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\title{$J$-invariant of linear algebraic groups}

\author{V.~Petrov, N.~Semenov, K.~Zainoulline
\footnote{
The paper is based on the PhD Thesis of the first author and
was partially supported by CNRS, DAAD A/04/00348, INTAS 03-51-3251,  
RTN HPRN-CT-2002-00287, SFB701.}}

%\date{July 17, 2006}

%%%%%%%%%%%%%%%%%%%% DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\maketitle


\begin{abstract}
Let $G$ be a linear algebraic group 
over a field $F$ and $X$ be a 
projective homogeneous $G$-variety such that $G$ splits
over the function field of $X$.
In the present paper we introduce an invariant of $G$ called
$J$-invariant 
which characterizes the motivic behaviour of $X$.
This generalizes the respective notion invented by A.~Vishik in the
context of quadratic forms.
As a main application
we obtain a uniform proof of
all known motivic decompositions of generically split
projective homogeneous varieties 
(Severi-Brauer varieties, Pfister quadrics, maximal orthogonal Grassmannians,
$\G$- and $\F$-varieties)
as well as provide new examples (exceptional varieties of types
$\E_6$, $\E_7$ and $\E_8$).
We also discuss relations with
torsion indices, canonical dimensions
and cohomological invariants of the group $G$.  
\end{abstract}

\section*{Introduction}

Let $G$ be a simple linear algebraic group over a field $F$
and $X$ be a projective homogeneous $G$-variety.
In the present paper we address the problem of computing 
the Grothendieck-Chow motive $\M(X)$ of $X$ or, in other words,
providing a direct sum decomposition of $\M(X)$.

When the group $G$ is split, i.e.,
contains a split maximal torus, the motive of $X$ has the simplest
decomposition -- it is isomorphic to
a direct sum of twisted Tate motives.
This was first observed by B.~K\"ock in \cite{Ko91}.
The next step was done by V.~Chernousov, 
S.~Gille and A.~Merkurjev (see \cite{CGM}).
They provided an algorithm of
expressing the motive of any rational projective homogeneous $G$-variety
in terms of motives of anisotropic $G$-varieties, i.e., those
which have no rational points.
Finally, P.~Brosnan (see \cite{Br05}) generalized this result
to the case of an isotropic group $G$ and possibly anisotropic $X$. 
In all these proofs one constructs
a (relative) cellular filtration on $X$, 
which allows to express the motive of the total space $X$ 
in terms of motives of the base. Since the latter  
consists of homogeneous varieties of anisotropic groups,
it reduces the problem to the case of anisotropic group $G$.

For anisotropic groups the situation becomes more complicated:
only very few partial results are known.
Observe that in this case the components 
of a motivic decomposition are expected to 
have a non-geometric nature, i.e., can not be identified
with (twisted) motives of some other varieties. 
The first examples of such decompositions were provided
by M.~Rost \cite{Ro98}. He proved that the motive
of a Pfister quadric decomposes as a direct sum
of twisted copies of a certain non-geometric
motive $\RR$ known in the literature as Rost motive.
Motives of Severi-Brauer varieties
were described by N.~Karpenko \cite{Ka96}.
For exceptional varieties examples of motivic decompositions 
were provided by J.-P. Bonnet \cite{Bo03} (varieties of type $\G$) and 
by S.~Nikolenko, N.~Semenov, K.~Zainoulline \cite{NSZ} 
(varieties of type $\F$).

In the present paper we provide a uniform proof of all these results.
Observe that in all examples the variety $X$ has the following property:
Over the generic point of $X$ the motive $\M(X)$ 
splits as a direct sum of twisted
Tate motives. Varieties which satisfy this condition
will be called {\em generically split}.
The main result of the paper says that (see Theorem~\ref{thmMain})

\begin{mainthm} 
Let $G$ be a simple linear algebraic group 
of inner type over
a field $F$ and $p$ be a prime integer. 
Let $X$ be a generically split projective homogeneous $G$-variety.
Then the Chow motive of $X$ with $\zz/p$-coefficients
is isomorphic to a direct sum
$$
\M(X;\zz/p)\simeq \bigoplus_{i\ge 0}\RR_p(G)(i)^{\oplus a_i}
$$
of twisted copies of a certain indecomposable motive $\RR_p(G)$,
where the integers $a_i$ are coefficients of the quotient
of the respective Poincar\'e polynomials.
\end{mainthm}

Observe that the motive $\RR_p(G)$ depends only on 
$G$ and $p$ but
not on the type of a parabolic subgroup defining $X$.
Moreover, considered with $\qq$-coefficients 
it always splits as a direct sum of Tate motives. 

Our proof is based on two different observations.
The first comes from the topology of compact Lie groups.
Namely, to compute Chow rings of compact Lie groups 
V.~Kac \cite{Kc85} invented the notion
of {\em $p$-exceptional degrees} -- numbers which encode the information
about the Chow ring of a group modulo $p$.
These numbers can be expressed in terms of
degrees of basic polynomial invariants and, therefore,
deal with the combinatorics of the Weyl group.
%%% FROM PROBLEM:
%>From the other hand the results of 
From the other hand the results of 
N.~Karpenko, A.~Merkurjev \cite{KM05} and K.~Zainoulline \cite{Za06}
concerning {\em canonical $p$-dimensions} of algebraic groups
tell us that there is a strong interrelation between 
$p$-exceptional degrees and the `size' of
the image of the restriction map
$\res\colon
\CH^*(X\times X)\to \CH^*(\BX\times \BX)$ to the separable closure of $F$.
To relate this image with a motivic decomposition of $X$
we use {\it Rost Nilpotence Theorem}. 
It was first proved for quadrics
by M.~Rost and lately generalized to arbitrary 
projective homogeneous varieties
by V.~Chernousov, S.~Gille and A.~Merkurjev (see \cite[Theorem~8.2]{CGM}).
Roughly speaking, it says that any decomposition which is given
by idempotent cycles from this image, indeed, comes from $F$.

All this together lead to the notion
of {\em $J$-invariant} $J_p(G)$
of a group $G$ modulo $p$ (see Definition~\ref{jinv}) which generalizes
the respective notion introduced by A.~Vishik \cite{Vi05}
in the case of quadrics. Our main observation is that $J_p(G)$ 
completely characterizes 
the motive $\RR_p(G)$ and, hence,  
the motivic decomposition of $X$. 
Observe that if the $J$-invariant takes its minimal possible non-trivial
value $J_p(G)=(1)$, then the motive $\RR_p(G)\otimes\qq$ has the following
recognizable decomposition (cf. \cite[\S5]{Vo03})
$$
\RR_p(G)\otimes \qq \simeq \bigoplus_{i=0}^{p-1} 
\qq(i\cdot \tfrac{p^{n-1}-1}{p-1})
$$
where $n=2$ or $3$ (see the last section).

Apart from the notion of $J$-invariant 
we generalize some of the results of paper \cite{CPSZ}:
Using the motivic version of a result
of D.~Edidin and W.~Graham \cite{EG97} 
on cellular fibrations we provide
a general formulae which
expresses the motive of the total space of a cellular fibration 
in terms of the motives of its base (see Theorem~\ref{thmf}). 
We also provide several criteria for the existence of liftings
of motivic decompositions via the reduction map $\zz\to \zz/m$.
We prove that such a lifting always exists if 
$m=2$, $3$, $4$ or $6$ (see Theorem~\ref{lem47}).

The paper is organized as follows.
In the first section we recall the definition of Chow motives and 
some properties of generically split varieties.
Rather technical section~\ref{IDLI} is devoted to lifting of idempotents.
In section~\ref{EDGR} we discuss motives of cellular fibrations.
The proof of the main result actually starts with 
section~\ref{JINV}, where we introduce the notion of $J$-invariant and
provide a motivic decomposition for the variety of complete flags.
In section~\ref{MRES} we finish the proof
and give some properties of the motive $\RR_p(G)$.
The last two sections are devoted 
to various applications of $J$-invariant
and examples of motivic decompositions. 




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\section{Motives of generically split varieties}

\begin{ntt} \label{chmot}
In the present paper we will work with Chow motives
of smooth projective varieties over a field $F$.
We will use the following notation (cf. \cite{Ma68}, \cite[\S 7]{CGM} or \cite[XII]{EKM})

Given a smooth projective variety $X$ over a field $F$ 
we denote by $\M(X)$ its {\em Chow motive}, and by 
$\M(X)(n)=\M(X)\otimes \zz(n)$ the respective {\em twist} by a Tate object.
A morphism of motives between $\M(X)(n)$ and $\M(Y)(m)$, 
where $X$ is irreducible, 
is given by a class $\phi$ of rationally equivalent cycles of dimension 
$\dim X+n-m$ on $X\times Y$.
Hence, the group of endomorphisms
$\End(\M(X))$ coincides 
with the Chow group $\CH_{\dim X}(X\times X)$.
The element $\phi$ is called a 
{\em correspondence} between $X$ and $Y$ of degree $n-m$.

Given a correspondence $\phi$ of degree $d$ and $k\in \zz$ the composite
$$
\xymatrix{
\CH_{k}(X)\ar[r]^-{(\pr_X)^*} & \CH_{k+\dim Y}(X\times Y)\ar[r]^-{- \cap \phi} &
\CH_{k+d}(X\times Y) \ar[r]^-{(\pr_Y)_*}& 
\CH_{k+d}(Y)
}
$$
of the pull-back $(\pr_X)^*$, intersection product with $\phi$ and the push-forward
$(\pr_Y)_*$ is called 
the {\em realization} of $\phi$ and is denoted by $\phi_\star$.
Given correspondences 
$\phi \in\CH_{\dim X +d}(X\times Y)$ and 
$\psi\in \CH_{\dim Y+e}(Y\times Z)$ of degrees $d$ and $e$ respectively
the correspondence of degree $d+e$
$$
(\pr_{X\times Z})_*((\pr_{Y\times Z})^*\cap (\pr_{X\times Y})^*)\in \CH_{\dim X +d+e}(X\times Z)
$$
is called the {\em correspondence product} of $\phi$ and $\psi$
and is denoted by $\psi\circ\phi$. By definition
$(\psi\circ\phi)_\star=\psi_\star\circ\phi_\star$.
Given a correspondence $\phi$ we denote by $\phi^t$ its transpose.


The correspondence
product endows the group $\End(\M(X))$
with the ring structure. The identity element of this ring
is the class of the diagonal $\Delta_X$.



Finally, by $\M(X;\Lambda)$, where $\Lambda$ is a commutative ring,
we denote an object of the category of motives with $\Lambda$-coefficients
obtained by taking correspondences $\phi$
with $\Lambda$-coefficients, i.e., 
replacing $\CH(X\times Y)$ by $\CH(X\times Y)\otimes_\zz \Lambda$.
\end{ntt}

\begin{dfn}\label{spvar} 
Let $L/F$ be a field extension.
We say $L$ is a {\em splitting field} of a smooth projective variety $X$ 
or, equivalently, a variety $X$ {\em splits} over $L$ if 
the motive $\M(X)$ splits over $L$ as 
a finite direct sum of twisted Tate motives, i.e.,
$$
\M(X)_L\simeq \bigoplus_{n} \zz_L(n)
$$
We say a variety $X$ is {\em generically split} if 
it splits over the field of rational functions $F(X)$.
\end{dfn}

\begin{ex}
A variety $X$ is called {\em generically cellular} if
it is cellular over its generic point, i.e.,
$X_{F(X)}=X\times_F F(X)$ has a proper descending filtration by
closed subvarieties $X_i$ such that each complement $X_i\setminus X_{i+1}$
is a disjoint union of affine spaces defined over $F(X)$. 
According to \cite[Corollary~67.2]{EKM} if $X$ is generically cellular,
then $X$ is generically split.
\end{ex}

\begin{ex}
Let $G$ be a semisimple linear algebraic group over a field
$F$ and $X$ be a projective 
homogeneous $G$-variety. Let $L/F$ be a field extension.
If the group $G$ splits over $L$, i.e.,
$G_L=G\times_F L$ contains a split maximal torus defined over $L$,
then $X_L$ is cellular and, hence, $L$ is a splitting field of $X$.
In particular, if $G$ splits
over the function field of $X$, then $X$ is generically split.
Many examples of such varieties can be provided investigating 
Tits indices of $G$ (see Example~\ref{intmot}).
\end{ex}




\begin{ntt}\label{remsp}
Assume $X$ has a splitting field $L$.
We will write
$\CH(\overline X;\Lambda)$ for $\CH(X_L;\Lambda)$
and $\CHO(X;\Lambda)$ for the image of the restriction map
$\CH(X;\Lambda) \to \CH(\overline X;\Lambda)$  
(cf. \cite[1.2]{KM05}). 
Similarly, by $\M(\overline X)$ we denote the motive 
of $X$ considered over $L$.
If $M$ is a direct summand of $\M(X)(n)$,
by $\overline M$ we denote the motive $M_L$.
Elements of $\CHO(X)$ 
will be called {\em rational} cycles on $X_L$ with respect
to the field extension $L/F$. 
If $L'$ is another splitting field of $X$, then
there is a chain of canonical isomorphisms
$\CH(X_L)\simeq \CH(X_{LL'})\simeq \CH(X_{L'})$, where $LL'$ is the composite
of $L$ and $L'$. Hence, the groups $\CH(\overline X)$ and $\CHO(X)$ do not
depend on the choice of $L$.


