because $\sigma$ is an homeomorphism.

**Remark 3.11.** How can we visualize the morphism $\sigma$ on cells? By Proposition 3.8, the Bruhat cell associated to $w \in W^I$ in $\tilde{X}$ is an affine space of dimension $l$, equal to the cardinality of $\Phi_w^I = \{\gamma_1, \dots, \gamma_l\}$. Let us consider the integers $n_i$, which we recall are associated to the roots in $\Phi_w^I$ via the equality

$$U_{-\gamma_i} \cap P = u_{-\gamma_i}(\alpha_{p^{n_i}}).$$

If we denote as $Y_i$ the coordinate on the affine line given by $U_{\gamma_i}$, then the morphism $\sigma$ acts on such a line as an $n_i$-th iterated Frobenius morphism, hence its behavior on the cell $Bw\tilde{o} = \tilde{X}_w^+$ can be summarized in the following diagram

$$ \begin{tikzcd} U_w^I \approx \tilde{X}_w^+ = \operatorname{Spec} k[Y_1, \dots, Y_l] \approx \mathbf{A}^l \arrow[r] \arrow[d, "\sigma"] & G/P_I \arrow[d, "\sigma"] \\ X_w^+ = \operatorname{Spec} k[Y_1^{p^{n_1}}, \dots, Y_l^{p^{n_l}}] \approx \mathbf{A}^l \arrow[r] \arrow[d, "\sigma"] & G/P \arrow[d, "\sigma"'] \end{tikzcd} $$

We reinterpret all the ingredients of Theorem 3.5 in order to specialize and state it in the case of flag varieties. First, $X = G/P$ is indeed smooth, projective and $G$-simple. Its unique $B$-fixed point is $x^- = o$ the base point, which gives as open cell $B^-o = Bw_0o = Bw_0^I o = X_{w_0}^+$. Moreover, the irreducible components of $X \setminus X_{w_0}^I$ are the closures of the strata of codimension one, i.e. the cells $Bwo$ with $w \in W^I$ of length $l(w) = l(w_0^I) - 1$. Those are exactly of the form $w = w_0s_\alpha w_{0,I}$ for $\alpha \in \Delta \setminus I$, since for $\alpha \in I$ we have that $w_0s_\alpha$ is in the same left coset as $w_0^I$. In particular, the divisors in the statement of Theorem 3.5 are

$$D_{\alpha} = \overline{Bw_0s_{\alpha}w_{0,I}o} = \overline{Bw_0s_{\alpha}o} = \overline{B^{-}s_{\alpha}o}, \quad \text{for } \alpha \in \Delta \setminus I,$$

hence the unique $T$-fixed point $x_\alpha^-$ such that $D_\alpha$ is the closure of $X^-(x_\alpha^−)$ is $x_\alpha^− = s_\alpha o$, and we are led to consider the $B$-invariant curves

$$C_{\alpha} = \overline{Bx_{\alpha}} = \overline{Bs_{\alpha}o}.$$

We are now able to reformulate the results of Section 3.2.1 in the following:

**Theorem 3.12.** Let us consider a sequence $G \supset P \supset P_{\text{red}} = P_I \supset B \supset T$ and let $X = G/P$ with base point $o$ and open cell $X^- = B^-o$. Then the following hold:

(1) The irreducible components of $X \setminus X^-$ are the closures $D_\alpha$ of the negative cells associated to the points $s_\alpha o$ for $\alpha \in \Delta \setminus I$. Moreover, they are globally generated Cartier divisors, whose linear equivalence classes form a basis for $\text{Pic}(X)$.

(2) Every ample (resp. nef) divisor on $X$ is linearly equivalent to a unique linear combination of the $D_\alpha$'s with positive (resp. non-negative) integer coefficients. In particular, the natural map $\text{Pic}(X) \to \mathbb{N}^1(X)$ is an isomorphism.

(3) Considering the $B$-invariant curves $C_\alpha$'s defined above, the intersection numbers satisfy $D_\alpha \cdot C_\beta = \delta_{\alpha\beta}$.

(4) The convex cone of curves $\text{NE}(X)$ is generated by the classes of the $C_\alpha$'s, which form a basis of the rational vector space $N_1(X)_Q$.

### 3.2.3. Contractions.
Theorem 3.12 tells us in particular that the Picard group of a flag variety $X = G/P$ is a free $\mathbb{Z}$-module of rank the number of simple roots not belonging to the root system of a Levi factor of $P_{\text{red}}$. This gives a motivation to the study, done in Section 2, of parabolic subgroups having maximal reduced part. In order to move on to higher ranks by exploiting the previous results in rank one, we adopt the following strategy : we define a finite
collection of morphisms which behave nicely, arise naturally from the variety $X$, and whose
targets are homogeneous spaces of Picard rank one. As a first step towards such a construction,
we recall the notion of a contraction between varieties and some of its properties.

**Definition 3.13.** Let $X$ and $Y$ be varieties over an algebraically closed field $k$. A contraction between them is a proper morphism $f: X \to Y$ such that $f^\# : \mathcal{O}_Y \to f_*\mathcal{O}_X$ is an isomorphism.

We will make use of the following results (stated here for reference).

**Theorem 3.14.** Let $f: X \to Y$ be a contraction between projective varieties over $k$. Then $f$ is uniquely determined, up to isomorphism, by the convex subcone NE($f$) of NE($X$) generated by the classes of curves which it contracts. Moreover, if $Y'$ is a third projective variety and $f': X \to Y'$ satisfies NE($f') \subset NE(f')$, then there is a unique morphism $\psi: Y \to Y'$ such that $f' = \psi \circ f$.

<figure><img src="image_6.png" /></figure>

*Proof.* See [8, Proposition 1.14]. □

**Theorem 3.15** (Blanchard's Lemma). Let $f: X \to Y$ be a contraction between projective varieties over $k$. Assume that $X$ is equipped with an action of a connected algebraic group $G$. Then there exists a unique $G$-action on $Y$ such that the morphism $f$ is $G$-equivariant.

*Proof.* See [5, 7.2]. □

The following construction is done here for any globally generated line bundle and is then applied to $\mathcal{O}_X(D_\alpha)$ to define the desired family of contractions.

**Lemma 3.16.** Let $X$ be a projective variety over $k$ and $L$ a line bundle over $X$ which is generated by its global sections. Then

(a) There is a well-defined contraction

$$f: X \to Y := \operatorname{Proj} \bigoplus_{n=0}^{\infty} H^0(X, L^{\otimes n}).$$

(b) A curve $C$ in $X$ is contracted by $f$ if and only if $L \cdot C = 0$.

*Proof.* (a) : Let us denote as $S$ the graded ring on the right hand side and denote as $S_d = H^0(X, L^{\otimes d})$ its homogeneous part of degree $d$. The schemes $X$ and $Y = \operatorname{Proj} S$ are covered by the open subset

$$D(t) = \operatorname{Spec} \left( \bigcup_{n=0}^{\infty} \frac{H^0(X, L^{\otimes n})}{t^n} \right) \quad \text{and} \quad X_t := \{x \in X, t_x \notin \mathfrak{m}_x L_x\} = X \setminus Z(t),$$

for $t$ homogeneous in $\bigoplus_{d \ge 1} S_d$, because by hypothesis $L$ is globally generated. This allows to define $f$ via the inclusion

$$(3.3) \qquad \bigcup_{n=0}^{\infty} \frac{H^0(X, L^{\otimes nd})}{t^n} \subset \mathcal{O}_X(X_t), \quad \text{for } t \in S_d.$$

Moreover, [12, II, Lemma 5.14] - applied to the coherent sheaf $\mathcal{O}_X$ and the line bundle $L^{\otimes nd}$ - implies that (3.3) is an equality, which gives the condition $f_*\mathcal{O}_X \simeq \mathcal{O}_Y$.

