On an asymptotic behavior of elements of order $p$\\ in irreducible representations of the classical algebraic groups\\ with large enough highest weights
I. D. Suprunenko
National Academy of Sciences of Belarus
Surganov str. 11
Minsk
220072
Belarus
The behavior of the images of a fixed element of order $p$ in irreducible representations of a classical algebraic group in characteristic $p$ with highest weights large enough with respect to $p$ and this element is investigated. More precisely, let $G$ be a classical algebraic group of rank $r$ over an algebraically closed field $K$ of characteristic $p>2$. Assume that an element $x\in G$ of order $p$ is conjugate to that of an algebraic group of the same type and rank $m<r$ naturally embedded into $G$. Next, an integer function $\sigma_x$ on the set of dominant weights of $G$ and a constant $c_x$ that depend only upon $x$, and a polynomial $d$ of degree one are defined. It is proved that the image of $x$ in the irreducible representation of $G$ with highest weight $\omega$ contains more than $d(r-m)$ Jordan blocks of size $p$ if $m$ and $r-m$ are not too small and $\sigma_x(\omega)\geq p-1+c_x$.
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\begin{document}

\title{On an
asymptotic behavior of elements
of order $p$\\ in irreducible representations
of the classical algebraic groups\\
with large enough highest weights}

\author{I. D. Suprunenko\\
Institute of Mathematics,  National Academy of Sciences of Belarus\\
  Surganov str. 11, Minsk, 220072, Belarus\\
suprunenko@im.bas-net.by}

\date{}
\maketitle

\begin{abstract}
The behavior of the images of a fixed element of order $p$ in
irreducible representations of a classical algebraic group in
characteristic $p$ with highest weights large enough with respect
to $p$ and this element is investigated. More precisely, let $G$
be a classical algebraic group of rank $r$ over an algebraically
closed field $K$ of characteristic $p>2$. Assume that an element
$x\in G$ of order $p$ is conjugate to that of an algebraic group
of the same type and rank $m<r$ naturally embedded into $G$. Next,
an integer function $\sigma_x$ on the set of dominant weights of
$G$ and a constant $c_x$ that depend only upon   $x$, and a
polynomial $d$ of degree one are  defined.  It is proved that the
image of $x$
 in the irreducible representation of $G$ with highest weight $\omega$
contains more than $d(r-m)$ Jordan blocks of size $p$ if $m$ and
$r-m$ are not too small and $\sigma_x(\omega)\geq p-1+c_x$.
\end{abstract}

\maketitle


Asymptotic lower estimates for the number of Jordan blocks of size
$p$ in the images of a fixed element of order $p$ in irreducible
representations of a classical algebraic group in characteristic
$p$ with highest weights large enough with respect to $p$ and this
element are obtained. More precisely, let $G$ be a classical
algebraic group of rank $r$ over an algebraically closed field $K$
of characteristic $p>2$. Assume that an element $x\in G$ of order
$p$ is conjugate to that of an algebraic group $G_m$ of the same
type and rank $m<r$ naturally embedded into $G$. Set
\begin{equation*}
d(r-m)=
\begin{cases}
r-m & \text{for $G=A_r(K)$},\\
2r-2m & \text{for other types.}
\end{cases}
\end{equation*}
Let $\Delta_x$ be the labelled Dynkin diagram of the conjugacy
class containing $x$ in the sense of Bala and Carter \cite{BC} and
let  $c_x$ be the sum of the labels at $\Delta_x$ for $G\neq
A_r(K)$ and the half of this sum for $G=A_r(K)$. For brevity,
throughout the article we refer to $\Delta_x$  as the labelled
Dynkin diagram of $x$. Next, an integer function $\sigma_x$ on the
set of dominant weights of $G$ that depends only upon $\Delta_x$
is defined. For $p$-restricted weights $\sigma_x$ coincides with
the canonical homomorphism determined by $\Delta_x$.  It is proved
that the image of $x$ in the irreducible representation of $G$
with highest weight $\om$ contains more than $d(r-m)$ Jordan
blocks of size $p$ if $m$ and $r-m$ are not too small and
$\sigma_x(\om)\geq p-1+c_x$.

We need some more notation to formulate the main results. Let $\om_i$
and $\alpha_i$ be  the fundamental weights and the simple roots of
$G$ (with respect to a fixed maximal torus $T$) labelled as in
\cite{B1}. Denote by $\delta_i$, $1\leq i\leq r$, the label on
$\Delta_x$ corresponding to its $i$th node. We have $0\leq
\delta_i\leq2$.  In what follows $\mathbb{Z}$ is the set of integers,
$\mathbf{X}=\mathbf{X}(G)$ is the set of weights of $G$,
$\mathbf{X^+}\subset\mathbf{X}$ is the set of dominant weights,
$\Irr=\Irr(G)$ is the set of irreducible rational representations
of $G$ (considered up to the equivalence) and $\om(\phi)$ is the
highest weight of a representation $\phi$. There exists a uniquely
determined homomorphism $\tau_x: \mathbf{X}\rightarrow\mathbb{Z}$
such that $\tau_x(\alpha_i)=\delta_i$. The weight $\mu\in\mathbf{X^+}$
is called $p$-restricted if $\mu=\sum^r_{i=1}a_i\om_i$ with all
$a_i<p$. Each weight $\om\in\mathbf{X^+}$ can be represented in
the form $\sum^s_{j=0}p^j\om^j$ where $\om^j$ are $p$-restricted.
Set $\sigma_x(\om)=\tau_x(\sum^s_{j=0}\om^j)$. Now we can state our
main result.
\begin{theorem}\label{mt} Let $\phi\in\Irr$ and $\sigma_x(\om(\phi))\geq p-1+c_x$.
 Assume that $m>1$ for $G=B_r(K)$ or $D_r(K)$, $r-m>1$ for $G=A_r(K)$,
and $>3$ for $G=B_r(K)$ or $D_r(K)$. Then the element $\phi(x)$ has
  more than $d(r-m)$ Jordan blocks of size $p$. \end{theorem}
Proposition~\ref{mp} below shows that one cannot weaken the
inequality for $\sigma_x(\om(\phi))$ in Theorem~\ref{mt} and that
the estimates obtained are asymptotically exact.
\begin{proposition}\label{mp} Let $\phi\in\Irr$ and $\om=\om(\phi)=a\om_1$ with $a<p$.
Assume that $m$ and $r$ are such as in Theorem~\ref{mt} and $x$ is
a regular unipotent element in $G_m$. Set $c=m$ for $G=A_r(K)$,
$2m$ for $G=B_r(K)$, $2m-1$ for $G=C_r(K)$, and $2m-2$ for
$G=D_r(K)$. Suppose that $p>c$. Then $\abs{x}=p$,
$\sigma_x(\om)=ac$ and $c_x=c$.   There exist constants
$N_G(a,m,p)$ and $Q_G(a,m,p)$ that depend upon the type of $G$,
$a$, $m$, and $p$ and do not depend upon $r$ such that $\phi(x)$
contains at most $N_G(a,m,p)$ Jordan blocks of size $p$ if
$p<ac<p+c-1$ and at most $d(r-m)+Q_G(a,m,p)$ such blocks if
$ac=p+c-1$. \end{proposition}

