F}}(n)$. Then ${\mathcal{M}}\simeq{\mathcal{L}}_{\mathcal{P}}(\mathfrak{p})\otimes_{\mathcal{P}}{\mathcal{L}}_{\mathcal{P}}(-\varpi_{2})={\mathcal{L}}_{\mathcal{P}}(\mathfrak{p})(-1)$. We have obtained

[TABLE]

$3^{\circ}$ Extensions

(3.1) Let $i\in\mathbb{N}$. One has

[TABLE]

which gives a long exact sequence

[TABLE]

with

[TABLE]

There follow isomorphisms

[TABLE]

(3.2) Together with (2.5) we find

(3.3) Let $\bar{\mathcal{D}}_{\mathcal{P}}^{(0)}={\mathcal{M}}\!{\it od}_{{\mathcal{O}}_{\mathcal{P}}^{(1)}}({\mathcal{O}}_{\mathcal{P}},{\mathcal{O}}_{\mathcal{P}})$ be the sheaf of rings of small differential operators on ${\mathcal{P}}$ with ${\mathcal{O}}_{\mathcal{P}}^{(1)}$ denoting the sheaf consisting of the $p$-th powers of the elements of ${\mathcal{O}}_{\mathcal{P}}$. This is the first term of the $p$-filtration [Haa] of the sheaf of rings of differential operators [EGA], and is a central reducion of the sheaf of rings of arithmetic differential operators of level 0 [Ber] which is called the sheaf of rings of cristaline differential operators in [BMR].

(3.4) With a little more efforts one can also show

[TABLE]

In particular, ${\mathcal{M}}$ is indecomposable as an ${\mathcal{O}}_{\mathcal{P}}$-modules.

**Proof: **Let $i\in\mathbb{N}$. We first show

[TABLE]

For

[TABLE]

Likewise,

[TABLE]

which gives $\mathrm{Ext}^{j}_{\mathcal{P}}({\mathcal{E}}(s_{2}),{\mathcal{M}})=0$ $\forall j\geq 2$, and an exact sequence

[TABLE]

On the other hand,

[TABLE]

which gives $\mathbf{Mod}_{\mathcal{P}}({\mathcal{E}}(s_{2}),{\mathcal{M}})\leq\nabla(\varpi_{2})=L(\varpi_{2})$. We must then have $\mathbf{Mod}_{\mathcal{P}}({\mathcal{E}}(s_{2}),{\mathcal{M}})=0=\mathrm{Ext}^{1}_{\mathcal{P}}({\mathcal{E}}(s_{2}),{\mathcal{M}})$ also.

Now, \mathrm{Ext}^{i}_{\mathcal{P}}({\mathcal{M}},{\mathcal{M}})=\mathrm{Ext}^{i}_{\mathcal{P}}({\mathcal{L}}_{\mathcal{P}}((-\varpi_{2})\otimes\ker(\nabla(\varpi_{2})\twoheadrightarrow\begin{tabular}[]{|c|}\hline\cr\varpi_{2}\\ \hline\cr\nabla^{P}(3\varpi_{1}-\varpi_{2})\\ \hline\cr\end{tabular})),{\mathcal{M}}) gives rise to a long exact sequence of $G$-modules

[TABLE]

with

[TABLE]

and hence \mathrm{Ext}^{i}_{\mathcal{P}}({\mathcal{M}},{\mathcal{M}})\simeq\mathrm{Ext}^{i+1}_{\mathcal{P}}({\mathcal{L}}_{\mathcal{P}}(\begin{tabular}[]{|c|}\hline\cr\varepsilon\\ \hline\cr\nabla^{P}(3\varpi_{1}-2\varpi_{2})\\ \hline\cr\end{tabular})),{\mathcal{M}}). There arises then a long exact sequence of $G$-modules

[TABLE]

with $\mathrm{Ext}^{\bullet}_{\mathcal{P}}({\mathcal{E}}(e),{\mathcal{M}})=0$ by (1), and hence

[TABLE]

One obtains then another long exact sequence

[TABLE]

with

[TABLE]

Thus the exact sequence above reads

[TABLE]

and isomorphisms $\mathrm{Ext}^{j}_{\mathcal{P}}({\mathcal{M}},{\mathcal{M}})\simeq\mathrm{H}^{j}({\mathcal{P}},{\mathcal{L}}_{\mathcal{P}}(\nabla^{P}(3\varpi_{1}-2\varpi_{2})\otimes\nabla^{P}(3\varpi_{1}-\varpi_{2})))$ $\forall j\geq 2$. Now,

[TABLE]

and hence the sequence (3) reads

[TABLE]

On the other hand, $\mathrm{Ext}^{i}_{\mathcal{P}}({\mathcal{M}},{\mathcal{M}})\simeq\mathrm{H}^{i+1}({\mathcal{P}},{\mathcal{L}}_{\mathcal{P}}(\nabla^{P}(3\varpi_{1}-2\varpi_{2})\otimes\mathfrak{p}))$ from (2) with \nabla^{P}(3\varpi_{1}-2\varpi_{2})\otimes\mathfrak{p}\simeq\nabla^{P}(3\varpi_{1}-2\varpi_{2})\otimes\begin{tabular}[]{|c|}\hline\cr\nabla^{P}(2\varpi_{1}-\varpi_{2})\\ \hline\cr\varepsilon\\ \hline\cr\nabla^{P}(3\varpi_{1}-2\varpi_{2})\\ \hline\cr-\varpi_{2}\\ \hline\cr\end{tabular} having a $P$-module filtration whose subquotients are $\nabla^{P}(5\varpi_{1}-3\varpi_{2})=\nabla^{P}(s_{2}\bullet(-\varpi_{1}+\varpi_{2}))$, $\nabla^{P}(3\varpi_{1}-2\varpi_{2})=\nabla^{P}(s_{2}\bullet 0)$ twice, $\nabla^{P}(\varpi_{1}-\varpi_{2})$, $\nabla^{P}(6\varpi_{1}-4\varpi_{2})=\nabla^{P}(s_{2}s_{1}\bullet\varpi_{1})$, $\nabla^{P}(4\varpi_{1}-3\varpi_{2})=\nabla^{P}(s_{2}s_{1}\bullet 0)$, $\nabla^{P}(2\varpi_{1}-2\varpi_{2})=\nabla^{P}(s_{2}\bullet(-\varpi_{1}))$, $\nabla^{P}(-\varpi_{2})=-\varpi_{2}$, $\nabla^{P}(3\varpi_{1}-3\varpi_{2})=\nabla^{P}(s_{2}s_{1}\bullet(\varpi_{1}-\varpi_{2}))$. It follows that the possible $G$-composition factors of $\mathrm{Ext}^{i}_{\mathcal{P}}({\mathcal{M}},{\mathcal{M}})$ are just $L(e)$ and $L(\varpi_{1})$, and hence the assertion.

Appendixes

A. Keep the notation from the main text. We assume, in particular, that $p\geq 11$. We will recover from [KY] the proof of the fact that the ${\mathcal{E}}(w)={\mathcal{L}}_{\mathcal{P}}(\mathrm{soc}_{\ell(w)+1}^{1})$, $w\in W^{P}$, form a Karoubian complete strongly exceptional sequence in the bounded derived category $\mathrm{D}^{b}(\mathrm{coh}{\mathcal{P}})$ of coherent sheaves on ${\mathcal{P}}$ such that $\mathbf{Mod}_{\mathcal{P}}({\mathcal{E}}(x),{\mathcal{E}}(y))\neq 0$ iff $x\geq y$ in the Chevalley-Bruat order. The present parametrization of the sheaves is twisted from the one in [KY] by the involution $w_{0}?w_{P}$ on $W^{P}$, which reverses the Chevalley-Bruhat order. Incorporating progress since, we also employ [AbK] to replace some ad hoc arguments using [AK89].

(A.1) In order to determine the $G_{1}P$-module structure on $\hat{\nabla}_{P}(\varepsilon)$, we first compute its $G_{1}T$-structure using the formula [AbK, 5.2]. As each $\mathbf{Mod}_{G_{1}}(L(w),\mathrm{soc}_{i}\hat{\nabla}_{P}(\varepsilon))$ is equipped with a structure of $P$-module, one readily finds in the notation of (1.3)

[TABLE]

with \mathrm{soc}_{3,s_{1}s_{2}}^{1}=\begin{tabular}[]{|c|}\hline\cr\nabla^{P}(2\varpi_{1}-2\varpi_{2})\\ \hline\cr\nabla^{P}(\varpi_{1}-2\varpi_{2})\\ \hline\cr\end{tabular} and \mathrm{soc}_{5,s_{1}s_{2}s_{1}s_{2}}^{1}=\begin{tabular}[]{|c|}\hline\cr\nabla^{P}(\varpi_{1}-2\varpi_{2})\\ \hline\cr\nabla^{P}(2\varpi_{1}-3\varpi_{2})\\ \hline\cr\end{tabular}.