There is the K\"unneth decomposition
$\End (\M(\overline X))=\CH(\overline X\times \overline X)=
\CH(\overline X)\otimes \CH(\overline X)$ 
and  Poincar\'e duality. The latter means that given a
basis of $\CH(\overline X)$ 
there is a dual one with respect to the pairing  
$(\alpha,\beta)\mapsto \deg(\alpha\cdot\beta)$, where $\deg$
is the degree map.
In view of the K\"unneth decomposition 
the correspondence product of cycles in
on $\CH(\overline X\times\overline X)$
is given by the formula
$(\alpha_1\times \beta_1)\circ (\alpha_2\times\beta_2)=
\deg(\alpha_1\beta_2)(\alpha_2\times\beta_1)$, the realization by
$(\alpha\times\beta)_\star(\gamma)=\deg(\alpha\gamma)\beta$ 
and the transpose by
$(\alpha\times\beta)^t=\beta\times\alpha$.
Since $\CH_*(\overline X)$ is a free graded $\zz$-module,
we may define its Poincar\'e polynomial as
$P(\CH_*(\overline X),t)=\sum_{i\ge 0}\rk_{\zz}
\CH_i(\overline X)\cdot t^i$.

Sometimes we will use contravariant notation $\CH^*$ 
for Chow groups, where
$\CH^k(X)=\CH_{\dim X-k}(X)$ for irreducible $X$.
\end{ntt}

\begin{lem}\label{gensp}
Let $X$ and $Y$ be two smooth projective varieties such that
$F(Y)$ is a splitting field of $X$ and $Y$ has a splitting field.
Consider the projection in the K\"unneth decomposition
$$
\pr_{0}\colon 
\CH^r(\overline X\times \overline Y)=
\bigoplus_{i=0}^r \CH^{r-i}(\overline X)\otimes \CH^i(\overline Y)
\to \CH^r(\overline X).
$$
Then for any $\rho \in \CH^r(\overline X)$ we have
$\pr_{0}^{-1}(\rho)\cap \overline{\CH^r}(X\times Y)\neq \emptyset$. 
\end{lem}

\begin{proof} Let $L$ be a common splitting field of $X$ and $Y$.
Lemma follows from the commutative diagram
$$
\xymatrix{
\CH^r(X\times_F Y) \ar[r]^{\res_{L/F}}\ar@{>>}[d]& 
\CH^r(X_L\times_L Y_L)\ar@{>>}[d] 
 \ar[dr]^{\pr_{0}}& \\
\CH^r(X_{F(Y)})\ar[r]^-\simeq & \CH^r((X_L)_{L(Y_L)}) 
\ar[r]^-\simeq & \CH^r(X_L)
}
$$
where the left square is obtained 
by taking the generic fiber of the base change morphism $X_L\to X$;
the vertical arrows are taken from 
the localization sequence for Chow groups
and, hence, are surjective; and
the bottom horizontal maps are isomorphisms since $L$ is a splitting field.
\end{proof}

We will extensively use the following version of Rost Nilpotence
Theorem.

\begin{lem}\label{rostnil}
Let $X$ be a smooth projective variety such that it splits over any field $K$ over which it has a rational point. Then for any $\alpha$ in the kernel of the natural map $\End(\M(X))\to\End(\M(\overline X))$
we have $\alpha^{\circ(\dim X+1)}=0$.
\end{lem}
\begin{proof}
See \cite[Theorem~68.1]{EKM}.
\end{proof}



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\section{Lifting of idempotents}\label{IDLI}

\begin{dfn}
Given a $\zz$-graded ring $A^*$ and two idempotents $\phi_1$, $\phi_2\in A^0$
we say $\phi_1$ and $\phi_2$ are orthogonal if $\phi_1\phi_2=\phi_2\phi_1=0$. 
We say an element $\theta_{12}$  provides an isomorphism of degree $d$ between
idempotents $\phi_1$ and $\phi_2$  if $\theta_{12}\in \phi_2 A^{-d} \phi_1$ and there exists $\theta_{21}\in \phi_1 A^d \phi_2$  such that
$\theta_{12} \theta_{21}=\phi_2$ and  $\theta_{21} \theta_{12}=\phi_1$.
\end{dfn}

\begin{ex}\label{motex} Let $\Lambda$ be a commutative ring.
Set $A^*=\End^*(\M(X;\Lambda))$, where
$$\End^{k}(\M(X;\Lambda))=\CH^{k}(X\times X;\Lambda),\quad k\in\zz $$ 
and
the multiplication is given by the correspondence product.
By definition $\End^0(\M(X;\Lambda))$ is the ring of endomorphisms
of the motive $\M(X;\Lambda)$.
Note that a direct summand of $\M(X;\Lambda)$ can be identified
with a pair $(X,\phi)$, where
$\phi$ is an idempotent, i.e., $\phi\circ\phi=\phi$ (see \cite[ch.~XII]{EKM}).
Then an isomorphism $\theta_{12}$ of degree $d$ 
between $\phi_1$ and $\phi_2$ can be identified with 
an isomorphism between the motives $(X,\phi_1)$ and $(X,\phi_2)(d)$.

\end{ex}

\begin{dfn}\label{pres} 
Let $f\colon A^*\to B^*$ be a homomorphism of $\zz$-graded rings. 
We say that $f$ is \emph{decomposition preserving} 
if given a family $\phi_i\in B^0$ of pair-wise orthogonal idempotents 
such that $\sum_i \phi_i=1_B$, 
there exists a family of pair-wise orthogonal idempotents $\varphi_i\in A^0$ 
such that  $\sum_i \varphi_i=1_A$ and 
each $f(\varphi_i)$ is isomorphic to $\phi_i$ 
by means of an isomorphism of degree $0$. 
We say $f$ is \emph{strictly decomposition preserving} if, 
moreover, one can choose $\varphi_i$ such that 
$f(\varphi_i)=\phi_i$.
 
We say $f$ is \emph{isomorphism preserving}
if for any idempotents $\varphi_1$ and $\varphi_2$ in $A^0$ and 
any isomorphism $\theta_{12}$ of degree $d$ 
between idempotents $f(\varphi_1)$ and $f(\varphi_2)$ in $B^0$  
there exists an isomorphism $\vartheta_{12}$ of degree $d$ 
between $\varphi_1$ and $\varphi_2$. 
We say $f$ is \emph{strictly isomorphism preserving} if, 
moreover, one can choose $\vartheta_{12}$ such that 
$f(\vartheta_{12})=\theta_{12}$.
\end{dfn}

\begin{ntt}\label{prope}
By definition we have the following properties of {\em (strictly) decomposition and isomorphism preserving} morphisms:
\begin{enumerate}
\item[(i)] 
Let $f\colon A^*\to B^*$ and 
$g\colon B^*\to C^*$ be homomorphisms such that
$g\circ f$ is {\em decomposition} (resp. {\em isomorphism}) {\em preserving}  
and $g$ is {\em isomorphism preserving}. 
Then $f$ is {\em decomposition} (resp. {\em isomorphism}) {\em preserving}.
\item[(ii)]
Assume we are given a commutative diagram with $\ker f'\subset\im i$
$$
\xymatrix{
A^*\ar@{>>}[r]^f\ar@{^{(}->}[d]_i & B^*\ar@{^{(}->}[d]^{i'}\\
A'^*\ar@{>>}[r]^{f'} & B'^*.
}
$$
If $f'$ is {\em strictly decomposition} 
(resp. {\em strictly isomorphism}) {\em preserving}, 
then so is $f$.
\end{enumerate}
\end{ntt}

\begin{prop}\label{nilpcor}
Let $f\colon A^*\to B^*$ be a surjective homomorphism 
such that the kernel of the restriction of $f$ to $A^0$ 
consists of nilpotent elements. 
Then $f$ is strictly decomposition (cf. \cite[Proposition~95.1]{EKM})
and strictly isomorphism preserving.
\end{prop}

\begin{proof}
The fact that $f$ is {\em strictly decomposition preserving} follows
from \cite[Proposition~27.4]{AnFu}.
The fact that $f$ is {\em strictly isomorphism preserving} 
follows from Lemma~\ref{izvrat} below.
\end{proof}

\begin{lem}\label{izvrat}
Let $A$, $B$ be two rings, $A^0$, $B^0$ be their subrings,
$f^0\colon A^0\to B^0$ be a ring homomorphism, $f\colon A\to B$ be a map of 
sets satisfying the following conditions:
\begin{itemize}
\item $f(\alpha)f(\beta)$ equals either $f(\alpha\beta)$ or $0$ for all
$\alpha,\beta\in A$;
\item $f^0(\alpha)$ equals $f(\alpha)$ if $f(\alpha)\in B^0$ or $0$ otherwise;
\item $\ker f^0$ consists of nilpotent elements.
\end{itemize}
Let $\varphi_1$ and $\varphi_2$ be two idempotents in $A^0$, $\psi_{12}$ and 
$\psi_{21}$ be elements in $A$ such that $\psi_{12}A^0\psi_{21}\subset A^0$, 
$\psi_{21}A^0\psi_{12}\subset A^0$,
$f(\psi_{21})f(\psi_{12})=f(\varphi_1)$, 
$f(\psi_{12})f(\psi_{21})=f(\varphi_2)$. Then there exist elements
$\vartheta_{12}\in\varphi_2 A^0\psi_{12}A^0\varphi_1$ and
$\vartheta_{21}\in\varphi_1 A^0\psi_{21}A^0\varphi_2$ such that
$\vartheta_{21}\vartheta_{12}=\varphi_1$, 
$\vartheta_{12}\vartheta_{21}=\varphi_2$,
$f(\vartheta_{12})=f(\varphi_2)f(\psi_{12})=f(\psi_{12})f(\varphi_1)$, 
$f(\vartheta_{21})=f(\varphi_1)f(\psi_{21})=f(\psi_{21})f(\varphi_2)$.
\end{lem}

\begin{proof}
Since $\ker f^0$ consists of nilpotents, $f^0$ sends non-zero idempotents in 
$A^0$ to non-zero idempotents in $B^0$; in particular, $f(\varphi_1)=f^0
(\varphi_1)\ne 0$,
$f(\varphi_2)=f^0(\varphi_2)\ne 0$.
Observe that
$$
f(\psi_{12})f(\varphi_1)=f(\psi_{12})f(\psi_{21})f(\psi_{12})=f(\varphi_2)f(\psi_{12})
$$
and, similarly, $f(\psi_{21})f(\varphi_2)=f(\varphi_1)f(\psi_{21})$.
Changing $\psi_{12}$ to $\varphi_2\psi_{12}\varphi_1$ and $\psi_{21}$ to 
$\varphi_1\psi_{21}\varphi_2$ we may assume that 
$\psi_{12}\in\varphi_2A\varphi_1$ and $\psi_{21}\in\varphi_1A\varphi_2$. We 
have
$$
f^0(\varphi_2)=f(\varphi_2)=f(\psi_{12})f(\psi_{21})=f(\psi_{12}\psi_{21})
=f^0(\psi_{12}\psi_{21});$$
therefore $\alpha=\psi_{12}\psi_{21}-\varphi_2\in A^0$ is nilpotent, say 
$\alpha^n=0$. Note that $\varphi_2\alpha=\alpha=\alpha\varphi_2$. Set 
$\alpha^\vee=\varphi_2-\alpha+\ldots+(-1)^{n-1}\alpha^{n-1}\in A^0$; then 
$\alpha\alpha^\vee=\varphi_2-\alpha^\vee$, 
$\varphi_2\alpha^\vee=\alpha^\vee=\alpha^\vee\varphi_2$ and $f(\varphi_2)=f^0
(\varphi_2)=f^0(\alpha^\vee)=f(\alpha^\vee)$. Therefore setting 
$\vartheta_{21}=\psi_{21}\alpha^\vee$ we have $\vartheta_{21}\in 
\varphi_1A\varphi_2$, $\psi_{12}\vartheta_{21}=\varphi_2$ and 
$f(\vartheta_{21})=f(\psi_{21})$.

