\documentstyle[12pt]{article} %\topmargin-2cm %\oddsidemargin-1cm %\evensidemargin-5cm %\addtolength{\textwidth}{4cm} %\addtolength{\textheight}{5cm} \newcommand{\proofhead}[1]{\par\pagebreak[1]\noindent{\bf#1.\ }} \newcommand{\pf}{\proofhead{Proof}} \newcommand{\qed}{{\unskip\nolinebreak[1]\hspace{1.5em}\mbox{}\nolinebreak \hfill$\Box$\parfillskip=0pt\finalhyphendemerits=0\par\pagebreak[1]}} \newenvironment{proof}{\pf}{\qed} \author{Corneliu Hoffman \\ {\normalsize Department of Mathematics and Statistics}\\ {\normalsize Bowling Green State University}\\ {\normalsize Bowling Green, OH 43403 }} \title{ Projective Representations in Cross Characteristics for some Classical Groups} \date{} \newtheorem{lemma}{Lemma} \newtheorem{th}{Theorem} \newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \begin{document} \maketitle \begin{abstract} Let $G(p^n)$ a finite simple group of Lie type and $V$ a projective irreducible representation of $G(p^n)$ over a field of characteristic different from $p$. Lower bounds for the dimension of $V$ have been computed in \cite{ls} and \cite{sz} and recently improved in the case of linear groups by \cite{gt} and in the case of characteristic zero representations in \cite{tz}. The purpose of this paper is to improve the bounds in \cite{sz} for the case of orthogonal groups and to provide estimates for the gap between the smallest representations for orthogonal and unitary groups. \footnote{This work was part of my Ph.D. thesis at University of Southern California. I am very grateful to my advisor Robert Guralnick for the help and support throughout this period.}\end{abstract} \section{Introduction} Let $G(q)$ be a finite simple group of Lie type over a field of order $q$ where $q=p^e$. We denote by $l(G(q))$ the degree of the smallest nontrivial projective irreducible representation of $G(q)$ over a field of characteristic $r \neq p$. The results of \cite{ls} and \cite{sz} provide lower bounds for the numbers $l(G(q))$ and these bounds are the best possible in many cases. However the results of \cite{gt} for linear groups and those of \cite{tz} for representations in characteristic zero suggest that the bounds of \cite {sz} can be improved. We will improve the bounds for the case of the orthogonal groups and, we will say something about the gap between the smallest possible representations and the next family of such representations in the case of orthogonal and unitary groups. Note that for our purpose one can assume that the representations are defined over an algebraically closed field because increasing the field will decrease the dimensions of the representations. We will use the results of \cite{sz} together with the better bound for linear groups provided in \cite{gpps} to prove: \begin{th} \label{t1} Let $V$ be a nontrivial irreducible projective representation of $P\Omega _{2n}^\epsilon (q)$ where $\epsilon =\pm $ or of $P\Omega _{2n+1}(q)$ over a field of characteristic $r$ not dividing $q$. Then: \begin{enumerate} \item If $G=P\Omega _{2n}^{+}(q),n\geq 4, q>3$ then $$ \frac{(q^n-1)(q^{n-1}+q)}{q^2-1}-2\leq \dim V\leq \,\frac{(q^n-1)(q^{n-1}+q) }{q^2-1}+1 $$ or $$\dim V\geq \frac{(q^n-1)(q^{n-1}+q)}{q^2-1}+\frac{q^n-1}{q-1}-4$$ Moreover the smallest of these cases will not hold if $r$ does not divide $ (q^n-1)/(q-1)$. \item If $G=P\Omega _{2n}^{+}(q),n\geq 5, q\leq 3$ then $$ \frac{(q^n-1)(q^{n-1}-1)}{q^2-1}\leq \dim V\leq \,\frac{(q^n-1)(q^{n-1}-1)}{ q^2-1}+1 $$ or $$\dim V\geq \frac{(q^n-1)(q^{n-1}-1)}{q^2-1}+\frac{(q^n-1)}{(q-1)}-2$$ Moreover if $r$ does not divide $(q^n-1)/(q-1)$, the last bound is \\ $\frac{ (q^n-1)(q^{n-1}-1)}{q^2-1}+\frac{q^n-1}{q-1}-1$. \item If $G=P\Omega _{2n}^{-}(q),n\geq 4$, $(n,q)\neq (4,2),(5,2),(4,4),(5,3)$ then $$ \frac{(q^n+1)(q^{n-1}-q)}{q^2-1}-1\leq \dim V\leq \,\frac{(q^n+1)(q^{n-1}-q) }{q^2-1} $$ or $$\dim V\geq \frac{(q^n+1)(q^{n-1}-q)}{q^2-1}+\frac{q^{n-1}-1}{q-1}-3$$ Moreover the smallest of these cases will hold if and only if $r$ divides $ (q^{n-1}-1)/(q-1)$. \item If $G=P\Omega _{2n+1}(q),n\geq 3, q>3, q$ odd, then $$ \frac{q^{2n}-1}{q^2-1}-2\leq \dim V\leq \,\frac{q^{2n}-1}{q^2-1}+q-2 $$ or $$\dim V\geq \frac{q^{2n}-1}{q^2-1}+\frac{q^n-1}{q-1}-4$$ Moreover the smallest of these cases will not hold if $r$ does not divide $(q^n-1)/(q-1)$. \end{enumerate} \end{th} We note that our bounds are very close to the dimensions of the representations in characteristic zero obtained in \cite{tz} hence after passing to positive characteristic these representations will either stay irreducible or there will be at most two trivial modules among the composition factors. In particular in the case $G=P\Omega _{2n}^{-}(q)$, we can determine exactly how this representation will factor. For convenience we include a complete table with the bounds for $l(G(q))$. In this table we mark with a star the cases where the bound is known to be sharp. Note also that the bounds in the cases $E_i, i=6, 7, 8$ were obtained in \cite{h}. \begin{table}[b] \label{ta1} $$\begin{array}{|c||l|c|} \hline \mbox{Group} & \mbox{Bound} & \mbox{Exceptions} \\ \hline \mbox{PSL}_2(q) \ *& (q-1)/\mbox{gcd}(2,q-1) & \begin{array}{l} l(PSL_2(4))=2, \\ l(PSL_2(9))=3 \end{array} \\ \hline \mbox{PSL}_{n}(q) , n\geq 3 \ *& (q^n-1)/(q-1)-2 & \begin{array}{l} l(PSL_3(2))=2, \\ l(PSL_3(4))=4, \\ l(PSL_4(2))=7, \\ l(PSL_4(3))=26 \end{array} \\ \hline \mbox{PSp}_{2n}(q), n \geq 2 \ *& \begin{array}{l} (q^n-1)/2,\ \ q \ \mbox{odd}\\ (q^n-1)(q^n-q)/(2(q+1)),\ \ q \ \mbox{even} \end{array} & l(\mbox{PSp}_4(2)')=2 \\ \hline P\Omega_{2n+1}(q), n\geq 3 & \begin{array}{l} (q^{2n}-1)/(q^2-1)-2, \ \ q \neq 3 \\ (q^n-1)(q^n-q)/(q^2-1), \ \ q=3 \end{array} & l(P\Omega_7(3))=27 \\ \hline P\Omega^{+}_{2n}(q), n\geq 4 & \begin{array}{l} (q^n-1)(q^{n-1}+q)/(q^2-1)-2, \ \ q > 3 \\ (q^n-1)(q^{n-1}-1)/(q^2-1), \ \ q \leq 3 \end{array} & l(P\Omega_8^{+}(2))=8 \\ \hline P\Omega^{-}_{2n}(q), n\geq 4 \ * & (q^n+1)(q^{n-1}-q)/(q^2-1)-1& \begin{array}{l} l(P\Omega^{-}_{8}(2))\geq 32,\\ l(P\Omega^{-}_{8}(4))\geq 1026, \\ l(P\Omega^{-}_{10}(2))\geq 151, \\ l(P\Omega^{-}_{10}(3))\geq 2376, \end{array} \\ \hline \mbox{PSU}_{2n}(q), n\geq 2 & (q^{2n}-1)/(q+1) & \begin{array}{l} l(\mbox{PSU}_4(2))=4, \\ l(\mbox{PSU}_4(3))=6, \end{array} \\ \hline \mbox{PSU}_{2n+1}(q) & (q^{2n+1}-q)/(q+1) & \\ \hline ^{2}B_{2}(q) & (q-1)\sqrt{q/2} & l(^{2}B_{2}(8))=8 \\ \hline ^{2}G_{2}(q) & q(q-1) & \\ \hline G_{2}(q) & q(q^2-1) & \begin{array}{l} l(G_2(3))=14 \\ l(G_2(4))=12 \end{array} \\ \hline ^{3}D_{4}(q) & q^3(q^2-1) & \\ \hline ^{2}F_{4}(q) & q^4(q-1)\sqrt{q/2} & \\ \hline F_{4}(q) & \begin{array}{l} q^4(q^6-1), \ \ q \ \mbox{odd} \\ q^7(q^3-1)(q-1)/2, \ \ q\ \mbox{even} \end{array} & \\ \hline ^{2}E_{6}(q) & q^9(q^2-1) & \\ \hline E_6(q) & q\Phi_8(q)\Phi_9(q)-2 & \\ \hline E_7(q) & q \Phi_7(q)\Phi_{12}(q)\Phi_{14}(q)-3 & \\ \hline E_8(q) &q\Phi_4(q)^2 \Phi_8(q) \Phi_{12}(q)\Phi_{20}(q)\Phi_{24}(q)-4 & \\ \hline \end{array} $$ \caption{ Bounds for dimensions of non trivial projective representations for finite Chevalley groups in non defining characteristic} \end{table} \section{On linear groups} Before considering the orthogonal groups, we will investigate the small dimensional indecomposable representations for the linear group of rank $n>2$. We will only consider modules $N$ such that $G= SL_n(q)$ will act on both $N$ and $N^*$ without fixing more than a