(b) : Let us consider the sheaf $\mathcal{O}_Y(1)$ defined as in [12, II, Proposition 5.11], fix some global
section $s \in H^0(X, L)$ and assume the open set on which it does not vanish is affine i.e. $X_s = \text{Spec} \mathcal{O}_X(X_s)$. Recall that we have the trivialization $L|_{X_s} \simeq s\mathcal{O}_{X_s}$, so considering sections over $X_s$ gives

$$H^0(X_s, f^*\mathcal{O}_Y(1)) = \left(\bigcup_{n=0}^{\infty} \frac{S_{n+1}}{s^n}\right) \otimes \bigcup_{n=0}^{\infty} \frac{s^n}{s^n} \mathcal{O}_X(X_s) = \frac{s^{n+1}\mathcal{O}_X(X_s)}{s^n} = H^0(X_s, L).$$

By covering $X$ with the open sets $X_s$ as $s$ ranges over the global sections, this gives the condition $f^*\mathcal{O}_Y(1) = L$, hence

$$ \begin{aligned} H^0(X, L) &= H^0(X, f^*\mathcal{O}_Y(1)) = H^0(Y, f_*f^*\mathcal{O}_Y(1)) = H^0(Y, \mathcal{O}_Y(1) \otimes_{\mathcal{O}_Y} f_*\mathcal{O}_X) \\ &= H^0(Y, \mathcal{O}_Y(1)), \end{aligned} $$

where the last equality comes from $f$ being a contraction. In particular, $\mathcal{O}_Y(1)$ is ample over $Y$, thus it must have strictly positive intersection with any effective 1-cycle by Kleinman's criterion. In other words, given a nonzero class $C \in \text{NE}(X)$, $f_*C = 0$ if and only if

$$0 = \mathcal{O}_Y(1) \cdot f_*C = f^*\mathcal{O}_Y(1) \cdot C = L \cdot C,$$

by the projection formula, and we are done. $\square$

Before going back to our particular case, let us prove a criterion for a morphism between homogeneous spaces to be a contraction.

**Lemma 3.17.** Consider a chain of algebraic groups $H \subset H' \subset G$ over $k$. The morphism $f: G/H \to G/H'$ is a contraction if and only if $H'/H$ is proper over $k$ and $\mathcal{O}(H'/H) = k$.

*Proof.* Let us consider $q: G \to G/H$ and $q': G \to G/H'$ to be the quotient maps and $m: G \times H/H' \to G/H$ the morphism given by the group multiplication and then by quotienting by $H$ by [19, Proposition 7.15] we have a cartesian square

$$ \begin{array}{ccc} G \times H'/H & \xrightarrow{pr_G} & G \\ \bigdownarrow_{m} & & \bigdownarrow_{q'} \\ G/H & \xrightarrow{f} & G/H' \end{array} $$

Since $q'$ is faithfully flat and $pr_G$ is obtained as base change of $f$ via such a morphism, $f$ being proper is equivalent to $pr_G$ being proper; now, the latter is obtained as base change of $H'/H \to \text{Spec } k$ via the structural morphism of $G$, which is also fppf, hence it is proper if and only if $H'/H$ is proper over $k$. This shows the first condition.

Moreover, the formation of the direct image of sheaves also commutes with fppf extensions: more precisely, applying this to the structural sheaves in our case yields

$$(q')^* f_* \mathcal{O}_{G/H} = (pr_G)_* \mathcal{O}_{G \times H'/H} = \mathcal{O}_G \otimes \mathcal{O}_{H'/H}(H'/H),$$

hence by taking $q'_*$ on both sides one gets

$$f_* \mathcal{O}_{G/H} = \mathcal{O}_{G/H'} \iff \mathcal{O}_{H'/H}(H'/H) = k,$$

which gives the second condition. $\square$

**Remark 3.18.** Let us consider again a fixed parabolic subgroup $P$. We now construct a collection of morphisms $f_\alpha: X \to G/Q_\alpha$, for $\alpha \in \Delta \setminus I$, such that

(1) the target $G/Q_\alpha$ is defined in a concrete geometrical way,

(2) each $f_\alpha$ is a contraction,

(3) the stabilizer $Q_\alpha$ coincides with the smallest subgroup scheme of $G$ containing both $P$ and $P^\alpha$: in particular, $(Q_\alpha)_{\text{red}}$ is a maximal reduced parabolic subgroup,
(4) the collection $(f_\alpha)_{\alpha \in \Delta \setminus I}$ "tells us a lot" about the variety $X$.

The reason why $Q_\alpha$ is not directly defined as being the algebraic subgroup generated by $P$ and $P^\alpha$ is that this notion does not behave well since $P$ is nonreduced in general.

Let us apply Lemma 3.16 to the variety $X = G/P$ and the line bundle $L = \mathcal{O}_X(D_\alpha)$, which can be done thanks to Theorem 3.12. This gives a contraction

$$ (3.4) \qquad f_\alpha: X \to Y_\alpha := \operatorname{Proj} \bigoplus_{n=0}^\infty H^0(X, \mathcal{O}_X(nD_\alpha)). $$

By Theorem 3.15, there is a unique $G$-action on $Y_\alpha$ such that $f_\alpha$ is equivariant. Moreover, since $f_\alpha$ is a dominant morphism between projective varieties, it is surjective, hence the target must be of the form $Y_\alpha = G/Q_\alpha$ for some subgroup scheme $P \subseteq Q_\alpha \subseteq G$. We take this construction as the definition of the subgroup $Q_\alpha$, so that conditions (1) and (2) are already satisfied. Moreover, by Theorem 3.12 and Lemma 3.16, a curve $C$ is contracted by $f_\alpha$ if and only if $D_\alpha \cdot C = 0$, meaning that this map contracts all $C_\beta$ for $\beta \neq \alpha$ while it restricts to a finite morphism on $C_\alpha$. This leaves one more condition to show.

**Lemma 3.19.** The smallest subgroup scheme of $G$ containing both $P$ and $P^\alpha$ is $Q_\alpha$.

*Proof.* By definition of $Y_\alpha$ we have the inclusion $P \subset Q_\alpha$.

Let $H$ be the subgroup scheme of $G$ generated by $P$ and $P^\alpha$. Since

$$ P_{\mathrm{red}} = P_I = \bigcap_{\alpha \in \Delta \setminus I} P^\alpha, $$

the subgroup generated by $P_{\mathrm{red}}$ and $P^\alpha$ is just $P^\alpha$. Next, consider the quotient map $\tilde{\pi}: \tilde{X} \to G/P^\alpha$ and the composition $f_\alpha \circ \sigma: \tilde{X} \to G/Q_\alpha$: the latter contracts, by the above discussions, all curves $\tilde{C}_\beta$ for $\beta \neq \alpha$, hence $\mathrm{NE}(\tilde{\pi}) \subset \mathrm{NE}(f_\alpha \circ \sigma)$. Moreover, $\tilde{\pi}$ is a contraction by Lemma 3.17, because its fiber at the base point is $P^\alpha/P_I$ which is proper and has no nonconstant global regular functions. By Theorem 3.14, there exists a unique morphism $\varphi$ making the diagram

$$ \begin{tikzcd} \tilde{X} = G/P_{\mathrm{red}} \arrow[r, "\tilde{\pi}"] & G/P^\alpha \arrow[d, "\sigma"] \\ X = G/P \arrow[r, "f_\alpha"] & G/Q_\alpha \arrow[u, "\varphi"] \arrow[d, "\pi"] \end{tikzcd} $$

commute: this shows $P^\alpha \subset Q_\alpha$ hence $H \subset Q_\alpha$.