Now $\vartheta_{21}\psi_{12}$ is an idempotent. We have
$$
f^0(\varphi_1)=f(\varphi_1)=f(\vartheta_{21})f(\psi_{12})
=f(\vartheta_{21}\psi_{12})=f^0(\vartheta_{21}\psi_{12});
$$
therefore $\beta=\vartheta_{21}\psi_{12}-\varphi_1\in A^0$ is nilpotent. Note 
that $\beta\varphi_1=\beta=\varphi_1\beta$. Now
$\varphi_1+\beta=(\varphi_1+\beta)^2=\varphi_1+2\beta+\beta^2$ and therefore
$\beta(1+\beta)=0$. But $1+\beta$ is invertible and hence we have $\beta=0$. 
It means that $\vartheta_{21}\psi_{12}=\varphi_1$ and we can set 
$\vartheta_{12}=\psi_{12}$.
\end{proof}


\begin{cor}\label{karpcor}
The map $\End^*(\M(X;\zz/p^n))\to\End^*(\M(X;\zz/p))$ 
is strictly decomposition (cf. \cite[Corollary~95.3]{EKM})
and strictly isomorphism preserving.
\end{cor}

\begin{proof} Apply Proposition~\ref{nilpcor} to the case
$A^*=\End^*(\M(X;\zz/p^n))$, $B^*=\End^*(\M(X;\zz/p))$ and
 the reduction map $f\colon A^*\to B^*$.
\end{proof}


\begin{lem}\label{chinese}
Let $m=m_1m_2$ be a product of two coprime integers.
Then the map $\End^*(\M(X;\zz/m))\to\End^*(\M(X;\zz/m_1))\times
\End^*(\M(X;\zz/m_2))$ is an isomorphism.
\end{lem}
\begin{proof}
Apply Chinese Remainder Theorem.
\end{proof}



\begin{ntt}
%%% FROM PROBLEM:
%>From now on assume that $X$ is a smooth projective variety 
From now on assume that $X$ is a smooth projective variety 
which has a splitting field such that the kernel of the restriction map
$$
\res_E \colon\End^*(\M(X_E;\Lambda))\to\End^*(\M(\overline X;\Lambda))
$$
consists of nilpotent elements for all extensions $E/F$ and 
all rings of coefficients $\Lambda$.
By Lemma~\ref{rostnil} the latter holds if 
$X$ splits over any field extension over which it has a rational point.
We denote by $\overline{\End^*}(\M(X;\Lambda))$ the image of $\res_F$.
\end{ntt}

\begin{cor}\label{rostcor}
The map $\End^*(\M(X_E;\Lambda))\to\overline{\End^*}(\M(X_E;\Lambda))$ 
is strictly decomposition and strictly isomorphism preserving
for any field extension $E/F$.
\end{cor}

\begin{proof}
Apply Proposition~\ref{nilpcor} to the homomorphism $\res_E\colon A^*\to B^*$
between the graded rings
$A^*=\End^*(\M(X_E;\Lambda))$ and
$B^*=\overline{\End^*}(\M(X_E;\Lambda))$.
\end{proof}


\begin{dfn}
We say that a field extension $E/F$ is {\em rank preserving} with respect to 
$X$ if the restriction map $\res_{E/F}\colon\CH(X)\to\CH(X_E)$ 
becomes an isomorphism after tensoring with $\mathbb{Q}$.
\end{dfn}

\begin{lem}\label{prat} 
Assume $X$ has a splitting field. Then for any rank preserving finite field 
extension $E/F$ we have
$[E:F]\cdot\CHO(X_E)\subset\CHO(X)$.
\end{lem}
\begin{proof} 
Let $L$ be a splitting field containing $E$.  Let $\gamma$ be any element in 
$\CHO(X_E)$. By definition there exists $\alpha\in\CH(X_E)$ 
such that  $\gamma=\res_{L/E}(\alpha)$.
Since $\res_{E/F}\otimes \mathbb{Q}$ is an isomorphism, there exists an 
element $\beta\in\CH(X)$ and  a non-zero integer $n$ such that $\res_{E/F}
(\beta)=n\alpha$. By projection formula 
$$
n\cdot \cores_{E/F}(\alpha)=\cores_{E/F}(\res_{E/F}(\beta))=[E:F]\cdot \beta.
$$
Applying $\res_{L/E}$ to both sides we obtain
$n(\res_{L/E}(\cores_{E/F}(\alpha)))=n[E:F]\cdot\gamma$. 
Therefore, $\res_{L/E}(\cores_{E/F}(\alpha))=[E:F]\cdot\alpha$. 
\end{proof}

\begin{cor}\label{transf}
Assume $X$ has a splitting field, $E/F$ is a field extension of degree coprime 
with $m$, which is rank preserving with respect to $X\times X$. Then the map 
$\End^*(\M(X;\zz/m))\to\End^*(\M(X_E;\zz/m))$ is decomposition and 
isomorphism preserving.
\end{cor}

\begin{proof}
By Lemma~\ref{prat} we have
$\overline{\End^*}(\M(X_E;\zz/m))=\overline{\End^*}(\M(X;\zz/m))$.
Now apply Corollary~\ref{rostcor} and \ref{prope}.(i) with
$A^*=\End^*(\M(X;\zz/m))$, $B^*=\End^*(\M(X_E;\zz/m))$ and
$C^*=\overline{\End^*}(\M(X_E;\zz/m))$.
\end{proof}

\begin{lem}\label{surjlem} 
The map $\SL_l(\zz)\to \SL_l(\zz/m)$ induced by the reduction
modulo $m$ is surjective.
\end{lem}
\begin{proof} Since $\zz/m$ is a semi-local ring, 
the group $\SL_l(\zz/m)$ is generated by elementary matrices 
(see \cite[Theorem~4.3.9]{HOM89}).
\end{proof}

Given a free graded $\zz$-module $V^*$ set $\End^{-d}(V^*)$, $d\in \zz$,
to be the group of endomorphisms of $V^*$ decreasing the degree by $d$.



\begin{prop}\label{lem46}(cf. \cite[\S 96]{EKM})
Consider a free graded $\zz$-module $V^*$ of finite rank and
the reduction map $f\colon \End^*(V^*)\to \End^*(V^*\otimes_{\zz}\zz/m)$.
Assume that the graded components of the respective
$\im\phi_i$ (see Definition~\ref{pres}) are free $\zz/m$-modules. 
Then $f$ is strictly decomposition preserving.
Moreover, if $(\zz/m)^\times=\{\pm 1\}$, then 
$f$ is strictly isomorphism preserving.
\end{prop}

\begin{proof} 
We are given a decomposition $V^k\otimes_{\zz}\zz/m=\oplus_i W_i^k$,
where $W_i^k$ is the $k$-graded component of $\im\phi_i$.
Present $V^k$ as a direct sum  $V^k=\bigoplus_i V_i^k$ of free $\zz$-modules
such that $\rk_\zz V_i^k=\rk_{\zz/m} W_i^k$. Fix a $\zz$-basis $\{v_{ij}^k\}_j$ of $V_i^k$. For each $W_i^k$ choose a basis $\{w_{ij}^k\}_j$ such that the linear transformation $D^k$ of $V^k\otimes_{\zz}\zz/m$ sending each $v_{ij}^k\otimes 1$ to $w_{ij}^k$ has  determinant $1$. By Lemma~\ref{surjlem} there is a lifting $\tilde D^k$ of  $D^k$ over $\zz$. So we obtain $V^k=\bigoplus_i \tilde W_i^k$, where  $\tilde W_i^k=\tilde D^k(V_i^k)$ satisfies
$\tilde W_i^k\otimes_{\zz}\zz/m=W_i^k$. Define $\varphi_i$ on each $V^k$ to be the projection onto $\tilde W_i^k$.

Now let $\varphi_1,\varphi_2$ be two idempotents in $\End^*(V^*)$. 
Denote by $V_i^k$ the $k$-graded component of $\im\varphi_i$. 
An isomorphism $\theta_{12}$ between $\varphi_1\otimes 1$ and $\varphi_2\otimes 1$ of degree $d$ can be identified with a family of isomorphisms $\theta_{12}^k\colon V_1^k\otimes\zz/m\to V_2^{k-d}\otimes\zz/m$. 
In the case $(\zz/m)^\times=\{\pm1\}$ all these isomorphisms are given
by matrices with determinants $\{\pm 1\}$ and, hence, can be 
lifted to isomorphisms $\vartheta_{12}^k\colon V_1^k\to V_2^{k-d}$
by Lemma~\ref{surjlem}.
\end{proof}

Now we are ready to formulate and prove the main result of this section

\begin{thm}\label{lem47}
Assume $X$ has a splitting field of degree $m$ which is rank preserving with 
respect to $X\times X$. Then the map $$\End^*(\M(X))\to\End^*(\M(X;\zz/m))$$
preserves decompositions with the property that $\im\res(\phi_i)$ are free 
$\zz/m$-modules, where
$$\res\colon\End^*(\M(X;\zz/m))\to\End^*(\M(\overline X;\zz/m))$$
is the restriction. If additionally $(\zz/m)^\times=\{\pm 1\}$ then this map 
is isomorphism preserving.
\end{thm}



\begin{proof}
Consider the diagram
$$
\xymatrix{
\End^*(\M(X))\ar@{>>}[r]^-f\ar@{>>}[d]&\End^*(\M(X;\zz/m))\ar@{>>}[d]\\
\overline{\End^*}(\M(X))\ar@{>>}[r]^-{\bar f}\ar@{^{(}->}[d]_{i}&\overline{\End^*}(\M(X;\zz/m))\ar@{^{(}->}[d]\\
\End^*(\M(\overline X))\ar@{>>}[r]^-{f'}&\End^*(\M(\overline X;\zz/m)).
}
$$
Note that using Poincar\'e duality (see \ref{remsp})
we can identify
$\End^{-d}(\M(\overline X))$ with the group of endomorphisms of 
$\CH^*(\overline X)$ which decrease the grading by $d$.
Applying Proposition~\ref{lem46} to the case $V^*=\CH^*(\overline X)$
we obtain that the map $f'$ is {\em strictly decomposition preserving}. 
Moreover, if 
$(\zz/m)^\times=\{\pm 1\}$ then $f'$ is {\em strictly isomorphism preserving}.
By Lemma~\ref{prat} $\ker f' \subset \im i$ and, therefore, 
applying \ref{prope}.(ii) we obtain that $\bar f$ is 
{\em strictly decomposition preserving} 
and, moreover, $\bar f$ is {\em strictly isomorphism preserving} 
if $(\zz/m)^\times=\{\pm 1\}$.  
Now by Corollary~\ref{rostcor}
the vertical arrows of the top square 
are {\em strictly decomposition} and {\em strictly isomorphism preserving}.
It remains to apply \ref{prope}.(i).
\end{proof}






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\section{Motives of fibered spaces}\label{EDGR}


\begin{dfn}\label{celfib}
Let $X$ be a smooth projective variety over a field $F$.
We say a smooth projective morphism $f\colon Y\to X$ is a 
{\em cellular fibration} if it is a locally trivial fibration
whose fiber $\mathcal{F}$ is cellular, i.e.,
has a decomposition into affine cells (see \cite[\S 67]{EKM}). 
\end{dfn}

\begin{lem}\label{edigrah} 
Let $f\colon Y\to X$ be a cellular fibration.
Then $\M(Y)$ is isomorphic to 
$\M(X)\otimes \M(\mathcal{F})$. 
\end{lem}

\begin{proof}
We follow the proof of \cite[Proposition~1]{EG97}. 
Define the morphism
$$
\varphi\colon \bigoplus_{i\in\mathcal{I}} \M(X)(\codim B_i) \to \M(Y)
$$
to be the direct sum $\varphi=\bigoplus_{i\in\mathcal{I}}\varphi_i$,
where each $\varphi_i$ is given by the cycle 
$[\pr_Y^*(B_i)\cdot \Gamma_f]\in\CH(X\times Y)$ 
produced from  the graph cycle $\Gamma_f$ and 
the chosen (non-canonical) basis $\{B_i\}_{i \in \mathcal{I}}$
of $\CH(Y)$ over $\CH(X)$.
The realization of $\varphi$ coincides exactly with
an isomorphism of abelian groups $\CH(X)\otimes\CH(\mathcal{F})\to 
\CH(Y)$ constructed in \cite[Proposition~1]{EG97}.
Then, by Manin's identity principle (see \cite[\S3]{Ma68}) 
$\varphi$ is an isomorphism.
\end{proof}



\begin{lem}\label{lemhomog}
Let $G$ be a linear algebraic group over a field $F$, 
$X$ be a projective homogeneous $G$-variety and $Y$ be a $G$-variety. 
Let $f\colon Y\to X$ be a $G$-equivariant projective morphism. 
Assume that the fiber of $f$ over $F(X)$ is 
isomorphic to $\mathcal{F}_{F(X)}$ for some variety $\mathcal{F}$ over $F$. 
Then $f$ is a locally trivial fibration with the fiber $\mathcal{F}$.
\end{lem}