We show that both of the last two extensions as $P$-modules are nonsplit and uniquely to yield isomorphisms of $P$-modules

[TABLE]

Just suppose the extension in $\mathrm{soc}_{3,s_{1}s_{2}}^{1}$ is split. Then there would be a $G_{1}B$-submodule $M_{1}$ of $\hat{\nabla}_{P}(\varepsilon)$ containing $\mathrm{soc}^{2}\hat{\nabla}_{P}(\varepsilon)$ such that $M_{1}/\mathrm{soc}^{2}\hat{\nabla}_{P}(\varepsilon)\simeq L(s_{1}s_{2})\otimes(-2\varpi_{1})^{[1]}$ as $G_{1}B$-modules. It would then induce an exact sequence of $G$-modules

[TABLE]

But $\mathrm{R}^{1}\mathrm{ind}_{G_{1}B}^{G}(\mathrm{soc}^{2}\hat{\nabla}_{P}(\varepsilon)\otimes 2p\varpi_{1})$ has no $G$-composition factor whose $G_{1}$-part is $L(s_{1}s_{2})$ while $\mathrm{ind}_{G_{1}B}^{G}(M_{1}\otimes 2p\varpi_{1})\leq\mathrm{ind}_{G_{1}B}^{G}(\hat{\nabla}_{P}(\varepsilon)\otimes 2p\varpi_{1})\leq\mathrm{ind}_{G_{1}B}^{G}(\hat{\nabla}(\varepsilon)\otimes 2p\varpi_{1})\simeq\nabla(2p\varpi_{1})$ with $\nabla(2p\varpi_{1})$ having no composition factor $L(s_{1}s_{2})$, absurd. Also,

[TABLE]

Likewise, just suppose the extension in $\mathrm{soc}_{5,s_{1}s_{2}s_{1}s_{2}}^{1}$ is split. Dualizing $\hat{\nabla}_{P}(\varepsilon)$, by the rigidity of $\hat{\nabla}_{P}(\varepsilon)^{*}\simeq\hat{\nabla}_{P}(3(p-1)\varpi_{2})$ [AbK] there would be a $G_{1}B$-submodule $M_{2}$ of $\mathrm{soc}^{2}\hat{\nabla}_{P}(3(p-1)\varpi_{2})$ containing $\mathrm{soc}\hat{\nabla}_{P}(3(p-1)\varpi_{2})\simeq L(w^{P})\otimes(2\varpi_{2})^{[1]}$ such that $M_{2}/\mathrm{soc}\hat{\nabla}_{P}(3(p-1)\varpi_{2})\simeq L(s_{1}s_{2}s_{1}s_{2})\otimes(-2\varpi_{1}+3\varpi_{2})^{[1]}$ as $G_{1}B$-modules. There would then be an exct sequence of $G$-modules

[TABLE]

with $\mathrm{R}^{1}\mathrm{ind}_{G_{1}B}^{G}(L(w^{P})\otimes p(2\varpi_{1}-\varpi_{2}))\simeq L(w^{P})\otimes\mathrm{R}^{1}\mathrm{ind}_{B}^{G}(2\varpi_{1}-\varpi_{2})^{[1]}=0$. But $\mathrm{ind}_{G_{1}B}^{G}(M_{2}\otimes p(2\varpi_{1}-3\varpi_{2}))\leq\mathrm{ind}_{G_{1}B}^{G}(\hat{\nabla}_{P}(3(p-1)\varpi_{2})\otimes p(2\varpi_{1}-3\varpi_{2}))\leq\mathrm{ind}_{G_{1}B}^{G}(\hat{\nabla}(3(p-1)\varpi_{2})\otimes p(2\varpi_{1}-3\varpi_{2}))\simeq\nabla(2p\varpi_{1}-3\varpi_{2})$ with $\nabla(2p\varpi_{1}-3\varpi_{2})$ having no composition factor $L(s_{1}s_{2}s_{1}s_{2})$, absurd. Also,

[TABLE]

(A.2) We set ${\mathcal{E}}(w)={\mathcal{L}}_{\mathcal{P}}(\mathrm{soc}_{\ell(w)+1}^{1})$ for each $w\in W^{P}$, and determine their mutual extensions $\mathrm{Ext}^{i}_{\mathcal{P}}({\mathcal{E}}(x),{\mathcal{E}}(y))$, $x,y\in W^{P}$, $i\in\mathbb{N}$, as $G$-modules.

We start with the computations involving ${\mathcal{E}}(e)$. We have

[TABLE]

(A.3) We compute next the extensions with ${\mathcal{E}}(s_{2})$. Let $i\in\mathbb{N}$. One has

[TABLE]

(A.4) Let $i\in\mathbb{N}$. We have

[TABLE]

As ${\mathcal{E}}(s_{1}s_{2})\simeq{\mathcal{L}}_{\mathcal{P}}((-\varpi_{2})\otimes\ker(\nabla(\varpi_{1})\twoheadrightarrow\nabla^{P}(\varpi_{1})))$ by (A.1.2), there is a long exact sequence

[TABLE]

with

[TABLE]

Thus, $\mathrm{Ext}^{i}_{\mathcal{P}}({\mathcal{E}}(s_{1}s_{2}),{\mathcal{E}}(s_{1}s_{2}))\simeq\mathrm{Ext}^{i+1}_{\mathcal{P}}({\mathcal{L}}_{\mathcal{P}}((-\varpi_{2})\otimes\nabla^{P}(\varpi_{1})),{\mathcal{E}}(s_{1}s_{2}))$, the right hand side of which fits into another long exact sequence

[TABLE]

with $\mathrm{Ext}^{\bullet}_{\mathcal{P}}({\mathcal{L}}_{\mathcal{P}}((-\varpi_{2})\otimes\nabla^{P}(\varpi_{1})),{\mathcal{L}}_{\mathcal{P}}((-\varpi_{2})\otimes\nabla(\varpi_{1})))\simeq\mathrm{H}^{\bullet}({\mathcal{P}},{\mathcal{L}}_{\mathcal{P}}(\nabla^{P}(\varpi_{1})^{*}))\otimes\nabla(\varpi_{1})\simeq\mathrm{H}^{\bullet}({\mathcal{P}},{\mathcal{L}}_{\mathcal{P}}(\nabla^{P}(\varpi_{1}-\varpi_{2})))\otimes\nabla(\varpi_{1})=0$. It follows that

[TABLE]

To find $\mathrm{Ext}^{i}_{\mathcal{P}}({\mathcal{E}}(s_{1}s_{2}),{\mathcal{E}}(s_{2}s_{1}s_{2}))$, consider the long exact sequence (3) with ${\mathcal{E}}(s_{1}s_{2})$ in the covariant entries replaced by ${\mathcal{E}}(s_{2}s_{1}s_{2})$. As $\mathrm{Ext}^{\bullet}_{\mathcal{P}}({\mathcal{L}}_{\mathcal{P}}((-\varpi_{2})\otimes\nabla(\varpi_{1})),{\mathcal{E}}(s_{2}s_{1}s_{2}))\simeq\mathrm{Ext}^{\bullet}_{\mathcal{P}}({\mathcal{E}}(s_{2}),{\mathcal{E}}(s_{2}s_{1}s_{2}))\otimes\nabla(\varpi_{1})^{*}=0$ by (A.3.4), one obtains

[TABLE]

Then

[TABLE]

the right hand side of which fits into the long exact sequence (3) with ${\mathcal{E}}(s_{1}s_{2})$ in the covariant entries replaced by ${\mathcal{L}}_{\mathcal{P}}(\nabla^{P}(2\varpi_{1}-3\varpi_{2}))$. As $\mathrm{Ext}^{\bullet}_{\mathcal{P}}({\mathcal{L}}_{\mathcal{P}}((-\varpi_{2})\otimes\nabla(\varpi_{1})),{\mathcal{L}}_{\mathcal{P}}(\nabla^{P}(2\varpi_{1}-3\varpi_{2})))\simeq\nabla(\varpi_{1})^{*}\otimes\mathrm{H}^{\bullet}({\mathcal{P}},{\mathcal{L}}_{\mathcal{P}}(\nabla^{P}(2\varpi_{1}-2\varpi_{2})))=0$, one obtains

[TABLE]

Finally, one has

[TABLE]

(A.5) Let $i\in\mathbb{N}$. We have

[TABLE]

By (A.1.2) there is a long exact sequence

[TABLE]

with

[TABLE]

and hence

[TABLE]

Finally,

[TABLE]

(A.6) Let $i\in\mathbb{N}$. We have

[TABLE]

One has

[TABLE]

with $\mathrm{Ext}^{i}_{\mathcal{P}}({\mathcal{E}}(s_{1}s_{2}s_{1}s_{2}),{\mathcal{L}}_{\mathcal{P}}(\nabla^{P}(\varpi_{1}-2\varpi_{2})))\simeq\delta_{i,0}\{\nabla(\varpi_{1})\oplus\varepsilon\}$ by (3) while