 Put $l=[(m+2)/2]$ for $G=A_r(K)$ and $l=m$
otherwise. By Lemma~\ref{lc} below,
$c_x=\sum^l_{i=1}\delta_i$ and $c_x\leq p-1$. Hence
$c_x\leq2l$.

For $\varphi\in\Irr$ define the weight $\bar{\omega}(\varphi)$ as
follows: write down the $p$-adic expansion for the weight
$\omega=\omega(\varphi)$ considered before the statement of
Theorem~\ref{mt} and set
$\bar{\omega}(\varphi)=\sum^s_{j=0}\om^j$. So
$\sigma_x(\omega(\varphi))=\tau_x(\bar{\omega}(\varphi))$.

 The study of an asymptotic behavior of elements of order $p$ in
representations of the classical groups in characteristic $p$ was
begun by the author in \cite{Supl} where a notion of a $p$-large
representation was introduced. In our present notation a
representation $\phi\in\Irr$ is $p$-large if and only if
$\sigma_x(\om(\phi))\geq p$ for a long root element $x\in G$ (an
equivalent definition from \cite{Supl}: the value of ${\bar
\om}(\phi)$ on the maximal root is $\geq p$). The common goal of
 \cite{Supl} and the present article is
 to investigate the behavior of elements of order $p$ in irreducible
representations in characteristic $p$ for a fixed $p$ and
$r\rightarrow\infty$ and to discover asymptotic regularities which
are specific for prime characteristics but do not (or almost do
not) depend upon $p$. Such properties can find applications in
recognizing representations and linear groups. According to
~\cite[Theorem~1.1]{Supl}, the image of any element of order $p$
in a $p$-large representation has at least $f(r)$ Jordan blocks of
size $p$ where
\begin{equation*}
f(r)=
\begin{cases}
2r-2 & \text{for $G=A_r(K)$},\\ 6r-7 & \text{for $G=B_r(K)$,
$p=3$},\\ 8r-10 & \text{for $G=B_r(K)$, $p>3$},\\ 4r-4 & \text{for
$G=C_r(K)$},\\ 4r-8 & \text{for $G=D_r(K)$, $p=2$},\\ 6r-10 &
\text{for $G=D_r(K)$, $p=3$},\\ 8r-16 & \text{for $G=D_r(K)$,
$p>3$}.
\end{cases}
\end{equation*}
In \cite[Theorem~1.3]{Supl} for $p>2$ and all  types of the
classical groups examples of representations $\phi\in\Irr$ such
that $\phi(x)$ has only one Jordan block of size $p$ for a long
root element $x$ and $\sigma_x(\om(\phi))=p-1$ are given. It is
also shown \cite[Theorem~1.4] {Supl} that the estimates in
\cite[Theorem~1.1]{Supl} are asymptotically exact for the groups
of type $A$, $B$, and $D$ provided $p>2$ in the last two cases.

In what follows $\mathbb{C}$ is the complex field,
$G_{\mathbb{C}}$ is  the simple algebraic group over  $\mathbb{C}$
of the same type and rank as $G$, and $\Irr_{\mathbb{C}}$ is the set
of irreducible rational representations of $G_{\mathbb{C}}$
(considered up to the equivalence).  For $\rho\in\Irr$ or
$\Irr_{\mathbb{C}}$  and a unipotent element $z\in G$ or
$G_{\mathbb{C}}$ denote by  $k_{\rho}(z)$ the degree of the
minimal polynomial of $\rho(z)$. It is well known that $k_{\rho}(z)$ is equal
to the maximal size of a Jordan block of $\rho(z)$. If $\phi\in\Irr$, then
$\phi_{\mathbb{C}}$ is   the irreducible representation of
$G_{\mathbb{C}}$ with highest weight ${\bar \om}(\phi)$. For
unipotent $x\in G$ put
$k_{\phi_{\mathbb{C}}}(x)=k_{\phi_{\mathbb{C}}}(y)$ where $y\in
G_{\mathbb{C}}$ is a unipotent element with the labelled Dynkin
diagram   $\Delta_x$ (this is correctly determined). Now let
$\abs{x}=p$. By the results \cite[Theorem 1.1,
Lemma~2.5, and Proposition~2.12]{Sump},
$k_{\phi_{\mathbb{C}}}(x)=\sigma_x(\om(\phi))+1$ and
$k_{\phi}(x)=\min\{p, k_{\phi_{\mathbb{C}}}(x)\}$. Hence if $z$ is
a long root element,  then $k_{\phi}(z)=k_{\phi_{\mathbb{C}}}(z)$
if and only if $\phi$ is not $p$-large. The results of \cite{Sump}
imply that for not very small $p$ and $r$ there exists a wide
class of representations $\phi\in\Irr$ such that
$k_{\phi}(z)=k_{\phi_{\mathbb{C}}}(z)<p$ for a long root element
$z\in G$, but $k_{\phi}(x)=p<k_{\phi_{\mathbb{C}}}(x)$ for many
other elements $x\in G$ of order $p$. In this connection in
\cite[Section 2]{SuKo} a notion of a $p$-large representation for
a given
 element $x$ of order $p$ was introduced. A representation $\phi\in\Irr$ was
called $p$-large for $x$ if $\sigma_x(\om(\phi))\geq p$. It has
been conjectured (\cite[Conjecture 1]{SuKo} that if $x\in G_m$,
$r$ is large enough with respect to $m$ and $\phi$ is $p$-large
for $x$, then $\phi(x)$ has at least $F(r)$ blocks of size $p$
where $F$ is an increasing function. Our Proposition~\ref{mp}
formally disproves this conjecture, but Theorem~\ref{mt} actually
proves a refined version of it with a stronger assumption on
$\sigma_x(\om(\phi))$. Thus for arbitrary elements $x$ there is a
gap between the class of representations $\phi\in\Irr$ with
$k_{\phi}(x)=k_{\phi_{\mathbb{C}}}(x)$ and that of representations
where asymptotic estimates for the number of Jordan blocks of size
$p$ in $\phi(x)$ hold. Perhaps for some classes of elements of
order $p$ stronger estimates than those of Theorem~\ref{mt} are
possible, but now it is not clear how to determine such classes.

