I guess U(n,F_q) - upper triangular matrices over F_q.
I am not an expert in the field, but was recently interested in similar question, so I'll put these remarks.
There is certain amount of quite recent works related to representations of U(n,F_q) and some "big names" involved. There are certain conjectures which are easy to state, but hard to prove, and some more conceptual problems and recent breakthroughs.
One of the origins of the modern interest are papers by A.A. Kirillov (1995-2005),
e.g. Variation on a triangular theme, Two more variations on a triangular theme,
where he considered a question whether the "orbit method" can be extended to finite Lie groups such as U(n,F_q). (Originally (60-ies) A.A. Kirillov proposed "orbit method" for Lie groups over R, and U(n,R) where the first groups where he demonstrated its work).
Kirillov's papers are always pleasure to read, he puts the accent not on technical details, but on new ideas, problems and insighting observations.
The moral that in certain cases "orbit method" can be applied, but there some problems,
which are deserved to be further studied to achieve further progress.
See e.g. Vipul Naik's site, where one can find lots of interesting and understandable
information, worked out examples and further references: orbit method for U(3,p), orbit method for finite Lazard groups.
One of the major breakthroughs in the subject is an idea of "supercharacters".
Let me quote from E. Arias-Castro P. Diaconis R. Stanley
The character theory of U(n,F_q) is a well known nightmare.
In recent work, Carlos Andre, Roger
Carter and Ning Yan have developed a
theory based on certain unions of
conjugacy classes (here called
super-classes) and sums of irreducible
characters (here called
The main point is that "classification of characters" is to certain extent "wild" problem, while "supercharacters" are quite manageable to classify.
See more comments in MO question- "Irreducible representations of the unitriangular group" .
Let me also quote from F. Ladisch answer on MO question - "Representation theory of p-groups in particular upper triangular matrices over F_p"
There are, by now, many papers about
the character theory of the upper
triangular group and related topics,
which is in part motivated by Higman's
conjecture that for every n, the
number of conjugacy classes of Un(Fq)
is a polynomial in q with integer
Let me also mention "23 page article with 28 authors :)" which is devoted to the subject: Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras