\title{Generalized Blocks of Unipotent Characters in the Finite General
Linear Group}
\author{Jean-Baptiste Gramain \\ \\Department of Mathematical Sciences\\
University of Aberdeen, Scotland}
\date{August, 2006}
\maketitle
\begin{abstract}
In a paper of 2003, B. K\"ulshammer, J. B. Olsson and G. R. Robinson
defined $\ell$-blocks for the symmetric groups, where $\ell >1$ is
an arbitrary integer, and proved that they satisfy an analogue of
the Nakayama Conjecture. Inspired by this work and the definitions
of generalized blocks and sections given by the authors, we give in
this paper a definition of $d$-sections in the finite general linear
group, and construct $d$-blocks of unipotent characters, where $d
\geq 1$ is an arbitrary integer. We prove that they satisfy one
direction of an analogue of the Nakayama Conjecture, and, in some
cases, prove the other direction. We also prove that they satisfy an
analogue of Brauer's Second Main Theorem.
\end{abstract}