Conversely, let us consider the projection $\pi: X \to G/H$. We already know by Theorem 3.12 that $\tilde{\pi}$ contracts all $\tilde{C}_\beta$ for $\beta \neq \alpha$; moreover, the square on the left in the following diagram is commutative and its horizontal arrows are both homeomorphisms. This implies that $\pi$ contracts all $C_\beta$ for $\beta \neq \alpha$. In other words, the inclusion $\mathrm{NE}(f_\alpha) \subset \mathrm{NE}(\pi)$ holds.

$$ \begin{tikzcd} \tilde{X} \arrow[r, "\sigma"] & X \arrow[r, "f_\alpha"] & G/Q_\alpha \arrow[d, "\pi"] \\ G/P^\alpha \arrow[r, "\tilde{\pi}"] & G/H \arrow[u, "\psi"] \arrow[d, "\pi"] \end{tikzcd} $$

Since $f_\alpha$ is a contraction by definition, this gives a factorisation by $\psi$ - again by Theorem 3.14 - which means that $Q_\alpha \subset H$. $\square$

**Remark 3.20.** The homogeneous space $X$ is now equipped with a finite number of contractions $f_\alpha$ such that the target of each morphism has Picard group $\mathbb{Z}$, with a unique canonical
ample generator, corresponding to the image of $D_\alpha$. The inclusion

$$ (3.5) \qquad P \subseteq \bigcap_{\alpha \in \Delta} Q_{\alpha} $$

holds by definition of $Q_\alpha$. If the characteristic is $p \ge 5$, by [25] there are nonnegative integers $m_\alpha$ for $\alpha \in \Delta \setminus I$ such that $P$ is the intersection of the $G_{m_\alpha} P^\alpha$, hence $P \subset Q_\alpha \subset G_{m_\alpha} P^\alpha$ and the inclusion (3.5) becomes an equality. Geometrically, this corresponds to saying that the product map

$$ f := \prod_{\alpha \in \Delta} f_{\alpha} : X \to \prod_{\alpha \in \Delta} G/Q_{\alpha} $$

is a closed immersion, realizing $X$ as the unique closed orbit of the $G$-action on the target.

**3.3. Examples in Picard rank two.** Let us consider a simple simply connected algebraic group $G$ over $k$, having Dynkin diagram with an edge of multiplicity equal to the characteristic $p \in \{2, 3\}$, so that the definitions and properties of Subsection 1.2.2 apply. In what follows, we call a parabolic subgroup of standard type if it is of the form $G_{m_1} P^{\alpha_1} \cap \dots \cap G_{m_r} P^{\alpha_r}$ for some integers $m_i$ and simple roots $\alpha_i$, while a homogeneous space is said to be of standard type its underlying variety is isomorphic to some $G'/P'$, where $P'$ is a parabolic subgroup of standard type.

The main result in this part is the following, which provides us with a first family of homogeneous projective varieties (in types $B_n$, $C_n$ and $F_4$) which are not of standard type.

**Proposition 3.21.** Let $p = 2$ and consider a simple, simply connected group $G$ and two distinct simple roots $\alpha$ and $\beta$ such that: either $G$ is of type $B_n$ or $C_n$ and the pair $(\alpha, \beta)$ is of the form $(\alpha_j, \alpha_i)$ with $i < j < n$ or $j = n$ and $i < n - 1$, or $G$ is of type $F_4$ and the pair $(\alpha, \beta)$ is one among

$$ (\alpha_1, \alpha_4), \ (\alpha_2, \alpha_1), \ (\alpha_2, \alpha_4), \ (\alpha_3, \alpha_1), \ (\alpha_3, \alpha_4), \ (\alpha_4, \alpha_1). $$

Then the homogeneous space $X = G/(N_r G P^\alpha \cap P^\beta)$ is not of standard type.

First, we give a motivation to the fact that we look for an example in rank two, then we prove Proposition 3.21 in two consecutive steps.

Let us fix a simple root $\alpha \in \Delta$. In order to find a parabolic subgroup not of standard type, the easiest and more natural idea is to consider the very special isogeny $\pi_G: G \to \bar{G}$ and the subgroup $P := N_G P^\alpha$. Its reduced part $P_{\text{red}} = P^\alpha$ is maximal, but $P$ is not of the form $G^m P^\alpha$ for any $m$. Indeed, its associated function $\varphi_P: \Phi^+ \to \mathbb{N} \cup \{\infty\}$ is given by

$$ \begin{align*} \gamma &\mapsto \infty & \text{if } \alpha \notin \text{Supp}(\gamma) \\ \gamma &\mapsto 0 & \text{if } \alpha \in \text{Supp}(\gamma) \text{ and } \gamma \in \Phi_> \\ \gamma &\mapsto 1 & \text{if } \alpha \in \text{Supp}(\gamma) \text{ and } \gamma \in \Phi_< \end{align*} $$

while the function associated to a parabolic subgroup of standard type satisfies $\varphi_{G^m P^\alpha}(\gamma) = m$ for all roots $\gamma$ containing $\alpha$ in their support, regardless of their length. There always exist both a short and a long root containing any simple root $\alpha$ in their support, namely

$$ (3.6) \qquad \bullet \text{ in type } B_n, \text{Supp}(\varepsilon_1) = \text{Supp}(\varepsilon_1 + \varepsilon_2) = \Delta; $$

$$ (3.7) \qquad \bullet \text{ in type } C_n, \text{Supp}(2\varepsilon_1) = \text{Supp}(\varepsilon_1 + \varepsilon_2) = \Delta; $$

$$ (3.8) \qquad \bullet \text{ in type } F_4, \text{Supp}(\alpha_1 + 2\alpha_2 + 3\alpha_3 + 2\alpha_4) = \text{Supp}(\alpha_1 + 2\alpha_2 + 4\alpha_3 + 2\alpha_4) = \Delta; $$

$$ (3.9) \qquad \bullet \text{ in type } G_2, \text{Supp}(2\alpha_1 + \alpha_2) = \text{Supp}(3\alpha_1 + 2\alpha_2) = \Delta. $$
Let us remark that the above roots can be constructed in a uniform way: they are respectively the highest short root and the highest (long) root. Thus, we can conclude that $\varphi_P \neq \varphi_{G^m Q}$ for all $m$, proving that $P$ is a parabolic subgroup not of standard type. However, $X = G/P$ is isomorphic as a variety to $\overline{G}/P_\alpha$, hence the homogeneous space $X$ is still of standard type. The same reasoning applies when one considers the product of a parabolic subgroup of standard type and of a kernel of a noncentral isogeny with source $G$: this might define a new parabolic subgroup, but an homogeneous space which is still of standard type. Together with Proposition 3.1, this implies that it is not possible to find examples of homogeneous spaces not of standard type having Picard rank one, when the characteristic satisfies the edge hypothesis (see Section 1.2.2). This provides a motivation to the study of the rank two case, which means considering parabolic subgroups whose reduced part is of the form $P^\alpha \cap P^\beta$ for two distinct simple roots $\alpha$ and $\beta$. In such a context we are able to find the desired class of examples.

**Lemma 3.22.** Let us consider a simple, simply connected group $G$ having Dynkin diagram with an edge of multiplicity $p$, fix two distinct simple roots $\alpha$ and $\beta$ and an integer $r \ge 0$. Both the parabolic

$$P := N_{r,G} P^{\alpha} \cap P^{\beta}$$

and its pull-back via the very special isogeny $\pi_{\overline{G}}: \overline{G} \to G$ are not of standard type if and only if one of the following conditions is satisfied:

(i) $G$ is of type $B_n$ or $C_n$ and the pair $(\alpha, \beta)$ is of the form $(\alpha_j, \alpha_i)$ with $i < j < n$ or $j = n$ and $i < n - 1$;

(ii) $G$ is of type $F_4$ and the pair $(\alpha, \beta)$ is one amongst

$$(\alpha_1, \alpha_4), \ (\alpha_2, \alpha_1), \ (\alpha_2, \alpha_4), \ (\alpha_3, \alpha_1), \ (\alpha_3, \alpha_4), \ (\alpha_4, \alpha_1).$$

In particular, this situation can only happen when $p = 2$.

