Wolfgang Willems and Alexander Zimmermann,

On Morita theory for self-dual modules

To appear in Quarterly Journal of Maths. 


\begin{abstract}
Let $G$ be a finite group and let $k$ be a field of characteristic
$p$. It is known that a $kG$-module $V$ carries a non-degenerate $G$-invariant
bilinear form $b$ if and only if $V$ is self-dual. We show that whenever
a Morita bimodule $M$ which induces an equivalence
between two blocks $B(kG)$ and $B(kH)$
of group algebras $kG$ and $kH$ is self-dual then
the correspondence preserves self-duality. Even more, if the bilinear form on
$M$ is symmetric then for $p$ odd the correspondence
preserves the geometric type of simple modules. In characteristic $2$  
this holds also true
for projective modules.
\end{abstract}