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
Previous Page Next Page