*Proof.* Let us take a look at the function $\varphi_P: \Phi^+ \to \mathbb{N} \cup \{\infty\}$ associated to the parabolic $P$ - recall that it is determined by the equality

$$U_{-\gamma} \cap P = u_{-\gamma}(\alpha_{p\varphi(\gamma)}), \quad \gamma \in \Phi^{+}$$

- and let us compare it to the one associated to some $Q = G^m P^\alpha \cap G^n P^\beta$ (i.e. a parabolic of standard type), which is necessarily of this form because $Q_{\text{red}} = P_{\text{red}} = P^\alpha \cap P^\beta$. Our aim is to find in which cases there is a contradiction with the equality $P = Q$. First of all, assuming $\varphi_P(\beta) = \varphi_Q(\beta)$ leads to $n = 0$. Now, let us write down the values that $\varphi_P$ and $\varphi_Q$ assume on all positive roots in the following table.

<table><thead><tr><th></th><th>α, β ∈ Supp(γ)</th><th>α ∈ Supp(γ),<br>β ∉ Supp(γ), γ short</th><th>α ∈ Supp(γ),<br>β ∉ Supp(γ), γ long</th><th>β ∈ Supp(γ)</th></tr></thead><tbody><tr><th>φ<sub>Q</sub>(γ)</th><td>∞</td><td>m</td><td>m</td><td>0</td></tr><tr><th>φ<sub>P</sub>(γ)</th><td>∞</td><td>r + 1</td><td>r</td><td>0</td></tr></tbody></table>

Thus, the two functions can never coincide if and only if there exist at least one long root and one short root containing $\alpha$ and not $\beta$ in their respective supports. Let us examine each root system to determine when this is the case.

* If $G$ is of type $G_2$ in characteristic $p = 3$, then all roots distinct from $\alpha_1$ and $\alpha_2$ contain both simple roots in their support, hence the desired condition is never satisfied. Thus from now on we can assume that $p = 2$.

* If $G$ is of type $B_n$, let $\alpha = \alpha_j$ and $\beta = \alpha_i$ for some $1 \le i, j \le n$. A positive short root is of the form $\varepsilon_m = \alpha_m + \dots + \alpha_{n-1} + 2\alpha_n$ for $m < n$ or $\varepsilon_n = \alpha_n$: hence if $j < i$ then a short root containing $\alpha$ in its support also contains $\beta$. Let us then assume $i < j$: in this case $\gamma = \varepsilon_j$ satisfies the condition. Moving on to long roots, if $i < j < n$ then
$\gamma = \alpha = \varepsilon_j - \varepsilon_{j+1}$ is as wanted, while if $j = n$ then $\gamma = \varepsilon_{n-1} + \varepsilon_n = \alpha_n + 2\alpha_{n-1}$ satisfies the condition when $i < n - 1$, while if $i = n - 1$ then there is no such $\gamma$.

* If $G$ is of type $C_n$, let $\alpha = \alpha_j$ and $\beta = \alpha_i$ for some $1 \le i, j \le n$. A positive long root is of the form $2\varepsilon_m = 2(\alpha_m + \dots + \alpha_{n-1} + \alpha_n)$ for $m < n$ or $2\varepsilon_n = \alpha_n$: hence if $j < i$ then a long root containing $\alpha$ in its support also contains $\beta$. Let us then assume $i < j$: in this case $\gamma = 2\varepsilon_j$ satisfies the condition. Moving on to short roots, if $i < j < n$ then $\gamma = \alpha = \varepsilon_j - \varepsilon_{j+1}$ is as wanted, while if $j = n$ then $\gamma = \varepsilon_{n-1} + \varepsilon_n = \alpha_n + \alpha_{n-1}$ satisfies the condition when $i < n - 1$, while if $i = n - 1$ then there is no such $\gamma$. This completes condition (i).

* If $G$ is of type $F_4$, there is no short root containing $\alpha_1$ (resp. $\alpha_1$, resp. $\alpha_2$) in its support and not containing $\alpha_2$ (resp. $\alpha_3$, resp. $\alpha_3$); moreover, there is no long root containing $\alpha_3$ (resp. $\alpha_4$, resp. $\alpha_4$) in its support and not containing $\alpha_2$ (resp. $\alpha_2$, resp. $\alpha_3$). This can be seen by directly looking at the list of positive roots in such a system, recalled at the beginning of Subsection 2.5. The remaining pairs are listed below, which gives condition (ii).

<table><thead><tr><th>α</th><th>β</th><th>a short γ: α ∈ Supp(γ), β ∉ Supp(γ)</th><th>a long γ: α ∈ Supp(γ), β ∉ Supp(γ)</th></tr></thead><tbody><tr><td>α₁</td><td>α₄</td><td>α₁ + α₂ + α₃</td><td>α₁</td></tr><tr><td>α₂</td><td>α₁</td><td>α₂ + α₃</td><td>α₂</td></tr><tr><td>α₂</td><td>α₄</td><td>α₂ + α₃</td><td>α₂</td></tr><tr><td>α₃</td><td>α₁</td><td>α₃</td><td>α₂ + 2α₃</td></tr><tr><td>α₃</td><td>α₄</td><td>α₃</td><td>α₂ + 2α₃</td></tr><tr><td>α₄</td><td>α₁</td><td>α₄</td><td>α₂ + 2α₃ + 2α₄</td></tr></tbody></table>

Up to this point we have only shown that the parabolic $P$ is not of standard type if and only if conditions (i) or (ii) are satisfied. Now, let us consider the pull-back

$$ \pi_{\overline{G}}^{-1}(P) = \pi_{\overline{G}}^{-1}(N_{r,G}P^{\alpha} \cap P^{\beta}) = \overline{G}_{r+1}P^{\overline{\alpha}} \cap N_{\overline{G}}P^{\overline{\beta}} $$

and compare it with $Q = \overline{G}_m P^{\overline{\alpha}} \cap \overline{G}_n P^{\overline{\beta}}$, analogously as before. This gives in particular, considering a root $\gamma \in \Phi^+$ satisfying $\overline{\alpha}, \overline{\beta} \in \text{Supp}(\gamma)$, that $\varphi_Q(\gamma) = \min(m, n)$ for all $\gamma$, while $\varphi_{\pi^{-1}(P)}(\gamma)$ is equal to 1 if $\gamma$ is short, and equal to 0 if $\gamma$ is long. To show that those two parabolics can never coincide it is enough to have both such a long and a short root. This is always the case, as recalled at the beginning of this Subsection in (3.6)-(3.8), hence this concludes the proof. □

**Lemma 3.23.** Keeping the above notations, consider two distinct simple positive roots $\alpha$ and $\beta$ satisfying one of the conditions of Lemma 3.22. Then the parabolic $P := N_{r,G}P^{\alpha} \cap P^{\beta}$ gives a variety $X := G/P$ which is not of standard type.

*Proof.* The reduced part of the parabolic subgroup $P$ is $P_{\text{red}} = P^{\alpha} \cap P^{\beta}$: by Theorem 3.12, the convex cone of curves of the variety $X$ is generated by the classes of the curves

$$ C_{\alpha} = \overline{Bs_{\alpha}o} \quad \text{and} \quad C_{\beta} = \overline{Bs_{\beta}o}. $$

Next, let us consider the two contractions

$$ f_{\alpha}: X \to G/Q_{\alpha} \quad \text{and} \quad f_{\beta}: X \to G/Q_{\beta} $$

defined by (3.4). Clearly, $Q_{\beta} = \langle Q, P^{\beta} \rangle = P^{\beta}$ is smooth because $P \subset P^{\beta}$. On the other hand, let us show that $Q_{\alpha} = N_{r,G}P^{\alpha}$. Since both $P$ and $P^{\alpha}$ are subgroups of the right hand term, the inclusion $Q_{\alpha} \subset N_{r,G}P^{\alpha}$ holds. To prove the other inclusion, let us notice that the
hypothesis on $\alpha$ and $\beta$, as shown in the proof of Lemma 3.22, guarantees the existence of some short positive root $\gamma$ containing $\alpha$ and not $\beta$ in its support. In particular, this implies that

$$P \cap U_{-\gamma} = (N_{r,G}P^{\alpha} \cap U_{-\gamma}) \cap (P^{\beta} \cap U_{-\gamma}) = u_{-\gamma}(\alpha_{p^{r+1}}),$$

hence $Q_{\alpha} \cap U_{-\gamma}$ is the image of a Frobenius kernel of height at least equal to $r+1$. By the factorisation of isogenies in Proposition 1.12, the only two possibilities are thus $Q_{\alpha} = G_{r+1}P^{\alpha}$ and $Q_{\alpha} = N_{r,G}P^{\alpha}$, which allows to conclude that $Q_{\alpha} = N_{r,G}P^{\alpha}$. This means that the product of the contractions

$$f = f_{\alpha} \times f_{\beta}: X \longrightarrow X_{\alpha} \times X_{\beta}$$

is a closed immersion, where $X_{\alpha}$ (resp. $X_{\beta}$) is the underlying variety of $G/N_{G}P^{\alpha}$ (resp. $G/P^{\beta}$). Moreover, these maps are - up to a permutation - uniquely determined by the variety $X$, because the monoid $NC_{\alpha} \oplus NC_{\beta} \subset N_{1}(X)$ of effective 1-cycles does not depend on the group action on it: the two contractions are uniquely determined by its two generators and by the fact that the first is a nonsmooth morphism while the second is smooth.