[TABLE]

It follows that

[TABLE]

One has

[TABLE]

the right hand side of which fits by (A.1.2) into the long exact sequence (A.5.5) with ${\mathcal{E}}(s_{2}s_{1}s_{2})$ replaced by ${\mathcal{L}}_{\mathcal{P}}(\nabla^{P}(2\varpi_{1}-3\varpi_{2}))$. As $\mathrm{Ext}^{\bullet}_{\mathcal{P}}({\mathcal{L}}_{\mathcal{P}}(\nabla^{P}(2\varpi_{1}-3\varpi_{2}),{\mathcal{L}}_{\mathcal{P}}((-2\varpi_{2})\otimes\nabla(\varpi_{1})))\simeq\mathrm{H}^{\bullet}({\mathcal{P}},{\mathcal{L}}_{\mathcal{P}}(\nabla^{P}(2\varpi_{1}-\varpi_{2})))\otimes\nabla(\varpi_{1})=0$, one obtains

[TABLE]

Finally, one has

[TABLE]

(A.7) Let $i\in\mathbb{N}$. We have

[TABLE]

(A.8) We have thus shown

[TABLE]

(A.9) To see that the ${\mathcal{E}}(w)$, $w\in W^{P}$, Karoubian generate $\mathrm{D}^{b}(\mathrm{coh}{\mathcal{P}})$, it is enough by a result attributed to Kontsevich by Positselskii [BMR02, Th. 3.5.1] to verify that all ${\mathcal{L}}_{\mathcal{P}}(-2n\rho_{P})$, $n\in\mathbb{N}^{+}$, are Karoubian generated by the ${\mathcal{E}}(w)$’s, where $2\rho_{P}=\sum_{\beta\in R^{+}\setminus\{\alpha_{1}\}}\beta=2\varpi_{2}$ in the present setting.

Let $\hat{\mathcal{E}}=\langle{\mathcal{E}}(w)\mid w\in W^{P}\rangle$ denote the triangulated subcategory of $\mathrm{D}^{b}(\mathrm{coh}{\mathcal{P}})$ Karoubian generated by ${\mathcal{E}}(w)$, $w\in W^{P}$. Let $\pi:{\mathcal{B}}\to{\mathcal{P}}$ be the natural morphism. Recall from [Or, 1.3.6] the projection formula $\mathrm{id}_{\mathrm{D}^{b}(\mathrm{coh}{\mathcal{P}})}\simeq(\mathbf{R}\pi_{*})\circ\pi^{*}:\mathrm{D}^{b}(\mathrm{coh}{\mathcal{P}})\to\mathrm{D}^{b}(\mathrm{coh}{\mathcal{P}})$. Recall also from [J, I.5.17] an isomorphism $\pi^{*}{\mathcal{L}}_{\mathcal{P}}(M)\simeq{\mathcal{L}}(M)$ for each $P$-module $M$, and from [J, I.5.19] an isomorphism $\mathbf{R}\pi_{*}{\mathcal{L}}(m\varpi_{1}+n\varpi_{2})\simeq{\mathcal{L}}_{\mathcal{P}}(\nabla^{P}(m\varpi_{1}+n\varpi_{2}))$ for each $m\in\mathbb{N}$ and $n\in\mathbb{Z}$. Then, setting $\tilde{\mathcal{E}}=\langle\pi^{*}{\mathcal{E}}(w)|w\in W^{P}\rangle$ to be the triangulated subcategory of $\mathrm{D}^{b}(\mathrm{coh}{\mathcal{B}})$ Karoubian generated by $\pi^{*}{\mathcal{E}}(w)$, $w\in W^{P}$, it is enough to show that

[TABLE]

For our purpose note also that whenever ${\mathcal{L}}(\nabla^{P}(m\varpi_{1}+n\varpi_{2}))\in\tilde{\mathcal{E}}$, $m\in\mathbb{N}$, $n\in\mathbb{Z}$, we may also allow ${\mathcal{L}}(m\varpi_{1}+n\varpi_{2})\in\tilde{\mathcal{E}}$, $m\in\mathbb{N}$, and conversely.

We thus start with $\tilde{\mathcal{E}}=\langle{\mathcal{O}}_{\mathcal{B}},{\mathcal{L}}(-\varpi_{2}),{\mathcal{L}}(\nabla^{P}(\varpi_{1}-2\varpi_{2})),{\mathcal{L}}(\varpi_{1}-2\varpi_{2}),{\mathcal{L}}(\nabla^{P}(2\varpi_{1}-2\varpi_{2})),{\mathcal{L}}(2\varpi_{1}-2\varpi_{2}),{\mathcal{L}}(\nabla^{P}(2\varpi_{1}-3\varpi_{2})),{\mathcal{L}}(2\varpi_{1}-3\varpi_{2}),{\mathcal{L}}(-2\varpi_{2})\rangle$. In view of the $B$-filtration on \nabla^{P}(\varpi_{1}-2\varpi_{2})=\begin{tabular}[]{|c|}\hline\cr\varpi_{1}-2\varpi_{2}\\ \hline\cr-\rho\\ \hline\cr\end{tabular}, as ${\mathcal{L}}(\nabla^{P}(\varpi_{1}-2\varpi_{2}))$ and ${\mathcal{L}}(\varpi_{1}-2\varpi_{2})\in\tilde{\mathcal{E}}$, one has also ${\mathcal{L}}(-\rho)\in\tilde{\mathcal{E}}$. Likewise from the $B$-filtration \nabla^{P}(2\varpi_{1}-2\varpi_{2})=\begin{tabular}[]{|c|}\hline\cr2\varpi_{1}-2\varpi_{2}\\ \hline\cr-\varpi_{2}\\ \hline\cr-2\varpi_{1}\\ \hline\cr\end{tabular}, get ${\mathcal{L}}(-2\varpi_{1})\in\tilde{\mathcal{E}}$. Then from \nabla^{P}(2\varpi_{1}-3\varpi_{2})=\begin{tabular}[]{|c|}\hline\cr2\varpi_{1}-3\varpi_{2}\\ \hline\cr-2\varpi_{2}\\ \hline\cr-2\varpi_{1}-\varpi_{2}\\ \hline\cr\end{tabular}, get ${\mathcal{L}}(-2\varpi_{1}-\varpi_{2})\in\tilde{\mathcal{E}}$.

We now explain our strategy. Recall from [HKR, 5.1.4] the Koszul resolution

[TABLE]

Tensoring entries of $\tilde{\mathcal{E}}$ with $\nabla(\varpi_{1})$ and $\nabla(\varpi_{2})$, we will find ${\mathcal{L}}(r\varpi_{1}-\varpi_{2})\in\tilde{\mathcal{E}}$, $r\in[-3,3]$. Then from (2) we obtain all ${\mathcal{L}}(n\varpi_{1}-\varpi_{2})\in\tilde{\mathcal{E}}$, $n\in\mathbb{Z}$, and hence also all ${\mathcal{L}}(\nabla^{P}(m\varpi_{1}-\varpi_{2}))\in\tilde{\mathcal{E}}$, $m\in\mathbb{N}$. That, in turn, will yield all ${\mathcal{L}}(r\varpi_{1})\in\tilde{\mathcal{E}}$, $r\in[-4,2]$, and hence by (2) again all ${\mathcal{L}}(n\varpi_{1})$, $n\in\mathbb{Z}$, in $\tilde{\mathcal{E}}$. Then get all ${\mathcal{L}}(r\varpi_{1}-2\varpi_{2})\in\tilde{\mathcal{E}}$, $r\in[-2,4]$, and hence all ${\mathcal{L}}(n\varpi_{1}-2\varpi_{2})\in\tilde{\mathcal{E}}$, $n\in\mathbb{Z}$. Thus, all the weights $\nu$ of $\nabla(\varpi_{2})\otimes(-\varpi_{2})$ will have ${\mathcal{L}}(\nu)\in\tilde{\mathcal{E}}$, and hence tensoring ${\mathcal{L}}(n\varpi_{1}-\varpi_{2})$ with $\nabla(\varpi_{2})\otimes{\mathcal{L}}(-\varpi_{2})$ will yield all ${\mathcal{L}}(n\varpi_{1}-3\varpi_{2}))\in\tilde{\mathcal{E}}$, $n\in\mathbb{Z}$. Repeat the procedure to obtain all ${\mathcal{L}}(n\varpi_{1}-m\varpi_{2})\in\tilde{\mathcal{E}}$, $n\in\mathbb{Z}$, $m\in\mathbb{N}$, and hence in particular (1).