The case $p=2$ is not considered here, but in this situation
$\phi\in\Irr$ is $2$-large if $\om(\phi)\neq 2^j\om_i$. For
$2$-large representations the estimates from
\cite[Theorem~1.1]{Supl} are available. For remaining
representations certain estimates could be obtained as well, but
this article does not seem a proper place for this. We plan to
handle this question in a subsequent paper which will be devoted to refining
some estimates in \cite{Supl}.

The results of this article as well as those of \cite{Supl} can be easily
transferred to irreducible $K$-representations of finite classical groups
in characteristic $p$.
\section{Notation and preliminary comments}
Throughout the article for a semisimple algebraic group $S$ the
symbols $\Irr(S)$, $\mathbf{X}(S)$, and $\mathbf{X^+}(S)$ mean the
same as the similar ones for $G$ introduced earlier; $R(S)$ is the
set of roots of $S$, $\langle S_1, \ldots, S_j\rangle$ is the
subgroup in $S$ generated by subgroups $S_1, \ldots, S_j$;
$\Irr_p(S)\subset\Irr(S)$ is the set of $p$-restricted
representations, i.e. irreducible representations with
$p$-restricted highest weights; $\mathbf{X}(\phi)$
($\mathbf{X}(M)$) is the set of weights of a representation
$\varphi$ (a module $M$); $\dim M$ is the dimension of $M$;
$M(\omega)$ is the irreducible $S$-module with highest weight
$\omega$; $L$ is the Lie algebra of $G$; $R=R(G)$, $R^+\subset R$
is the set of positive roots; $\Irr_p=\Irr_p(G)$;
$\mathcal{X}_{\beta}\subset G$ and $X_{\beta}\in L$ are the root
subgroup and the root element associated with $\beta\in R$,
$\mathcal{X}_{\pm i}=\mathcal{X}_{\pm\alpha_i}$, and  $X_{\pm
i}=X_{\pm\alpha_i}$. Set
$H(\beta_1,\ldots,\beta_j)=\langle\mathcal{X}_{\beta_1},
\mathcal{X}_{-\beta_1} \ldots, \mathcal{X}_{\beta_j},
\mathcal{X}_{-\beta_j}\rangle$. For  $\omega\in\mathbf{X}(S)$ and
$\alpha\in R(S)$  denote by $\langle\omega,\alpha\rangle$ the
value of the weight $\omega$ on the root $\alpha$. For an
$S$-module $M$ and a unipotent element $x\in S$ define $k_M(x)$
similarly to $k_{\phi}(x)$. If $\abs{x}=p$, then $n_{\phi}(x)$ is
the number of Jordan blocks of size $p$ of the matrix $\phi(x)$
for a representation $\phi$ of $S$ and $n_M(x)$ denotes the same
number for a module $M$ affording  $\phi$.

 An element $x\in G$ of order $p$  can be
 embedded into a closed connected subgroup $\Gamma$ of type $A_1$ whose
  labelled diagram coincides with $\Delta_x$ (see \cite[Theorem 4.2]{LaTe}).
Set $\mathbf{X}_1=\mathbf{X}(A_1(K))$ (the simply connected group
of this type) and identify $\mathbf{X}_1$ with $\mathbb{Z}$
mapping $a\om_1\in\mathbf{X}_1$ into $a\in\mathbb{Z}$. Then
$\mathbf{X}(\Gamma)$ can be identified with a subset of
$\mathbb{Z}$. The canonical homomorphism $\tau_x$ can be obtained
as the restriction of weights from a maximal torus $T\subset G$ to
a maximal torus $T_1\subset \Gamma$ such that $T_1\subset T$. From
now on we fix the tori $T$ and $T_1$, and all weights and roots of
$G$ and $\Gamma$ are considered with respect to $T$ and $T_1$.
Throughout the text  $\ep_i$ with $1\le i\le r+1$ for $G=A_r(K)$
and $1\le i\le r$ otherwise are weights of the standard
realization of $G$ labelled as in \cite[ch. VIII, \S 13]{B2}. Set
$e_i=\tau_x(\varepsilon_i)$. One can choose $\Gamma$, $T$ and
$T_1$ such that the restriction to $\Gamma$ of the natural
representation of $G$ is a direct sum of irreducible components
with $p$-restricted highest weights (see comments in \cite[Section
3]{Te});
 $e_i\geq e_j$ for $i<j$; $e_i\geq0$ if $G=A_r(K)$ and
$i\leq(r+1)/2$; and $e_i\geq0$ for all $i\le r$ if $G\neq A_r(K)$.
If $H\subset G$ is a semisimple subgroup generated by some root
subgroups, then $T_H=T\cap H$ is a maximal torus in $H$. If
$T_1\subset T_H$, we denote by the same symbol $\tau_x$ the
homomorphism $\mathbf{X}(H)\rightarrow\mathbb{Z}$ determined by
restricting weights from $T_H$ to $T_1$. This causes no confusion.
If an element  $v$ of  some $G$-module is an eigenvector for $T$,
we denote its weights with respect to $T$, $T_H$, and $T_1$ by
$\omega(v)$, $\omega_H(v)$, and $\omega_{\Gamma}(v)$. In what
follows $x$ is conjugate to an element of $G_m$, $\abs{x}=p$, $m$
and $r-m$ are such as in  the assertion of Theorem~\ref{mt}, and
$\delta_i=\tau_x(\alpha_i)$, $1\le i\le r$.
\begin{lemma}\label{lc} Set $l=[(m+2)/2]$ for $G=A_r(K)$ and $l=m$
otherwise. Then $c_x=\sum^l_{i=1}\delta_i$ and
 $c_x\leq p-1$.
\end{lemma}

\begin{proof} Put $k=l-1$ for $G=A_r(K)$, $m=2t$, and $k=l$ for $G=A_r(K)$,
$m=2t+1$.  Our assumptions on $e_i$, $m-r$, and $x$ imply that
$e_i=0$ for  $k<i<r+2-k$ if $G=A_r(K)$ and $e_i=0$ for $i>m$
otherwise; notice that $e_{k+1}=e_{k+2}=0$ for $G=A_r(K)$. Now it
follows from the definition of $c_x$ and the formulae in \cite[ch.
VIII, \S 13]{B2}  that
$c_x=\sum^l_{i=1}\delta_i=e_1-e_{l+1}=e_1$. As $e_1$ is a
weight of a $p$-restricted $\Gamma$-module, we have $e_1<p$. This
yields the lemma.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{mt}] Set
$\omega=\omega(\varphi)$ and let $\omega=\sum^r_{i=1}a_i\omega_i$.
It is clear that $\om\neq0$ as $\tau_x(\om)\neq0$. Define
subgroups $H_1$ and $H_2\subset G$ as follows. For $G=A_r(K)$ set
$u=r-t+2$ if $m=2t$ and $r-t+1$ if $m=2t+1$,
$\beta=\varepsilon_{t+1}-\varepsilon_u$,
  \begin{equation*}
H_1=H(\alpha_1,\ldots,\alpha_t, \beta, \alpha_u, \ldots,
\alpha_r), \quad H_2=H(\alpha_{t+2}, \ldots, \alpha_{u-2})
 \end{equation*}
(we have $H_1=H(\alpha_1, \varepsilon_2-\varepsilon_{r+1})$ for
$m=2$ and $H_1=H(\varepsilon_1-\varepsilon_{r+1})$ for $m=1$). For
$G=B_r(K)$, $C_r(K)$, or $D_r(K)$ put $\beta=\varepsilon_m$,
$2\varepsilon_m$, or $\varepsilon_{m-1}+\varepsilon_m$,
respectively, and
 \begin{equation*}
 H_1=H(\alpha_1,\ldots,\alpha_{m-1}, \beta).
 \end{equation*}
Next, set
\[H_2=H(\alpha_{m+1},\dotsc,\alpha_{r-1},\varepsilon_{r-1}+\varepsilon_r)\]
for $G=B_r(K)$ and
\[H_2=H(\alpha_{m+1},\dotsc,\alpha_r)\]
for $G=C_r(K)$ or $D_r(K)$
(here $H_1=H(\beta)$ for $G=C_r(K)$ and $m=1$).
 One easily observes that the sets of roots in brackets used to
 define $H_1$ and $H_2$ yield bases of the systems $R(H_1)$ and
 $R(H_2)$, respectively. Denote these bases by $\mathcal{B}_i$.  In all
cases $H_1$ is conjugate to $G_m$ in $G$.  We have
  $H_2\cong A_{r-m-1}(K)$, $D_{r-m}(K)$,
$C_{r-m}(K)$, or $D_{r-m}(K)$ for $G=A_r(K)$, $B_r(K)$, $C_r(K)$,
or $D_r(K)$, respectively.  It is clear
  that  the
 subgroups $H_1$ and $H_2$ commute. Set $H=H_1H_2$. Let
 $U_i=\langle\mathcal{X}_{\gamma}\mid \gamma\in R^+, \quad
 \mathcal{X}_{\gamma}\subset H_i\rangle$, $i=1,2$, and $U=U_1U_2$. It is
 not difficult to conclude that $U_i$ is a maximal unipotent
 subgroup in $H_i$ and  $U$ is such a subgroup in $H$. We can assume
 that $x\in U_1$, $\Gamma\subset H_1$ and $T_1\subset T_{H_1}$.
 We shall write a weight $\mu\in\mathbf{X}(H)$  in the form
 $(\mu_1, \mu_2)$ where $\mu_i\in\mathbf{X}(H_i)$ is the
 restriction of $\mu$ to $T_{H_i}$.  Set $M=M(\om)$.

It is clear that $n_V(x)=\dim(x-1)^{p-1}V$ for each $H$-module
$V$. Taking this into account, it is not difficult to conclude the
following. If $0\subset W_1\subset\ldots\subset W_t=V$ is a
filtration of $V$, $F_i=W_i/W_{i-1}$, $1\le i\le t$, and
$n_{F_i}(x)=n_i$, then
\begin{equation}\label{eqx}
n_V(x)\geq \sum^t_{i=1}n_i.
\end{equation}


First suppose that  $\phi\in\Irr_p$. Since passing to the dual
representation does not influence the Jordan form of $\varphi(x)$,
one can assume that $a_i\neq0$ for some $i\leq (r+1)/2$ if
$G=A_r(K)$. As for $p$-large representations the estimates of
\cite[Theorem~1.1]{Supl} hold; we also assume that $\phi$ is not
$p$-large. Hence $\langle\mu, \alpha\rangle<p$ for all
$\mu\in\mathbf{X}(\phi)$ and long roots $\alpha$ (for all $\alpha$
if $G=A_r(K)$ or $D_r(K)$). By the formulae for the maximal roots
of the classical groups in \cite[Tables 1-4]{B1}, this forces that
\begin{equation}\label{eqp}
\begin{split}
a_1+\ldots+a_r<p \quad  & \text{for}\quad
G=A_r(K)\quad\text{or}\quad C_r(K), \\
 a_1+2a_2+\ldots+2a_{r-1}+a_r<p \quad & \text{for} \quad
 G=B_r(K),\\
 a_1+2a_2+\ldots+2a_{r-2}+a_{r-1}+a_r<p \quad & \text{for} \quad
G=D_r(K).
\end{split}
\end{equation}
Now we proceed to construct
 two composition factors $M_1$ and $M_2$ of the restriction
 $M|H$ such that $n_{M_1}(x)\geq d(r-m)$ and $n_{M_2}(x)>0$.
This will be done for almost all $\omega$.
 In exceptional cases we shall find  one  factor $M_1$ such that
 $n_{M_1}(x)>d(r-m)$. By (\ref{eqx}), this would yield
 the assertion of the theorem.