The following step consists in studying the automorphisms of the varieties $X$ and $X_{\beta}$. First, we can apply Theorem 2.18 to the variety $X_{\beta} = G/P^{\beta}$ since its stabilizer is smooth and since by Lemma 3.22 the pair $(G_{\text{ad}}, P^{\beta}/Z(G))$ is not associated to any of the exceptional pairs, except in the case of $G = \text{Sp}_{2n}$ and $(\alpha, \beta) = (\alpha_{j}, \alpha_{1})$, which we treat later. This implies

$$\underline{\text{Aut}}_{X_{\beta}}^{0} = G_{\text{ad}}.$$

Next, let us consider the group $\underline{\text{Aut}}_{X}^{0}$: its natural action on $X$ gives, applying Theorem 3.15 to the contraction $f_{\beta}: X \longrightarrow X_{\beta}$, an action on $X_{\beta}$ i.e. a morphism

$$\underline{\text{Aut}}_{X}^{0} \xrightarrow{\xi} \underline{\text{Aut}}_{X_{\beta}}^{0} = G_{\text{ad}}.$$

In particular, the isogeny $\xi$ is a section of the natural morphism given by the action of $G_{\text{ad}}$ on $X$, thus giving a semidirect product $\underline{\text{Aut}}_{X}^{0} = G_{\text{ad}} \times \ker \xi$. Since $\underline{\text{Aut}}_{X}^{0}$ is reduced, $\ker \xi$ must be finite, smooth and connected, so it is trivial and we conclude that $\underline{\text{Aut}}_{X}^{0} = G_{\text{ad}}$.

Finally, let us consider another action of a semisimple, simply connected $G'$ onto the variety $X$; realizing it as a quotient $G'/P'$ for some parabolic subgroup $P'$. Since it is simply connected, $G'$ is either simple or the direct product $G_{(1)} \times \cdots \times G_{(l)}$ where each $G_{(i)}$ is simple.

* If $G'$ is simple, then its action on $X$ induces a morphism $G' \longrightarrow \underline{\text{Aut}}_{X}^{0} = G_{\text{ad}}$, which is in particular an isogeny. By Proposition 1.12, this morphism can be factorised as

$$G' \xrightarrow{F^m} G \longrightarrow \underline{\text{Aut}}_{X}^{0} \quad \text{or} \quad G' \xrightarrow{F^m \circ \pi} G \longrightarrow \underline{\text{Aut}}_{X}^{0},$$

where the second possibility only can happen whenever $G$ satisfies the edge hypothesis. The stabilizer of the $G'$-action is the preimage of the stabilizer of the $G$-action via such an isogeny, hence it is either of the form $G^m P$ for some $m$ or of the form $\overline{G_m \pi^{-1}(P)}$. Now, a parabolic $Q$ is of standard type if and only if $G^m Q$ is for any integer $m$, since the associated functions satisfy $\varphi_{Q}(\gamma) + m = \varphi_{G^m Q}(\gamma)$. This means that $P'$ is of standard type if and only if $P$ (resp. $\pi^{-1}(P)$) is. This remark, together with Lemma 3.22 allows us to conclude that, due to our choice of roots $\alpha$ and $\beta$, $P'$ is still a parabolic subgroup not of standard type.

If $G = \text{Sp}_{2n}$ and $P^{\beta} = P^{\alpha_{1}}$, then Theorem 2.18 yields $\underline{\text{Aut}}_{X_{\beta}}^{0} = \text{PGL}_{2n}$. Repeating the above reasoning implies that $\underline{\text{Aut}}_{X}^{0} = \text{PGL}_{2n}$ as well, hence the isogeny with source $G'$ is necessarily the composition of an iterated Frobenius and a central isogeny. This implies that the stabilizer of the $G'$-action is of the form $P' = G^m P$ hence still not of standard type.
• If $G' = G_{(1)} \times \cdots \times G_{(l)}$ is not simple, consider the morphism

$$G_{(1)} \times \cdots \times G_{(l)} \xrightarrow{\phi} G \dashrightarrow G_{\text{ad}}$$

determined by the action: then $H := \ker \phi$ is a normal subgroup of $G'$ and the image of $\phi$ is simple, thus $H$ is necessarily of the form

$$H = \prod_{i \neq i_0} G_{(i)} \times K, \quad \text{for some } K \subset Z(G_{(i_0)}),$$

thus $K$ is trivial because the quotient $G$ is also simply connected. In particular, denoting as $P_{(i_0)} := P' \cap G_{(i_0)}$, we have

$$X = G'/P' = G' / \left( \prod_{i \neq i_0} G_{(i)} \times P_{(i_0)} \right) = G_{(i_0)}/P_{(i_0)}$$

Applying the reasoning above to $G_{(i_0)}$ instead of $G'$ leads to the conclusion that the associated function of $P_{(i_0)}$ is not of standard type, hence the same is true for the stabilizer $P' = \prod_{i \neq i_0} G_{(i)} \times P_{(i_0)}$. $\square$

Notice that, except for the group of type $G_2$ in characteristic 2, Lemma 3.23 covers the classification of all homogeneous spaces of Picard rank two that "we know the existence of" i.e. those of the form $G/P$ with $P = (\ker \varphi)P^\alpha \cap (\ker \psi)P^\beta$ for a couple of isogenies $\varphi$ and $\psi$ with source $G$.

Indeed, Proposition 1.12 implies that one of the two kernels must be contained in the other, hence up to permuting $\alpha$ and $\beta$ the inclusion $\ker \psi \subset \ker \varphi$ holds. Taking the quotient by $\ker \psi$ allows to assume either $P = G^r P^\alpha \cap P^\beta$, which is the standard type case, or $P = N_{r,G} P^\alpha \cap P^\beta$ for some $r \ge 0$. The latter gives a variety not of standard type if and only if $p = 2$ and the above hypothesis on roots is satisfied.

**Problem:** Let us consider a simple group and a parabolic subgroup $P \subset G$ with reduced part $P_{\text{red}} = P_I$ which is not maximal. The associated family of contractions give a natural inclusion

$$P \subset \bigcap_{\alpha \in \Delta \setminus I} Q_{\beta}.$$

The question whether there exist a parabolic subgroup $P$ for which the inclusion is strict is still open. At this point, we are neither able to exclude their existence nor to exhibit an explicit example.
## 4. APPENDIX

Let us resume here a short description of the Chevalley embedding of the group of type $G_2$, which holds in any characteristic. We will then specialize to characteristic two which is the interesting one for our purposes. First we describe its action on the algebra of octonions, then we use it to compute some of the root subgroups of such a group, which are fundamental in order to study the parabolic subgroups $P_\eta$ and $P_\iota$ (see Definition 2.25).