one dimensional space. Define $e(n,q):= \frac{q^n-1}{q-1}$ If $N$ is such a module and the length is more than 4, then at least 2 of the factors are nontrivial irreducible modules hence the dimension is more that $2e(n,q)-4$. Therefore consider only those modules that have length at most 3 and only one nontrivial factor. In fact after considering their radicals, we need to discuss the length 2 indecomposable with one nontrivial factor. We claim: \begin{lemma} \label{l0} Let $G=SL_n(q), n> 2, (n,q) \neq (3,2),(3,4), (4,2), (4,3)$ and $N$ is an indecomposable $kG$ module of dimension smaller than $2e(n,q)-4$ and such that $\dim C_N(G) \leq 1, \dim C_N^*(G) \leq 1$. Then $N$ is either irreducible or has exactly one nontrivial factor of dimension $e(n,q)-2$ and at most two trivial factors. In particular the dimension $d$ of any such module satisfies $e(n,q)-2\leq d\leq e(n,q)$. \end{lemma} \begin{proof} By Theorem 9.1.5 of \cite{gpps}, we know that there are only three possibilities for the dimensions of the nontrivial irreducible factors of $N$. These are $e(n,q)-2,\,e(n,q)-1,\,e(n,q)$. Moreover for the last two dimensions, the corresponding modules are well determined by Theorem 9.1.6 of \cite{gpps}. More precisely if $P$ is the stabilizer of a 1 space in the natural action of $G$ and $M$ an irreducible module then $\dim M=e(n,q)-1$ implies $ M\oplus k=k_P^G$, and $\dim M=e(n,q)$ implies $M=\lambda _P^G$,($\lambda $ nontrivial). By the preceding remarks, we need to investigate indecomposable modules of length 2 having small dimensional factors, that is having M and the trivial module as factors. Consider in each case $Ext^1(k,M)=H^1(G, M)$ and $Ext^1(M, k)=H^1(G, M^*)$. By Shapiro lemma, $H^1(G,\lambda_{P}^{G})=H^1(P,\lambda)=0$ (respectively $H^1(G, k_{P}^{G})=H^1(P, k)=0$) and if $\dim M = e(n,q)-1$ (respectively $e(n,q)$) then $M^*=M$ (respectively $M^*=\tau_P^G$ for some $\tau$). Assume $M\oplus k=k_P^G$, then $H^1(G,k_{P}^{G})=H^1(G,k)\oplus H^1(G,M)$ so $Ext^1(k,M)=Ext^1(M,k)=H^1(G,M)=0$. The same result holds if $M=\lambda_P^G$ hence any module of length two having $M$ and the trivial module as its factors is in fact the direct sum of these factors and so any indecomposable module of length $\geq 1$ and dimension smaller than $2e(n,q)-4$ has an irreducible factor of dimension $e(n,q)-2$ and by the remarks preceding the lemma it has length $\leq 3$ and the result follows. \end{proof} In \cite{gt} Guralnick and Tiep proved that if $V$ is an irreducible representation of $SL_n(q), n>2$ of dimension smaller than $\delta(n,q)$ then it is one of the three types of representations mentioned above. Here $$\delta(n,q)= \left\{ \begin{array}{l} (q-1)(q^2-1)/(3,q-1)\ \mbox{if} \ n=3 \\ (q-1)(q^3-1)/(2,q-1)\ \mbox{if} \ n=4 \\ (q^{n-1}-1)((q^{n-2}-q)/(q-1)-k_{n-2})\ \mbox{if}\ n \geq 5 \end{array} \right. $$ where $k_n=1$ if $r$ divides $(q^n-1)/(q-1)$ and 0 otherwise. We will use this result in what follows. \section{Unitary Groups} Before dealing with the proof of Theorem \ref{t1} we prove some estimates for the gap between the smallest representations for unitary groups. Recently Malle and Hiss obtained much better results in this special case but the methods presented here are entirely elementary. \begin{prop}\label{unitary} If $G=SU_{2n}(q)$ , $n\geq 2$ and $V$ is a nontrivial irreducible $kG$ module (char $k $ does not divide $q$), then: $$\ dim V = \frac{q^{2n}-1}{q+1},\ \frac{q^{2n}-1}{q+1}+1$$ or $$\dim V \geq 2 \frac{q^{2n}-1}{q+1}$$ \end{prop} \begin{proof} Let $P$ be a stabilizer of a maximal totally singular space for the natural module of $G$, $L$ the Levi complement and $Q$ the unipotent radical of $P$. Then $Q$ is