\begin{proof}
By the assumptions, we have $Y\times_X\Spec F(X)\simeq(\mathcal{F}\times 
X)\times_X\Spec F(X)$ as schemes over $F(X)$. Since $F(X)$ is a direct limit 
of $\mathcal{O}(U)$ taken over all non-empty affine open subsets $U$ of $X$, 
by [EGA IV, Corollaire~8.8.2.5] there exists $U$ such that $f^{-1}
(U)=Y\times_X U$ is isomorphic to $(\mathcal{F}\times X)\times_X 
U\simeq\mathcal{F}\times U$ as a scheme over $U$. Since $G$ acts transitively 
on $X$ and $f$ is $G$-equivariant, the map $f$ is a locally trivial 
fibration.
\end{proof}


\begin{cor}\label{corprod}
Let $X$ be a projective $G$-homogeneous variety, $Y$ be a projective variety 
such that $Y_{F(X)}\simeq\mathcal{F}_{F(X)}$ for some variety $\mathcal{F}$. 
Then the projection map $X\times Y\to X$ is a locally trivial fibration with 
the fiber $\mathcal{F}$.
Moreover, if $\mathcal{F}$ is cellular, 
then $\M(X\times Y)\simeq\M(X)\otimes\M(\mathcal{F})$.
\end{cor}

\begin{proof}
Apply Lemma~\ref{lemhomog} to the projection map $X\times Y\to X$
and use Lemma~\ref{edigrah}.
\end{proof}


\begin{ntt}\label{nota}
Let $G$ be a simple (connected) linear algebraic group over a field $F$, 
$X$ be a projective homogeneous $G$-variety. Denote by $\mathcal D$ the 
Dynkin diagram of $G$. According to \cite{Ti66}
one can always choose a quasi-split 
group $G_0$ over $F$ with the same Dynkin diagram, 
a parabolic subgroup $P$ of $G_0$ and 
a cocycle $\xi\in H^1(F,G_0)$ such that 
$G$ is isogenic to ${}_\xi G_0$ and 
$X$ is isomorphic to ${}_\xi(G_0/P)$.
We will use the following standard notation:
If $G_0$ is split, then $G$ 
is called to be of 
\emph{inner type} over $F$.
\end{ntt}

\begin{lem}\label{mainthmf}
Let $G$ be a semisimple linear algebraic group over $F$, $X$ and $Y$ be 
projective homogeneous $G$-varieties 
corresponding to parabolic subgroups $P$ 
and $Q$ of $G_0$, $Q\subseteq P$. Denote by $f\colon Y\to X$ the map induced 
by the quotient map $G_0/Q\to G_0/P$. If $G$ becomes 
split over $F(X)$ then $f$ is a cellular fibration with the fiber
$\mathcal{F}=P/Q$.
\end{lem}
\begin{proof}
Since $G$ splits over $F(X)$, the fiber of $f$ over $F(X)$ is isomorphic to 
$(P/Q)_{F(X)}=\mathcal{F}_{F(X)}$. Now apply Lemma~\ref{lemhomog} and note 
that $\mathcal{F}$ is cellular.
\end{proof}

\begin{ex}\label{intmot} 
Let $P=P_\Theta$ be the standard parabolic subgroup of a split group 
$G_0$, corresponding to a subset $\Theta$ of the respective 
Dynkin diagram $\mathcal{D}$ (enumeration of roots follows Bourbaki). In this 
notation the Borel subgroup corresponds to the empty set. Let $\xi$ be a 
cocycle in $H^1(F,G_0)$. 
Set $G={}_\xi G_0$ and $X={}_\xi(G_0/P)$. Denote 
by $q$ the degree of a splitting field of $G_0$ and 
by $d$ the index of associated Tits
algebra (for groups of classical types $d$ is given in 
\cite[Table~II]{Ti66}). 
Analyzing Tits indices of $G$
we see that $G$ becomes 
split over $F(X)$ and, therefore, 
$X$ becomes {\em generically split} over $F$
if the subset 
$\mathcal D\setminus\Theta$ contains one of the following vertices $k$ 
(cf. \cite[\S 7]{KR94}): 

\medskip

\noindent
\begin{tabular}{l|l|l|l|l}
$G_0$ & ${}^1\A_n$ & $\B_n$            & $\C_n$ & ${}^1\D_n$ \\\hline
$k$ & $\gcd(k,d)=1$ & $k=n$;     & $k$ is odd; & $k=n-1$; \\
&  & any $k$ in the &  & $k=n$ if $2\nmid n$ or $d=1$;\\
& & Pfister case &  &  any $k$ in the Pfister case
\end{tabular}
 
\medskip

\noindent
\begin{tabular}{l|l|l|l|l|l}
$G_0$ & $\G$ & $\F$ & ${}^1\E_6$ & $\E_7$ & $\E_8$ \\\hline
$k$ & any & $k=1,2,3$; & $k=3,5$; & $k=2,5$; & $k=2,3,4,5$; \\
& & any $k$ if  & $k=2,4$ if $d=1$; & $k=3,4$ if $d=1$; & any $k$ if \\
& & $q=3$ & $k=1,6$ if $q$ is odd & $k\neq 7$ if $q=3$ & $q=5$\\
\end{tabular}

\medskip

\noindent
(here by the Pfister case we mean the case when the cocycle $\xi$ corresponds
to a Pfister form or its maximal neighbor)
\end{ex}




Case-by-case arguments of  paper \cite{CPSZ} show that 
under certain conditions the Chow motive
of a twisted flag variety $X$ can be expressed
in terms of the motive of a minimal flag.
These conditions cover almost all twisted flag varieties 
corresponding to groups of types $\A_n$ and $\B_n$ 
together with some examples of types $\C_n$, $\G$ and $\F$.
Using the following theorem we provide a uniform proof of these results
as well as extend it to some other types. 

\begin{thm}\label{thmf} Let $Y$ and $X$ be taken as in Lemma~\ref{mainthmf}.
Then the Chow motive $\M(Y)$ of $Y$ is isomorphic
to a direct sum of twisted copies of the motive $\M(X)$, i.e.,
$$
\M(Y) \simeq \bigoplus_{i\ge 0} \M(X)(i)^{\oplus c_i},
$$
where $\sum c_it^i=P(\CH_*(\overline Y),t)/P(\CH_*(\overline X),t)$.
\end{thm}
\begin{proof} Apply Lemmas~\ref{mainthmf} and \ref{edigrah}.
\end{proof}

\begin{rem}\label{expldec} 
The explicit formula for $P(\CH_*(\overline X),t)$ involves the degrees
of basic polynomial invariants of $G_0$ and is provided in
\cite[Ch.~IV, Cor.~4.5]{Hi82}.
\end{rem}




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\section{Varieties of complete flags}\label{JINV}

\begin{ntt}\label{sdef}
Let $G_0$ be a split simple linear algebraic group with a 
maximal split torus $T$ and 
a Borel subgroup $B$ containing $T$.
Let $G={}_\xi G_0$ be a twisted form of $G_0$ 
given by a cocycle $\xi\in H^1(F,G_0)$ and 
$X={}_\xi(G_0/B)$ be the corresponding \emph{variety of complete flags}. 
Observe that the group $G$ splits over any field $K$ 
over which $X$ has a rational point, in particular, over
the function field $F(X)$. According to \cite{De74} the Chow ring 
$\CH(\BX)$ can be expressed in purely combinatorial terms and, 
therefore, depends only on the type of $G$ but not on the base field $F$.
\end{ntt}


\begin{ntt}\label{notgen}
Let $p$ be a prime integer.
To simplify the notation
we denote by $\Ch(X)$ the
Chow ring of $X={}_\xi(G_0/B)$ with $\zz/p$-coefficients and by 
$\ChO(X)$ the image of the restriction map 
$\CH(X;\zz/p)\to \CH(\overline X;\zz/p)$. 
Let $\hat T$ denote the group of characters of $T$ and $S(\hat T)$
be the symmetric algebra. 
By $R$ we denote the image of the characteristic map 
$c\colon S(\hat T) \to \Ch(\BX)$
(see \cite[(4.1)]{Gr58}).
According to \cite[Thm.6.4]{KM05} there is an embedding
\begin{equation}\label{ratemb}
R\subseteq \ChO(X),
\end{equation}
where the equality holds if the cocycle $\xi$ 
corresponds to a generic torsor.
\end{ntt}

\begin{ntt}
Let $\Ch(\BG)$ denote the Chow ring with $\zz/p$-coefficients
of the group $G_0$.
Consider the pull-back induced by the quotient map
$$
\pi\colon \Ch(\BX)\to \Ch(\BG)
$$
According to \cite[Rem.~$2^\circ$]{Gr58} 
$\pi$ is surjective with the kernel
generated by $R^+$, 
where $R^+$ stands for the subgroup of non-constant elements of $R$.

The explicit presentation of $\Ch(\BG)$  
is known for all types of $G$ and all torsion primes $p$ of $G$
(see \cite[Definition~3]{Gr58}). 
Namely, by \cite[Theorem~3]{Kc85} it is a quotient of the polynomial ring
in $r$ variables $x_1,\ldots,x_r$ of codimensions 
$d_1\le d_2\le\ldots\le d_r$ coprime to $p$, 
modulo an ideal generated by certain $p$-powers 
$x_1^{p^{k_1}},\ldots,x_r^{p^{k_r}}$ ($k_i\ge 0$, $i=1,\ldots,r$)
\begin{equation}\label{kcpres}
\Ch^*(\BG)=(\zz/p)[x_1,\ldots,x_r]/(x_1^{p^{k_1}},\ldots,x_r^{p^{k_r}}).
\end{equation}
In the case $p$ is not a torsion prime of $G$ we have $\Ch^*(\BG)=\zz/p$, i.e.,
$r=0$.

Note that the complete list of
numbers $\{d_ip^{k_i}\}_{i=1\ldots r}$ 
called \emph{$p$-exceptional degrees} of $G_0$ 
was provided in \cite[Table II]{Kc85}.  
Taking the $p$-primary and $p$-coprimary parts of each $p$-exceptional degree 
one immediately restores the respective $k_i$ and $d_i$.
\end{ntt}

\begin{ntt} We introduce two orders on the set of additive generators 
of $\Ch(\BG)$, i.e., monomials 
$x_1^{m_1}\ldots x_r^{m_r}$. 
To simplify the notation, we will denote
the monomial $x_1^{m_1}\ldots x_r^{m_r}$ by $x^M$, where $M$ is an $r$-tuple
of integers $(m_1,\ldots,m_r)$. The codimension of $x^M$ will be denoted
by $|M|$. Observe that $|M|=\sum_{i=1}^r d_im_i$.
\begin{itemize}
\item
Given two $r$-tuples $M=(m_1,\ldots,m_r)$
and $N=(n_1,\ldots,n_r)$
we say $x^M\preccurlyeq x^N$ (or equivalently $M\preccurlyeq N$)
if $m_i\le n_i$ for all $i$. This gives a partial ordering on the
set of all monomials ($r$-tuples).
\item
Given two $r$-tuples $M=(m_1,\ldots,m_r)$
and $N=(n_1,\ldots,n_r)$ we say
$x^M\le x^N$ (or equivalently $M\le N$) if either $|M|<|N|$, or $|M|=|N|$ and 
$m_i\le n_i$ for the greatest $i$ such that $m_i\ne n_i$.
This gives a well-ordering on the set of all monomials ($r$-tuples)
known also as {\em DegLex} order. 
\end{itemize}
\end{ntt}



\begin{dfn}\label{jinv} Let $X={}_\xi (G_0/B)$ be the twisted
form of the variety of complete flags by means of a cocycle 
$\xi\in H^1(F,G_0)$.
Let $\ChO(G)$ denote the image of the composite
$$
\Ch(X)\xra{\res} \Ch(\BX) \xra{\pi} \Ch(\BG)
$$
Since both maps are ring homomorphisms, $\ChO(G)$ is a subring of $\Ch(\BG)$.