(A.10) In (A.9) we have added ${\mathcal{L}}(-\rho),{\mathcal{L}}(-2\varpi_{1})$ and ${\mathcal{L}}(-2\varpi_{1}-\varpi_{2})$ to $\tilde{\mathcal{E}}$, so that $\tilde{\mathcal{E}}=\langle{\mathcal{O}}_{\mathcal{B}},{\mathcal{L}}(-\varpi_{2}),{\mathcal{L}}(\nabla^{P}(\varpi_{1}-2\varpi_{2})),{\mathcal{L}}(\varpi_{1}-2\varpi_{2}),{\mathcal{L}}(\nabla^{P}(2\varpi_{1}-2\varpi_{2})),{\mathcal{L}}(2\varpi_{1}-2\varpi_{2}),{\mathcal{L}}(\nabla^{P}(2\varpi_{1}-3\varpi_{2})),{\mathcal{L}}(2\varpi_{1}-3\varpi_{2}),{\mathcal{L}}(-2\varpi_{2}),{\mathcal{L}}(-\rho),{\mathcal{L}}(-2\varpi_{1}),{\mathcal{L}}(-2\varpi_{1}-\varpi_{2})\rangle$.

As ${\mathcal{L}}(-\varpi_{2})\in\tilde{\mathcal{E}}$, so does

[TABLE]

As ${\mathcal{L}}(\nabla^{P}(\varpi_{1}-2\varpi_{2}))$ and ${\mathcal{L}}(\nabla^{P}(2\varpi_{1}-2\varpi_{2}))\in\tilde{\mathcal{E}}$, one has ${\mathcal{L}}(\nabla^{P}(\varpi_{1}-\varpi_{2}))\in\tilde{\mathcal{E}}$, and hence also ${\mathcal{L}}(\varpi_{1}-\varpi_{2})\in\tilde{\mathcal{E}}$. Then, from \nabla^{P}(\varpi_{1}-\varpi_{2})=\begin{tabular}[]{|c|}\hline\cr\varpi_{1}-\varpi_{2}\\ \hline\cr-\varpi_{1}\\ \hline\cr\end{tabular}, get also ${\mathcal{L}}(-\varpi_{1})\in\tilde{\mathcal{E}}$.

Likewise, as ${\mathcal{L}}(-2\varpi_{2})\in\tilde{\mathcal{E}}$, get from \nabla(\varpi_{1})\otimes(-2\varpi_{2})\simeq\begin{tabular}[]{|c|}\hline\cr\nabla^{P}(\varpi_{1}-2\varpi_{2})\\ \hline\cr\nabla^{P}(2\varpi_{1}-3\varpi_{2})\\ \hline\cr\nabla^{P}(\varpi_{1}-3\varpi_{2})\\ \hline\cr\end{tabular}) that ${\mathcal{L}}(\nabla^{P}(\varpi_{1}-3\varpi_{2}))$ and ${\mathcal{L}}(\varpi_{1}-3\varpi_{2})\in\tilde{\mathcal{E}}$. Then, from \nabla^{P}(\varpi_{1}-3\varpi_{2})=\begin{tabular}[]{|c|}\hline\cr\varpi_{1}-3\varpi_{2}\\ \hline\cr-\varpi_{1}-2\varpi_{2}\\ \hline\cr\end{tabular}, get also ${\mathcal{L}}(-\varpi_{1}-2\varpi_{2})\in\tilde{\mathcal{E}}$.

One has $\nabla(\varpi_{1})\otimes{\mathcal{L}}(\varpi_{1}-2\varpi_{2})\in\tilde{\mathcal{E}}$. As all weights $\nu$ of $\nabla(\varpi_{1})\otimes(\varpi_{1}-2\varpi_{2})$ except $3\varpi_{1}-3\varpi_{2}$ have ${\mathcal{L}}(\nu)\in\tilde{\mathcal{E}}$, ${\mathcal{L}}(3\varpi_{1}-3\varpi_{2})\in\tilde{\mathcal{E}}$, and hence also ${\mathcal{L}}(\nabla^{P}(3\varpi_{1}-3\varpi_{2}))\in\tilde{\mathcal{E}}$. Then, as \nabla^{P}(3\varpi_{1}-3\varpi_{2})=\begin{tabular}[]{|c|}\hline\cr3\varpi_{1}-3\varpi_{2}\\ \hline\cr\varpi_{1}-2\varpi_{2}\\ \hline\cr-\varpi_{1}-\varpi_{2}\\ \hline\cr-3\varpi_{1}\\ \hline\cr\end{tabular}, get also ${\mathcal{L}}(-3\varpi_{1})\in\tilde{\mathcal{E}}$.

Now, $\nabla(\varpi_{2})\otimes{\mathcal{L}}(-\varpi_{2})\in\tilde{\mathcal{E}}$ with

[TABLE]

As all components of $\nabla(\varpi_{2})\otimes{\mathcal{L}}(-\varpi_{2})$ except ${\mathcal{L}}(\nabla^{P}(3\varpi_{1}-2\varpi_{2}))$ belong to $\tilde{\mathcal{E}}$, so does ${\mathcal{L}}(\nabla^{P}(3\varpi_{1}-2\varpi_{2}))$, and hence also ${\mathcal{L}}(3\varpi_{1}-2\varpi_{2})$. As \nabla^{P}(3\varpi_{1}-2\varpi_{2})=\begin{tabular}[]{|c|}\hline\cr3\varpi_{1}-2\varpi_{2}\\ \hline\cr\varpi_{1}-\varpi_{2}\\ \hline\cr-\varpi_{1}\\ \hline\cr-3\varpi_{1}+\varpi_{2}\\ \hline\cr\end{tabular} has all its weights $\nu$ but $-3\varpi_{1}+\varpi_{2}$ such that ${\mathcal{L}}(\nu)\in\tilde{\mathcal{E}}$, ${\mathcal{L}}(-3\varpi_{1}+\varpi_{2})\in\tilde{\mathcal{E}}$.

We have seen above that ${\mathcal{L}}(\varpi_{1}-\varpi_{2})\in\tilde{\mathcal{E}}$, and hence also $\nabla(\varpi_{1})\otimes{\mathcal{L}}(\varpi_{1}-\varpi_{2})\in\tilde{\mathcal{E}}$. As all weights $\nu$ of $\nabla(\varpi_{1})\otimes(\varpi_{1}-\varpi_{2})$ except $2\varpi_{1}-\varpi_{2}$ have ${\mathcal{L}}(\nu)\in\tilde{\mathcal{E}}$, ${\mathcal{L}}(2\varpi_{1}-\varpi_{2})\in\tilde{\mathcal{E}}$, and hence also ${\mathcal{L}}(\nabla^{P}(2\varpi_{1}-\varpi_{2}))\in\tilde{\mathcal{E}}$. Then, as \nabla^{P}(2\varpi_{1}-\varpi_{2})=\begin{tabular}[]{|c|}\hline\cr2\varpi_{1}-\varpi_{2}\\ \hline\cr\varepsilon\\ \hline\cr-2\varpi_{1}+\varpi_{2}\\ \hline\cr\end{tabular}, get also ${\mathcal{L}}(-2\varpi_{1}+\varpi_{2})\in\tilde{\mathcal{E}}$.

One has $\nabla(\varpi_{1})\otimes{\mathcal{O}}_{\mathcal{B}}\in\tilde{\mathcal{E}}$ with \nabla(\varpi_{1})=\begin{tabular}[]{|c|}\hline\cr\nabla^{P}(\varpi_{1})\\ \hline\cr\nabla^{P}(2\varpi_{1}-\varpi_{2})\\ \hline\cr\nabla^{P}(\varpi_{1}-\varpi_{2})\\ \hline\cr\end{tabular}. As all components except ${\mathcal{L}}(\nabla^{P}(\varpi_{1}))$ belong to $\tilde{\mathcal{E}}$, so does ${\mathcal{L}}(\nabla^{P}(\varpi_{1}))$, and hence also ${\mathcal{L}}(\varpi_{1})$. Then, as \nabla^{P}(\varpi_{1})=\begin{tabular}[]{|c|}\hline\cr\varpi_{1}\\ \hline\cr-\varpi_{1}+\varpi_{2}\\ \hline\cr\end{tabular}, ${\mathcal{L}}(-\varpi_{1}+\varpi_{2})\in\tilde{\mathcal{E}}$ also.

As $\nabla(\varpi_{1})\otimes{\mathcal{L}}(2\varpi_{1}-2\varpi_{2})\in\tilde{\mathcal{E}}$ and as all weights $\nu$ of $\nabla(\varpi_{1})\otimes(2\varpi_{1}-2\varpi_{2})$ except $4\varpi_{1}-3\varpi_{2}$ have ${\mathcal{L}}(\nu)\in\tilde{\mathcal{E}}$, ${\mathcal{L}}(4\varpi_{1}-3\varpi_{2})\in\tilde{\mathcal{E}}$, and hence also ${\mathcal{L}}(\nabla^{P}(4\varpi_{1}-3\varpi_{2}))\in\tilde{\mathcal{E}}$. Then, as all weights $\nu$ of $\nabla^{P}(4\varpi_{1}-3\varpi_{2})$ except $-4\varpi_{1}+\varpi_{2}$ have ${\mathcal{L}}(\nu)\in\tilde{\mathcal{E}}$, ${\mathcal{L}}(-4\varpi_{1}+\varpi_{2})\in\tilde{\mathcal{E}}$.