abelian and can be viewed as the group of skew hermitian $n\times n$ matrices over a field with $q^2$ elements. $L=GL_n(q^2)$ and acts on $Q$ as $A \rightarrow gA\bar{g}$. Since $r:=char k$does not divide the order of $Q$, the irreducible $G$ module $V$ decomposes as $V=[Q,V]\oplus C_V(Q)$ when regarded as a $Q$ module (and so the same is true for $V$ as a $P$ module). Note that the smallest orbit of the action of $L$ on $Q$ has length $(q^{2n}-1)/(q+1)$ (it corresponds to the rank one skew hermitian matrices) and therefore the same is true about the action of $L$ on the characters of $Q$. The length of the next orbit is at least twice as large. The conclusion is that $\dim [Q,V] = \frac{q^{2n}-1}{q+1}$ or is at least twice that much. Assume that the $\dim [Q,V]= \frac{q^{2n}-1}{q+1}$ and that $C=C_V(Q)\neq 0$. We will prove that $C$ is a trivial module for the derived group $L'$. To see that we first note that \cite{s} constructs a $G$ module $W$ of dimension $ \frac{q^{2n}-1}{q+1)$. The above argument shows that $W=[Q,W]$. Let $x\in L'$ a transvection and $y\in Q$ a conjugate of $x$. Note that $M=[Q,V]=\alpha_S^{L'}$ as an $L'$ module where $S$ is the stabilizer of an eigenspace of $Q$ and $\alpha$ a linear character of $S$, hence $\alpha$ is trivial on transvections of $S$. In this case the value of the Brauer character $t_M(x)$ does not depend on $\alpha$ and it is the number of $x$ invariant characters of $Q$ occurring in $M$. A similar argument gives that $t_{[Q,W]}(x)$ has the same value. Next $y$ is a rational element in $P$ so $t_M(y)$ depends only on the dimension of $C$ so not on the structure of $M$. Also $W=[Q,W]$ so $t_W(x)=t_W(y)$. Therefore $t_M(x)=t_W(x)=t_W(y)=t_M(y)$ hence $t_C(x)=t_C(y)$ and since $y$ acts trivially on $C$ so does $x$. Since $L'$ is generated by transvections it follows that $L'$ acts trivially on $C$. $M$ is a transitive permutation module for $L'$ so $\dim C_M(L')=1$. If $\bar{Q}$ is the opposite unipotent radical, $V=[\bar{Q},V]\oplus C_V(\bar{Q})$ and a similar argument gives $\dim C=\dim C_V(\bar{Q})$. In particular both $C$ and $C_V(\bar{Q})$ are hyperplanes in $C_V(L')$ and cannot intersect so they must have dimension at most one hence the conclusion. \end{proof} Note that in fact this give us similar estimates on the irreducible projective representations of the simple groups of type $PSU_{2n}(q)$ with the exceptions of $(n,q)=(2,2), (2,3)$ in which cases there are exceptional Schur multipliers. \section{Proof of Theorem \ref{t1}} Let $G(q)$ be one of $P\Omega _{2n}^\epsilon (q)$ where $\epsilon =\pm $ or $P\Omega _{2n+1}(q)$, $q$ odd and $\bar{G}$ a perfect central extension of $G(q)$ acting irreducibly on $V$. If $S$ is a subgroup of $G(q)$ we will denote with $\bar{S}$ the subgroup of $\bar{G}$ such that $\bar{S} / Z(\bar{G})=S$. We will denote by $P=JU$ the subgroup of $G(q)$ that is the stabilizer of a maximal totally singular subspace for the natural module, $U$ the unipotent radical of $P$ and $J$ the Levi complement ($J$ will be a linear group). Then since $U$ is a $p$ group, we have $\bar{U}= U_0 \times Z(\bar{G})$ where $U_0$ is $P$ isomorphic to $U$ by assumption on $(n,q)$. Let $V=[U_0, V]\oplus C_V(U_0)$ as a $P$ module. First we note that in fact $[U_0,V]$ is the sum of nontrivial representations of $U_0$ that occur in $V$. Therefore one can bound from below the dimension of this module by $l\cdot d$ where $l$ is the length of the smallest orbit of the action of $J$ on the characters of $U$ and $d$ is the degree of the corresponding characters. We will use the bound in \cite{sz} to see that in fact if $[U_0,V]$ is small therefore $M=C_V(U_0)$ must be quite large. This