Characters of algebra groups and unitriangular groups

In this work we study the irreducible complex characters of finite algebra groups, with a special interest in the groups of unitriangular matrices over finite fields. By an algebra group we mean a group of the form 1 + J, where J is the Jacobson radical of a finite associative algebra over a commutative ring R and the additive group of J is a p-group. We start with an exposition of Kirillov's method adapted to our setting, which yields a description of the irreducible characters of the algebra group 1 + J when J(p) = 0. Although this method can be applied immediately to the unitriagular groups when the size of the matrices does not exceed the characteristic of the field, it fails in general to describe all the irreducible characters. However, we shall show that the method is still useful to describe some of the irreducible characters, namely, those whose degree is small or large enough.