**4.1. The Chevalley embedding of $G_2$.** Let $G$ be the simple group of type $G_2$ in characteristic $p > 0$. It can be viewed - as illustrated in [21], from which we will keep most of the notation - as the automorphism group of an octonion algebra. The latter is the algebra

$$ \mathcal{O} = \{(u, v) : u, v \text{ are } 2 \times 2 \text{ matrices}\}, $$

with basis

$$ e_{11} = \begin{pmatrix} (1 & 0) \\ (0 & 0) \end{pmatrix}, \quad (0 & 0) \\ (0 & 0) \end{pmatrix}, $$

$$ e_{12} = \begin{pmatrix} (0 & 1) \\ (0 & 0) \end{pmatrix}, \quad (0 & 0) \\ (0 & 0) \end{pmatrix}, $$

$$ e_{21} = \begin{pmatrix} (0 & 0) \\ (1 & 0) \end{pmatrix}, \quad (0 & 0) \\ (0 & 0) \end{pmatrix}, $$

$$ e_{22} = \begin{pmatrix} (0 & 0) \\ (0 & 1) \end{pmatrix}, \quad (0 & 0) \\ (0 & 0) \end{pmatrix}, $$

$$ f_{11} = \begin{pmatrix} (0 & 0) \\ (0 & 0) \end{pmatrix}, \quad (1 & 0) \\ (0 & 0) \end{pmatrix}, $$

$$ f_{12} = \begin{pmatrix} (0 & 0) \\ (0 & 0) \end{pmatrix}, \quad (0 & 1) \\ (0 & 0) \end{pmatrix}, $$

$$ f_{21} = \begin{pmatrix} (0 & 0) \\ (0 & 0) \end{pmatrix}, \quad (0 & 0) \\ (1 & 0) \end{pmatrix}, $$

$$ f_{22} = \begin{pmatrix} (0 & 0) \\ (0 & 0) \end{pmatrix}, \quad (0 & 0) \\ (0 & 1) \end{pmatrix}, $$

unit $e = (1, 0) = e_{11} + e_{22}$, and which is equipped with a norm

$$ q(u, v) = \det(u) - \det(v). $$

Let us write here for reference a table of products of the basis vectors :

$$ \begin{array}{c|ccccccccc} \diagup & e_{11} & e_{21} & e_{12} & e_{22} & f_{11} & f_{21} & f_{12} & f_{22} \\ \hline e_{11} & e_{11} & 0 & e_{12} & 0 & f_{11} & f_{21} & 0 & 0 \\ e_{21} & e_{21} & 0 & e_{22} & 0 & 0 & 0 & f_{11} & f_{21} \\ e_{12} & 0 & e_{11} & 0 & e_{12} & f_{12} & f_{22} & 0 & 0 \\ e_{22} & 0 & e_{21} & 0 & e_{22} & 0 & 0 & f_{12} & f_{22} \\ f_{11} & 0 & 0 & -f_{12} & f_{11} & 0 & -e_{21} & 0 & e_{11} \\ f_{21} & 0 & 0 & -f_{22} & f_{21} & e_{21} & 0 & -e_{11} & 0 \\ f_{12} & f_{12} & -f_{11} & 0 & 0 & 0 & -e_{22} & 0 & e_{12} \\ f_{22} & f_{22} & -f_{21} & 0 & 0 & e_{22} & 0 & -e_{12} & 0 \end{array} \qquad (4.1) $$

An embedding of the group $G_2$ into $SO_7$ - which gives an irreducible representation in all characteristics but two - can be seen as follows: let us consider its action on the vector space

$$ (4.2) \qquad V := e^{\perp} = \{(u, v) : \det(1+u) - \det(u) = 1\} = \{(u, v) : u_{11} + u_{22} = 0\}. $$

By [21, Lemma 2.3.1], a maximal torus of $G$ - with respect to the basis $(e_{12}, e_{21}, f_{11}, e_{11} - e_{22}, -f_{12}, f_{21}, f_{22})$ of $W$ - acts on $V$ as

$$ \mathbf{G}_m^2 \ni (\xi, \eta) \mapsto \mathrm{diag}(\xi\eta, \xi^{-1}\eta^{-1}, \eta^{-1}, 1, \xi, \xi^{-1}, \eta) \in \mathrm{GL}_7 $$

Let us re-parameterize it with $\xi = a, \eta = ab$, this gives the torus

$$ \mathbf{G}_m^2 \ni (a, b) \mapsto \mathrm{diag}(a^2b, a^{-2}b^{-1}, a^{-1}b^{-1}, 1, a, a^{-1}, ab) =: t \in \mathrm{GL}_7, $$
and the basis of simple roots we fix is $\alpha_1(t) := a$ and $\alpha_2(t) := b$. Such a torus acts on $V$ with the following weight spaces :

$$ 
\begin{aligned}
V_0 &= k(e_{11} - e_{22}), V_{\alpha_1} = kf_{12}, V_{-\alpha_1} = kf_{21}, V_{\alpha_1+\alpha_2} = kf_{22}, \\
V_{-\alpha_1-\alpha_2} &= kf_{11}, V_{2\alpha_1+\alpha_2} = ke_{12}, V_{-2\alpha_1-\alpha_2} = ke_{21},
\end{aligned}
$$

which correspond to 0 and the short roots. Re-arranging $V$ as

$$ (4.3) \qquad V = k(-f_{12}) \oplus kf_{11} \oplus ke_{12} \oplus k(e_{11} - e_{22}) \oplus ke_{21} \oplus kf_{22} \oplus kf_{21} $$

gives the maximal torus $T$

$$ (4.4) \qquad \mathbf{G}_m^2 \ni (a, b) \longmapsto \mathrm{diag}(a, a^{-1}b^{-1}, a^2b, 1, a^{-2}b^{-1}, ab, a^{-1}) = t \in T \subset \mathrm{GL}_7. $$

This way, $T$ can be identified with the maximal torus in [13, page 13]: in his description of the embedding $G \subset \mathrm{GL}_7$, the group $G$ is generated by the two following copies of $\mathrm{GL}_2$,

$$ \theta_1: A \mapsto \left( \begin{array}{cc} A & \mathrm{Sym}^2(A) \det A^{-1} \\ & A \end{array} \right) \quad \text{and} \quad \theta_2: B \mapsto \left( \begin{array}{ccc} \det B^{-1} & & \\ & \tilde{B} & \\ & & B \end{array} \right), $$

where

$$ \tilde{A} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} t A^{-1} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}. $$

However, in characteristic $p=2$, due to the fact that $e \in V$ and that $G$ acts on the quotient $W = V/ke$, these become the following two copies embedded in $\mathrm{GL}(W) = \mathrm{GL}_6$:

$$ \theta_1: A \mapsto \left( \begin{array}{cc} A & A^{(1)} \det A^{-1} \\ & A \end{array} \right) \quad \text{and} \quad \theta_2: B \mapsto \left( \begin{array}{ccc} \det B^{-1} & & \\ & B & \\ & & B \end{array} \right), $$

where $A^{(1)}$ denotes the Frobenius twist applied to $A$.

**Lemma 4.1.** The subgroups $\theta_1(\mathrm{GL}_2)$ and $\theta_2(\mathrm{GL}_2)$ have root system with positive root respectively $\beta_1 := 2\alpha_1 + \alpha_2$ and $\beta_2 := -3\alpha_1 - 2\alpha_2$.

*Proof.* See the computation of the root homomorphisms associated respectively to $\beta_1$ and $\beta_2$, done in Remark 4.2: these are respectively the intersection of $\theta_1(\mathrm{GL}_2)$ and $\theta_2(\mathrm{GL}_2)$ with the upper triangular matrices of $\mathrm{GL}_7$. $\square$

Let us remark that $\{\beta_1, \beta_2\}$ is indeed a basis for the root system of type $G_2$, with corresponding set of positive roots being

$$ -3\alpha_1 - 2\alpha_2, \alpha_1 - \alpha_2, -\alpha_2, \alpha_2, 3\alpha_1 + \alpha_2, 2\alpha_1 + \alpha_2 $$

and with Borel subgroup given by the intersection of $G$ with the upper triangular matrices in $\mathrm{GL}_7$.
4.2. **Root subgroups.** Let us move on to the explicit computation of some of the root subgroups in type $G_2$. As before, we will do everything considering the action on a 7-dimensional vector space - the orthogonal of the identity element of $\mathbb{O}$ - so that the computations hold in any characteristic, then at the end we will summarize what we get in characteristic 2.

