Proof. Similar to the proof of [Reference Gille and ZainoullineGZ12, Proposition 5.1]. Namely, let $\chi _1,\ldots ,\chi _n$ stand for a basis of the character group of $T$, and let $L(\chi _i)$ denote ${\mathbb {A}}^1$ with the action of $T$ via $\chi _i$. Then $T$ embeds $T$-equivariantly into $\oplus _i L(\chi _i)\simeq {\mathbb {A}}^n$ with the complement equal to the union of the coordinate hyperplanes. Twisting by $E$ produces an open embedding of $E$ into a vector bundle $V={}_E{\mathbb {A}}^n_X$ over $X$ with the complement equal to the union of zero loci of $\chi _i$ considered as regular maps on $V$. Applying homotopy invariance and the localization axiom, we obtain an isomorphism

\[ A^*(E)\simeq A^*(X)/(c^A_1({}_EL(\chi_1)),\ldots,c^A_1({}_EL(\chi_n))), \]

as claimed.

Proof. Indeed, $E$ is a $T$-torsor over $E/T$, and the natural map $E/T\to E/B$ is an affine bundle, hence gives an isomorphism $A^*(E/B)\simeq A^*(E/T)$, and all that remains is to apply Lemma 4.1.

Proof. Since $X\times G$ is a trivial $G$-torsor over $X$, we have

\begin{align*} A^*(X\times G)&\simeq A^*(X\times G/B)\otimes_{A^*(\mathrm{B} T)}A^*(\mathrm{pt})\quad\text{by Corollary 4.2}\\ &\simeq A^*(X)\otimes_{A^*(\mathrm{pt})}A^*(G/B)\otimes_{A^*(\mathrm{B} T)}A^*(\mathrm{pt})\quad\text{since $G/B$ is cellular}\\ &\simeq A^*(X)\otimes_{A^*(\mathrm{pt})}A^*(G)\quad\text{by Corollary {4.2}.} \end{align*}

Proof. By definition $J$ is contained in $\operatorname {Ker}\varepsilon$. Take any element $\bar e$ from

\[ \mathrm{Im}(A^*(E)\xrightarrow{\mathop{\mathrm{res}}\nolimits} A^*(G))\cap\operatorname{Ker}(A^*(G)\xrightarrow{\varepsilon} A^*(\mathrm{pt})). \]

By diagram (4.4) we can write $\Delta (\bar e)=1\otimes {\bar e}+\sum \bar e_i\otimes a_i$ with $\bar e_i$ from

\[ \mathrm{Im}(A^*(E)\xrightarrow{\mathop{\mathrm{res}}\nolimits} A^*(G))\cap\operatorname{Ker}(A^*(G)\xrightarrow{\varepsilon} A^*(\mathrm{pt})) \]

and $a_i$ from $A^*(G)$. But this sum belongs to $A^*(G)\otimes _{A^*(\mathrm {pt})}J+J\otimes _{A^*(\mathrm {pt})}A^*(G)$. Since $\Delta$ is a ring homomorphism, the lemma follows.

Proof. For pullbacks the claim is obvious, for $\rho$ is defined in terms of pullbacks. For a projective morphism $f\colon X\to Y$ we have a Cartesian square

whose horizontal maps are flat. Therefore, this square is transversal (see [Reference Levine and MorelLM07, Definition 1.1.1]) and thus induces the following commutative diagram in cohomology:

and the claim follows.

Proof. Consider the natural projection map $E\times X\to {}_E X$. Then the following diagram commutes.

Now we can write $\rho (\bar \alpha )=1\otimes {\bar \alpha }+\sum {\bar e_i\otimes a_i}$ with $\bar e_i$ from

\[ \mathrm{Im}(A^*(E)\xrightarrow{\mathop{\mathrm{res}}\nolimits} A^*(G))\cap\operatorname{Ker}(A^*(G)\xrightarrow{\varepsilon} A^*(\mathrm{pt})) \]

and $a_i$ from $A^*(\bar X)$, and the claim follows.

Theorem 4.14 Let $X$, $Y$ be smooth projective cellular $G$-varieties over $F$. Let ${}_E X$, ${}_E Y$ be the respective twisted forms of $X$ and $Y$. Let $\alpha$ be a correspondence from $A^*({}_E X\times {}_E Y)$. Then $\bar \alpha _\star \colon A^*(\bar X)\to A^*(\bar Y)$ is a homomorphism of $H^*$-comodules.