 Let $v\in M$ be a nonzero highest weight
 vector. Put $\mu_i=\omega_{H_i}(v)$. The vector $v$ generates an
 indecomposable $H$-module $V_1$ with highest weight
 $\mu=(\mu_1, \mu_2)$. Using (\ref{eqp}), one can deduce that
 $\langle\mu_1, \beta\rangle<p$ for all $\beta\in\mathcal{B}_1$.
 Here for $G=B_r(K)$ we take into account that $m>1$. Hence
 $\mu_1$ is $p$-restricted. Now assume that either $G\neq B_r(K)$, or
 $a_i\neq0$ for some $i<r$. For such representations  we construct another
  weight vector $w\in M$ that is fixed by $U$. Set $l=t+1$ for $G=A_r(K)$,
$m=2t$; otherwise take $l$ as in
Lemma~\ref{lc}. First suppose that $a_j\neq0$ for some $j\le l$ (Case 1).
Choose maximal such $j$ and put $w=X_{-l}\ldots X_{-(j+1)}X_{-j}v$. Now let
$a_j=0$ for all $j\le l$ (Case 2). Our assumptions on $a_i$ imply that
$a_i\neq0$ for some $i>l$; furthermore, one
can take $i\le(r+1)/2$  for $G=A_r(K)$ and $i<r$ for $G=B_r(K)$. Choose
minimal such $i$ and set $w=X_{-l}\ldots X_{-(i-1)}X_{-i}v$ if $G\neq D_r(K)$
 or $i<r$ and $w=X_{-l}\ldots X_{-(r-3)}X_{-(r-2)}X_{-r}v$ for $G=D_r(K)$ and
$i=r$. It follows from \cite[Lemma 2.1(iii) and Lemma 2.9]{Supl} that in all
cases $w\neq0$. Using
\cite[Lemma~72]{St-b} and analyzing the roots in $\mathcal{B}_1$
and $\mathcal{B}_2$ and the weight system $\mathbf{X}(\phi)$, we
get that $U$ fixes $w$ in all situations. Here it is essential
that the case $G=B_r(K)$ with $\om=a_r\om_r$ is excluded. In the
latter case we cannot assert that $\mathcal{X}_{\beta}$ fixes $w$.
Set $\lambda_i=\om_{H_i}(w)$, $i=1,2$. Now it is clear that $w$
generates an indecomposable $H$-module $V_2$ with highest weight
$\lambda=(\lambda_1, \lambda_2)$. We claim that $\lambda_1$ is
$p$-restricted. Write down all the situations where
$\langle\lambda_1, \gamma\rangle\neq \langle\mu_1, \gamma\rangle$
for some $\gamma\in\mathcal{B}_1$.  We have
$\langle\lambda_1, \beta\rangle=\langle\mu_1, \beta\rangle-1$ in Case 1 if
$j=l$ and $G\neq B_r(K)$ or $j=l-1$ and $G=D_r(K)$ and in Case 2 for
$G\neq B_r(K)$ and all $i$; and $\langle\lambda_1, \beta\rangle=\langle\mu_1,
\beta\rangle-2$ for $G=B_r(K)$ both in Case 1 with $j=l$ and in Case 2. In
Case 1 we also have
$\langle\lambda_1,\alpha_{j-1}\rangle=\langle\mu_1, \alpha_{j-1}\rangle+1$
if $j>1$  and
$\langle\lambda_1, \alpha_j\rangle=\langle\mu_1,\alpha_j\rangle-1$
if $j<l$. In Case 2 one gets
$\langle\lambda_1,\alpha_{l-1}\rangle=\langle\mu_1, \alpha_{l-1}\rangle+1$
if $l>1$.
In all other situations we have
$\langle\lambda_1, \gamma\rangle=\langle\mu_1, \gamma\rangle$. Now
apply (\ref{eqp}) to conclude that $\lambda_1$ is $p$-restricted.

Set $M_1=M(\mu)$, $M_2=M(\lambda)$, $M^j_1=M(\mu_j)$, and
$M^j_2=M(\lambda_j)$, $j=1,2$. Obviously, $M_i$ is a composition
factor of $V_i$. It is well known that $M_i=M^1_i\otimes M^2_i$.
It is clear that $\tau_x(\mu_1)=\tau_x(\omega)\geq p$. Since $x\in
H_1$, we have $\delta_t=0$ if $\alpha_t\in\mathcal{B}_2$.
So by Lemma~\ref{lc},
 \begin{equation*}
\tau_x(\lambda_1)=\tau_x(\omega(w))\geq
\tau_x(\omega)-\sum^l_{i=1}\delta_i=\tau_x(\omega)-c_x\geq p-1.
 \end{equation*}
 It follows from \cite[Theorem~1.1, Lemma~2.5,  and Proposition~2.12]{Sump}
that $k_{M^1_i}(x)=p$. Hence $n_{M_i}(x)\geq  \dim M^2_i$.  One
easily observes that $M^2_1$ and $M^2_2$ cannot both be trivial
$H_2$-modules. Our assumptions on $r-m$ and
\cite[Proposition~5.4.13]{KL} imply that the dimension of a
nontrivial irreducible $H_2$-module is at least $d(r-m)$. In the
exceptional case where $G=B_r(K)$ and $\om=a_r\om_r$ we need to
evaluate $\dim M^2_1$.
First let $a_r\not=1$. As above, $X_rv\not=0$. This implies that
$\mathbf{X}(M_1^2)$ contains a dominant weight $\mu_2-\alpha_r$ and $\dim
M_1^2$ is greater than the size of the orbit of $\mu_2$ under the action of the
Weyl group of $H_2$. The latter is equal to $2^{r-m-1}\ge d(r-m)$ for our
values of $r-m$. By (1), this yields the assertion of the theorem for almost
all $\varphi\in\operatorname{Irr}_p$. It remains to consider the case where
$G=B_r(K)$ and $\omega=\omega_r$. It is well known that then the restriction
$M\downarrow H_1$ is a direct sum of $2^{r-m}$ $H_1$-modules $N=M(\omega_m)$.
Since $k_m(x)=p$, we get $k_N(x)=p$ and $n_M(x)\ge 2^{r-m}>d(r-m)$.