Let us consider the group $G$ acting on the vector space $V$ arranged as in (4.3). Denoting as $x_0, \dots, x_6$ the coordinates on $V$, the norm becomes

$$ (4.5) \qquad q(x) = -x_3^2 - x_2x_4 - x_1x_5 - x_0x_6, $$

while the maximal torus $T$ given in (4.4) acts on $V$ through this table of characters

$$ \begin{array}{c|cccccc} 1 & a^2b & a^{-1}b^{-1} & a & a^3b & b^{-1} & a^2 \\ \hline a^{-2}b^{-1} & 1 & a^{-3}b^{-2} & a^{-1}b^{-1} & a & a^{-2}b^{-2} & b^{-1} \\ \hline ab & a^3b^2 & 1 & a^2b & a^4b^2 & a & a^3b \\ \hline a^{-1} & ab & a^{-2}b^{-1} & 1 & a^2b & a^{-1}b^{-1} & a \\ \hline a^{-3}b^{-1} & a^{-1} & a^{-4}b^{-2} & a^{-2}b^{-1} & 1 & a^{-3}b^{-2} & a^{-1}b^{-1} \\ \hline b & a^2b^2 & a^{-1} & ab & a^3b^2 & 1 & a^2b \\ \hline a^{-2} & b & a^{-3}b^{-1} & a^{-1} & ab & a^{-2}b^{-1} & 1 \end{array} \qquad (4.6) $$

The idea is the following: we know that - for any root $\gamma \in \Phi$ - the root subgroup $U_\gamma \subset G$ is determined by being the unique subgroup of $GL(V)$ (resp. $GL(W)$ in characteristic 2), which is smooth, unipotent, is acted on by $T$ via the character $\gamma$, and whose elements are automorphisms of octonions. We will impose some of these necessary condition - such as $u_\gamma(\lambda)$ being an isometry for any $\lambda \in G_a$ - to determine the root homomorphism $u_\gamma: G_a \to U_\gamma$.

• First, let us consider the root $\alpha_1$. By (4.6) and the condition for $u_{\alpha_1}$ to be a group homomorphism, there exist some constants $\eta_1, \dots, \eta_5 \in k$ such that for any $\lambda \in G_a$, $u_{\alpha_1}(\lambda)$ acts on $V$ as

$$ \begin{pmatrix} 1 & 0 & 0 & \eta_1\lambda & 0 & 0 & \eta_5\lambda^2 \\ 0 & 1 & 0 & 0 & \eta_2\lambda & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & \eta_3\lambda & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & \eta_4\lambda \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix}. $$

Moreover, $u_{\alpha_1}(\lambda)$ being an isometry means, by (4.5), that

$$ \begin{aligned} q(x) &= q(u_{\alpha_1}(\lambda) \cdot x) = q(x_0 + \eta_1 \lambda x_3 + \eta_5 \lambda^2 x_6, x_1 + \eta_2 \lambda x_4, x_2 + \eta_3 \lambda x_5, x_3 + \eta_4 \lambda x_6, x_4, x_5, x_6) \\ &= q(x) + -(2\eta_4 + \eta_1)\lambda x_3 x_6 + -(\eta_5 + \eta_4^2)\lambda^2 x_6^2 - (\eta_3 + \eta_2)\lambda x_4 x_5, \end{aligned} $$

hence $\eta_1 = -2\eta_4$, $\eta_5 = -\eta_4^2$ and $\eta_2 = -\eta_3$. This still leaves two independent parameters $\eta_3$ and $\eta_4$ instead of one, so let us also impose the condition of $u_{\alpha_1}(\lambda)$ respecting the product $e_{12}f_{21} = f_{22}$ - see (4.1) :

$$ \begin{aligned} (u_{\alpha_1}(\lambda) \cdot e_{12})(u_{\alpha_1}(\lambda) \cdot f_{21}) &= u_{\alpha_1}(\lambda) \cdot (f_{22}) \\ e_{12}(\eta_4^2 \lambda^2 f_{12} + \eta_4(e_{11} - e_{22}) + f_{21}) &= \eta_3 \lambda e_{12} + f_{22} \\ -\eta_4 \lambda e_{12} + f_{22} &= \eta_3 \lambda e_{12} + f_{22}, \end{aligned} $$

implying $\eta_3 = -\eta_4$. Let us reparametrise the root homomorphism such that $\eta_4 = 1$: this, together with an analogous computation for $-\alpha_1$, gives the desired representations, of the
form

$$ u_{\alpha_1} : \lambda \mapsto \begin{pmatrix} 1 & 0 & 0 & -2\lambda & 0 & 0 & -\lambda^2 \\ 0 & 1 & 0 & 0 & \lambda & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & -\lambda & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & \lambda \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix}, \quad u_{-\alpha_1} : \lambda \mapsto \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ \lambda & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & -\lambda & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & \lambda & 0 & 0 & 1 & 0 \\ -\lambda^2 & 0 & 0 & -2\lambda & 0 & 0 & 1 \end{pmatrix}. $$

• Let us consider the root $\alpha_2$. By (4.6) and the condition for $u_{\alpha_2}$ to be a group homomorphism, there exist some constants $\eta_1$ and $\eta_2 \in k$ such that for any $\lambda \in G_a$, $u_{\alpha_2}(\lambda)$ acts on $V$ as

$$ u_{\alpha_2}(\lambda) \cdot x = (x_0, x_1, x_2, x_3, x_4, \eta_1 \lambda x_0 x_5, \eta_2 \lambda x_1 + x_6). $$

Moreover, the isometry condition means that

$$ \begin{aligned} q(x) &= q(u_{\alpha_2}(\lambda) \cdot x) = -x_3^2 - x_2 x_4 - \eta_1 \lambda x_0 x_1 - x_1 x_5 - \eta_2 \lambda x_0 x_1 - x_0 x_6 \\ &= q(x) - (\eta_2 + \eta_1) \lambda x_0 x_1, \end{aligned} $$

hence $\eta_1 = -\eta_2$. As before, we can conclude that the associated root subgroups are of the form

$$ u_{\alpha_2} : \lambda \mapsto \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ -\lambda & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & \lambda & 0 & 0 & 0 & 0 & 1 \end{pmatrix}, \quad u_{-\alpha_2} : \lambda \mapsto \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & \lambda & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & -\lambda \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix}. $$

• Let us consider the root $2\alpha_1 + \alpha_2$. By (4.6) and the condition for $u_{2\alpha_1+\alpha_2}$ to be a group homomorphism, there exist some constants $\eta_1, \dots, \eta_5 \in k$ such that for any $\lambda \in G_a$, $u_{2\alpha_1+\alpha_2}(\lambda)$ acts on $V$ as

$$ \begin{pmatrix} 1 & \eta_1 \lambda & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & \eta_2 \lambda & \eta_5 \lambda^2 & 0 & 0 \\ 0 & 0 & 0 & 1 & \eta_3 \lambda & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & \eta_4 \lambda \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix}. $$