Proof. We have to show that the diagram

commutes. We have

\begin{align*} \rho\circ\bar\alpha_\star&=\rho\circ(\operatorname{\mathrm{pr}}_{\bar Y})_*\circ\mu_{\bar\alpha}\circ\operatorname{\mathrm{pr}}_{\bar X}^*\quad\text{by Definition 4.13}\\ &=(\mathrm{id}\otimes(\operatorname{\mathrm{pr}}_{\bar Y})_*)\circ\mu_{\rho(\bar\alpha)}\circ(\mathrm{id}\otimes\operatorname{\mathrm{pr}}_{\bar X}^*)\circ\rho\quad\text{by Lemma 4.11}\\ &=(\mathrm{id}\otimes(\operatorname{\mathrm{pr}}_{\bar Y})_*)\circ(\mathrm{id}\otimes\mu_{\bar\alpha})\circ(\mathrm{id}\otimes\operatorname{\mathrm{pr}}_{\bar X}^*)\circ\rho\quad\text{by Lemma 4.12}\\ &=(\mathrm{id}\otimes\bar\alpha_\star)\circ\rho. \end{align*}

Proof. Consider the expansion of an element $a$ from $\operatorname {Ker}(A^*(G/B)\xrightarrow {\tilde \varepsilon } A^*(\mathrm {pt}))$ in the standard basis $Z_w$ (see § 3.1). Note that

\[ \tilde\varepsilon(Z_w)=\begin{cases} 1, & w=1,\\ 0, & w\ne 1. \end{cases} \]

Therefore, $\tilde \varepsilon (a)=0$ is the coefficient of the element $Z_1\in A^*(G/B)$. All other elements $Z_w$, $w\ne 1$, from the standard basis are of positive codimension and come from the algebraic cobordism $\Omega ^*(G/B)$, and hence are nilpotent. Therefore, the ideal $I$ is nilpotent.

Proof. We have

\begin{align*} A^*(G)&\simeq A^*(G/B)\otimes_{A^*(\mathrm{B} T)}A^*(\mathrm{pt})\quad\text{by Corollary 4.2}\\ &\simeq A^*(G/B)\otimes_{A^*(E/B)}\big(A^*(E/B)\otimes_{A^*(\mathrm{B} T)}A^*(\mathrm{pt})\big)\\ &\simeq A^*(G/B)\otimes_{A^*(E/B)}A^*(E)\quad\text{by Corollary 4.2}. \end{align*}

It follows that

\[ A^*(G)\otimes_{A^*(E)}A^*(\mathrm{pt})\simeq A^*(G/B)\otimes_{A^*(E/B)}A^*(\mathrm{pt}), \]

where the right-hand side is $A^*(G/B)/IA^*(G/B)$, and the left-hand side is $H^*$.

Proof. Indeed, by Lemma 5.3 the map $\theta$ becomes an isomorphism after tensoring with $A^*(\mathrm {pt})$ over $R^*$. But this tensoring is the same as taking the quotient modulo the ideal $I$. So, the cokernel $M$ of $\theta$ satisfies $MI=M$, and $I$ is nilpotent by Lemma 5.2, so $M=0$.

To prove the injectivity of $\theta$ we localize the coefficient ring $A^*(\mathrm {pt})$ at a prime ideal and so assume that $H^*$ is free. Choose a basis $e_1,\ldots e_n$ of $H^*$.

Note that $R^*$ is stable under the Demazure operators $\Delta _i$ (see § 3.7) because they are defined on $A^*(E/B)$. Consider the filtration on $R^*$ whose $k$th term $R^{(k)}$ is the intersection of the kernels of all linear functions of the form $\tilde \varepsilon \circ A_1\circ \cdots \circ A_l$ such that each $A_j$ is either $\Delta _{i_j}$ or $s_{i_j}$, and $\Delta$s appear at most $k$ times.

By Lemma 3.10 it suffices to show that if $a\in H^*\otimes _{A^*(\mathrm {pt})} R^{(k)}$ lies in the kernel of $\theta$, then $a\in H^*\otimes _{A^*(\mathrm {pt})} R^{(k+1)}$. Write $a$ as $\sum e_i\otimes y_i$ with $y_i\in R^{(k)}$. Now applying $D=A_1\circ \cdots \circ A_l$ containing $k+1$ $\Delta$s to the image of $a$, using Lemma 3.8 $l$ times and collecting the leftmost summands of each sum separately, we obtain an expression of the form

\[ \sum_i w(\sigma(e_i))D(y_i)+\sum_{i,j} D'_j(\sigma(e_i))D_j(y_i), \]

where each $D_j$ contains at most $k$ $\Delta$s, $D'_j$ are of the same form as $D$ (i.e. $D'_j$ is a composition of Demazure operators and simple reflections), and $w$ is a product of some simple reflections. On the other hand, this expression must be zero. Taking it modulo $I$ and applying $w^{-1}$, we see that each $D(y_i)$ belongs to $I$ and so $y_i$ belongs to $R^{(k+1)}$.

Proof. Recall that a cofree comodule over an $A^*(\mathrm {pt})$-module $R^*$ is the module $H^*\otimes _{A^*(\mathrm {pt})}R^*$ with the coaction $H^*\otimes _{A^*(\mathrm {pt})}R^*\xrightarrow {\Delta \otimes \mathrm {id}} H^*\otimes _{A^*(\mathrm {pt})}H^*\otimes _{A^*(\mathrm {pt})}R^*$.

