  The general linear group GL(n, K) over a field K contains a particularly
prominent subgroup U(n, K), consisting of all the upper triangular unipotent
elements. In this paper we are interested in the case when K is the finite
field F_q, and our goal is to better understand the representation theory of
U(n, F_q). The complete classification of the complex irreducible
representations of this group has long been known to be a difficult task. The
orbit method of Kirillov, famous for its success when K has characteristic 0,
is a natural source of intuition and conjectures, but in our case the relation
between coadjoint orbits and complex representations is still a mystery. Here
we introduce a natural variant of the orbit method, in which the central role
is played by certain clusters of coadjoint orbits. This "method of clusters"
leads to the construction of a subring in the representation ring of U(n, F_q)
that is rich in structure but pleasantly comprehensible. The cluster method
also has many of the major features one would expect from the philosophy of
orbit method.