Now suppose that $\phi\in\Irr\backslash\Irr_p$. By the Steinberg
tensor product theorem \cite[Theorem~1.1]{St}, $\phi$ can be
represented in the form $\bigotimes^s_{j=1}\phi_j\Fr^j$ where $\Fr$
is the Frobenius morphism of $G$ associated with raising elements
of $K$ to the $p$th power and all $\phi_j\in\Irr_p$. It is clear
that the morphism $\Fr$ does not influence the Jordan form of
$\phi(x)$. Hence one can assume that $\phi=\psi\otimes\theta$
where $\theta=\phi_j\Fr^j$ for some $j$ and both $\psi$ and
$\theta$ are nontrivial. Set $a=\sigma_x(\om(\psi))$,
$\nu=\om(\phi_j)$, $b=\tau_x(\nu)$ and define by $\mu$ the
restriction of $\nu$ to $T_H$. Now it follows from the definitions
of $\sigma_x$ and $\tau_x$ that $\sigma_x(\om)=a+b$. By
\cite[Theorem~1.1, Lemma~2.5 and Proposition~2.12]{Sump},
$k_{\psi}(x)=\min\{a+1,p\}$ and $k_{\theta}(x)=\min\{b+1,p\}$.
First suppose that $a$ or $b\geq p-1$. Set $\rho=\psi$ if $a\geq
p-1$ and $\rho=\theta$ otherwise and denote by $\pi$ the remaining
representation from the pair $(\psi, \theta)$. Then
$k_{\rho}(x)=p$ and  \cite[ch. VIII, Lemma~2.2]{Fe}  implies that
$n_{\phi}(x)\geq \dim\pi$. Let $d(r)$ be the value of $d(r-m)$ if
one formally sets $m=0$. Then by \cite[Proposition~5.4.13]{KL},
$\dim\pi\geq d(r)>d(r-m)$ which settles the case under
consideration.