Lemma 4.12 implies that the $I$-adic filtration on $A^*(G/B)$ is compatible with the coaction by $H^*$. There is an isomorphism

\[ H^*\otimes_{A^*(\mathrm{pt})}\big(I^kR^*/I^{k+1}R^*\big)\to I^kA^*(G/B)/I^{k+1}A^*(G/B) \]

induced by the map $\theta$ of Lemma 5.4. It is compatible with the coaction (where the left-hand side is considered as a cofree $H^*$-comodule), since the diagram

commutes, and the kernel of the composite vertical map on the left is generated by $I$ by Lemma 5.3. Since the quotients of the filtration are cofree, it (non-canonically) splits, so we have an isomorphism of comodules

\[ A^*(G/B)\simeq H^*\otimes_{A^*(\mathrm{pt})}\bigg(\bigoplus_{k\ge 0} I^kR^*/I^{k+1}R^*\bigg). \]

But by Assumption 5.1, $R^*\simeq \bigoplus _{k\ge 0} I^kR^*/I^{k+1}R^*$ as $A^*(\mathrm {pt})$-modules, and the claim follows.

Proof. Theorem 4.14 implies that the image of $A^*(E/B\times E/B)$ in

\[ A^*(G/B\times G/B)\simeq\operatorname{End}_{A^*(\mathrm{pt})}(G/B) \]

actually lies in $\operatorname {End}_{H^*}A^*(G/B)$. The latter is isomorphic to $H^\vee \otimes _{A^*(\mathrm {pt})}\operatorname {End}_{A^*(\mathrm {pt})}R^*$ by Lemma 5.5, and since by Assumption 5.1 all $A^*(\mathrm {pt})$-modules are projective, it remains to compare the ranks over $A^*(\mathrm {pt})$. But it follows from [Reference Chernousov and MerkurjevCM06] and [Reference Nenashev and ZainoullineNZ06] that

\[ A^*(E/B\times E/B)\simeq A^*(E/B)\otimes_{A^*(\mathrm{pt})}A^*(G/B) \]

as $A^*(E/B)$-modules, and the isomorphism is compatible with the restriction map. So the rank of the image is equal to

\[ \operatorname{rk}_{A^*(\mathrm{pt})}R^*\cdot\operatorname{rk}_{A^*(\mathrm{pt})}A^*(G/B)=(\operatorname{rk}_{A^*(\mathrm{pt})}R^*)^2\cdot\operatorname{rk}_{A^*(\mathrm{pt})}H^*\quad\text{by Lemma 5.4}, \]

which is the same as the rank of $H^\vee \otimes _{A^*(\mathrm {pt})}\operatorname {End}_{A^*(\mathrm {pt})}R^*$.

Theorem 5.7 (Two layers of motivic decompositions) In the above notation assume additionally that the theory $A^*$ is free. Then there is a one-to-one correspondence between motivic decompositions of the $A^*$-motive of $E/B$ and direct sum decompositions of $A^*(G/B)$ as an $H^\vee$-module.

Assume further that $R^*$ is graded free as an $A^*(\mathrm {pt})$-module:

(5.8)\begin{equation} R^*\simeq\bigoplus_{i\in\mathcal{I}} A^*(\mathrm{pt})(i) \end{equation}

for some multiset of non-negative integers $\mathcal {I}$. Then there exists an $A^*$-motive $\mathcal {R}$ such that the $A^*$-motive of $E/B$ decomposes as follows:

(5.9)\begin{equation} M(E/B)\simeq\bigoplus_{i\in\mathcal{I}}\mathcal{R}\{i\}. \end{equation}

There is a one-to-one correspondence between motivic decompositions of $\mathcal {R}$ in the category of $A^*$-motives and direct sum decompositions of $H^\vee$ as a module over itself.

Moreover, $A^*(\bar {\mathcal {R}})\simeq H^*$ as $H^*$-comodules.

Proof. By Rost nilpotence (see § 2.4) motivic decompositions of $M(E/B)$ correspond to full systems of mutually orthogonal idempotents in

\[ \mathrm{Im}\big(A^*(E/B\times E/B)\xrightarrow{\mathop{\mathrm{res}}\nolimits} A^*(G/B\times G/B)\big). \]

Applying Lemma 5.6 and Morita equivalence, we see that these decompositions correspond to direct sum decompositions of $H^*\otimes _{A^*(\mathrm {pt})}R^*\simeq A^*(G/B)$ as an $H^\vee$-module.

The second claim is now clear: decomposition (5.8) of $R^*$ as an $A^*(\mathrm {pt})$-module implies the respective decomposition of $A^*(G/B)$ as an $H^\vee$-module and hence decomposition (5.9). Since $H^*$ is finitely generated projective over $A^*(\mathrm {pt})$, the structure of an $H^\vee$-module determines the structure of an $H^*$-comodule and vice versa. So, decompositions of $H^*$ as an $H^\vee$-module correspond to decompositions of $H^\vee$ as a module over itself. Finally, $A^*(\bar {\mathcal {R}})$ is isomorphic to $H^*$ as an $H^\vee$-module, so it is isomorphic to $H^*$ as an $H^*$-comodule.

