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\title{On duality-inducing automorphisms and sources of simple
modules in finite classical groups}
\author{Radha Kessar}
\date{February 7, 2008}
\begin{document}
\maketitle
\section{Vertices and sources}
Let $p$ be a prime number,
$k = \ov\FM_p$,
$G$ a finite group,
and $V$ an indecomposable (left) $kG$-module.
A \emph{vertex-source pair} of $V$ is a pair
$(P,W)$, where $P$, is a $p$-subgroup of $G$,
$W$ an indecomposable $kP$-module such that
$V \mid \Ind_P^G W$ and such that $P$ is minimal
for this property. Then $P$ is called a vertex,
and $W$ is called a source.
Let us recall that $V$ is projective if and only
if $(1,k)$ is a vertex-source pair of $V$.
Vertex-source pairs of $V$ form a complete
$G$-conjugacy class.
$(S,k)$ is a vertex-source pair of $V = k$ where
$S \in \Syl_p(G)$.
Question: what are the vertex-source pairs of
simple $kG$-modules, for $G$ a finite group ?
\begin{conj}[Feit 1979, block version] Let
$P$ be a finite $p$-group. Up to isomorphism
there exist at most finitely many pairs
$(Q,W)$ which are vertex-source pairs of simple
modules of finite groups lying in blocks with
defect group isomorphic to $P$.
\end{conj}
Actually, this is a weaker form of the original conjecture.
The conjecture is trivial in the case where $P$ is cyclic,
and open in all other cases\ldots
\section{Endo-permutation modules}
Let $P$ be a $p$-group.
Let $W$ be an (indecomposable) $kP$-module.
Then $W^* := \Hom_k(W,k)$ is again a left
$kP$-module: for $x\in P$, $f\in W^µ$, $w\in W$,
we set $xf(w) = f(x^{-1}w)$. Thus
$\End_k(W) = W \otimes_k W^*$ is a left $kP$-module
via the diagonal action of $P$.
We say that $W$ is an endo-permutation $kP$-module
if $W \otimes_k W^*$ is a permutation module,
i.e. there is a $k$-basis of $W\otimes_k W^*$
stable under $P$. If moreover $(P,W)$ is a
vertex-source pair for $W$, we say that
$W$ is a capped endo-permutation $kP$-module.
\begin{example}
The trivial module $k$ is a capped endo-permutation
$kP$-module.
\end{example}
\begin{example}
\[
0 \longto \O(k) \longto kP \longto k \longto 0
\]
$\O(k)$ is a \cep $kP$-module.
\end{example}
If $W_1$ and $W_2$ are two \cep $kP$-modules,
then up to isomorphism there is a unique \cep
$kP$-module $S_{W_1,W_2}$ such that $S_{W_1,W_2}$
appears as a summand of $W_1\otimes_k W_2$.
\begin{theo}[Dade 1978]
Let $D_k(P)$ be the set of isomorphism classes
$[W]$ of \cep $kP$-modules.
For $[W_1]$ and $[W_2]$ in $D_k(P)$, set
$[W_1] + [W_2] = [S_{W_1,W_2}]$.
Then $(D_k(P),+)$ is an abelian group, with identity
element $k$, and $-[W] = W^*$.
\end{theo}
\begin{theo}[Puig 1990]
$D_k(P)$ is a finitely generated group.
\end{theo}
In 2007, the complete elucidation of the structure
of $D_k(P)$ was achieved, thanks to the work of
Alperin, Bouc, Carlson, Dade, Mazza, Puig and
Thévenaz.
\begin{example}
\label{ex:rank 2}
Let $P = C_p \times C_p$. Then we have
\[
D_k(P) =
\begin{cases}
\ZM & p = 2\\
\ZM \oplus (\ZM/2\ZM)^{p + 1}
\end{cases}
\]
and $[\O(k)]$ is a generator for
$D_k(P)/D_k^\text{tor}(P)$
\end{example}
\section{Endo-permutation modules as sources}
If $G$ is a $p$-solvable group, then the vertex-source
pairs of every simple $kG$-module are of the form
$(Q,W)$, $[W] \in D_k(G)$.
If $P \simeq C_2 \times C_2$ and $G$ has a
$2$-block $B$ with defect group $P$, then
$B$ has a simple module with vertex-source pair
$(P,W)$, $[W] \in D_k(P)$.
Let $N \trianglelefteq G$ such that $G/N$ is
a $p$-group, $V$ is a simple $kG$-module
such that $\Res^G_N V$ is a simple projective
$kN$-module. Then $V$ has vertex-source pair
$(P,W)$, $[W] \in D_k(P)$.