For each $1\le i\le r$ set $j_i$ to be the smallest non-negative
integer such that the subring $\ChO(G)$ contains an element $a$ 
with the greatest monomial $x_i^{p^{j_i}}$ 
with respect to the {\em DegLex} order on $\Ch(\BG)$, i.e.,
of the form 
$$
a=x_i^{p^{j_i}}+\sum_{x^M\lneq x_i^{p^{j_i}}} c_M x^M.
$$
The $r$-tuple of integers $(j_1,\ldots,j_r)$ will be called
{\em $J$-invariant of $G$ modulo $p$} and will be denoted by $J_p(G)$.
\end{dfn}

\begin{ex} From presentation \eqref{kcpres} we have 
$j_i\le k_i$ for all $i=1,\ldots,r$. 
According to \eqref{ratemb} the $J$-invariant
takes its maximal value $J_p(G)=(k_1,\ldots,k_r)$ if
the cocycle $\xi$ corresponds to a generic torsor.
Later on (see Corollary~\ref{trivJ}) it will
be shown that the $J$-invariant takes its minimal possible value 
$J_p(G)=(0,\ldots,0)$ if and only if 
the group $G$ splits by a finite field extension of degree coprime to $p$.
\end{ex}

\begin{ex} If the Chow ring 
$\Ch(\BG)$ has only one generator, i.e., $r=1$, then
the $J$-invariant is equal to the smallest
non-negative integer $j_1$ such that $x_1^{p^{j_1}}\in \ChO(G)$.
\end{ex}

The next example explains the terminology `$J$-invariant'

\begin{ex}
Let $\phi$ be a quadratic form with trivial discriminant. 
In \cite[Definition~5.11]{Vi05} 
A.~Vishik introduced the notion of 
$J$-invariant of $\phi$, the tuple of integers,  
which describes the subgroup of rational
cycles on the respective maximal orthogonal Grassmannian.
This invariant is an important tool in studying
algebraic cycles on quadrics.
An equivalent but `dual' 
(in terms of non-rationality of cycles) definition of 
$J(\phi)$ was provided in \cite[\S~88]{EKM}. 
Using Theorem~\ref{thmf} one can show that $J(\phi)$ introduced in \cite{EKM}
can be expressed in terms of $J_2(\OO(\phi))=(j_1,\ldots,j_r)$ as follows:
$$
J(\phi)=\{2^ld_i\mid i=1,\ldots,r,\,0\le l\le j_i-1\}.
$$
Since all $d_i$ are odd, $J_2(\OO(\phi))$ is uniquely determined by $J(\phi)$ as well.
\end{ex}


Now we are ready to formulate and prove the main result of this section

\begin{thm}\label{thmKac}
Given  $G$ and $p$ with $J_p(G)=(j_1,\ldots,j_r)$ 
the motive of $X$ is isomorphic to the direct sum
$$
\M(X;\zz/p)\simeq \bigoplus_{i\ge 0}\RR_p(G)(i)^{\oplus c_i}, 
$$
where the motive $\RR_p(G)$ is indecomposable, 
its Poincar\'e polynomial over a splitting field is equal to
\begin{equation}\label{poincr}
P(\overline{\RR_p(G)},t)=\prod_{i=1}^r \frac{1-t^{d_ip^{j_i}}}{1-t^{d_i}},
\end{equation}
and the integers $c_i$ are the coefficients of the polynomial 
$$
\sum_{i\ge 0} c_i t^i=P(\Ch^*(\BX),t)/P(\overline{\RR_p(G)},t).
$$
\end{thm}

Fix preimages $e_i$ of $x_i$ in $\Ch(\BX)$. For an $r$-tuple 
$M=(m_1,\ldots,m_r)$
set $e^M=\prod_{i=1}^r e_i^{m_i}$. 
Set $K=(k_1,\ldots,k_r)$ and $N=p^K-1=(p^{k_1}-1,\ldots,p^{k_r}-1)$.
Set $d=\dim X-|N|=\deg(P(R^*,t))$.

\begin{lem}\label{Grothlem}
The Chow ring $\Ch(\BX)$ is a free $R$-module
with a basis $\{e^M\}$, $M\preccurlyeq N$.
\end{lem}
\begin{proof}
Note that $R^+$ is a nilpotent ideal in $R$.
Applying Nakayama Lemma we obtain that
$\{e^M\}$ generate $\Ch(\BX)$.
By \cite[(2)]{Kc85} $\Ch(\BX)$ is a free $R$-module, hence,
for the Poincar\'e polynomials we have
$$
P(\Ch^*(\BX),t)=P(\Ch^*(\BG),t)\cdot P(R^*,t).$$
Substituting $t=1$ we obtain that 
$$
\rk \Ch(\BX)=\rk \Ch(\BG)\cdot \rk R.
$$
To finish the proof  observe that 
$\rk \Ch(\BG)$ 
coincides with the number of generators $\{e^M\}$.
\end{proof}

\begin{prop}\label{pairclm}
The pairing $R\times R \to \zz/p$ given by 
$(\alpha,\beta)\mapsto \deg(e^N\alpha\beta)$ is non-degenerated, i.e.,
for any element $\alpha\in R$ 
there exists $\beta$ such that $\deg(e^N\alpha\beta)\neq 0$. 
\end{prop}

\begin{proof}
Choose a homogeneous basis of $\Ch(\BX)$.
Let $\alpha^\vee$ be the Poincar\'e dual of $\alpha$ with respect to this
basis. 
By Lemma~\ref{Grothlem} $\Ch(\BX)$ is a free $R$-module with a basis 
$\{e^M\}$, hence,
expanding $\alpha^\vee$
we obtain
$$
\alpha^\vee=\sum_{M\preccurlyeq N} e^M\beta_M, \text{ where } \beta_M\in R.
$$
Note that if $M\neq N$ then $\codim \alpha\beta_M > d$, therefore,
$\alpha\beta_M=0$. So we can set $\beta=\beta_N$.
\end{proof}

%%% FROM PROBLEM:
%>From now on we fix a homogeneous $\zz/p$-basis $\{\alpha_i\}$ of $R$ and the
From now on we fix a homogeneous $\zz/p$-basis $\{\alpha_i\}$ of $R$ and the
dual basis $\{\alpha_i^{\#}\}$ 
with respect to the pairing introduced in Proposition~\ref{pairclm}.

\begin{cor} \label{cbasis} 
For $|M|\le |N|$ we have
$$
\deg(e^M\alpha_i\alpha_j^{\#})=\begin{cases} 1, & M=N \text{ and } i=j \\
0, & \text{ otherwise}
\end{cases}
$$
\end{cor}

\begin{proof} 
If $M=N$, then it follows from the definition of the dual basis.
Assume $|M|<|N|$.  
If $\deg(e^M\alpha_i\beta_j)\neq 0$, 
then $\codim(\alpha_i\beta_j)>d$,
a contradiction with the fact that $\alpha_i\beta_j\in R$. 
Hence, we reduced to the case $M\neq N$ and $|M|= |N|$.
Since $|M|=|N|$, $\codim(\alpha_i\beta_j)=d$ and, hence, 
$R^+\alpha_i\beta_j=0$.
%%% FROM PROBLEM:
%>From the other side 
From the other side 
there exists $i$ such that $m_i\ge p^{k_i}$ and
$e^{p^{k_i}}\in \Ch(\BX)\cdot R^+$. Hence, $e^M\alpha_i\beta_j=0$.
\end{proof}

\begin{dfn}\label{grfil}
Given two pairs $(L,l)$ and $(M,m)$, 
where $L,M$ are $r$-tuples and $l,m$ are integers, 
we say $(L,l)\le (M,m)$ if either $L\lneq M$, or in the case $L=M$ 
we have $l\le m$.
We introduce the filtration on the ring $\Ch(\BX)$ as follows: 
\begin{quote}
The $(M,m)$-th term $\Ch(\BX)_{M,m}$ 
is the subring generated by the elements 
$e^I\alpha$ with $I\le M$, $\alpha\in R$, $\codim\alpha\le m$.
\end{quote}
Define the \emph{associated graded ring} as follows:
$$
A^{*,*}=\bigoplus_{(M,m)}A^{M,m},
\text{ where }
A^{M,m}=\Ch(\BX)_{M,m}/\bigcup_{(L,l)\lneq(M,m)}\Ch(\BX)_{L,l}.
$$
By Lemma~\ref{Grothlem} if $M\preccurlyeq N$ the graded component $A^{M,m}$ 
consists of the classes of elements 
$e^M\alpha$ with $\alpha\in R$ and $\codim\alpha= m$.
In particular, $\rk A^{M,m}=\rk R^m$.
Comparing the ranks we see that
$A^{M,m}$ is trivial when $M\not\preccurlyeq N$. 

Consider the subring $\ChO(X)$ of rational cycles 
with the induced filtration. 
The associated graded subring will be denoted by 
$A^{*,*}_{rat}$.
%%% FROM PROBLEM:
%>From the definition of $J$-invariant it follows that
From the definition of $J$-invariant it follows that
the elements $e_i^{p^{j_i}}$, $i=1,\ldots, r$, belong to $A^{*,*}_{rat}$.

Similarly, we introduce the filtration 
on the ring $\Ch(\BX\times\BX)$ as follows:
\begin{quote} 
The $(M,m)$-th term is the subring generated 
by the elements $e^I\alpha\times e^L\beta$ with $I+L\le M$, 
$\alpha,\beta\in R$ and $\codim\alpha+\codim\beta\le m$.
\end{quote} 
The associated graded ring will be denoted by $B^{*,*}$.
By definition 
$B^{*,*}$ is isomorphic to the
product of graded rings $A^{*,*}\otimes_{\zz/p} A^{*,*}$. 
The graded subring associated to $\ChO(X\times X)$ will be denoted by 
$B^{*,*}_{rat}$.
\end{dfn}

\begin{ntt}\label{keyobserv}
The key observation is that due to Corollary~\ref{cbasis} we have
$$
\Ch(\BX\times\BX)_{M,m}\circ\Ch(\BX\times\BX)_{L,l}\subset
\Ch(\BX\times\BX)_{M+L-N,m+l-d}\text{ and }
$$
$$
(\Ch(\BX\times\BX)_{M,m})_\star(\Ch(\BX)_{L,l})\subset
\Ch(\BX)_{M+L-N,m+l-d}
$$
and, therefore, we have the correctly defined composition law
$$
\circ\colon B^{M,m}\times B^{L,l}\to B^{M+L-N,m+l-d}
$$
and realization map (see \ref{remsp})
$$
\star\colon B^{M,m}\times A^{L,l}\to A^{M+L-N,m+l-d}
$$
In particular, $B^{N+*,d+*}$ 
can be viewed as a graded ring with respect to the composition
such that $(\alpha\circ\beta)_\star=\alpha_\star\circ\beta_\star$.
Note also that both operations preserve rationality of cycles.
\end{ntt}

The proof of  the following result is based on the fact that the variety
$X$ is generically split

\begin{lem}\label{sigrat}
The elements $e_i\times 1-1\times e_i$, $i=1,\ldots,r$, 
belong to $B^{*,*}_{rat}$.
\end{lem}

\begin{proof} The proof is based on the fact that the variety
$X$ is generically split. Fix an $i$.
Since $X$ splits over $F(X)$, by Lemma~\ref{gensp}
there exists a cycle in $\overline{\Ch^{d_i}}(X\times X)$ of the form
$$
\xi=e_i\times 1+\sum_s \mu_s\times \nu_s+1\times\mu,
$$
where $\codim\mu_s,\codim\nu_s<d_i$. Then the cycle
$$
\pr_{13}^*(\xi)-\pr_{23}^*(\xi)=(e_i\times 1-1\times e_i)\times 1+
\sum_s (\mu_s\times 1-1\times\mu_s)\times\nu_s
$$
belongs to $\ChO(X\times X\times X)$. 
Applying Corollary~\ref{corprod} 
to the projection 
$\pr_3\colon X\times X\times X\to X\times X$ 
we conclude that the pull-back 
$\pr_3^*\colon\Ch(X\times X)\to\Ch(X\times X\times X)$ 
has a (non-canonical) section, say, $\delta_3$. 
Since the construction of this section preserves base change
it preserves rationality of cycles.
Hence, passing to a splitting field we obtain a rational cycle
$$
\delta_3(\pr_{13}^*(\xi)-\pr_{23}^*(\xi))=e_i\times 1-1\times e_i+
\sum_s (\mu_s\times 1-1\times\mu_s)\delta_3(1\times1\times\nu_s)
$$
whose image in $B^{*,*}_{rat}$ is $e_i\otimes 1-1\otimes e_i$.
\end{proof}



We will write $(e\times 1-1\times e)^M$ for the product
$\prod_{i=1}^r (e_i\times 1-1\times e_i)^{m_i}$.
and ${M\choose L}$ for the product of binomial coefficients
$\prod_{i=1}^r{m_i\choose l_i}$.
In the computations we will extensively use the following two
formulas
(the first follows directly from Corollary~\ref{cbasis} and the second
is a well-known binomial identity).

\begin{ntt}\label{karpform}
Let $\alpha$ be an element of $R^*$ and $\alpha^\#$ be its dual
with respect to the non-degenerate pairing from~\ref{pairclm}, 
i.e., $\deg(e^N\alpha\alpha^\#)=1$. Then we have
$$
((e\times 1-1\times e)^M(\alpha^\#\times 1))_\star(e^L\alpha)=
{M \choose M+L-N}(-1)^{M+L-N}e^{M+L-N}
$$
\end{ntt}


\begin{ntt}[Lucas' Theorem]\label{lucas}
The following identity holds
$$
{n\choose k}\equiv \prod_{i\ge 0} {n_i \choose k_i} \mod p,
$$
where $k=\sum_{i\ge 0}k_ip^i$ and $n=\sum_{i\ge 0} n_i p^i$ are
base $p$ presentations of $k$ and $n$.
\end{ntt}

Set for brevity $J=J_p(G)=(j_1,\ldots,j_r)$ and 
recall that $K=(k_1,\ldots,k_r)$.

\begin{prop}\label{allratX}
Let $\{\alpha_i\}$ be a homogeneous $\zz/p$-basis of $R$.
Then the set of elements $\mathcal{B}=\{e^{p^JL}\alpha_i\mid 
L\preccurlyeq p^{K-J}-1\}$ 
forms a $\zz/p$-basis of $A^{*,*}_{rat}$.
\end{prop}

\begin{proof}
According to Lemma~\ref{Grothlem} 
the elements from $\mathcal{B}$ are linearly independent. 
Assume $\mathcal{B}$ does not generate $A^{*,*}_{rat}$.
Choose an element $\omega\in A^{M,m}_{rat}$
of the smallest index $(M,m)$
which is not in the linear span of $\mathcal{B}$.
By definition of $A^{M,m}$ (see Definition~\ref{grfil}) 
$\omega$ can be written as 
$\omega=e^M\alpha$, where $M\preccurlyeq N$, $\alpha\in R^m$ and
$M$ can not be presented as $M=p^JL'$ for an $r$-tuple $L'$.
The latter means that in the decomposition of $M$ into $p$-primary
and $p$-coprimary components $M=p^SL$, where 
$S=(s_1,\ldots,s_r)$, $L=(l_1,\ldots,l_r)$ and $p\nmid l_k$ 
for $k=1,\ldots,r$, 
we have 
$J\not\preccurlyeq S$.
Choose an $i$ such that $s_i<j_i$. Denote $M_i=(0,\ldots,0,m_i,0,\ldots,0)$
and $S_i=(0,\ldots,0,s_i,0,\ldots,0)$,
where $m_i$ and $s_i$ stand at the $i$-th place.