Proof. We have $H^0=\mathbb {F}_p$, so there are no non-trivial decompositions of $H^\vee$ as an $H^\vee$-module. By the last assertion of Theorem 5.7 we see that the Poincaré polynomial of $\mathcal {R}$ is the same as the Poincaré polynomial of $H^*$, which in view of Remark 4.8 is given by formula (3.6).

Proof. Assume that the motive $R_p(E)\{m\}$ is a direct summand of the motive $M(E/P)$ for some integer $m$. Consider the following diagram which is commutative by Theorem 4.14:

where $\iota$ is induced by the embedding of the motivic summand $R_p(E)\to M(E/P)\{-m\}$.

By the last assertion of Theorem 5.7 we can identify $H^*\simeq \mathop {\mathrm {Ch}}\nolimits ^*(\overline {R_p(E)})$.

We can take $\alpha$ to be the image of $1$ and $\beta$ to be the image of $e_J$ under the comodule map $H^*\xrightarrow {\iota }\mathop {\mathrm {Ch}}\nolimits ^*(M(G/P)\{-m\})$. This implies the first claim.

To prove the converse statement consider the following commutative diagram.

The torus $T$ acts on $G/P$ naturally on the left, on $E\times G/P$ by the rule ${t(e,gP)=(et^{-1},tgP)}$ and on $E\times E/P$ on the first factor only. The last action is obviously free, and so the $T$-equivariant Chow ring can be identified with the Chow ring of the quotient (see [Reference Edidin and GrahamEG98, Proposition 8(a)]). The vertical maps from the top to the middle row are the forgetful maps, and the vertical maps from the middle row to the bottom row are the restriction maps.

The second map in each row is induced by the isomorphism $E\times G/P\to E\times E/P$ (respectively, $G\times G/P\to G\times G/P$ for the bottom row) sending $(e,gP)$ to $(e,egP)$ (respectively, $(h,gP)$ to $(h,hgP)$). An immediate check shows that these maps are $T$-equivariant (note that we let $T$ act on $G\times G/P$ in two different ways). The composite bottom map is induced by the map $G\times G/P\to G/P$ sending $(h,gP)$ to $h^{-1}gP$ and so coincides with $(S\otimes \mathrm {id})\circ \tilde \rho$, where $S$ denotes the antipode in the Hopf algebra $\mathop {\mathrm {Ch}}\nolimits ^*(G)$ and $\tilde \rho$ is the coaction map.

For the convenience of the reader let us simplify the following diagram.

(6.5)

We know that in $\mathop {\mathrm {Ch}}\nolimits ^*(G)$,

\[ \Delta(e_i)=e_i\otimes 1+1\otimes e_i+\sum u_j\otimes v_j, \]

with $\operatorname {codim} u_j>1$ and $\operatorname {codim} v_j>1$, so

\[ S(e_i)={-}e_i+\text{lower-degree terms,} \]

where the order of terms is as in Definition 3.5. For dimensional reasons this implies that the image of $S(e_J)$ in $H^*$ is $\pm e_J$.

Recall that in § 5 we denoted by $\sigma \colon H^*\to \mathop {\mathrm {Ch}}\nolimits ^*(G/B)$ a section of the canonical homomorphism $\mathop {\mathrm {Ch}}\nolimits ^*(G/B)\to H^*$. Consider now any preimage of $\pm \beta$ in the top left corner. It goes to some cycle $x$ in the top right corner, that is in $\mathop {\mathrm {Ch}}\nolimits ^*(E/B\times E/P)$. Using the commutativity of the diagram and Lemma 5.3, we have

\[ \bar x=\sigma(e_J)\times\alpha+\sum a'_i\times b_i+\sum \delta_i c_i\times d_i \]

with $\delta _i\in R^{>0}$, some cycles $a'_i$, $c_i$, $d_i$, and the same $b_i$ as in the statement of the theorem.

By Lemma 5.4 we can write the class of a rational point $[\mathrm {pt}]$ in $\mathop {\mathrm {Ch}}\nolimits ^*(G/B)$ as $\gamma \sigma (e_J)$ for some rational $\gamma \in R^*$ of the maximal possible degree (in particular, $\gamma \delta =0$ for every $\delta \in R^{>0}$). Then

\[ f=(\gamma\times 1)\bar x=[\mathrm{pt}]\times\alpha+\sum a''_i\times b_i, \]

with $\operatorname {codim} a''_i<\dim G/B$, is a rational cycle in $\mathop {\mathrm {Ch}}\nolimits ^*(G/B\times G/P)$.

Denote by $\alpha ^\vee$ a Poincaré dual to $\alpha$, meaning that $\deg (\alpha \alpha ^\vee )=1$. Using [Reference Edidin and GrahamEG97, Proposition 1] applied to the projection map $E/B\times E/P\to E/B$ (or diagram (6.5) again), we see that there is a rational cycle in $\mathop {\mathrm {Ch}}\nolimits ^*(G/P\times G/B)$ of the form

\[ g=\alpha^\vee{\times} 1+\sum l_j\times m_j \]

with $\operatorname {codim} m_j>0$.