will actually give us that $M\neq C_M(\bar{J'})$, which in particular means that $M$ as a $\bar{J'}$ module will have a nontrivial irreducible factor and so we can use the bound in \cite{gt} for the dimension of $M$. \begin{prop}\label{l1}If $G(q)=P\Omega _{2n}^{+}(q)$ with $n\geq 4$ and $(n,q)\neq (4,2),\\ (3,4)$ or $(4,3)$, then Theorem \ref{t1} holds. \end{prop} \begin{proof} As above, pick $P$ the stabilizer of a maximal totally singular subspace of the natural orthogonal module, $U$ its unipotent radical and $J$ the Levi complement. Then $J=GL_n(q)$ and $U$ can be regarded as the space of $n\times n$ antisymmetric matrices on which $ J$, acts in the natural way $(T^A=A\;T\;^tA)$\footnote{Note that if $q$ is even then this is not exactly true. Nevertheless the structure of the module will be exactly the same so all the following arguments will work.}. The orbits are the matrices of a given rank $i$ ( only if $i$ is even) hence the size of such an orbit is $q^{i(i-2)}(q^n-1)\cdots (q^{n-i+1}-1)/(q^2-1)\cdots (q^i-1)$. Therefore the smallest orbit has length $l=(q^n-1)(q^{n-1}-1)/(q^2-1)$ and so $\dim [U_0,V]=l$ or is at least twice as big. If $q=2,3$ then the lower bound in Theorem \ref{t1} is obtained. (Note that in fact this is also the bound in \cite{sz} for $ q=3$). If $q>3$ assume that $\dim [U_0,V]=l$ . We know from \cite{sz} that $\dim V\geq (q^n-1)(q^{n-1}+q)/(q^2-1)-n$ so $M=C_V(U_0)$ must have dimension larger than $(q^n-1)/(q-1)-n$. Note that $[U_0,V]$ is a transitive permutation module for $\bar{J'}$ hence $C_{[U_0,V]}(\bar{J'})$ is one dimensional and so $ C_M(\bar{J'})$ is a hyperplane in $C_V(\bar{J'})$. If we repeat the same argument for the opposite parabolic group we get that if $C_M(\bar{J'})$ has dimension at least 1, then $C_V(\bar{G})\neq 0$ which contradicts the hypothesis. Therefore $\dim C_M(\bar{J'})\leq 1$ and so since $n\geq 4$, we get that $C_V(U_0)$ has a nontrivial factor as a $\bar{J'}$ module. The dimension of this factor must be at least $(q^n-1)/(q-1)-2$ from \cite{gt} hence the lower bound. For the rest of the theorem we consider once again $V=[U_0,V]\oplus M$, we consider the next possibility for the dimensions of the two modules. The next possible dimension of $[U_0,V]$ is very large and will already solve the problem. If we regard $M$ as a $\bar{J'}$ module we see that the maximum dimension of $C_M(\bar{J'})$ is 1. Therefore using Lemma \ref{l0} we get that an indecomposable factor of $M$ has dimension $e(n,q)-2, e(n,q)-1, e(n,q)$ or $ \geq 2e(n,q)-4$ and the result follows. \end{proof} \begin{prop} \label{l2} Let $G(q)=P\Omega _{2n}^{-}(q)$ with $n\geq 4$ and $(n,q)\neq (4,2), (5,2), \\ (4,4)$ or $(5,3)$, $V$ a projective irreducible representation in characteristic $r$ not dividing $q$ then $$\dim V \geq \frac{(q^n+1)(q^{n-1}-q)}{q^2-1}-1$$ \end{prop} \begin{proof} Consider $P_n=JU$ the stabilizer of a maximal totally singular subspace of the natural module. Then $J=GL_{n-1}(q)\times D_{2(q+1)}$ and $U$ is nonabelian solvable with $U^{\prime }=\Phi (U)$ being the space of antisymmetric $(n-1)\times (n-1)$ matrices with the above mentioned action of $J$ \footnote{As before there is a slight variation for $q$ even}. As before $\bar{U}=U_0\times Z(\bar{G})$. $V$ regarded as an $U_0$ module has an irreducible component on which $Z(U_0)$ acts non-trivially or else the representation is not faithful. Note also that the degree of any nonlinear character of $U_0$ is a multiple of $q^2$(this is because $U/Z(U)=N\oplus N$ as a $J'$ module where $N$ is the natural module and $J$ acts as $(A,x)v=Avx$ where $v$ is a $2\times (n-1)$ matrix and we view $x \in O_2^-(q)$. The conclusion is that