Set $T=N-M+M_i$.                    
By Lemma~\ref{sigrat} and \ref{karpform} together with
 observation~\ref{keyobserv}
the element
$$
((e\times 1-1\times e)^T(\alpha^\#\times 1))_\star(e^M\alpha)=
{p^{k_i}-1 \choose m_i}(-1)^{m_i}e^{m_i}
$$
belongs to $A^{M_i,0}_{rat}$. By \ref{lucas} we have
$p\nmid {p^{k_i}-1\choose m_i}$ and, therefore, this element is non trivial.
Moreover, since $s_i<j_i$, this element is not in the span of $\mathcal{B}$.
Since $(M,m)$ was chosen to be the smallest one
and $(M_i,0)\le (M,m)$ we obtain that $(M,m)=(M_i,0)$.
Repeating the same arguments 
for $T=N-M_i+p^{S_i}$ we obtain that
$M_i=p^{S_i}$, i.e., $l_i=1$. 

Now let
$\gamma$ be a representative of $\omega=e_i^{p^{s_i}}$ in $\ChO(X)$. 
Then its image $\pi(\gamma)$ in $\ChO(G)$
has the leading term $x_i^{p^{s_i}}$ 
with $s_i<j_i$. This contradicts to the definition of $J$-invariant.
\end{proof}

\begin{cor}\label{allratXX}
The elements 
$$
\{(e\times 1-1\times e)^S(e^{p^JL}\alpha_i\times e^{p^J(p^{K-J}-1-M)}\alpha_j^{\#})\mid 
L, M\preccurlyeq p^{K-J}-1,\; S\preccurlyeq p^J-1\}
$$
form a $\zz/p$-basis of $B^{*,*}_{rat}$. In particular,
they form a basis of $B^{N,d}_{rat}$
if and only if $S=p^J-1$ and $L=M$.
\end{cor}

\begin{proof}
According to \ref{Grothlem} these elements are linearly independent 
and their number is $p^{|2K-J|}(\rk R)^2$. They are rational 
according to Definition~\ref{grfil} and
Lemma~\ref{sigrat}.
Applying Corollary~\ref{corprod} we obtain that
$$
\rk B^{*,*}_{rat} =
\rk \ChO(X\times X)= 
\rk\ChO(X)\cdot\rk\Ch(\BX),
$$
where the latter coincides with $\rk A^{*,*}_{rat}\cdot p^{|K|}\rk R =
 p^{|2K-J|}(\rk R)^2$ by Lemma~\ref{Grothlem} and Proposition~\ref{allratX}.
\end{proof}

\begin{lem}\label{clidem}
The elements
$$
\theta_{L,M,i,j}=
(e\times 1-1\times e)^{p^J-1}
(e^{p^JL}\alpha_i\times e^{p^J(p^{K-J}-1-M)}\alpha_j^{\#}),\; 
L,M\preccurlyeq p^{K-J}-1,
$$
belong to $B^{*,*}_{rat}$ and
satisfy the relation
$\theta_{L,M,i,j}\circ\theta_{L',M',i',j'}=\delta_{LM'}\delta_{ij'}
\theta_{L',M,i',j}$.
\end{lem}

\begin{proof}
Follows from Corollary~\ref{cbasis}.
\end{proof}


\begin{proof}[Proof of Theorem~\ref{thmKac}]
Consider the projection map
$$
f^0\colon\ChO(X\times X)_{N,d}\to B^{N,d}_{rat}.
$$
By Lemma~\ref{clidem} 
the elements $\theta_{L,L,i,j}$ 
form a family of pairwise-orthogonal
idempotents whose sum is the identity.
The kernel of $f^0$ is nilpotent and, therefore, by Proposition~\ref{nilpcor} 
there exist pair-wise orthogonal idempotents $\varphi_{L,i}$ in 
$\ChO(X\times X)$ which are mapped to $\theta_{L,L,i,i}$ 
and whose sum is the identity. 
Their components of codimension $|N|+d=\dim X$ 
have the same properties and, hence, we may
assume that 
$\varphi_{L,i}$ belong to $\overline{\Ch^{\dim X}}(X\times X)$.

We show that $\varphi_{L,i}$ are indecomposable. 
By Corollary~\ref{allratXX} and Lemma~\ref{clidem} 
the ring $(B^{N,d}_{rat},\circ)$ can be identified
with a product of matrix rings over $\zz/p$
$$
B^{N,d}_{rat}\simeq\prod_{s=0}^d\End((\zz/p)^{{p^{|K-J|}\rk R^s}}).
$$
By means of this identification $\theta_{L,L,i,i}\colon 
e^{p^JM}\alpha_j\mapsto \delta_{L,M}\delta_{i,j}e^{p^JL}\alpha_i$ 
is an idempotent of rank $1$ and, therefore, is indecomposable. 
Since $f^0$ preserves isomorphisms, 
$\varphi_{L,i}$ are indecomposable as well.


We show that $\varphi_{L,i}$ is isomorphic to $\varphi_{M,j}$. In the ring 
$B^{*,*}_{rat}$ mutually inverse isomorphisms between them are given by
$\theta_{L,M,i,j}$ and $\theta_{M,L,j,i}$. Let
$$
f\colon\ChO(X\times X)\to B^{*,*}_{rat}
$$
be the \emph{leading term} map; it means that for any $\gamma\in
\ChO(X\times X)$ we find the smallest degree $(I,s)$ such that
$\gamma$ belongs to $\ChO(X\times X)_{I,s}$ and set $f(\gamma)$ to be
the image of $\gamma$ in $B^{I,s}_{rat}$. 
Note that $f$ is not a homomorphism 
but satisfies the condition that $f(\xi)\circ f(\eta)$ equals either 
$f(\xi\circ\eta)$ or $0$. Choose preimages $\psi_{L,M,i,j}$ and
$\psi_{M,L,j,i}$ of $\theta_{L,M,i,j}$ and $\theta_{M,L,j,i}$ 
by means of $f$. 
Applying Lemma~\ref{izvrat} we obtain mutually inverse isomorphisms 
$\vartheta_{L,M,i,j}$ and $\vartheta_{M,L,j,i}$ between 
$\varphi_{L,i}$ and 
$\varphi_{M,j}$. It remains to take their homogeneous components of the 
appropriate degrees.


Now applying Lemma~\ref{rostnil} and Corollary~\ref{rostcor} 
to the restriction map
$$
\res_F\colon \End(\M(X;\zz/p))=\Ch(X\times X)\to \ChO(X\times X)=
\overline{\End^*}(\M(X;\zz/p))
$$
and the family of idempotents $\varphi_{L,i}$
we obtain the family of pair-wise orthogonal idempotents
$\phi_{L,i}\in\End(\M(X;\zz/p))$ 
such that 
$$
\Delta_X=\sum_{L,i} \phi_{L,i}.
$$
Since $\res_F$ is isomorphism preserving, for the respective motives we have
$(X,\phi_{L,i})\simeq (X,\phi_{0,0})(|L|+i)$ for all $L$ and $i$ 
(see Example~\ref{motex}).
Denoting $\RR_p(G)=(X,\phi_{0,0})$
we obtain the desired motivic decomposition.
\end{proof}

As a direct consequence of the proof we obtain

\begin{cor}\label{remKrull}
Any direct summand of $\M(X;\zz/p)$ 
is isomorphic to a direct sum of twisted copies of $\RR_p(G)$.
\end{cor}

\begin{proof}
Indeed, in the ring $B^{N,d}_{rat}$ any idempotent is isomorphic 
to a sum of idempotents $\theta_{L,L,i,i}$, 
and the map $f^0$ preserves isomorphisms. 
\end{proof}

\begin{rem}
Note that Corollary~\ref{remKrull} can be viewed as a particular
case of the Krull-Schmidt Theorem proven by V.~Chernousov
and A.~Merkurjev (see \cite[Corollary~9.7]{CM06}).
\end{rem}

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\section{Motivic decompositions}\label{MRES}


In the present section we prove the main result of this paper

\begin{thm}\label{thmMain}
Let $G$ be a simple linear algebraic group of inner type
over a field $F$ and $p$ be a prime integer.
Let $X$ be a generically split
projective homogeneous $G$-variety. 
Then the motive of $X$ with $\zz/p$-coefficients is 
isomorphic to the direct sum
$$
\M(X;\zz/p)\simeq \bigoplus_{i\ge 0}\RR_p(G)(i)^{\oplus a_i},
$$
where $\RR_p(G)$ is an indecomposable motive, whose Poincar\'e
polynomial $P(\overline{\RR_p(G)},t)$ is given by \eqref{poincr} and,
hence, depends only on the $J$-invariant of $G$,
and $a_i$ are the coefficients of the quotient polynomial 
$$
\sum_{i\ge 0} a_i t^i=P(\CH^*(\BX),t)/P(\overline{\RR_p(G)},t).
$$
\end{thm}

\begin{proof}
The variety $X$ is generically split means that the group $G$ becomes
split over $F(X)$. Let $Y$ be the variety of complete $G$-flags.
According to Theorem~\ref{thmf} the motive of $Y$ is isomorphic to
a direct sum of twisted copies of the motive of $X$.
To finish apply Theorem~\ref{thmKac} and Corollary~\ref{remKrull}.
\end{proof}

\begin{lem}\label{surj} 
Let $G$ be a group of inner type, 
$X$ be a projective homogeneous $G$-variety. 
Then any field extension $E/F$ is rank preserving with respect to $X$ and 
$X\times X$.
\end{lem}

\begin{proof}
By  \cite[Theorem~2.2 and 4.2]{Pa94} the restriction map
$K_0(X) \to K_0(X_E)$  becomes an isomorphism after tensoring with 
$\qq$. Now the Chern character
$ch\colon K_0(X)\otimes\qq\to \CH^*(X)\otimes\qq$
is an isomorphism and respects pull-backs, hence $E$ is rank preserving with 
respect to $X$. It remains to note that $X\times X$ is $G\times 
G$-homogeneous variety.
\end{proof}

Now we provide several
properties of
$\RR_p(G)$ which will be extensively used in the applications:

\begin{prop}\label{thmComp}
Let $G$ and $G'$ be two simple groups of inner type, $X$ and $X'$ be 
the corresponding varieties of complete flags.
\begin{itemize}
\item {\rm\bf (base change)} For any field extension $E/F$ we have
$$\RR_p(G)_E\simeq\bigoplus_{i\ge 0}\RR_p(G_E)(i)^{\oplus a_i},$$
where $\sum a_it^i=P(\overline{\RR_p(G)},t)/P(\overline{\RR_p(G_E)},t)$.
\item {\rm\bf (transfer argument)} If $E/F$ is a field extension of degree 
coprime to $p$ then $J_p(G_E)=J_p(G)$ and $\RR_p(G_E)=\RR_p(G)_E$. Moreover, 
if $\RR_p(G_E)\simeq\RR_p(G'_E)$ then $\RR_p(G)\simeq\RR_p(G')$.
\item {\rm\bf (comparison lemma)} If $G$ splits over $F(X')$ and $G'$ splits 
over $F(X)$ then $\RR_p(G)\simeq\RR_p(G')$.
\end{itemize}
\end{prop}



\begin{proof}
The first claim follows from Theorem~\ref{thmKac} and 
Corollary~\ref{remKrull}. 
To prove the second claim note that $E$ is rank preserving with respect to 
$X$ and $X\times X$ by Lemma~\ref{surj}. Now $J_p(G_E)=J_p(G)$ by 
Lemma~\ref{prat}, and hence $\RR_p(G_E)=\RR_p(G)_E$ by the first claim. The 
remaining part of the claim follows from Corollary~\ref{transf} applied to the 
variety $X\coprod X'$.