Now assume that both $a$ and $b<p-1$. Then $k_{\psi}(x)=a+1$ and
$k_{\theta}(x)=b+1$. Since $\sigma_x(\om)\geq p-1+c_x$, we have
$b>c_x$. Arguing as for $p$-restricted $\phi$, we can and shall suppose
that $\langle\nu, \alpha_i\rangle\neq0$ for some $i\leq (r+1)/2$ if
$G=A_r(K)$. Put $M'=M(\nu)$ and construct the  composition factors
$M_i$, $i=1,2$, of the restriction $M'|H$ as for  $p$-restricted
$M$ before. Transfer the notation $\mu_1$, $\lambda_1$, and
$M^i_j$, $i,j=1,2$, to $M'$. Again we have the exceptional case
$G=B_r(K)$ and $\nu=a_r\om_r$ where we do not construct $M_2$ and
consider $M_1$ only. Obviously, $\tau_x(\mu_1)=b$. As before, we
deduce that $\tau_x(\lambda_1)\geq b-c_x$. By
 \cite[Theorem~1.1, Lemma~2.5, and Proposition~2.12]{Sump}, $k_{M_1}(x)=b+1$ and
$k_{M_2}(x)\geq b+1-c_x$.
Let $n_i$ be the number of Jordan blocks of the maximal size in the canonical
form of $x$ as an element of $\operatorname{End} M_i$, $i=1,2$. Looking at the
realizations of $M_i$ as tensor products, one easily observes that $n_i\ge\dim
M_i^2$.
Set $F_1=M(\omega(\psi))\otimes M_1$,
$F_2=M(\omega(\psi))\otimes M_2$ and consider $F_i$ as $H$-modules
in the natural way. In the general case the $H$-module $M$ has a
filtration two of whose quotients are isomorphic to $F_1$ and
$F_2$, respectively. In the exceptional case $F_1$ is a quotient
of a submodule in $M$. Observe that $a+k_{M_2}(x)\geq p$. Using
\cite[ch. VIII, Theorem~2.7]{Fe} that describes the canonical
Jordan form  of a tensor product of unipotent blocks, we obtain
that $k_{M_i}(x)=p$ and $n_{F_i}(x)\geq \dim M^2_i$. As for
$p$-restricted $M$, we show that $n_1\ge 2^{r-m}$ if $G=B_r(K)$ and
$\nu=\omega_r$ and conclude that $\dim M_1^2+\dim M_2^2>d(r-m)$ in the general
case and $\dim\,M_1^2>d(r-m)$ in the exceptional cases with $a_r\not=1$.
Now
(\ref{eqx}) completes the proof. \end{proof}
\begin{proof}[Proof of Proposition~\ref{mp}] Let $a$, $x$, $m$, and
$c$ be such as in the assertion of the proposition. Assume that
$p<ac\leq p+c-1$. Therefore
 we have $(a-1)c\leq p-1$. Set $M=M(\omega)$ and denote by
$M_t$ the weight subspace of
weight $t\in\mathbb{Z}$ in the $\Gamma$-module $M$. It is clear
that the Weyl group of $\Gamma$ interchanges $M_t$ and $M_{-t}$;
hence $\dim M_t=\dim M_{-t}$. Put  $e=(a-1)c$,
$V_1=\bigoplus_{t>e}M_t$, $V_2=\bigoplus_{t>e}M_{-t}$, and $V=M_e$. Set
$f=[(m+1)/2]$ for $G=A_r(K)$, $f=m$ for $G=B_r(K)$ or $C_r(K)$,
and $f=m-1$ for $G=D_r(K)$. Let $v\in M$ be a nonzero highest
weight vector and put $w=X_{-f}\ldots X_{-2}X_{-1}v$. By
\cite[Lemma 2.9]{Supl}, $w\neq0$.  We need a
subgroup $S$ which can be defined as follows. Put $I=\{i\mid
1\le i\le r,\quad \delta_i=0\}$ and $S=\langle\mathcal{X}_i,
\mathcal{X}_{-i}\mid i\in I\rangle$. The canonical Jordan forms of $x$
in the standard realizations of $G_m$ and $G$ are well known. We have
$\abs{x}=p$ since the dimension of the first realization is at most $p$
due to our assumptions. Taking into
account these Jordan forms, one easily obtains the values of $\delta_i$,
$1\le i\le r$, and using Lemma~\ref{lc}, deduces the following facts:
$I=\{i\mid f+1\leq i\leq r-f\}$ for $G=A_r(K)$ and
$m=2f$, $I=\{i\mid f+1\leq i\leq r\}$ for $G=B_r(K)$ and $D_r(K)$,  and $S=H_2$
in all other cases where $H_2$ is the subgroup defined in the
proof of Theorem~\ref{mt}; $c_x=\sum^f_{i=1}\delta_i=c$,
$\tau_x(\omega)=ac$; and $w\in V$.   Next, observe that  $S\cong A_{r-m}$ for
$G=A_r(K)$ and $m=2f$, $S\cong B_{r-m}(K)$ for $G=B_r(K)$,
and $S\cong D_{r-m+1}$ for $G=D_r(K)$. Our
construction of the vector $w$ shows that $\mathcal{X}_i$ fixes
$w$ if $i\in I$. This forces that $w$ generates an indecomposable
$S$-module $M_S$ with highest weight $\omega_S(w)$. Then one
immediately concludes that $M_S\cong M(\omega_1)$. This
 yields that $\dim M_S=r-m+1=d(r-m)+1$ for $G=A_r(K)$ and $m=2f$,
 $\dim M_S=2(r-m)+1=d(r-m)+1$ for $G=B_r(K)$,
$\dim M_S=2(r-m+1)=d(r-m)+2$ for $G=D_r(K)$, and $\dim M_S=d(r-m)$
 otherwise. It is clear that $M_S\subset V$. Denote by
$\mathbf{X}_f\subset\mathbf{X}(M)$ the subset of weights of the
form $\omega-\sum^f_{i=1}b_i\alpha_i$ and by $M_A$ the irreducible
$A_f(K)$-module with highest weight $a\omega_1$. By Smith's theorem
\cite{Sm}, for each $\mu\in\mathbf{X}_f$ the dimension of the
weight subspace $M_{\mu}\subset M$ coincides with that of the
weight subspace in $M_A$ whose weight differs from $a\omega_1$ by
the same linear combination of the simple roots. Hence $\dim
M_{\mu}$ does not depend upon $r$. Set
$W=\bigoplus_{\mu\in\mathbf{X}_f}M_{\mu}$. Since $M$ is an
irreducible $L$-module and $p>2$, observe that $M$ is a linear
span of vectors of the form $X_{-i_s}\ldots X_{-i_2}X_{-i_1}v$.
Now, analyzing the weight structure of $M$, we conclude that
$V_1\subset W$ and $V=(V\cap W)\oplus M_S$. This implies that
$\dim V_1$ ($=\dim V_2$) and $\dim (V\cap W)$ do not depend upon
$r$.

 It follows from \cite[Lemma 72]{St-b} that
 \begin{equation}\label{eqpo}
 (x-1)^{p-1}M_t\subset\bigoplus_{i\geq t+2p-2} M_i.
 \end{equation}
  Let $M_t\not\subset V_2$. Then $t\geq -e$. Obviously, $e<p-1$ if
  $ac<p-1+c_x$ and $e=p-1$ for $ac=p-1+c_x$. Thus (\ref{eqpo}) implies
  that
  \begin{equation*}
  (x-1)^{p-1}M_t\subset\bigoplus_{t>p-1}M_t\subset V_1
  \end{equation*}
  in the first case and
  \begin{equation*}
(x-1)^{p-1}M_t\subset\bigoplus_{t\geq p-1}M_t\subset V_1\oplus V
\end{equation*} in the second case. This forces that
 $n_M(x)\leq\dim V_2+\dim V_1=2\dim V_1$ in the first case and
$n_M(x)\leq 2\dim V_1+\dim(V\cap W)+\dim M_S$ in the second case.
We have seen before that $\dim M_S=d(r-m)+u$ with $u=0$, $1$, or
$2$. Hence one can take $N_G(a,m,p)=2\dim V_1$ and
$Q_G(a,m,p)=2\dim V_1+\dim(V\cap W)+u$ to complete the proof.
\end{proof}

\begin{remark} For $G=A_r(K)$ or $C_r(K)$
we could give a shorter proof of Proposition~\ref{mp} using the
realization of $\phi$ in the $a$th symmetric power of the standard
module (see \cite[1.14 and  8.13]{Se}), but we need the proof
above for $B_r(K)$ and $D_r(K)$. \end{remark}
 \bigskip

This research  has been supported by the Institute of Mathematics
of the National Academy of Sciences of Belarus in the framework of
the State program ``Mathematical structures'' and by the Belarus
Basic Research  Foundation, Project F\,98-180.

\begin{thebibliography}{99}

\bibitem{BC} P. Bala and R.W. Carter, \textit{Classes of unipotent
elements in simple algebraic groups. II}, Math. Proc. Cambridge
Philos. Soc. \textbf{80} (1976),  no 1,  1--17. MR54:5363b.

\bibitem{B1} N. Bourbaki, \textit{Groupes et algebres de Lie. Ch.
IV--VI}, Hermann, Paris, 1968.