Now we have

\begin{align*} &f\circ g=\alpha^\vee{\times}\alpha+\sum_{i,j}\deg(a_i''m_j)l_j\times b_i,\\ &g\circ f=[\mathrm{pt}]\times 1+\sum_{i,j}\deg(l_jb_i)a_i''\times m_j. \end{align*}

Since $\mathop {\mathrm {Ch}}\nolimits ^*(G/B\times G/B)$ and $\mathop {\mathrm {Ch}}\nolimits ^*(G/P\times G/P)$ are finite, there is a positive integer $M$ such that $(f\circ g)^{\circ M}$ and $(g\circ f)^{\circ M}$ are idempotents, and $g\circ (f\circ g)^{\circ (M-1)}$ and $f$ are rational isomorphisms between the corresponding summands. Applying the Rost nilpotence principle (see § 2.4). we get the result.

Proof. Follows from [Reference KacKac85, Table II] (cf. [Reference VishikVis05, Proposition 3.1]).

Proof. We use the method described in § 6.1 We proceed by induction on $m$. The base $m=1$ is clear: $\operatorname {\mathrm {SO}}_n/P_1$ is either the projective line or the product of two projective lines. If $m>1$ consider the parabolic subgroup $Q=P_1$. Let $E$ be a generic $C$-torsor, where $C$ is the commutator subgroup of the Levi subgroup of $P_1$. Then the Chow motive of ${}_E(\operatorname {\mathrm {SO}}_n/P_1)$ decomposes as follows (see [Reference RostRos98, Proposition 2]):

\[ M({}_E(\operatorname{\mathrm{SO}}_n/P_1))=\mathbb{F}_2\oplus M({}_E(\operatorname{\mathrm{SO}}_{n-2}/P'_1))\{1\}\oplus \mathbb{F}_2\{n-2\}, \]

where the middle summand is a quadric of smaller dimension (in this summand $P'_1$ stands for the maximal parabolic subgroup of $\operatorname {\mathrm {SO}}_{n-2}$ of type $1$). By the induction hypothesis we have

(7.3)\begin{equation} \rho(l)=\sum_{i=1}^{m-1} e_i\otimes h^{m-i}+1\otimes l\mod ( e_m)\otimes\mathop{\mathrm{Ch}}\nolimits^*(\operatorname{\mathrm{SO}}_n/P_1), \end{equation}

where $(e_m)$ is the ideal of $\mathop {\mathrm {Ch}}\nolimits ^*(\operatorname {\mathrm {SO}}_n)$ generated by $e_m$.

We have the following commutative diagram.

(7.4)

The image of $l$ in $\mathop {\mathrm {Ch}}\nolimits ^*(\operatorname {\mathrm {SO}}_n)$ is $e_m$, so

(7.5)\begin{equation} \rho(l)=e_m\otimes 1+1\otimes l\mod \mathop{\mathrm{Ch}}\nolimits^*(\operatorname{\mathrm{SO}}_n)\otimes \operatorname{Ker}\pi. \end{equation}

Moreover, $\operatorname {Ker}\pi =(h)$. All that remains is to combine formulae (7.3) and (7.5) together.

Theorem 7.7 If $j\not \in J(q)$, then the Chow motive of $Q$ has connections $m-j$ and $m$, $m-j+1$ and $m+1,\ldots , m-1$ and $m+j-1$, and, if $n$ is even, $m'$ and $m+j$.

Proof. The condition $j\not \in J(q)$ means that the generator $e_j$ of the orthogonal group is not zero in the bialgebra $H^*$ corresponding to $q$.

Consider an indecomposable summand $N$ of the motive $M(Q)$ whose realization contains $l$ or $l+h^m$. By Theorem 4.14 the realization of $N$ is a subcomodule of $\mathop {\mathrm {CH}}\nolimits ^*(\bar Q)$ under the coaction of $H^*$, so Lemma 7.2 implies that the realization contains $h^{m-j}$. This means that there is a connection $m-j$ and $m$. The other connections can be obtained similarly considering the cycles $h^k l$, $k=1,\ldots , m$, and using the identity $\rho (h^k)=1\otimes h^k$ (see Lemma 4.12).

Proof. The description of the Chow ring follows from [Reference KacKac85, Table II] and the Steenrod algebra action is described in [Reference Ishitoya, Kono and TodaIKT76, Proposition 5.1]. Note that [Reference Ishitoya, Kono and TodaIKT76] deals with the singular cohomology instead of Chow rings. But the multiplicative structure and the action of the Steenrod operations on $\mathop {\mathrm {Ch}}\nolimits ^*(X)$ and on $\mathop {\mathrm {Ch}}\nolimits ^*(G)$, where $X$ denotes the variety of Borel subgroups of $\mathrm {E}_7^{\rm sc}$, can be described in purely combinatorial terms (see [Reference Duan and ZhaoDZ07] and [Reference Garibaldi, Petrov and SemenovGPS16, § 5]) and, hence, does not depend on the choice of the base field $F$.

