PREFACE xi

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

“categoriﬁcation”.

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