\title[Consequences of a Perfect Isometry]{Extended Conditions for the Consequences of a Perfect Isometry
}
\author{Morton E.~Harris}

\address{Department of Mathematics, Statistics,
and Computer Science (M/C 249),
University of Illinois at Chicago,
851 South Morgan Street, 
Chicago, IL 60607-7045}


\begin{abstract} 
In an important paper \cite{HI1},
M.~Brou\'e defines the fundamental
concept of a perfect isometry between
two $p$-blocks $f$ and~$e$ of finite groups
$H$ and $G$ that posits numerical conditions
on a generalized character of $G\times H$.
Then, in one of the fundamental results
of~\cite{HI1}, he shows, in \cite[Th\'eor\`eme~1.5]{HI1}, that
such a perfect isometry implies
several equivalences between basic
invariants of the blocks $f$ and $e$. In this
paper, we extend the conditions for 
a perfect isometry (that includes
Brou\'e's definition) and demonstrate that
all of the conclusions of \cite[Th\'eor\`eme~1.5]{HI1}
hold in our more general setting. We
seek applications of these results.
\end{abstract}