the dimension of $[U,V]$ is at least $q^2\cdot l$ where $l$ is the length of the smallest orbit of the action of $\bar{P_n}$ on $Z(U_0)$. As before $l=(q^{n-1}-1)(q^{n-2}-1)/(q^2-1)$ hence $\dim [U_0,V]=q^2(q^{n-1}-1)(q^{n-2}-1)/(q^2-1)$ or it is at least twice as large. We know by \cite{sz} that $\dim V\geq (q^n+1)(q^{n-1}-q)/(q^2-1)-n+2$. Let $M=C_V(\bar{J'})$ and assume $\dim [U_0,V]=q^2(q^{n-1}-1)(q^{n-2}-1)/(q^2-1)$. It follows that $\dim M\geq q(q^{n-2}-1)/(q-1)-n+2\geq q^2+1$. Also $C_{[U_0,V]}(\bar{J'})$ has dimension $q^2$ and so repeating the argument for the opposite parabolic subgroup $\widetilde{P}$ and making the same assumptions on the dimension of $[\widetilde{U_0},V]$ we get that if $\dim C_M(\bar{J'})>q^2$, then $ C_V(P)\cap C_V(\widetilde{P})\neq \emptyset $ which leads to a contradiction. Therefore $\dim C_M(\bar{J'})\leq q^2$ and so $M$ has a nontrivial factor as a $\bar{J'}$ module and the dimension of $M$ is at least $( q^{n-1}-1)/(q-1)-2$. If we add the lower bounds for $[U_0,V]$ respectively $M$ we get that $\dim V\geq (q^n+1)(q^{n-1}-q)/(q^2-1)-1$ hence the lower bound. \end{proof} For the rest of the theorem note that $C_M(\bar{J'})$ is at most $q^2$ dimensional and using lemma \ref{l0}, any indecomposable factor of $M$ has dimension $ e(n-1, q)-2, e(n-1 ,q)-1,e(n-1 ,q)$ or $\geq 2e(n-1,q)-4 $. Therefore the proof of the theorem is complete once we proved the following: \begin{prop} Let $G=P\Omega_{2n}^-(q)$ and $V, P_n$ be as before. Then either $$\dim V = \frac{(q^n+1)(q^{n-1}-q)}{(q^2-1)}-1+k \ (k=0,1,2) \mbox{ and } C_V(\bar{P'_n})=0,$$ or $$\dim V \geq \frac{(q^n+1)(q^{n-1}-q)}{(q^2-1)}+\frac{q^n-1}{q-1}-3$$ \end{prop} \begin{proof} We will proceed by induction. Let $G=P\Omega^-_{8}(q)$, $q \geq 3$ and assume $V$ is an irreducible representation of $\bar{G}$ such that $\dim V =(q^4+1)(q^3-q)/(q^2-1)-1+k=q^5+q-1+k$. If $\dim C_V(\bar{P'_4})=d$, then by the proof of Proposition \ref{l2}, $d\leq q^2$ and $k-2\leq d \leq k$. Let $P=LQ$ be the stabilizer of a line in the natural module of $G$. As before $\bar{Q} = Q_0 \times Z(\bar{G})$ where $Q_0\cong Q$ under the action of $P$. Then $V=[V, Q_0]\oplus C_V(Q_0)$ and since $Q_0\subset \bar{P'_4}$, $C_V(\bar{P'_4})\subset C_V(Q_0)=N$. $\dim [Q_0, V]= (q^3+1)(q^2-1)$ or it is at least twice as large and that will be more than the dimension of $V$. Therefore $\dim [Q_0, V]= (q^3+1)(q^2-1)$ and $\dim N =q^3-q^2+q+k$. Regard $N$ as an $\bar{L'}$ module and since $L'=\Omega_6^-(q)=U_4(q)$, the minimal dimension of a nontrivial irreducible factor of $N$ is (cf. \cite{sz}) $q^3-q^2+q-1$. so $N$ has exactly one nontrivial factor. Since $\dim C_{[Q_0,V]}(\bar{L'})=1$, we get that $\dim C_N(\bar{L'})\leq 1$ (otherwise we can consider the opposite parabolic and get a submodule of $V$) hence $N'$, the irreducible factor of $N$ contains a $d$ dimensional vector space $W$ on which $\bar{P'_4}$ acts trivially (this is because $C_N(\bar{L'})\cap C_V(\bar{P'_4})= \emptyset $), and $$q^3-q^2+q+k-2 \leq \dim N' \leq q^3-q^2+q+k.$$ Using Proposition \ref{unitary}, we get that $k \leq 2$ and so $0\leq d \leq 2$.Let us consider $S$ the stabilizer of a maximal totally singular space, $H$ its unipotent radical and $R$ its Levi complement. It follows that $H \subset P_4'$ so $W \subset C_{N'}(H)$. From the proof of Proposition \ref{unitary} we can see that $C_{N'}(H)$ is a trivial $R'$ module and it is at most one dimensional. It follows that $R'$ fixes $W$ and since $P_4'$ and $R'$ generate $G$, $W$ is a $G$ submodule of $V$ hence it is zero. If $n>4$, assume that $\dim V = (q^n+1)(q^{n-1}-q)/(q^2-1)-1+s $ and, as before $\dim C_V(\bar{P'_n}) =d$, $d\leq q^2$, $s-2 \leq