Now, $\nabla(\varpi_{2})\otimes{\mathcal{L}}(\varpi_{1}-\varpi_{2})\in\tilde{\mathcal{E}}$. As all weights $\nu$ of $\nabla(\varpi_{2})\otimes(\varpi_{1}-\varpi_{2})$ except $4\varpi_{1}-2\varpi_{2}$ have ${\mathcal{L}}(\nu)\in\tilde{\mathcal{E}}$, ${\mathcal{L}}(4\varpi_{1}-2\varpi_{2})\in\tilde{\mathcal{E}}$. Likewise, as ${\mathcal{L}}(-\rho)\in\tilde{\mathcal{E}}$, so does $\nabla(\varpi_{2})\otimes{\mathcal{L}}(-\rho)$. As all weights $\nu$ of $\nabla(\varpi_{2})\otimes(-\rho)$ except $-4\varpi_{1}$ have ${\mathcal{L}}(\nu)\in\tilde{\mathcal{E}}$, get ${\mathcal{L}}(-4\varpi_{1})\in\tilde{\mathcal{E}}$.

One has $\nabla(\varpi_{1})\otimes{\mathcal{L}}(-2\varpi_{1}-\varpi_{2})\in\tilde{\mathcal{E}}$. As all weights $\nu$ of $\nabla(\varpi_{1})\otimes(-2\varpi_{1}-\varpi_{2})$ except $-3\varpi_{1}-\varpi_{2}$ have ${\mathcal{L}}(\nu)\in\tilde{\mathcal{E}}$, so does ${\mathcal{L}}(-3\varpi_{1}-\varpi_{2})$. Likewise, we have seen ${\mathcal{L}}(2\varpi_{1}-\varpi_{2})\in\tilde{\mathcal{E}}$, and hence $\nabla(\varpi_{1})\otimes{\mathcal{L}}(2\varpi_{1}-\varpi_{2})\in\tilde{\mathcal{E}}$. As all weights $\nu$ of $\nabla(\varpi_{1})\otimes(2\varpi_{1}-\varpi_{2})$ except $3\varpi_{1}-\varpi_{2}$ have ${\mathcal{L}}(\nu)\in\tilde{\mathcal{E}}$, so does ${\mathcal{L}}(3\varpi_{1}-\varpi_{2})$. At this point we have all ${\mathcal{L}}(r\varpi_{1}-\varpi_{2})\in\tilde{\mathcal{E}}$, $r\in[-3,3]$. Then, using (A.9.2), one obtains

[TABLE]

Then, as \nabla(\varpi_{2})=\begin{tabular}[]{|c|}\hline\cr\varpi_{2}\\ \hline\cr\nabla^{P}(3\varpi_{1}-\varpi_{2})\\ \hline\cr\nabla^{P}(2\varpi_{1}-\varpi_{2})\\ \hline\cr\varepsilon\\ \hline\cr\nabla^{P}(3\varpi_{1}-2\varpi_{2})\\ \hline\cr-\varpi_{2}\\ \hline\cr\end{tabular}, all components of $\nabla(\varpi_{2})\otimes{\mathcal{O}}_{\mathcal{B}}\in\tilde{\mathcal{E}}$ except ${\mathcal{L}}(\varpi_{2})$ belong to $\tilde{\mathcal{E}}$, so does ${\mathcal{L}}(\varpi_{2})$. In turn, $\nabla(\varpi_{1})\otimes{\mathcal{L}}(\varpi_{1})\in\tilde{\mathcal{E}}$. As all weights $\nu$ of $\nabla(\varpi_{1})\otimes\varpi_{1}$ except $2\varpi_{1}$ have ${\mathcal{L}}(\nu)\in\tilde{\mathcal{E}}$, one has ${\mathcal{L}}(2\varpi_{1})\in\tilde{\mathcal{E}}$. Thus, all ${\mathcal{L}}(r\varpi_{1})\in\tilde{\mathcal{E}}$, $r\in[-4,2]$. Then, using (A.9.2) again, get

[TABLE]

By (1) one has $\nabla(\varpi_{1})\otimes{\mathcal{L}}(-3\varpi_{1}-\varpi_{2})\in\tilde{\mathcal{E}}$. As all weights $\nu$ of $\nabla(\varpi_{1})\otimes(-3\varpi_{1}-\varpi_{2})$ except $-2\varpi_{1}-2\varpi_{2}$ have ${\mathcal{L}}(\nu)\in\tilde{\mathcal{E}}$, one has ${\mathcal{L}}(-2\varpi_{1}-2\varpi_{2})\in\tilde{\mathcal{E}}$. Then all ${\mathcal{L}}(r\varpi_{1}-2\varpi_{2})\in\tilde{\mathcal{E}}$, $r\in[-2,4]$. It follows from (A.9.2) that

[TABLE]

Then, $\nabla(\varpi_{1})\otimes{\mathcal{L}}(-\varpi_{1}-2\varpi_{2})\in\tilde{\mathcal{E}}$. As all weights $\nu$ of $\nabla(\varpi_{1})\otimes(-\varpi_{1}-2\varpi_{2})$ except $-3\varpi_{2}$ have ${\mathcal{L}}(\nu)\in\tilde{\mathcal{E}}$ by (1) and (3), one has ${\mathcal{L}}(-3\varpi_{2})\in\tilde{\mathcal{E}}$. Likewise, $\nabla(\varpi_{1})\otimes{\mathcal{L}}(n\varpi_{1}-2\varpi_{2})\in\tilde{\mathcal{E}}$, $n\in\mathbb{Z}$, will yield

[TABLE]

In order to get all ${\mathcal{L}}(n\varpi_{1}-4\varpi_{2})\in\tilde{\mathcal{E}}$, $n\in\mathbb{Z}$, consider $\nabla(\varpi_{2})\otimes{\mathcal{L}}(n\varpi_{1}-2\varpi_{2})\in\tilde{\mathcal{E}}$ by (3). All the weights $\nu$ of $\nabla(\varpi_{2})\otimes(n\varpi_{1}-2\varpi_{2})$ except $(n+3)\varpi_{1}-4\varpi_{2}$ have ${\mathcal{L}}(\nu)\in\tilde{\mathcal{E}}$ by (1), (3) and (4), and hence also ${\mathcal{L}}((n+3)\varpi_{1}-4\varpi_{2})\in\tilde{\mathcal{E}}$. To see all ${\mathcal{L}}(n\varpi_{1}-5\varpi_{2})\in\tilde{\mathcal{E}}$, use $\nabla(\varpi_{2})\otimes{\mathcal{L}}(n\varpi_{1}-3\varpi_{2})\in\tilde{\mathcal{E}}$ from (4) to obtain ${\mathcal{L}}((n+3)\varpi_{1}-5\varpi_{2})\in\tilde{\mathcal{E}}$. Repeat to get all ${\mathcal{L}}(\nabla^{P}(n\varpi_{1}-m\varpi_{2}))\in\tilde{\mathcal{E}}$, $m\in\mathbb{N}$, $n\in\mathbb{Z}$, as desired.

(A.10) We have thus obtained

B. Let us also write down an easier case of the parabolic $P$ associated to the long simple root; parametrization of the sheaves is, as in A, different from the one in [KY] twisted by $w_{0}?w_{P}$. We assume that $p\geq 11$.

(B.1) Let $\mathrm{soc}^{i}\hat{\nabla}_{P}(\varepsilon)$ denote the $i$-th $G_{1}T$-socle of $\hat{\nabla}_{P}(\varepsilon)=\mathrm{ind}_{G_{1}P}^{G}(\varepsilon)$, and put $\mathrm{soc}_{i}\hat{\nabla}_{P}(\varepsilon)=\mathrm{soc}^{i}\hat{\nabla}_{P}(\varepsilon)/\mathrm{soc}^{i-1}\hat{\nabla}_{P}(\varepsilon)$. From its $G_{1}T$-structure we readily obtain

[TABLE]

where $\nabla^{P}=\mathrm{ind}_{B}^{P}$ and $w^{P}=s_{1}s_{2}s_{1}s_{2}s_{1}$.

For $w\in W^{P}$ we let $L(w)\otimes(\mathrm{soc}^{1}_{i,w})^{[1]}$ denote the $L(w)$-isotypic part of $\mathrm{soc}_{i}\hat{\nabla}_{P}(\varepsilon)$. We show that 3\varpi_{1}\otimes\mathrm{soc}^{1}_{4,s_{1}s_{2}s_{1}}=\begin{tabular}[]{|c|}\hline\cr\varpi_{1}\\ \hline\cr\nabla^{P}(-\varpi_{1}+\varpi_{2})\\ \hline\cr\varepsilon\\ \hline\cr\end{tabular} is an indecomposable $P$-module isomorphic to the quotient of $\nabla(\varpi_{1})$ by a $P$-submodule generated by a vector of weight $-2\varpi_{1}+\varpi_{2}$.