Question: Given $P$, identify the elements
$[W] \in D_k(P)$ such that
$(P,W)$ is a vertex-source pair of some simple
$kG$-module for some finite group $G$.
\begin{theo}[Burger-Feit, Puig, Mazza]
If $G$ is $p$-solvable and $(P,W)$ is a
vertex-source pair of simple $kG$-modules,
then $[W] \in D_k^\text{tor}(P)$.
\end{theo}
The proof depends on the classification of the
finite simple groups (some knowledge of the
automorphism groups of finite simple groups).
\section{Classical groups}
\begin{theo}[Kessar]
\label{th:K1}
Let $P = C_p \times C_p$, $[W] \in D_k(P)$.
Let $L = GL_n(q)$, $GU_n(q)$, $O^\pm_n(q)$,
$Sp_{2n}(q)$, where $q$ is some prime power,
or the symmetric group $S_n$.
Let $G$ be a group such that
$[L,L] \subset G \subset L$. Let
$\ov G = G/Z$ where $Z \leqslant Z(G)$.
Suppose that $k\ov G$ has a simple module
$V$ with $(P,W)$ as vertex-source pair.
Then $[W] \in D_k^\text{tor}(P)$.
\end{theo}
\begin{theo}[Kessar]
\label{th:K2}
Let $p$ be odd. Let $P = C_p \times C_p$,
$[W] \in D_k(P)$, $q$ a prime power not dividing
$q$. Suppose
\[
1 \longto N \longto G \longto P \longto 1
\]
is such that $N \simeq SL_n(q)/Z$,
$SU_n(q)/Z$. Suppose $V$ is a simple
$kG$-module such that
$\Res^G_N V$ is simple and projective.
Then if $(P,W)$ is a vertex-source pair of $V$,
then $[W] \in D_k^\text{tor}(P)$.
\end{theo}
\subsection*{Intended applications}
(i) We have the following theorem:
\begin{theo}[Eaton-Kessar-Linckelmann]
\label{th:EKL}
Let $P \simeq C_2 \times C_2$, then the
Block-Feit conjecture is true is it is true for
all groups $G$ of the form $G = NS$, where
$N\trianglelefteq G$ and $S$ is of order $1$, $2$
or $4$ and $N$ is a quasi-simple group.
\end{theo}
Combining Theorem \ref{th:EKL} with Theorem \ref{th:K1}
gives:
\begin{cor}
In order to check the block version of the Feit conjceture
for $P = C_2 \times C_2$, we nedd only check it for groups of the form
$G = NS$, where $N$ is quasi-simple and either $S$ is of order $2$ or $4$,
or $S$ is trivial and $N$ is a quasi-simple group not of classical type.
\end{cor}
\noindent (ii) Combining the Theorem \ref{th:K2} with the reduction
theorem of Adam Salminen gives:
\begin{cor}
Let $p\geqslant 5$, $G$ a finite group,
$N \trianglelefteq G$ with
$[G:N] = p^a$. If $G$ has a simple module
$V$ such that $\Res^G_N V$ is simple and projective
and $(Q,W)$ is a vertex-source pair of $V$,
then $[W] \in D_k^\text{tor}(P)$.
\end{cor}
\bigskip
\subsection*{Proofs}
To prove the theorems above, the following notion turned out to be crucial.
\begin{defi}
Let $H$ be a finite group, $M$ a $kH$-module.
Then $M$ is automorphically dual if there is an
automorphism $\phi:H\to H$ such that
$\Res_\phi M \simeq M^*$.
\end{defi}
Let us now give a sketch of the proof of Theorem \ref{th:K1}.
\begin{proof}
Let $G$, $V$, $P$, $W$ be as in the theorem.
The proof goes through the following steps.
First, we show that $V$ is automorphically dual
(this relies on some linear algebra related to
the classical groups).
Secondly, we make the easy observation that
the sources of automorphically dual modules
are automorphically dual.
Thirdly, from the structure of the Dade group
when $P\simeq C_p\times C_p$ (see Example \ref{ex:rank 2}),
a module $[W] \in D_k(P)$ is automorphically
dual if and only if it is self-dual
($W \simeq W^*$).
\end{proof}
For Theorem \ref{th:K2}, one cannot show that $V$ is automorphically
dual. We have to replace $(G,V)$ by a pair $(G',V')$ such that the
corresponding blocks are \emph{source algebra equivalent} (we will not
define this notion), and such that $V'$ is automorphically dual. The
proof is quite technical: for example, we need to use the description
of blocks and characters given by Deligne-Lusztig theory.
\end{document}