d \leq s$. Let $P, L, Q$ as above and notice that $Q$ is abelian, $L'=\Omega_{2(n-1)}^-(q)$. If $V=[Q_0,V]\oplus C_V(Q_0)$, then $C_V(\bar{P'_n})\subset C_V(Q_0)$ and the smallest orbit of the action of $L'$ on $Q$ has length $(q^{(n-1)}+1)(q^{n-2}-q)$ this being the only one that can occur. Consequently $\dim C_V(Q_0)= (q^{n-1}+1)(q^{n-2}-q)/(q^2-1)+s$ and so cf. \cite{sz} $C_V(Q)$ has only one nontrivial factor as a $\bar{L'}$ module. After possibly passing to the dual of the module and factoring out by a trivial $\bar{L'}$ module, we can obtain an irreducible $\bar{L'}$ module that contains a $d$ dimensional vector space $W$ on which $\bar{P'_n}$ acts trivially. Since $\bar{P'_n} \cap \bar{L'}$ is the corresponding parabolic of $\bar{L'}$, the induction hypothesis gives that $d=0$ proving the proposition. \end{proof} \begin{cor} \begin{enumerate} \item If $r$ divides $\frac{q^{n-1}-1}{q-1}$ then $P\Omega^-_{2n}(q)$ has an irreducible projective representation of dimension $(q^n+1)(q^{n-1}-q)/(q^2-1)-1$ over $\mbox{\bf F}_r$. This representation is a factor of the reduction modulo $r$ of the smallest irreducible projective representation in characteristic 0. \item If $r$ does not divide $\frac{q^{n-1}-1}{q-1}$ then the reduction mod $r$ of the smallest irreducible projective representation of $P\Omega^-_{2n}(q)$ in characteristic 0 is irreducible and there are no representations of dimension $(q^n+1)(q^{n-1}-q)/(q^2-1)-1$ \end{enumerate} \end{cor} \begin{proof} Note that part 2) follows immediately from Proposition \ref{l2} and the fact that by \cite{gt} there are no nontrivial representations of the linear group of degree smaller than $(q^{n-1}-1)/(q-1)-1$ if $r$ does not divide $q^{n-1}-1$. For 1), let $M$ be the reduction modulo $r$ of the $(q^n+1)(q^{n-1}-q)/(q^2-1)$ dimensional module in characteristic zero. If this is irreducible, then in the notation above $\dim C_V(U_0) =(q^{n-1}-1)/(q-1)-1$ and $C_M(\bar{P'_n})=0$. Since $r/q^{n-1}-1$, there are no irreducible $SL_{n-1}(q)$ modules of dimension $(q^{n-1}-1)/(q-1)-1$ by \cite{gt} so we get a contradiction. \end{proof} \begin{prop} \label{l3} If $G(q)=P\Omega _{2n+1}(q)$ with $n\geq 3$ , $q>3$ and odd then the lower bound of Theorem \ref{t1} holds. \end{prop} \begin{proof} Again let $P=JU$ be the stabilizer of a maximal totally singular subspace of the natural module. It follows that $J=GL_n(q)$ and $U$ will have two composition factors that are elementary $p$ groups, $U^{\prime }$ is the space of $n\times n$ antisymmetric matrices and $U/U^{\prime }$ is isomorphic to the natural $J$ module.As before $\bar{U}=U_0\times Z(\bar{G})$ As in the proof of Proposition \ref{l2} we can see that there is a factor in the decomposition of $V$ as a $U_0$ module on which $U_0^{\prime }$ acts non-trivially. Also the dimension of any nonlinear $U_0$ module is at least $q$. The conclusion is that the dimension of $[U_0, V]\;$ is $l=q(q^n-1)(q^{n-1}-1)/(q^2-1)$ or it is at least twice as large. By \cite{sz}, $\dim V\geq (q^{2n}-1)/(q^2-1)-n$. Assuming $\dim [U_0,V]=l$, it will follow that the dimension of $M=C_V(U_0)$ is at least $ (q^n-1)/(q-1)-n>q+1$. $C_{[U_0,V]}(\bar{J'})$ has dimension $q$ and so repeating the argument for the opposite parabolic subgroup $\widetilde{P}$ and making the same assumptions on the dimension of $[\widetilde{U_0},V]$ we get that if $\dim C_M(\bar{J'})>q$, then $C_V(\bar{P})\cap C_V(\bar{\widetilde{P}} )\neq \emptyset $ which leads to a contradiction. Therefore $\dim C_M(\bar{J'})\leq q$ and so $M$ has a nontrivial factor as a $\bar{J'}$ module. In particular this means that $\dim M\geq (q^{n-1}-1)/(q-1)-2$ and so $\dim V\geq (q^{2n}-1)/(q^2-1)-2$. \end{proof} For