(B.2) Just suppose $(-3\varpi_{1})\otimes\nabla^{P}(-\varpi_{1}+\varpi_{2})\simeq\nabla^{P}(-4\varpi_{1}+\varpi_{2})$ is a $P$-submodule of $\mathrm{soc}^{1}_{4,s_{1}s_{2}s_{1}}$. Then, $-3\varpi_{1}$ would be a direct summand of $\mathrm{soc}^{1}_{4,s_{1}s_{2}s_{1}}$ as $\mathrm{Ext}^{1}_{P}(-2\varpi_{1},-3\varpi_{1})=0$. Dualizing, $L(s_{1}s_{2}s_{1})\otimes(3\varpi_{1})$ would be a direct summand of the third subquotient $\mathrm{rad}_{3}(\hat{\nabla}_{P}(\varepsilon)^{*})=\mathrm{rad}^{3}(\hat{\nabla}_{P}(\varepsilon)^{*})/\mathrm{rad}^{4}(\hat{\nabla}_{P}(\varepsilon)^{*})$ in the $G_{1}T$ radical series of $\hat{\nabla}_{P}(\varepsilon)^{*}\simeq\hat{\nabla}_{P}(5(p-1)\varpi_{1})$. By [AbK] one has $\mathrm{rad}^{i}\hat{\nabla}_{P}(5(p-1)\varpi_{1})=\mathrm{soc}^{6-i}\hat{\nabla}_{P}(5(p-1)\varpi_{1})$. It would follow that there is a $P$-submodule $M$ of $\mathrm{soc}^{3}\hat{\nabla}_{P}(5(p-1)\varpi_{1})$ containing $\mathrm{soc}^{2}\hat{\nabla}_{P}(5(p-1)\varpi_{1})$ such that $M/\mathrm{soc}^{2}\hat{\nabla}_{P}(5(p-1)\varpi_{1})\simeq L(s_{1}s_{2}s_{1})\otimes 3p\varpi_{1}$. That would induce an exact sequence of $G$-modules

[TABLE]

But the $G_{1}P$-components of $\mathrm{soc}^{2}\hat{\nabla}_{P}((2p-5)\varpi_{1})$ are just $L(w^{P})\otimes p\varpi_{1}$, $L(e)\otimes p\varpi_{1}$, and $L(s_{2}s_{1}s_{2}s_{1})$ [KY, 1.6.12], and hence $\mathrm{R}^{1}\mathrm{ind}_{G_{1}P}^{G}(\mathrm{soc}^{2}(\hat{\nabla}_{P}((2p-5)\varpi_{1})))=0$. Also,

[TABLE]

with $\nabla((2p-5)\varpi_{1})$ not having a $G$-composition factor $L(s_{1}s_{2}s_{1})$ [A86, p. 149], absurd.

Now,

[TABLE]

Thus, $3\varpi_{1}\otimes\mathrm{soc}^{1}_{4,s_{1}s_{2}s_{1}}$ has, up to isomorphism, a unique indecomposable $P$-submodule $E$ extending $\nabla^{P}(-\varpi_{1}+\varpi_{2})$ by $\varepsilon$.

Next, just suppose $-2\varpi_{1}\leq\mathrm{soc}^{1}_{4,s_{1}s_{2}s_{1}}$. Then there would be a $P$-submodule $M^{\prime}$ of $\mathrm{soc}^{4}\hat{\nabla}_{P}(\varepsilon)$ containing $\mathrm{soc}^{3}\hat{\nabla}_{P}(\varepsilon)$ such that $M^{\prime}/\mathrm{soc}^{3}\hat{\nabla}_{P}(\varepsilon)\simeq L(s_{1}s_{2}s_{1})\otimes(-2\varpi_{1})^{[1]}$, which would induce an exact sequence of $G$-modules

[TABLE]

But $\mathrm{ind}_{G_{1}P}^{G}(M^{\prime}\otimes 2p\varpi_{1})\leq\mathrm{ind}_{G_{1}P}^{G}(\hat{\nabla}_{P}(\varepsilon)\otimes 2p\varpi_{1})\simeq\nabla(2p\varpi_{1})$ with $\nabla(2p\varpi_{1})$ having no composition factor $L(s_{1}s_{2}s_{1})$ [A86, p. 149], and neither $\mathrm{R}^{1}\mathrm{ind}_{G_{1}P}^{G}(\mathrm{soc}^{3}\hat{\nabla}_{P}(\varepsilon)\otimes 2p\varpi_{1})$ has $G_{1}$-composition factor $L(s_{1}s_{2}s_{1})$, absurd. Thus, $3\varpi_{1}\otimes\mathrm{soc}^{1}_{4,s_{1}s_{2}s_{1}}$ is $P$-indecomposable. Finally,

[TABLE]

as above, and hence $3\varpi_{1}\otimes\mathrm{soc}^{1}_{4,s_{1}s_{2}s_{1}}$ is, up to isomorphism, a unique $P$-extension of $\varpi_{1}$ by $E$. It follows that 3\varpi_{1}\otimes\mathrm{soc}^{1}_{4,s_{1}s_{2}s_{1}}\simeq\nabla(\varpi_{1})/\begin{tabular}[]{|c|}\hline\cr\nabla^{P}(-2\varpi_{1}+\varpi_{2})\\ \hline\cr-\varpi_{1}\\ \hline\cr\end{tabular}.

(B.3) We now set ${\mathcal{E}}(w)={\mathcal{L}}_{\mathcal{P}}(\mathrm{soc}^{1}_{\ell(w)+1,w})$ with ${\mathcal{P}}=G/P$.

(i) $\forall i\geq 1$, $\mathrm{Ext}^{i}_{\mathcal{P}}({\mathcal{E}}(x),{\mathcal{E}}(y))=0$.

(ii) $\mathbf{Mod}_{\mathcal{P}}({\mathcal{E}}(x),{\mathcal{E}}(y))\neq 0$ iff $x\geq y$ in the Chevalley-Bruhat order.

**Proof: **The assertion is immediate if $s_{1}s_{2}s_{1}\not\in\{x,y\}$. Let us compute the extensions involving ${\mathcal{E}}(s_{1}s_{2}s_{1})$. We make use of the $P$-structure on $\mathrm{soc}^{1}_{4,s_{1}s_{2}s_{1}}$:

[TABLE]

Let $i\in\mathbb{N}$. One has

[TABLE]

[TABLE]

There is then a long exact sequence

[TABLE]

with

[TABLE]

and hence by the linkage principle

[TABLE]

One has

[TABLE]

and hence a long exact sequence

[TABLE]

with $\mathrm{H}^{i}({\mathcal{B}},{\mathcal{L}}(s_{1}\bullet 0))\simeq\delta_{i,1}L(e)$. On the other hand,

[TABLE]

It follows that

[TABLE]

Finally,

[TABLE]

(B.4) We show that ${\mathcal{E}}(w)$, $w\in W^{P}$, Karoubian generate $\mathrm{D}^{b}(\mathrm{coh}{\mathcal{P}})$ as in (A.9). Let $\hat{\mathcal{E}}=\langle{\mathcal{E}}(w)|w\in W^{P}\rangle$ denote the triangulated subcategory of $\mathrm{D}^{b}(\mathrm{coh}{\mathcal{P}})$ Karoubian generated by the ${\mathcal{E}}(w)$, $w\in W^{P}$.

As $\Lambda_{P}=\mathbb{Z}\varpi_{1}$, it is enough to show that all ${\mathcal{L}}_{\mathcal{P}}(n\varpi_{1})\in\hat{\mathcal{E}}$, $n\in\mathbb{Z}$. For that we may transfer to ${\mathcal{B}}=G/B$ and show that all ${\mathcal{L}}(n\varpi_{1})\in\pi^{*}\hat{\mathcal{E}}$, $n\in\mathbb{Z}$. Put $\tilde{\mathcal{E}}=\pi^{*}\hat{\mathcal{E}}$. For our purpose we may also assume that, whenever ${\mathcal{L}}(\nabla^{P}(M))\in\tilde{\mathcal{E}}$ for a $P$-module $M$, ${\mathcal{L}}(M)\in\tilde{\mathcal{E}}$, and vice versa. In particular, if ${\mathcal{L}}(\nabla^{P}(n\varpi_{1}+\varpi_{2}))\in\tilde{\mathcal{E}}$, $n\in\mathbb{Z}$, then ${\mathcal{L}}(n\varpi_{1}+\varpi_{2})\in\tilde{\mathcal{E}}$, and hence also ${\mathcal{L}}((n+3)\varpi_{1}-\varpi_{2})\in\tilde{\mathcal{E}}$.