\bibitem{B2} N.~Bourbaki, \textit{Groupes et algebres de  Lie.
Ch. VII--VIII}, Hermann, Paris, 1975.

\bibitem{Fe} W.~Feit, \textit{The representation theory of finite
groups}, North-Holland, Amsterdam, 1982. MR83g:20001.

\bibitem{KL} P.~Kleidman and M.~Liebeck, \textit{The  subgroup  structure
  of the finite classical groups},  London Math. Soc. Lecture Note
Ser.  vol. 129, Cambridge Univ. Press, Cambridge, 1990. MR91g:20001.

\bibitem{LaTe} R. Lawther and D.M. Testerman, \textit{$A_1$ subgroups of
exceptional algebraic groups}, Mem. Amer. Math.
Soc. \textbf{141} (1999), no 674 , AMS, Providence. MR2000b:20059.

\bibitem{Se} G.M.~Seitz, \textit{The maximal subgroups of classical
algebraic groups},   Mem. Amer. Math. Soc. \textbf{67} (1987), no 365, AMS,
Providence. MR88g:20092.

\bibitem{Sm} S.~Smith, \textit{Irreducible modules and  parabolic
subgroups},  J. Algebra  \textbf{75} (1982), 286--289. MR83g:20043.

\bibitem{St} R.~Steinberg, \textit{Representations of algebraic
groups}, Nagoya Math. J. \textbf{22} (1963), 33--56. MR27:5870.

\bibitem{St-b} R.~Steinberg,  \textit{Lectures on Chevalley groups},
 Yale University, New Haven,  1968. MR57:6215.

\bibitem{Sump} I.D.~Suprunenko, \textit{The minimal polynomials of
elements  of order $p$ in irreducible representations of Chevalley
groups over fields of characteristic $p$}, Siberian Adv. Math.
\textbf{6} (1996), no 4, 97--150. Translated from Problems
of Algebra and Logic. Trudy Inst. Math. Sib. Otdel. RAN.
Novosibirsk. Vol. 30 (1996), 126--163.

\bibitem{Supl} I.D.~Suprunenko, \textit{On Jordan blocks of elements of
order $p$ in irreducible representations of classical groups with
$p$-large highest weights},  J. Algebra  \textbf{191} (1997), no
2, 589--627. MR98c:20085.

\bibitem{SuKo} I.D.~Suprunenko, \textit{$p$-large representations and
asymptotics: a survey and conjectures},  Algebra \textbf{12}. J. Math. Sci. (New
York) \textbf{100} (2000), no. 1, 1861--1870.

 \bibitem{Te} D.M. Testerman, \textit{$A_1$-type overgroups of elements
 of order $p$ in semisimple algebraic groups and related finite groups},
  J. Algebra \textbf{177} (1995), 34--76. MR96j:20067.
\end{thebibliography}
\end{document}

math0003223.extracted_bib
\begin{thebibliography}{99}

\bibitem{BC} P. Bala and R.W. Carter, \textit{Classes of unipotent
elements in simple algebraic groups. II}, Math. Proc. Cambridge
Philos. Soc. \textbf{80} (1976),  no 1,  1--17. MR54:5363b.

\bibitem{B1} N. Bourbaki, \textit{Groupes et algebres de Lie. Ch.
IV--VI}, Hermann, Paris, 1968.

\bibitem{B2} N.~Bourbaki, \textit{Groupes et algebres de  Lie.
Ch. VII--VIII}, Hermann, Paris, 1975.

\bibitem{Fe} W.~Feit, \textit{The representation theory of finite
groups}, North-Holland, Amsterdam, 1982. MR83g:20001.

\bibitem{KL} P.~Kleidman and M.~Liebeck, \textit{The  subgroup  structure
  of the finite classical groups},  London Math. Soc. Lecture Note
Ser.  vol. 129, Cambridge Univ. Press, Cambridge, 1990. MR91g:20001.

\bibitem{LaTe} R. Lawther and D.M. Testerman, \textit{$A_1$ subgroups of
exceptional algebraic groups}, Mem. Amer. Math.
Soc. \textbf{141} (1999), no 674 , AMS, Providence. MR2000b:20059.

\bibitem{Se} G.M.~Seitz, \textit{The maximal subgroups of classical
algebraic groups},   Mem. Amer. Math. Soc. \textbf{67} (1987), no 365, AMS,
Providence. MR88g:20092.

\bibitem{Sm} S.~Smith, \textit{Irreducible modules and  parabolic
subgroups},  J. Algebra  \textbf{75} (1982), 286--289. MR83g:20043.

\bibitem{St} R.~Steinberg, \textit{Representations of algebraic
groups}, Nagoya Math. J. \textbf{22} (1963), 33--56. MR27:5870.

\bibitem{St-b} R.~Steinberg,  \textit{Lectures on Chevalley groups},
 Yale University, New Haven,  1968. MR57:6215.

\bibitem{Sump} I.D.~Suprunenko, \textit{The minimal polynomials of
elements  of order $p$ in irreducible representations of Chevalley
groups over fields of characteristic $p$}, Siberian Adv. Math.
\textbf{6} (1996), no 4, 97--150. Translated from Problems
of Algebra and Logic. Trudy Inst. Math. Sib. Otdel. RAN.
Novosibirsk. Vol. 30 (1996), 126--163.

\bibitem{Supl} I.D.~Suprunenko, \textit{On Jordan blocks of elements of
order $p$ in irreducible representations of classical groups with
$p$-large highest weights},  J. Algebra  \textbf{191} (1997), no
2, 589--627. MR98c:20085.

\bibitem{SuKo} I.D.~Suprunenko, \textit{$p$-large representations and
asymptotics: a survey and conjectures},  Algebra \textbf{12}. J. Math. Sci. (New
York) \textbf{100} (2000), no. 1, 1861--1870.

 \bibitem{Te} D.M. Testerman, \textit{$A_1$-type overgroups of elements
 of order $p$ in semisimple algebraic groups and related finite groups},
  J. Algebra \textbf{177} (1995), 34--76. MR96j:20067.
\end{thebibliography}