the rest of the theorem note lemma \ref{l0} gives that any indecomposable factor of $M$ has dimension $1, e(n,q)-2,e(n,q)-1,e(n,q)$ or $\geq 2e(n-1,q)-4$ hence the result follows from the following : \begin{prop} With the notation of Proposition \ref{l3},$$ \dim C_M(\bar{J'})= \left\{ \begin{array}{l} 0\ \mbox{if char} \ k \neq 2 \\ \leq 1\ \mbox{if char}\ k = 2 \end{array}\right. $$ \end{prop} \begin{proof} Once again in the notation of Proposition \ref{l3}, if $V_\alpha$ is a nonlinear irreducible factor of $V$ as a $U_0$ module then $\dim V_\alpha =q$. The idea is to estimate the dimension of $C_{[U,V]}(\bar{J'})$ because that will give a bound on the dimension of $C_M(\bar{J'})$. To do that we need to find the dimension of the fixed space of the normalizer of $V_\alpha$ in $\bar{J'}$. This group will have a factor isomorphic to $SL_2(q)$ acting non-trivially on $V_\alpha$. In fact using \cite{s} we see that in fact the $SL_2(q)$ module $V_\alpha$ has two nontrivial factors $\xi_1, \xi_2$ of dimensions $(q-1)/2$ and respectively $(q+1)/2$. The smallest one will remain irreducible in any characteristic. using the character table for $SL_2(q)$ (for example \cite{d}) one can see that $$ \chi_{\xi_1}(-I_2)= -\epsilon(q-1)/2$$ $$ \chi_{\xi_2}(-I_2)= \epsilon(q+1)/2$$ here $\epsilon=(-1)^{\frac{q-1}{2}}$ and $\chi_{\xi_i}$ is the character assoc to $\xi_i$. Therefore unless the characteristic of the base field is 2, $\chi_{\xi_1}+\chi_{1_G}\neq \chi_{\xi_2}$ and so the representations of dimension $(q+1)/2$ cannot factor. Therefore the normalizer of $V_\alpha$ does not fix any subspace if the characteristic of $k$ is odd and it could fix at most a one dimensional space in characteristic 2. \end{proof} Note that some choices for $(n,q)$ have been excluded from our treatment. It turns out that in these cases the corresponding bounds in \cite{sz} will be better. That is because our method uses bounds for $l(L_n(q))$ and for those choices of $(n,q)$ the corresponding linear groups have exceptional multipliers hence the bounds are much worse. \begin{thebibliography}{G-P-P-S} \bibitem[ModAt]{moat} C. Jansen, K. Lux, R. Parker, R. Wilson {\em An atlas of Brauer Characters} Oxford University Press 1995 \bibitem[C-R]{cr} C. W. Curtis, I. Reiner {\em Methods of Representation Theory}, Vol I-II , Willey-Interscience, 1994 \bibitem[D]{d} L Dornhoff {\em Group representation theory}, Marcel Dekker, 1971 \bibitem[GAP]{gap} M. Sch\"onert, (ed.), {\em Gap--$3.4$, Manual}, RWTH Aachen, 1994. \bibitem[GPPS]{gpps} R. Guralnick, T. Pentilla, C. Praeger and J. Saxl, {\em Linear groups with orders having certain primitive prime divisors,} Proc. London Math. Soc.(3), 78 (1999), no. 1, 167-214 \bibitem[GT]{gt} R. Guralnick and P. H. Tiep {\em Low-dimensional representations of special linear groups in cross characteristics}, Proc. London Math. Soc.(3), 78 (1999), no. 1, 116-138 \bibitem[H]{h} Corneliu Hoffman {\em Low dimensional projective representations for some exceptional finite groups of Lie type}, Preprint \bibitem[LS]{ls} V. Land\'azuri and G. M. Seitz {\em On the minimal degrees of projective representations of the finite Chevalley groups,} J. Algebra 32 (1974), 418-443 \bibitem[S]{s} G. M. Seitz, {\em Some representations of classical groups}, J. London Math. Soc. (2) 10 (1975), 115-120 \bibitem[SZ]{sz} G. M. Seitz and A. E. Zalesskii, {\em On the minimal degrees of projective representations of the finite Chevalley groups, II}, J. Algebra 158(1993),233-243 \bibitem[TZ]{tz} P. H. Tiep and A. Zalesskii, {\em Minimal characters of the finite classical groups,} Comm. Algebra 24 (1996), 2093-2167 \end{thebibliography} \end{document}