Now we prove the last claim. The variety $X\times X'$ is the variety of 
complete $G\times G'$-flags. Applying Corollary~\ref{corprod} we can express 
$\M(X\times X';\zz/p)$ in terms of $\RR_p(G)$ and $\RR_p(G')$. Now apply 
Corollary~\ref{remKrull}.
\end{proof}

\begin{cor}
We have $\RR_p(G)\simeq\RR_p(G_{an})$, where $G_{an}$ is the anisotropic 
kernel of $G$.
\end{cor}


Let $m$ be a positive integer.
We say a polynomial $g(t)$ is $m$-{\it positive}, 
if $g\neq 0$, $P(\overline{\RR_p(G)},t)\mid g(t)$ 
and the quotient polynomial $g(t)/P(\overline{\RR_p(G)},t)$
has non-negative coefficients
for all primes $p$ dividing $m$.

\begin{prop}\label{intd}
Let $G$ be a simple linear algebraic group of inner type
over a field $F$ and $X$ be a generically split
projective homogeneous $G$-variety. Assume that 
$X$ splits by a field extension of degree $m$.
Let $f(t)$ be an $m$-positive polynomial dividing $P(\M(\BX),t)$ which
can not be presented as a sum of two $m$-positive polynomials.
Then the motive of $X$ with integer coefficients
splits as a direct sum
$$
\M(X)\simeq \bigoplus_{i}\RR_i(c_i),\quad c_i\in\zz,
$$
where $\RR_i$ is indecomposable and
$P(\overline{\RR}_i,t)=f(t)$ for all $i$.
Moreover, if $m=2,3,4$ or $6$, then
all motives $\RR_i$ are isomorphic up to twists.
\end{prop}

\begin{proof} First, we apply Corollary~\ref{karpcor} and
Lemma~\ref{chinese}
to obtain a decomposition with $\zz/m$-coefficients. 
By Lemma~\ref{surj} our field extension is rank preserving so
we can apply Theorem~\ref{lem47}
to lift the decomposition over $\zz$.
\end{proof}

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\section{Properties of $J$-invariant}

\begin{ntt}
Recall (see \cite{Br03}) that if the characteristic 
of the base field $F$ is different from $p$ then 
one can construct \emph{Steenrod $p$-th power operations}
$$
S^l\colon\Ch^*(X)\to\Ch^{*+l(p-1)}(X),\qquad l\ge 0
$$
such that $S^0=\id$, $S^l$ restricted to $\Ch^l(X)$ 
coincides with taking to the $p$-th power, 
and the total operation 
$S^\bullet=\sum_{l\ge 0}S^l$ 
is a homomorphism of $\zz/p$-algebras 
compatible with pull-backs. 
In the case of varieties over the field of complex numbers 
$S^l$ is compatible with its topological counterparts: 
reduced power operation $\mathcal{P}^l$ if $p\ne 2$ and 
Steenrod square $Sq^{2l}$ if $p=2$ 
(in this case $\Ch^*(X)$ can be viewed as a subring in $\HH^{2*}(X,\zz/p)$).

When $X$ is the variety of complete $G$-flags 
the action of Steenrod operations on $\Ch^*(\BX)$ 
can be described in purely combinatorial terms (see \cite{DZ07}) and, hence,
doesn't depend on the choice of a base field. 
Since Steenrod operations respect 
pull-back they respect rationality as well.

Over the field of complex numbers $\Ch^*(\BG)$ 
may be identified with the image of the pull-back map 
$\HH^{2*}(\BX,\zz/p)\to \HH^{2*}(\BG,\zz/p)$. 
An explicit description of this image and 
formulae describing the action of $\mathcal{P}^l$ 
on $\HH^*(\BG,\zz/p)$ can be found in \cite{MT91}.
\end{ntt}

The following lemma provides an important technical tool
for computing the possible values of $J$-invariant of $G$.

\begin{lem}\label{strestr}
Assume that in $\Ch^*(\BG)$ we have $S^l(x_i)=x_m^{p^s}$ and $S^l(x_{i'})<x_m^{p^s}$  if $i'<m$ with respect to the {\em DegLex} order.
Then $j_m\le j_i+s$. 
\end{lem}
\begin{proof}
By definition there exists a cycle $\alpha\in\ChO(X)$ 
such that the leading term of $\pi(\alpha)$ is $x_i^{p^{j_i}}$. 
For the total operation we have
$$
S(x_i^{p^{j_i}})=S(x_i)^{p^{j_i}}=S^0(x_i)^{p^{j_i}}+S^1(x_i)^{p^{j_i}}+
\ldots+S^{d_i}(x_i)^{p^{j_i}}.
$$
In particular, $S^{lp^{j_i}}(x_i^{p^{j_i}})=S^l(x_i)^{p^{j_i}}$.
Applying $S^{lp^{j_i}}$ to $\alpha$
we obtain a rational cycle whose image 
under $\pi$ has the leading term $x_m^{p^{j_i+s}}$.
\end{proof}

\begin{ntt}\label{bigtab}
We summarize information about restrictions on $J$-invariant which can be obtained using Lemma~\ref{strestr} into the following table (numbers $d_i$ and $k_i$ are taken from \cite[Table~II]{Kc85}).

\medskip
\noindent
\begin{tabular}{|l|l|l|l|l|l|}
\hline
$G_0$ & $p$ & $r$ & $d_i$ & $k_i$ & $j_i$\\\hline
$\SL_n/\mu_m,\,m\mid n$ & $p\mid m$ & $1$ & $1$ & $p^{k_1}\parallel n$ & any \\
$\PGSp_n,\,2\mid n$ & $2$ & $1$ & $1$ & $2^{k_1}\parallel n$ & any \\
$\SO_n$ & $2$ & $[\frac{n+1}{4}]$ & $2i-1$ & $[\log_2\frac{n-1}{2i-1}]$ & $j_i\ge j_{i+l}$ if $2\nmid{i-1\choose l},$\\
& & & & & $j_i\le j_{2i-1}+1$\\
$\Spin_n$ & $2$ & $[\frac{n-3}{4}]$ & $2i+1$ & $[\log_2\frac{n-1}{2i+1}]$ & $j_i\ge j_{i+l}$ if $2\nmid{i\choose l},$\\
& & & & & $j_i\le j_{2i}+1$\\
$\PGO_{2n}$, $n>1$ & $2$ & $[\frac{n+2}{2}]$ & $1,\,i=1$ & $2^{k_1}\parallel n$ & $j_i\ge j_{i+l}$ if $2\nmid{i-2\choose l},$ \\
&&& $2i-3,\,i\ge 2$&$[\log_2\frac{2n-1}{2i-3}]$&  $j_i\le j_{2i-2}+1$\\
$\Ss_{2n},\,2\mid n$ & $2$ & $\frac{n}{2}$ & $1,\,i=1$ &$2^{k_1}\parallel n$&$j_i\ge j_{i+l}$ if $2\nmid{i-1\choose l}$\\
&&& $2i-1,\,i\ge 2$& $[\log_2\frac{2n-1}{2i-1}]$ &  $j_i\le j_{2i-1}+1$\\
$\G,\,\F,\,\E_6$ & $2$ & $1$ & $3$ & $1$ &\\
$\F,\,\E_6^{sc},\,\E_7$ & $3$ & $1$ & $4$ & $1$ &\\
$\E_6^{ad}$ & $3$ & $2$ & $1,\,4$ & $2,\,1$ &\\
$\E_7^{sc}$ & $2$ & $3$ & $3,\,5,\,9$ & $1,\,1,\,1$ & $j_1\ge j_2\ge j_3$\\
$\E_7^{ad}$ & $2$ & $4$ & $1,\,3,\,5,\,9$ & $1,\,1,\,1,\,1$ & $j_2\ge j_3\ge j_4$\\
$\E_8$ & $2$ & $4$ & $3,\,5,\,9,\,15$ & $3,\,2,\,1,\,1$ & $j_1\ge j_2\ge j_3,$\\
& & & & & $j_1\le j_2+1,\,j_2\le j_3+1$\\
$\E_8$ & $3$ & $2$ & $4,\,10$ & $1,\,1$ & $j_1\ge j_2$\\
$\E_8$ & $5$ & $1$ & $6$ & $1$ &\\
\hline
\end{tabular}
\medskip
\end{ntt}

We give some applications of $J$-invariant. First, as a by-product of the proof of Theorem~\ref{thmKac} we obtain the following expression for the canonical $p$-dimension of the variety of complete flags (cf. \cite[Theorem~90.3]{EKM} for the case of quadrics).

\begin{prop} In the notation of Theorem~\ref{thmKac} we have
$$
cd_p(X)=\sum_{i=1}^r d_i(p^{j_i}-1).
$$
\end{prop}

\begin{proof}
Follows from Proposition~\ref{allratX} and \cite[Theorem~5.8]{KM05}.
\end{proof}

Let $X$ be a smooth projective variety which has a splitting field.

\begin{lem}\label{trlem}
For any $\phi,\psi\in\CH^*(\BX\times\BX)$ one has
$$
\deg((\pr_2)_*(\phi\cdot\psi^t))=\tr((\phi\circ\psi)_*).
$$
\end{lem}
\begin{proof}
Choose a homogeneous basis $\{e_i\}$ of $\CH^*(\BX)$. 
Let $\{e_i^\vee\}$ be its Poincar\'e dual. 
Since both sides are bilinear, 
it suffices to check the assertion for 
$\phi=e_i\times e_j^\vee$ and $\psi=e_k\times e_l^\vee$. 
In this case both sides are equal to $\delta_{il}\delta_{jk}$.
\end{proof}

Denote by $d(X)$ the greatest common divisor 
of the degrees of all zero cycles on $X$ and 
by $d_p(X)$ its $p$-primary component.

\begin{cor}\label{trcor}
For any $\phi\in\CHO(X\times X;\zz/m)$ we have $$g.c.d.(d(X),m)\mid\tr(\phi_*).$$
\end{cor}
\begin{proof}
Set $\psi=\Delta_{\bar X}$ and apply Lemma~\ref{trlem}.
\end{proof}

\begin{cor}\label{degr}
Assume that $\M(X;\zz/p)$ has a direct summand $M$. Then
\begin{enumerate}
\item $d_p(X)\mid P(\overline{M},1)$;
\item if $d_p(X)=P(\overline{M},1)$ and the kernel of the restriction 
$\End(\M(X))\to\End(\M(\BX))$ consists of nilpotents, 
then $M$ is indecomposable.
\end{enumerate}
\end{cor}

\begin{proof}
Set $q=d_p(X)$ for brevity. Let
$M=(X,\phi)$. 
By Corollary~\ref{karpcor} there exists an idempotent
$\varphi\in\End(\M(X);\zz/q)$ such that $\varphi\mod p=\phi$. 
Then $\res(\varphi)\in\End(\M(\BX);\zz/q)$ is a rational idempotent. 
Since every projective module over $\zz/q$ is free, we have
$$
\tr(\res(\varphi)_*)=\rk_{\zz/q}(\res(\varphi)_*)=
\rk_{\zz/p}(\res(\phi)_*)=P(\overline{M},1)\mod q,
$$
and the first claim follows by Corollary~\ref{trcor}. 
The second claim follows from the first, 
since the second assumption implies 
that for any nontrivial direct summand $M'$ of $M$ 
we have $P(\overline{M'},1)<P(\overline{M},1)$.
\end{proof}

\begin{ntt}
Let $G$ be a group of inner type. Denote by $n(G)$ the greatest common divisor of degrees of all finite splitting fields of $G$ and by $n_p(G)$ its $p$-primary component. Note that $n(G)=d(X)$ and $n_p(G)=d_p(X)$, where $X$ is the variety of complete $G$-flags.
\end{ntt}

We obtain the following estimate on $n_p(G)$ in terms of $J$-invariant 
(cf. \cite[Prop.~88.11]{EKM} in the case of quadrics).
\begin{prop}\label{npG}
For a group $G$ of inner type with $J_p(G)=(j_1,\ldots,j_r)$ we have
$$
n_p(G)\le p^{\sum_i j_i}.
$$
\end{prop}

\begin{proof}
Follows from Theorem~\ref{thmKac} and Corollary~\ref{degr}.
\end{proof}

\begin{cor}\label{trivJ} The following statements are equivalent:
\begin{itemize}
\item $J_p(G)=(0,\ldots,0)$;
\item $n_p(G)=1$;
\item $\RR_p(G)=\zz/p$.
\end{itemize}
\end{cor}
\begin{proof}
If $J_p(G)=(0,\ldots,0)$ then $n_p(G)=1$ by 
Proposition~\ref{npG}. If $n_p(G)=1$ 
then there exists a splitting field $L$ of degree $m$ prime to $p$ and, 
therefore, $\RR_p(G)=\zz/p$ by the transfer argument (see 
Proposition~\ref{thmComp}). The remaining implication is obvious.
\end{proof}

Finally, we obtain the following reduction formula 
(cf. \cite[Cor.~88.7]{EKM} in the case of quadrics).

\begin{prop}\label{thmKarp}
Let $G$ be a group of inner type, $X$ be the variety of complete $G$-flags, $Y$ be a projective variety such that the map
$\CH^l(Y)\to\CH^l(Y_{F(x)})$ is surjective for all $x\in X$ and $l\le n$.
Then $j_i(G)=j_i(G_{F(Y)})$ for all $i$ such that
$d_ip^{j_i(G_{F(Y)})}\le n$.
\end{prop}

\begin{proof}
Indeed, by \cite[Lemma~88.5]{EKM} the map
$\CH^l(X)\to\CH^l(X_{F(Y)})$ is surjective for all $l\le n$, and therefore
$j_i(G)\le j_i(G_{F(Y)})$. The converse inequality is obvious.
\end{proof}
\begin{cor}
$J_p(G)=J_p(G_{F(t)})$.
\end{cor}
\begin{proof}
Take $Y=\mathbb{P}^1$ and apply Proposition~\ref{thmKarp}.
\end{proof}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Examples}\label{mainexam}

In the present section we provide
examples of motivic decompositions of projective homogeneous varieties
obtained by applying Theorem~\ref{thmMain}. 

