will continue to prove fruitful. Another significant feature of the modular repre-
sentation theory of the finite groups of Lie type and the associated algebraic and
quantum groups is the existence of a rich accompanying homological theory. Homo-
logical problems emerge immediately because of the failure of complete reducibility.
In the equal characteristic case, the homological theory has been extensively de-
veloped, for the finite groups of Lie type, quantum enveloping algebras at roots of
unity, restricted Lie algebras and infinitesimal group schemes, as well as other set-
tings. Geometric ideas enter via the theory of support varieties, which associate to
each finite-dimensional module for a restricted Lie algebra (or finite group scheme)
an algebraic variety. In the cross-characteristic case, much less in known about the
cohomology. In the equal characteristic case, there is a considerable body of work
involving the homological algebra of the infinitesimal groups, and relations between
the cohomology of G, its infinitesimal subgroups, and its finite subgroups.
Finally, we mention that the modular representation theory of general finite
groups itself has a strong Lie-theoretic flavor. In part, this is due to the famous
Alperin conjecture, suggesting that the irreducible modular representations of gen-
eral finite group should be classified in a “weight theoretic” way, much like irre-
ducible modules for a complex semisimple Lie algebra are classified by their highest
weights. Another notable conjecture, the Brou´ e conjecture, has been recently ver-
ified for symmetric groups by Chuang and Rouquier using a the new method of
In 2009, the three editors established a network of Lie theorists in the south-
eastern region of the U.S. and proposed an annual regional workshop series of 3 to
4 days in Lie theory. The aim of these workshops was to bring together senior and
junior researchers as well as graduate students to build and foster cohesive research
groups in the region. With support from the National Science Foundation and the
affiliated universities in the region, three successful workshops were held at North
Carolina State University, the University of Georgia and the University of Virginia
in 2009, 2010 and 2011 respectively. Each of these workshops was attended by over
70 participants. The workshops included expository talks by senior researchers and
afternoon AIM style discussion sessions with a goal to educate graduate students
and junior researchers in the early part of their study for research in different as-
pects of Lie theory. In the third workshop at the University of Virginia, Professor
Leonard Scott was honored on the eve of his retirement for his lifetime contributions
to many of the aforementioned topics.
The plenary speakers in the three workshops were invited to contribute to this
proceedings. Most of the articles presented in this book are self-contained, and
several survey articles, by Jon Carlson, Jie Du, Bob Griess, and David Hemmer are
accessible to a wide audience of readers.
The editors take this opportunity to acknowledge the conference participants,
the contributors, and the editorial offices of the American Mathematical Society
for making this volume possible.
Kailash C. Misra
Daniel K. Nakano
Brian J. Parshall
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