Department of Mathematics
503 Boston Avenue
Medford, MA 02155
The structure and representations of algebraic groups
Mathematical problems often lead to the study of collections
of symmetries. The resulting axiomatic notion is that of a group,
and groups have come to play a fundamental role in algebra and
in many other parts of mathematics.
In differential geometry -- and in physical sciences such as
physics and chemistry -- one is often interested in Lie groups;
a Lie group is simultaneously an algebraic object -- a collection
of symmetries -- and a geometric object -- a smooth manifold.
The compact, connected Lie groups admit a classification; more
precisely, each simple, compact, connected Lie group appears in
one of a handful of families described by some "book-keeping data"
(their root system or Dynkin diagram).
I study the analogues of Lie groups arising in algebraic geometry
known as linear algebraic groups. Like Lie groups, these algebraic
groups are again simultaneously algebraic and geometric; they arise
as certain subgroups of the group of all n × n invertible matrices
for varying n, but the coefficients of the matrices may now be any
field of scalars -- one gets Lie groups when the scalars are the real
or complex numbers. Remarkably, over any coefficient field, the reductive
algebraic groups are described by essentially the same "book-keeping"
as the compact, connected Lie groups. Even more remarkably, a great
achievement of 20th century mathematics shows that the simple algebraic
groups over finite fields account for "most" finite simple groups.
The linear algebraic groups I study may all be viewed as collections
of linear symmetries of a vector space; a linear representation --
or a module -- is (roughly speaking) a choice of such a vector space.
Representations thus provide a practical tool -- linear algebra
-- for studying the abstract, algebraic notion of a group. Moreover,
in many applications in mathematics and science, one desires a description
of the space of functions on a group. In this regard, representation theory
may be viewed as a generalization of Fourier analysis. Representations
play a large role in geometry through invariant theory, and they are
significant in number theory e.g. due to their relationship with automorphic forms.
I study the structure and representations of linear algebraic groups,
and most of the questions he considers involve the "modular setting"
where the coefficient field contains some finite field - thus, the
characteristic of the coefficient field is a prime number p>0.
(In contrast, the fields of real or complex numbers have characteristic 0).
In this modular setting, my work has focused mainly on linear
representations and complete reducibility, on conjugacy classes,
and on Levi decompositions and related cohomological issues.