\paragraph{The case $r=d_1=1$.}
According to Table~\ref{bigtab} this corresponds to the case when
$G$ is of type $\A_n$ or $\C_n$. Let
$A$ be a central simple algebra corresponding to $G$. 
We have $A=\mathrm{M}_m(D)$, 
where $D$ is a division algebra of index $d\ge 1$ over a field $F$. 
Let $p$ be a prime divisor of $d$ ($p=2$ in the case of $\C_n$). 
Observe that according to Table~\ref{bigtab} $J_p(G)=(j_1)$ for some
$j_1\ge 0$.
Let $X_\Theta$ be the projective homogeneous $G$-variety 
given by a subset $\Theta$ of vertices of the respective Dynkin diagram 
such that $p\nmid j$ for some $j\notin\Theta$ (cf.~Example~\ref{intmot}).
Then by Theorem~\ref{thmMain} we obtain that
\begin{equation}\label{sbdec}
\M(X_\Theta;\zz/p)\simeq\bigoplus_{i\ge 0}\RR_p(G)(i)^{\oplus a_i},
\end{equation}
where $\RR_p(G)$ is indecomposable and 
$$
\overline{\RR_p(G)}\simeq\bigoplus_{i=0}^{p^{j_1}-1}(\zz/p)(i).
$$

Now we identify $\RR_p(G)$.
Using the comparison lemma (see Proposition~\ref{thmComp})
we conclude that 
$\RR_p(G)$ depends only on $D$, so we may assume $m=1$. 
By Table~\ref{bigtab} we have $p^{j_1}\mid d$, but on the other hand by Proposition~\ref{npG} we have
$n_p(G)\le p^{j_1}$. 
Therefore, $p^{j_1}$ is a $p$-primary part of $d$.
 
We have $D\simeq D_p\otimes_F D'$, where $p^{j_1}=\ind(D_p)$
and $p\nmid\ind(D')$. Passing to a splitting field of $D'$
of degree prime to $p$ and using Proposition~\ref{thmComp} 
we conclude that the motives of $X_\Theta$ 
and $\SB(D_p)$ are direct sums of twisted $\RR_p(G)$.
Comparing the Poincar\'e polynomials we conclude that 

\begin{lem}\label{idsb}
$\M(\SB(D_p);\zz/p)\simeq \RR_p(G)$.
\end{lem}

Applying Proposition~\ref{intd} to $X=\SB(D)$ and comparing
the Poincar\'e polynomials of $\M(X)$ and $\RR_i$ we obtain
that 
\begin{cor}
The motive of $\SB(D)$ with
integer coefficients is indecomposable.
\end{cor}

\begin{rem} Indeed, we provided
a uniform proof of the results
of paper \cite{Ka96}. Namely, the decomposition of $\M(\SB(A);\zz/p)$
(see \cite[Cor.~1.3.2]{Ka96})
and indecomposability of $\M(\SB(D);\zz)$ (see \cite[Thm.~2.2.1]{Ka96}).
\end{rem}





\paragraph{The case $r=1$ and $d_1>1$.} According
to Table~\ref{bigtab} this holds if
\begin{itemize}
\item[$p=2$:] $G$ is a group of type
$\Spin_n$, $n=7,8,9,10$, $\G$, $\F$ or $\E_6$;
\item[$p=3$:] $G$ is a group of type $\F$, $\E_7$ or strongly
inner form of type $\E_6$;
\item[$p=5$:] $G$ is a group of type $\E_8$.
\end{itemize}
Observe that in all these cases $J_p(G)=(0)$ or $(1)$.
Let $X$ be a generically split projective homogeneous $G$-variety
(cf. Example~\ref{intmot}).
By Theorem~\ref{thmMain} we obtain the decomposition
\begin{equation}\label{mndec}
\M(X;\zz/2)\simeq \bigoplus_{i\ge 0} \RR_p(G)(i)^{\oplus a_i},
\end{equation}
where the motive $\RR_p(G)$ is indecomposable and 
(cf. \cite[(5.4-5.5)]{Vo03})
$$
\overline{\RR_p(G)}\simeq 
\bigoplus_{i=1}^{p-1} (\zz/p)(i\cdot (p+1)).
$$

Now we identify $\RR_p(G)$.
Let $\rr$ be the Rost invariant as defined
in \cite{Me03} and $\rr_p$ denote its $p$-part.

\begin{lem}
Let $G$ be a simple linear algebraic group over $F$
satisfying $r=1$ and $d_1>1$ and $p$ be its torsion prime. 
Then $\rr_p(G)$ is trivial iff $\RR_p(G)\simeq\zz/p$.
\end{lem}

\begin{proof}
According to \cite[Theorem~0.5]{Ga01}, \cite{Ch94} and 
\cite[Theoreme~10]{Gi00}
the invariant $\rr_p(G)$ is trivial iff 
there exists a field extension $E/F$ of degree coprime with $p$
such that the group $G$ splits over $E$. By Corollary~\ref{trivJ}
the latter is equivalent to the fact that $\RR_p(G)\simeq\zz/p$.
\end{proof}



\begin{lem}\label{roinv} Let $G$ and $G'$ be simple linear algebraic
groups over $F$
satisfying $r=1$ and $d_1>1$ and $p$ be its common torsion prime. 
For groups of type $\E_8$ we assume in addition
that they are twisted forms by means of
cocycles taken from the image of the twisted map
defined in \cite[13.12]{Ga06} (examples of such groups are provided
in \cite[14.7]{Ga06}).

If $\rr_p(G)=\rr_p(G')c$ for some $c\in(\zz/p)^\times$, 
then $\RR_p(G)\simeq \RR_p(G')$.
\end{lem}

\begin{proof} By transfer arguments (see Proposition~\ref{thmComp})
it is enough to prove this over a $p$-primary closure of $F$. 
Let $X$ and $X'$ be the respective varieties of complete flags.
Observe that the invariant $\rr_p(G)$ becomes trivial over 
the function field $F(X)$. Since $\rr_p(G)=\rr_p(G')c$, it becomes
trivial over $F(X')$ as well. 
By triviality of the kernel of the Rost invariant 
(see \cite[Theorem~0.5]{Ga01} and \cite[Theorem~13.14]{Ga06})
this implies that $G$ splits over $F(X')$.
To finish the proof we apply the comparison lemma
(see Proposition~\ref{thmComp}) to the groups $G$ and $G'$.
\end{proof}


\paragraph{\it $\zz$-coefficients.}
Let $G$ be a group of type $\F$ or strongly inner form of type $\E_6$
which doesn't split 
by field extensions of degrees $2$ and $3$. Observe that such
a group splits by an extension of degree $6$.
Let $X$ be a generically split projective homogeneous $G$-variety.
Then according to Proposition~\ref{intd}
the Chow motive of $X$ with integer coefficients 
splits as a direct sum
of twisted copies of an indecomposable  motive $\RR(G)$ such that
\begin{align*}
\RR(G)\otimes \zz/2&=\bigoplus_{i=0,1,2,6,7,8} \RR_2(G)(i),\qquad 
P(\overline{\RR_2(G)},t)=1+t^3,
\\
\RR(G)\otimes \zz/3&=\bigoplus_{i=0,1,2,3}\RR_3(G)(i),\qquad
P(\overline{\RR_3(G)},t)=1+t^4+t^8,
\\
P(\overline{\RR(G)},t)&=1+t+t^2+\ldots+t^{11}.
\end{align*}



\begin{rem} In particular, we provided a uniform
proof of the main results of papers \cite{Bo03} and \cite{NSZ}, where
the cases of some $\G$- and $\F$-varieties are considered.
\end{rem}




\paragraph{The case $r>1$.}
According to Table~\ref{bigtab} this holds for groups $G$ of
types $\B_n$ and $\D_n$ and exceptional types 
$\E_7$, $\E_8$ for $p=2$ and
$\E_6^{ad}$, $\E_8$ for $p=3$.

\paragraph{\it Pfister case.}
Let $G=\OO(\phi)$, where $\phi$ is a $k$-fold Pfister form or 
its maximal neighbor. 
Assume $J_2(G)\ne(0,\ldots,0)$. In view of Corollary~\ref{trivJ}
this holds iff $n_2(G)\neq 1$. By Springer Theorem the latter holds
iff $\phi$ is not split.
According to Theorem~\ref{thmMain} we obtain the decomposition
$$
\M(X;\zz/2)\simeq \bigoplus_{i\ge 0} \RR_2(G)(i)^{\oplus a_i}
$$
where $\RR_2(G)$ is indecomposable. Moreover,
according to Theorem~\ref{lem47} the similar decomposition
exists with $\zz$-coefficients. 

Now we compute $J_2(G)$. 
Let $Y$ be a projective quadric corresponding to $\phi$. 
Then $G$ splits over $F(Y)$ and $Y$ splits over $F(x)$ for any $x\in X$. 
It is known that $\CH^l(\overline{Y})$ for $l<2^{k-1}-1$ 
is generated by $\CH^1(\overline{Y})$ and, therefore, is rational. 
By Proposition~\ref{thmKarp} and Table~\ref{bigtab}
we see that $j_i(G)=0$ for $0\le i<r$, where $r=2^{k-2}$. 
Therefore, $J_2(G)=(0,\ldots,0,1)$ and 
$P(\overline{\RR_2(G)},t)=1+t^{2^{k-1}-1}$.
Finally, by Corollary~\ref{remKrull} the motive $\RR_2(G)$
coincides with the motive introduced in \cite{Ro98} 
which is called Rost motive. 

In this way we obtain Rost decomposition
of the motive of a Pfister quadric and its maximal neighbor.

\paragraph{\it Maximal orthogonal Grassmannian}
Let $G=\OO(q)$, where 
$q\colon V\to F$ 
is an arbitrary anisotropic regular quadratic form and $X$ is
the respective maximal orthogonal Grassmannian. 
The variety $X$ is generically split, hence, 
by Theorem~\ref{thmMain} we have the decomposition
$$
\M(X;\zz/2)\simeq \bigoplus_{i\ge 0} \RR_2(G)(i)^{\oplus a_i},
$$
where the motive $\RR_2(G)$ is indecomposable. 
Comparing the Poincar\'e polynomials of $\M(X;\zz/2)$ and $\RR_2(G)$ we 
obtain the following particular cases:
\begin{itemize} 
\item If the group $G$ corresponds to
a generic cocycle (see \ref{notgen}), 
the motive $\M(X;\zz/2)$ is isomorphic to $\RR_2(G)$ and, hence, 
is indecomposable. This corresponds to the maximal value of the $J$-invariant.
\item If $q$ is a Pfister form or its maximal neighbor, by the previous example
$\RR_2(G)$ coincides
with the Rost motive. This corresponds to the minimal
non-trivial value of the $J$-invariant.
\end{itemize}




\paragraph{Acknowledgements} 
We are very grateful to Fabien Morel and Ivan Panin
for an encouraging attention to our work.
We would like to thank Skip Garibaldi, Stefan Gille, Nikita Karpenko and 
Alexander Vishik for many useful comments on the subject of the paper.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




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\end{document}