Now, using ${\mathcal{E}}(s_{1}s_{2}s_{1}),{\mathcal{E}}(s_{2}s_{1})$ and ${\mathcal{E}}(s_{2}s_{1}s_{2}s_{1})$, we see that ${\mathcal{L}}_{\mathcal{P}}(\nabla^{P}(-4\varpi_{1}+\varpi_{2}))\in\hat{\mathcal{E}}$, and hence ${\mathcal{L}}(-4\varpi_{1}+\varpi_{2})$ and ${\mathcal{L}}(-\rho)\in\tilde{\mathcal{E}}$. As

[TABLE]

${\mathcal{L}}_{\mathcal{P}}(\nabla^{P}(-3\varpi_{1}+\varpi_{2}))\in\hat{\mathcal{E}}$, and hence ${\mathcal{L}}(-3\varpi_{1}+\varpi_{2}))$ and ${\mathcal{L}}(-\varpi_{2})\in\tilde{\mathcal{E}}$. As

[TABLE]

${\mathcal{L}}(-2\varpi_{1}+\varpi_{2})$ and ${\mathcal{L}}(\varpi_{1}-\varpi_{2})\in\tilde{\mathcal{E}}$. As

[TABLE]

${\mathcal{L}}(-5\varpi_{1}+\varpi_{2}))$ and ${\mathcal{L}}(-2\varpi_{1}-\varpi_{2})\in\tilde{\mathcal{E}}$. As

[TABLE]

${\mathcal{L}}(-5\varpi_{1}+2\varpi_{2}))$ and ${\mathcal{L}}(\varpi_{1}-2\varpi_{2})\in\tilde{\mathcal{E}}$. As $\nabla(\varpi_{1})\otimes{\mathcal{L}}(-\varpi_{2})\in\tilde{\mathcal{E}}$, ${\mathcal{L}}(2\varpi_{1}-2\varpi_{2})\in\tilde{\mathcal{E}}$. As $\nabla(\varpi_{1})\otimes{\mathcal{L}}(-\rho)\in\tilde{\mathcal{E}}$, ${\mathcal{L}}(-2\varpi_{2})\in\tilde{\mathcal{E}}$. As $\nabla(\varpi_{1})\otimes{\mathcal{L}}(-3\varpi_{1}+\varpi_{2})\in\tilde{\mathcal{E}}$, ${\mathcal{L}}(-4\varpi_{1}+2\varpi_{2})\in\tilde{\mathcal{E}}$. As $\nabla(\varpi_{2})\otimes{\mathcal{L}}(-\varpi_{1})\in\tilde{\mathcal{E}}$, ${\mathcal{L}}(\nabla^{P}(-\varpi_{1}+\varpi_{2}))\in\tilde{\mathcal{E}}$. Then ${\mathcal{L}}(-\varpi_{1}+\varpi_{2})\in\tilde{\mathcal{E}}$, and hence also ${\mathcal{L}}(2\varpi_{1}-\varpi_{2})\in\tilde{\mathcal{E}}$. As $\nabla(\varpi_{1})\otimes{\mathcal{O}}_{\mathcal{B}}\in\tilde{\mathcal{E}}$, ${\mathcal{L}}(\varpi_{1})\in\tilde{\mathcal{E}}$. As $\nabla(\varpi_{1})\otimes{\mathcal{L}}(-4\varpi_{1}+\varpi_{2})\in\tilde{\mathcal{E}}$, ${\mathcal{L}}(-6\varpi_{1}+2\varpi_{2})\in\tilde{\mathcal{E}}$. As $\nabla(\varpi_{2})\otimes{\mathcal{L}}(-3\varpi_{1})\in\tilde{\mathcal{E}}$, ${\mathcal{L}}(\nabla^{P}(-6\varpi_{1}+\varpi_{2}))\in\tilde{\mathcal{E}}$. Then ${\mathcal{L}}(-6\varpi_{1}+\varpi_{2})\in\tilde{\mathcal{E}}$, and hence also ${\mathcal{L}}(-3\varpi_{1}-\varpi_{2})\in\tilde{\mathcal{E}}$. As $\nabla(\varpi_{1})\otimes{\mathcal{L}}(-4\varpi_{1})\in\tilde{\mathcal{E}}$, ${\mathcal{L}}(-5\varpi_{1})\in\tilde{\mathcal{E}}$. Thus

[TABLE]

As $\dim\nabla(\varpi_{1})=7$, one now obtains all ${\mathcal{L}}(n\varpi_{1})\in\tilde{\mathcal{E}}$, $n\in\mathbb{Z}$, from the exact sequence

[TABLE]

Thus, we have obtained

[TABLE]

with the ${\mathcal{E}}(w)$, $w\in W^{P}$, forming a Karoubian complete strongly exceptional poset such that $\mathbf{Mod}_{\mathcal{P}}({\mathcal{E}}(x),{\mathcal{E}}(y))\neq 0$ iff $x\geq y$ in the Chevalley-Bruhat order.

C. We also append an explicit imbedding of $G$ into $\mathrm{SO}_{7}(\Bbbk)$, which is essentially the same as Hée’s [Hée, 13.6]. We will allow $\Bbbk$ to be any algebraically closed field of characteristic not 2. The author is grateful to Tanisaki and Testerman for references.

(C.1) We show first the imbedding of a $\mathbb{Z}$-form $\mathfrak{g}_{\mathbb{Z}}$ of the Lie algebra of $G$ into a $\mathbb{Z}$-form $\mathfrak{g}^{\prime}_{\mathbb{Z}}$ of the Lie algebra of $\mathrm{SO}_{7}(\Bbbk)$. For that we recall imbeddings of $\mathbb{Q}$-Lie algebras, a classical result called the principle of triality, using folding [T]. Let $\tilde{\mathfrak{g}}$ be the Lie algebra of $\mathrm{SO}_{8}(\mathbb{Q})$ with Dynkin diagram

[TABLE]

Let $A$ be the associated Cartan matrix, and let $\tilde{e}_{i},\tilde{h}_{i},\tilde{f}_{i}$, $i\in[1,4]$, be the standard Chevalley generators of $\tilde{\mathfrak{g}}$ such that $[\tilde{e}_{i},\tilde{f}_{j}]=\delta_{i,j}\tilde{h}_{i}$, $[\tilde{h}_{i},\tilde{e}_{j}]=A_{ij}\tilde{e}_{i}$, $[\tilde{h}_{i},\tilde{f}_{j}]=-A_{ij}\tilde{f}_{j}$ $\forall i,j\in[1,4]$. Let $\sigma$ ba an automorphism of the Dynkin diagram; $A_{\sigma(i)\sigma(j)}=A_{ij}$ $\forall i,j\in[1,4]$. By the same letter $\sigma$ we let it also denote the induced automorphism of $\tilde{\mathfrak{g}}$ such that $\tilde{e}_{i}\mapsto\tilde{e}_{\sigma(i)}$, $\tilde{f}_{i}\mapsto\tilde{f}_{\sigma(i)}$, $\forall i\in[1,4]$. Let $\tilde{\mathfrak{g}}^{\sigma}=\{x\in\tilde{\mathfrak{g}}|\sigma(x)=x\}$ be the fixed point subalgebra of $\tilde{\mathfrak{g}}$ under $\sigma$. Let ${\mathcal{O}}$ be a $\langle\sigma\rangle$-orbit in the index set $[1,4]$. We divide into the following 2 cases.

Case 1: Either $|{\mathcal{O}}|=1$ or $A_{ji}=0$ for any distinct $i,j\in{\mathcal{O}}$,

Case 2: ${\mathcal{O}}=\{i,j\}$ with $i\neq j$ such that $A_{ij}=-1=A_{ji}$.