Moreover, the isometry condition implies

$$ \begin{aligned} q(u_{2\alpha_1+\alpha_2}(\lambda) \cdot x) &= q(x_0 + \eta_1 \lambda x_1, x_1, x_2 + \eta_2 \lambda x_3 + \eta_5 \lambda^2 x_4, x_3 + \eta_3 \lambda x_4, x_4, x_5 + \eta_4 \lambda x_6, x_6) \\ &= q(x) - (\eta_1 + \eta_4) \lambda x_1 x_6 - (\eta_3^2 + \eta_5) \lambda^2 x_4^2 - (2\eta_3 + \eta_2) \lambda x_3 x_4 = q(x), \end{aligned} $$

hence $\eta_1 = -\eta_4$, $\eta_5 = -\eta_3^2$ and $\eta_2 = -2\eta_3$. This still leaves two independent parameters $\eta_3$ and $\eta_4$ instead of one, so let us also impose the condition of $u_{2\alpha_1+\alpha_2}(\lambda)$ respecting the product $f_{22}e_{21} = -f_{21}$:

$$ \begin{aligned} (u_{2\alpha_1+\alpha_2}(\lambda) \cdot f_{22})(u_{2\alpha_1+\alpha_2}(\lambda) \cdot e_{21}) &= u_{2\alpha_1+\alpha_2}(\lambda) \cdot (-f_{21}) \\ f_{22}(-\eta_3^2 \lambda^2 e_{12} + \eta_3 \lambda (e_{11} - e_{22}) + e_{21}) &= -\eta_4 \lambda f_{22} - f_{21} \\ \eta_3 \lambda f_{22} - f_{21} &= -\eta_4 \lambda f_{22} - f_{21}, \end{aligned} $$
implying $\eta_3 = -\eta_4$, so we can conclude that the associated root subgroups are of the form

$$ u_{2\alpha_1+\alpha_2}: \lambda \mapsto \begin{pmatrix} 1 & -\lambda & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 2\lambda & -\lambda^2 & 0 & 0 \\ 0 & 0 & 0 & 1 & -\lambda & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & \lambda \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix}, \quad u_{-2\alpha_1-\alpha_2}: \lambda \mapsto \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ \lambda & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -\lambda & 1 & 0 & 0 & 0 \\ 0 & 0 & -\lambda^2 & 2\lambda & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -\lambda & 1 \end{pmatrix}. $$

• Let us consider the root $\alpha_1 + \alpha_2$. By (4.6) and the condition for $u_{\alpha_1+\alpha_2}$ to be a group homomorphism, there exist some constants $\eta_1, \dots, \eta_5 \in k$ such that for any $\lambda \in G_a$, $u_{\alpha_1+\alpha_2}(\lambda)$ acts on $V$ as

$$ \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ \eta_1\lambda & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & \eta_2\lambda & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & \eta_5\lambda^2 & 0 & \eta_3\lambda & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & \eta_4\lambda & 0 & 1 \end{pmatrix}. $$

Moreover, the isometry condition implies

$$ \begin{aligned} q(x) &= q(u_{\alpha_1+\alpha_2}(\lambda) \cdot x) = q(x_0, x_1, \eta_1\lambda x_0 + x_2, \eta_2\lambda x_1 + x_3, x_4, \eta_5\lambda^2 x_1 + \eta_3\lambda x_3 + x_5, \eta_4\lambda x_4 + x_6) \\ &= q(x) - (2\eta_2 + \eta_3)\lambda x_1 x_3 - (\eta_4 + \eta_1)\lambda x_0 x_4 - (\eta_2^2 + \eta_5)\lambda^2 x_1^2, \end{aligned} $$

hence $\eta_3 = -2\eta_2$, $\eta_1 = -\eta_4$ and $\eta_5 = -\eta_2^2$. Reasoning as in the above cases, let us also impose the condition of $u_{\alpha_1+\alpha_2}(\lambda)$ respecting the product $f_{11}f_{21} = -e_{21}$:

$$ \begin{aligned} & (u_{\alpha_1+\alpha_2}(\lambda) \cdot f_{11})(u_{\alpha_1+\alpha_2}(\lambda) \cdot f_{21}) = u_{\alpha_1+\alpha_2}(\lambda) \cdot (-e_{21}) \\ & (f_{11} + \eta_2\lambda(e_{11} - e_{22}) - \eta_2^2\lambda^2 f_{22})f_{21} = -e_{21} - \eta_4\lambda f_{21} \\ & -e_{21} + \eta_2\lambda f_{21} = -e_{21} - \eta_4\lambda f_{21}, \end{aligned} $$

implying $\eta_2 = -\eta_4$. Reparametrizing and doing an analogous computation for the negative root allows to conclude that the root subgroups are as follows:

$$ u_{\alpha_1+\alpha_2}: \lambda \mapsto \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ -\lambda & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & -\lambda & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & -\lambda^2 & 0 & 2\lambda & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & \lambda & 0 & 1 \end{pmatrix}, \quad u_{-\alpha_1-\alpha_2}: \lambda \mapsto \begin{pmatrix} 1 & 0 & \lambda & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 2\lambda & 0 & -\lambda^2 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & -\lambda & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & -\lambda \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix}. $$

• As last computation, let us consider the root $-3\alpha_1 - 2\alpha_2$. By (4.6) and the condition for $u_{-3\alpha_1-2\alpha_2}$ to be a group homomorphism, there exist some constants $\eta_1$ and $\eta_2 \in k$ such that for any $\lambda \in G_a$, $u_{-3\alpha_1-2\alpha_2}(\lambda)$ acts on $V$ as

$$ u_{-3\alpha_1-2\alpha_2}(\lambda) \cdot x = (x_0, x_1 + \eta_1\lambda x_2, x_2, x_3, x_4 + \eta_2\lambda x_5, x_5, x_6). $$

The isometry condition implies

$$ \begin{aligned} q(x) &= q(u_{-3\alpha_1-2\alpha_2}(\lambda) \cdot x) = -x_3^2 - x_2x_4 - \eta_2\lambda x_2x_5 - x_1x_5 - \eta_1\lambda x_2x_5 - x_0x_6 \\ &= q(x) - (\eta_2 + \eta_1)\lambda x_2x_5, \end{aligned} $$
hence $\eta_2 = -\eta_1$ and we can conclude that the root subgroups have the following form :
(4.7)

$$
u_{-3\alpha_1-2\alpha_2} : \lambda \mapsto \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & \lambda & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & -\lambda & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix}, \quad u_{3\alpha_1+2\alpha_2} : \lambda \mapsto \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & -\lambda & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \lambda & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix}.
$$

**Remark 4.2.** Let us recall that in characteristic 2 the group *G* acts on *W* = *V*/*ke*, giving an embedding *G* ⊂ *Sp*<sub>6</sub>: we list below what the root subspaces we need become in that case.

$$
u_{\alpha_1}(\lambda) = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & \lambda^2 \\ 0 & 1 & 0 & \lambda & 0 & 0 \\ 0 & 0 & 1 & 0 & \lambda & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix}, \quad u_{-\alpha_1}(\lambda) = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & \lambda & 0 & 1 & 0 & 0 \\ 0 & 0 & \lambda & 0 & 1 & 0 \\ \lambda^2 & 0 & 0 & 0 & 0 & 1 \end{pmatrix}
$$

$$
u_{\alpha_2}(\lambda) = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ \lambda & 0 & 0 & 0 & 1 & 0 \\ 0 & \lambda & 0 & 0 & 0 & 1 \end{pmatrix}, \quad u_{-\alpha_2}(\lambda) = \begin{pmatrix} 1 & 0 & 0 & 0 & \lambda & 0 \\ 0 & 1 & 0 & 0 & 0 & \lambda \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix}
$$

$$
u_{2\alpha_1+\alpha_2}(\lambda) = \begin{pmatrix} 1 & \lambda & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & \lambda^2 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & \lambda \\ 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix}, \quad u_{-2\alpha_1-\alpha_2}(\lambda) = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ \lambda & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & \lambda^2 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & \lambda & 1 \end{pmatrix}
$$

$$
u_{\alpha_1+\alpha_2}(\lambda) = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ \lambda & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & \lambda^2 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & \lambda & 0 & 1 \end{pmatrix}, \quad u_{-\alpha_1-\alpha_2}(\lambda) = \begin{pmatrix} 1 & 0 & \lambda & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & \lambda^2 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & \lambda \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix}
$$

$$
u_{3\alpha_1+2\alpha_2}(\lambda) = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & \lambda & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & \lambda & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix}, \quad u_{-3\alpha_1-2\alpha_2}(\lambda) = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & \lambda & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & \lambda & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix}
$$
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Email address: matilde.maccan@univ-rennes.fr