x PREFACE

and important topic of research, the representation theory of which parallels that

of the corresponding aﬃne Lie algebras. But the theory is much deeper and richer

than its classical counterpart, providing a clearer picture of connections with the

other areas mentioned above.

After the classiﬁcation of the ﬁnite simple groups (now complete), a full un-

derstanding of the representation theory of ﬁnite simple groups over ﬁelds k of

arbitrary characteristic provides a major problem for the 21st century. The spo-

radic Fischer-Griess monster (mentioned above) gives one important example of a

ﬁnite simple group closely related to Lie theory. Apart from the alternating groups

and the 26 sporadic simple groups, the ﬁnite simple groups come in inﬁnite families

closely related to the ﬁnite groups of rational points G(q) of simple algebraic groups

G over algebraically closed ﬁelds k of positive characteristic p 0. (The ﬁnite Ree

and Suzuki groups are variations on this theme.) The representation theory of these

ﬁnite groups of Lie type thus form a key area of investigation. One can consider a

ﬁeld F , algebraically closed for simplicity, having characteristic , and investigate

the category of FG(q)-representations. There are three cases to consider.

First, in case = 0, take F = C, the complex numbers. This theory is the

so-called ordinary representation theory of G(q). As a result of work of Deligne,

Lusztig, and many other mathematicians over the past 35 years, the ordinary theory

is quite well understood in comparison to the cases in which 0.

Second, if = p (the equal characteristic case), take F = k. By work of Stein-

berg, the irreducible kG(q)-modules all lift to irreducible rational representations

of the algebraic group G. This fact has provided strong motivation for the study of

the modular representation theory of the semisimple algebraic groups G over the

past 30 years. For example, a famous conjecture due to Lusztig posits the charac-

ters of the irreducible representations when the characteristic p is large (bigger than

the Coxeter number). For each type, this conjecture has been proved for p “large

enough” by Andersen-Jantzen-Soergel. The proof follows a path from characteristic

p to quantum groups at a root of unity to aﬃne Lie algebras and perverse sheaves.

Thus, it ultimately involves the inﬁnite dimensional Lie theory discussed above.

Although this approach fails to provide eﬀective bounds on the size of the prime p,

a new avenue via a related combinatorial category has been recently investigated

by Fiebig. As a result of Fiebig’s work, very large eﬀective bounds for Lusztig’s

conjecture are now known. In addition, the determination of the characters for

small p (i.e., less than the Coxeter number) remains largely uninvestigated.

Third, when 0 = p (the cross-characteristic case), much less is known

in general. When G is a general linear group GLn(k), the determination of the

decomposition numbers for the ﬁnite groups GLn(q) can be determined in terms

of decomposition numbers for q-Schur algebras and then for quantum groups over

ﬁelds of positive characteristic. This is the so-called Dipper-James theory. There are

close connections with the representation theory of Hecke algebras and symmetric

groups. In other types, much less is known; for example, the classiﬁcation of the

irreducible representations is incomplete. A major problem for these other types

would be to replace the quantum groups used for GLn(q) by some suitable structure.

The modular representation theory has provided a crucial interface with the

theory of ﬁnite dimensional algebras (especially, the theory of quasi-hereditary al-

gebras introduced by Cline, Parshall and Scott). It seems likely that this direction