The Chow ring $\mathop {\mathrm {CH}}\nolimits ^*(\mathrm {E}_7^{\rm sc})\otimes \mathbb {F}_2$ can be identified with the image of the natural pullback homomorphism $H^{2*}_{\mathrm {sing}}(X,\mathbb {F}_2) \to H^{2*}_{\mathrm {sing}}(\mathrm {E}_7^{\rm sc}, \mathbb {F}_2)$. This image and formulae describing the action of Steenrod operations are described in [Reference Ishitoya, Kono and TodaIKT76] and [Reference Mimura and TodaMT78] and imply the formulae in the statement of the present lemma.

Proof. The multiplicative structure is described in [Reference Duan and ZhaoDZ10, Theorem 5]. (Alternatively one can compute the multiplicative structure using the algorithm described in [Reference Petrov and SemenovPS10, § 2] or using the equivariant algorithm described in [Reference Garibaldi, Petrov and SemenovGPS16, § 5]).

By Lemma 6.2 (applied to the case where $G=\mathrm {E}_7^{\rm sc}$, $Q=P_7$ and $C=\mathrm {E}_6^{\rm sc}$) the generator $x_5$ goes to $e_5$ and $x_9$ goes to $e_9$ under the natural map $\mathop {\mathrm {Ch}}\nolimits ^*(\mathrm {E}_7^{\rm sc}/P_7)\to \mathop {\mathrm {Ch}}\nolimits ^*(\mathrm {E}_7^{\rm sc})$. Indeed, by [Reference KacKac85, Table II], $\mathop {\mathrm {Ch}}\nolimits ^*(\mathrm {E}_6^{\rm sc})=\mathbb {F}_2[e_3]/(e_3^2)$, and thus the generators $e_5$ and $e_9$ in $\mathop {\mathrm {Ch}}\nolimits ^*(\mathrm {E}_7^{\rm sc})$ map to $0$ under the natural homomorphism $\mathop {\mathrm {Ch}}\nolimits ^*(\mathrm {E}_7^{\rm sc})\to \mathop {\mathrm {Ch}}\nolimits ^*(\mathrm {E}_6^{\rm sc})$, and therefore come from $\mathop {\mathrm {Ch}}\nolimits ^*(\mathrm {E}_7^{\rm sc}/P_7)$ by Lemma 6.2. But since $h\in \mathop {\mathrm {Ch}}\nolimits ^*(\mathrm {E}_7^{\rm sc}/P_7)$ goes to $0$ in $\mathop {\mathrm {Ch}}\nolimits ^*(\mathrm {E}_7^{\rm sc})$, for dimensional reasons $e_5$ and $e_9$ must come from $x_5$ and $x_9$, respectively.

Note that we can adjust $x_9$ by adding $x_5h^4$ if necessary so that $S^4(x_5)=x_9$.

Formula (8.4) follows from the fact that $h$ is a rational cycle and from Lemma 4.12 applied to a generic $\mathrm {E}_7^{\rm sc}$-torsor.

To prove formula (8.5) we use the method described in § 6.1. Consider the parabolic subgroup $Q=P_1$ of our split group of type $\mathrm {E}_7$. Then the commutator subgroup $C$ of the Levi part of $Q$ is the group $\operatorname {\mathrm {Spin}}_{12}$ of type $\mathrm {D}_6$, and by [Reference KacKac85, Table II],

\[ \mathop{\mathrm{Ch}}\nolimits^*(C)=\mathbb{F}_2[e_3,e_5]/(e_3^2,e_5^2). \]

Let $E$ be a generic $C$-torsor over $F$. The Chow motive of ${}_E(\mathrm {E}_7^{\rm sc}/P_7)$ decomposes by [Reference BrosnanBro05] as follows:

(8.7)\begin{equation} M({}_E(\mathrm{E}_7^{\rm sc}/P_7))= M({}_E(\mathrm{D}_6/P'_1))\oplus M({}_E(\mathrm{D}_6/P'_1))\{17\}\oplus M({}_E(\mathrm{D}_6/P'_6))\{6\}, \end{equation}

where the $P'_i$ on the right-hand side denote the respective maximal parabolic subgroups for the split group of type $\mathrm {D}_6$ (the enumeration of simple roots follows Bourbaki).

By Lemma 7.2 there is an element $y_5\in \mathop {\mathrm {Ch}}\nolimits ^5(\mathrm {D}_6/P'_1)$ such that

\[ \rho(y_5)=e_5\otimes 1+e_3\otimes h^2+1\otimes y_5, \]

and under decomposition (8.7), $y_5$ corresponds to either $x_5$ or $x_5+h^5$. Note that the homomorphism $\mathop {\mathrm {Ch}}\nolimits ^*(\mathrm {E}_7^{\rm sc})\to \mathop {\mathrm {Ch}}\nolimits ^*(C)$ is an isomorphism in codimensions up to $5$. Therefore, in both cases formula (8.5) holds.