For each orbit ${\mathcal{O}}$ define elements of $\tilde{\mathfrak{g}}^{\sigma}$ by

[TABLE]

Now, let $\mathfrak{g}_{\mathbb{Q}}=\mathfrak{g}_{\mathbb{Z}}\otimes_{\mathbb{Z}}\mathbb{Q}$ and let $e_{1},e_{2},f_{1},f_{2}$ be the Chevalley generators corresponding to the simple roots $\alpha_{1}$ and $\alpha_{2}$. Taking $\sigma$ of order 3, one obtains from [T, Th. B.4] an isomorphism of $\mathbb{Q}$-Lie algebras $\theta_{1}:\mathfrak{g}_{\mathbb{Q}}\to\tilde{\mathfrak{g}}^{\sigma}$ such that

[TABLE]

Let next $V$ be a 7-dimensional $\Bbbk$-linear space with basis $v_{1},v_{2},v_{3},v_{0},v_{-3},v_{-2},v_{-1}$ equipped with a quadratic form $Q(\sum_{k=-3}^{3}\xi_{k}v_{k})=\xi_{0}^{2}+\sum_{k=1}^{3}\xi_{k}\xi_{-k}$ for $\xi_{i},\xi_{-i}\in\Bbbk$, $i\in[0,3]$. Thus the associated Gram matrix is

[TABLE]

We regard $G^{\prime}=\mathrm{SO}(V;\mathbb{B})=\{g\in\mathrm{SL}(V)|\mathbb{B}(gv,gv^{\prime})=\mathbb{B}(v,v^{\prime})\ \forall v,v^{\prime}\in V\}$ as our orthogonal group $\mathrm{SO}_{7}(\Bbbk)$. Let $T^{\prime}=\{\mathrm{diag}(\zeta_{1},\zeta_{2},\zeta_{3},1,\zeta_{3}^{-1},\zeta_{2}^{-1},\zeta_{1}^{-1})|\zeta_{1},\zeta-2\in\Bbbk^{\times}\}$ be a maximal torus of $G^{\prime}$ with simple coroots $\alpha_{1}^{\vee}=\varepsilon_{1}^{\vee}-\varepsilon_{2}^{\vee},\alpha_{2}^{\vee}=\varepsilon_{2}^{\vee}-\varepsilon_{3}^{\vee}$, and $\alpha_{3}^{\vee}=2\varepsilon_{3}^{\vee}$, where $\varepsilon_{1}^{\vee}:\zeta\mapsto\mathrm{diag}(\zeta,1,1,1,1,1,\zeta^{-1})$, $\varepsilon_{2}^{\vee}:\zeta\mapsto\mathrm{diag}(1,\zeta,1,1,1,1,\zeta^{-1},1)$, and $\varepsilon_{3}^{\vee}:\zeta\mapsto\mathrm{diag}(1,1,\zeta,1,\zeta^{-1},1,1)$. If $\varepsilon_{k}:$ $\mathrm{diag}(\zeta_{1},\zeta_{2},\zeta_{3},\zeta_{0},\zeta_{3}^{-1},\zeta_{2}^{-1},\zeta_{1}^{-1})\mapsto\zeta_{k}$, $k\in[1,3]$, the corresponding simple roots are $\alpha^{\prime}_{1}=\varepsilon_{1}-\varepsilon_{2},\alpha^{\prime}_{2}=\varepsilon_{2}-\varepsilon_{3}$, and $\alpha^{\prime}_{3}=\varepsilon_{3}$. If we let $E$ denote the identity matrix and $E(i,j)$, $i,j\in[-3,3]$, denote the square matrix of degree 7 with 1 at the $(i,j)$-th entry and 0 elsewhere, the root subgroups of $G^{\prime}$ are given by, for $i,j\in[1,3]$ with $i<j$,

[TABLE]

and, for $k\in[1,3]$,

[TABLE]

[TABLE]

Then $e^{\prime}_{1}=E(1,2)-E(-2,-1),e^{\prime}_{2}=E(2,3)-E(-3,-2),e^{\prime}_{3}=2E(3,0)-E(0,-3)$, and $f^{\prime}_{1}=E(2,1)-E(-1,-2),f^{\prime}_{2}=E(3,2)-E(-2,-3),f^{\prime}_{3}=E(0,3)-2E(-3,0)$ form Chevalley generators of the $\mathbb{Q}$-Lie algebra $\mathfrak{g}^{\prime}_{\mathbb{Q}}=\mathfrak{g}^{\prime}_{\mathbb{Z}}\otimes_{\mathbb{Z}}\mathbb{Q}$. If $\sigma$ is of order 2, one obtains by [T, Th. B.4] an isomorphism of Lie algebras $\theta_{2}:\mathfrak{g}^{\prime}_{\mathbb{Q}}\to\tilde{\mathfrak{g}}^{\sigma}$ such that

[TABLE]

It follows that $\theta_{1}$ factors through $\theta_{2}$ to yield an imbedding $\theta_{\mathbb{Q}}:\mathfrak{g}_{\mathbb{Q}}\hookrightarrow\mathfrak{g}^{\prime}_{\mathbb{Q}}$ of $\mathbb{Q}$-Lie algebras.

In $\mathfrak{g}_{\mathbb{Q}}$ one can take along with $e_{i},f_{i}$ and $[e_{i},f_{i}]$, $i\in[1,2]$,

[TABLE]

to form a Chevalley basis of $\mathfrak{g}_{\mathbb{Z}}$ [Ca, Th. B.2.1]. Also, along with $e^{\prime}_{i},f^{\prime}_{i}$ and $[e^{\prime}_{i},f^{\prime}_{i}]$, $i\in[1,3]$,

[TABLE]

form a Chevalley basis of $\mathfrak{g}^{\prime}_{\mathbb{Z}}$ [Ca, 11.2.4]. Then

[TABLE]

Thus,

[TABLE]

(C.2) Using the representation $\theta_{\mathbb{Z}}$ of $\mathfrak{g}_{\mathbb{Z}}$ on $\mathbb{Z}^{\oplus_{7}}$, we exponentiate to obtain a realization of $G$ in $\mathrm{GL}_{7}(\Bbbk)$ factoring through $G^{\prime}$ [St], which is essentially the same as [Hée, 13..6]. Let $y_{\alpha}$, $\alpha\in R$, denote the root vectors of the Chevalley basis (C.1) of $\mathfrak{g}_{\mathbb{Z}}$, and put $y^{\prime}_{\alpha}=\theta(y_{\alpha})$, $x_{\alpha}(\xi)=\exp(\xi y^{\prime}_{\alpha})$.

(C.3) Explicitly, the root subgroups $U_{\alpha}$, $\alpha\in R$, of $G$ are realized in $G^{\prime}$ as follows:

[TABLE]

In particular, if we let $x_{\alpha^{\prime}_{i}}(\xi)=\exp(\xi e^{\prime}_{i})$ and $x_{-\alpha^{\prime}_{i}}(\xi)=\exp(\xi f^{\prime}_{i})$, $i\in[1,3]$,

[TABLE]

$\forall\alpha\in R^{s}$ $\forall\zeta\in\Bbbk^{\times}$, set after [St, p. 43]

[TABLE]

As $[e^{\prime}_{1},e^{\prime}_{3}]=0$ in $\mathfrak{g}^{\prime}$, one has in $G^{\prime}$

[TABLE]

It follows that the fundamental weights $\varpi^{\prime}_{1}=\varepsilon_{1}$, $\varpi^{\prime}_{2}=\varepsilon_{1}+\varepsilon_{2}$, and $\varpi^{\prime}_{3}=\frac{1}{2}(\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3})$ for $T^{\prime}$ read as $T$-characters

[TABLE]

and that the $T$-weight $\mathrm{wt}(v_{k})$ of $v_{k}$, $k\in[-3,3]$, are

[TABLE]

(C.4) Remarks: (i) The realization of $G$ in $\mathrm{GL}_{7}(\Bbbk)$ as above holds, of course, over any field $\Bbbk$.

(ii) The ambient space $V$ as a $G$-module affords $\nabla(\varpi_{1})$, which remains simple over any field of odd characteristic. In characteristic 2, however, $G$ stabilizes $\Bbbk v_{0}$, and hence $V$ is rather isomorphic to the Weyl module $\Delta(\varpi_{1})$ over any field. By a base change and modulo sign changes in the Chevalley basis the presentation of the root subgroups coincides with the one given in [Te, p. 43].

(C.5) Let $B^{\prime}$ be the Borel subgroup of $G^{\prime}$ consisting of lower triangular matrices. By the unicity of parabolic subgroups the stabilizer in $G^{\prime}$ (resp. $G$) of the line $\Bbbk v_{-1}$ is the standard parabolic subgroup $P^{\prime}_{\{\alpha^{\prime}_{2},\alpha^{\prime}_{3}\}}$ (resp. $P_{\alpha_{2}}$) of $G^{\prime}$ (resp. $G$). One thus obtains an injective morphism $\phi:G/P_{\alpha_{2}}\to G^{\prime}/P^{\prime}_{\{\alpha_{2},\alpha_{3}\}}$. If $B^{+}$ (resp. ${B^{\prime}}^{+}$) is the Borel subgroup of $G$ (resp. $G^{\prime}$) opposite to $B$ (resp. $B^{\prime}$), using $\mathrm{d}({\eta^{\prime}}^{-1}):\mathfrak{g}\hookrightarrow\mathfrak{g}^{\prime}$, one sees that $\phi$ induces an isomorphism $B^{+}P_{\alpha_{2}}/P_{\alpha_{2}}\to{B^{\prime}}^{+}P^{\prime}_{\{\alpha_{2},\alpha_{3}\}}/P^{\prime}_{\{\alpha_{2},\alpha_{3}\}}$. As the latter is open in $G^{\prime}/P^{\prime}_{\{\alpha_{2},\alpha_{3}\}}$ and as $G/P_{\alpha_{2}}$ is complete, $\phi$ must itself be an isomorphism. Likewise the stabilizer of $\Bbbk v_{-2}\oplus\Bbbk v_{-1}$ in $G$ is $P=P_{\alpha_{1}}$. Thus