Finally, formula (8.6) follows from formula (8.5) by applying the Steenrod operation $S^4$.

Proof. By [Reference KarpenkoKar13, Theorem 3.5] the only possible summands of $M(E/P_7)$ up to Tate twists are upper motives of the form $U(E/P)$ for parabolic subgroups $P$ such that the Tits index of $(_E\mathrm {E}_7^{\rm sc})_{F(E/P)}$ contains the Tits index of $({_E\mathrm {E}_7^{\rm sc}})_{F(E/P_7)}$. Analysing the Tits indices for groups of type $\mathrm {E}_7$ at the prime $2$ (see [Reference TitsTit66, p. 59]), one can see that there are at most two candidates for possible summands of $M(E/P_7)$ up to Tate twist, namely $U(E/P_7)$ and the motive $R_2(E)$ (see § 3.4).

However, using Theorem 6.4 and Lemma 8.3, we see that there are no summands of type $R_2(E)$.

We will apply the shell technique from [Reference Garibaldi, Petrov and SemenovGPS16, § 4]. To this end we need to compute the first shell. We have already established that $E/P_7$ is not generically split (see [Reference Petrov and SemenovPS10, Theorem 5.7]), so, passing to its function, field we get that the semisimple anisotropic kernel of $_E\mathrm {E}_7^{\rm sc}$ over $F(E/P_7)$ is of type $\mathrm {D}_4$ or $\mathrm {E}_6$ and, therefore, since by [Reference Petrov, Semenov and ZainoullinePSZ08, Corollary 5.19] the $J$-invariants of a group and of its semisimple anisotropic kernel are equivalent, ${J_2(E_{F(E/P_7)})=(1,0,0)}$.

Using Lemma 8.3 we see that the only cycles over $F(E/P_7)$ satisfying the relation $\rho (x)=1\otimes x$ are spanned by $h^i$, $x_9h^i$, $x_5h^{12}$, $x_5h^{13}$, $x_5x_9h^{12}$, $x_5x_9h^{13}$. On the other hand, it follows from the algorithm of [Reference Chernousov, Gille and MerkurjevCGM05] that the motive of $E/P_7$ contains over $F(E/P_7)$ exactly eight Tate motives, namely

\[ \mathbb{F}_2,\,\mathbb{F}_2\{1\},\,\mathbb{F}_2\{9\},\,\mathbb{F}_2\{10\},\,\mathbb{F}_2\{17\}, \mathbb{F}_2\{18\},\,\mathbb{F}_2\{26\},\,\mathbb{F}_2\{27\}. \]

By the same arguments the only cycles in $\mathop {\mathrm {Ch}}\nolimits ^9(\mathrm {E}_7^{\rm sc}/P_7)$ rational over $F$ are spanned by $h^9$, and this cycle is not Poincaré dual (we say that two cycles $\alpha$ and $\beta$ are Poincaré dual if $\deg (\alpha \beta )=1$) to any cycles rational over $F(E/P_7)$ in the dual codimension $18$, namely $0$, $x_9h^9$, $x_5h^{13}$ or $x_9h^9+x_5h^{13}$. Therefore, $h^9$ does not belong to the first shell, that is, the Tate motive $\mathbb {F}_2\{9\}$ from the list above is not a generic point of a direct summand of $M(E/P_7)$ over $F$.

Similarly, there are no cycles in $\mathop {\mathrm {Ch}}\nolimits ^{17}(\mathrm {E}_7^{\rm sc}/P_7)$ and $\mathop {\mathrm {Ch}}\nolimits ^{26}(\mathrm {E}_7^{\rm sc}/P_7)$ rational over $F$ in the case where ${J_2(E)=(1,1,1)}$. Therefore, the Tate motives $\mathbb {F}_2\{17\}$ and $\mathbb {F}_2\{26\}$ are also not generic points of a direct summand of $M(E/P_7)$ over $F$.

On the other hand, the cycle $h$, which is rational over $F$, belongs to the first shell. All that remains is to apply [Reference Garibaldi, Petrov and SemenovGPS16, Theorem 4.10] with $b=1$ and $\alpha =h$.

Proof. Follows from [Reference KacKac85, Table II].

Proof. We proceed similarly to the proof of Lemma 8.3. The multiplicative structure is described in [Reference Duan and ZhaoDZ10, Theorem 7]. Formula (8.12) follows from Lemma 4.12.

Consider now the parabolic subgroup $Q=P_{7,8}$ corresponding to the subset $\{1,2,3,4,5,6\}$ of the Dynkin diagram of type $\mathrm {E}_8$. Then the commutator subgroup $C$ of the Levi part of $Q$ is of type $\mathrm {E}_6^{\rm sc}$, and by [Reference KacKac85, Table II],

\[ \mathop{\mathrm{Ch}}\nolimits^*(C)=\mathbb{F}_3[e_4]/(e_4^3). \]

Let $E$ be a generic $C$-torsor over $F$. The Chow motive of ${}_E(\mathrm {E}_8^{\rm sc}/P_8)$ decomposes by [Reference BrosnanBro05] and [Reference Chernousov, Gille and MerkurjevCGM05] as follows:

(8.15)\begin{align} M({}_E(\mathrm{E}_8/P_8))&=\mathbb{F}_2\oplus \mathbb{F}_2\{1\}\oplus \mathbb{F}_2\{28\}\oplus \mathbb{F}_2\{29\} \nonumber\\ &\quad \oplus \mathbb{F}_2\{56\}\oplus \mathbb{F}_2\{57\}\oplus M({}_E(\mathrm{E}_6^{\rm sc}/P'_6))\{2\} \nonumber\\ &\quad \oplus M({}_E(\mathrm{E}_6^{\rm sc}/P'_6))\{29\}\oplus M({}_E(\mathrm{E}_6^{\rm sc}/P'_6))\{30\}\oplus M({}_E(\mathrm{E}_6^{\rm sc}/P'_1))\{11\} \nonumber\\ &\quad \oplus M({}_E(\mathrm{E}_6^{\rm sc}/P'_1))\{12\}\oplus M({}_E(\mathrm{E}_6^{\rm sc}/P'_1))\{39\}\oplus M({}_E(\mathrm{E}_6^{\rm sc}/P'_2))\{18\}, \end{align}

where the $P'_i$ on the right-hand side denote the respective maximal parabolic subgroups for the split simply connected group of type $\mathrm {E}_6$ (the enumeration of simple roots follows Bourbaki).

Similarly to the proof of Lemma 8.3 we use Lemma 6.2 (applied to $G=\mathrm {E}_6^{\rm sc}$, $Q=P'_6$ and $C=\mathrm {D}_5^{\rm sc}$) to obtain an element $y_4\in \mathop {\mathrm {Ch}}\nolimits ^4(\mathrm {E}_6^{\rm sc}/P'_6)$ mapping to $e_4$, so

\[ \rho(y_4)=e_4\otimes 1+1\otimes y_4. \]

Under the decomposition above, $1$ corresponds to $h^2$ and $y_4$ corresponds to either $x_6$ or to $x_6\pm h^6$. In both cases formula (8.13) holds.

Furthermore, we have

\[ \rho(y_4^2)=e_4^2\otimes 1-e_4\otimes y_4+1\otimes y_4^2. \]

Under the decomposition above, $y_4^2$ corresponds to $z=x_{10}+\alpha x_6h^4+\beta h^{10}$ for some $\alpha ,\beta \in \mathbb {F}_3$. Note that the coefficient of $x_{10}$ in $z$ is non-zero, since otherwise we would come to a contradiction with formula (8.13). Namely, we would get

\[ \alpha(e_4\otimes h^6+1\otimes x_6h^4)+\beta\otimes h^{10}=\rho(\alpha x_6h^4+\beta h^{10})=e_4^2\otimes h^2+\cdots\mod (e_{10})\otimes\mathop{\mathrm{Ch}}\nolimits^*(\mathrm{E}_8/P_8) \]

which is impossible.

It follows that

(8.16)\begin{equation} \rho(x_{10})=e_4^2\otimes h^2-e_4\otimes x_6+1\otimes x_{10}+\gamma e_4\otimes h^6\mod (e_{10})\otimes\mathop{\mathrm{Ch}}\nolimits^*(\mathrm{E}_8/P_8) \end{equation}

for some $\gamma \in \mathbb {F}_3$, where $(e_{10})$ denotes the ideal of $\mathop {\mathrm {Ch}}\nolimits ^*(\mathrm {E}_8)$ generated by $e_{10}$.

We have the following commutative diagram.

(8.17)

The image of $x_{10}$ in $\mathop {\mathrm {Ch}}\nolimits ^*(\mathrm {E}_8)$ is $e_{10}$, so

(8.18)\begin{equation} \rho(x_{10})=e_{10}\otimes 1+1\otimes x_{10}\mod \mathop{\mathrm{Ch}}\nolimits^*(\mathrm{E}_8)\otimes \operatorname{Ker}\pi. \end{equation}

Moreover, $\operatorname {Ker}\pi =(h,x_6)$. Changing $x_{10}$ to $x_{10}-\gamma x_6h^4$ (this does not affect the relations) and combining formulae (8.16) and (8.18), we obtain formula (8.14).

Proof. We proceed similarly to the proof of Proposition 8.8.

By [Reference KarpenkoKar13, Theorem 3.5] the only possible summands of $M(E/P_8)$ up to Tate twists are upper motives of the form $U(E/P)$ for parabolic subgroups $P$ such that the Tits index of $({_E\mathrm {E}_8})_{F(E/P)}$ contains the Tits index of $({_E\mathrm {E}_8})_{F(E/P_8)}$. Analysing the Tits indices for groups of type $\mathrm {E}_8$ at the prime $3$ (see [Reference TitsTit66, p. 60]) one can see that there are at most two candidates for possible summands of $M(E/P_8)$ up to Tate twist, namely $U(E/P_8)$ and the motive $R_3(E)$ (see § 3.4).