Lie groups are continuous groups such as SL(2,R), the 2 by 2 matrices over R with determinant 1. A representation of a Lie group is a continuous linear action on a (possibly infinite dimensional) vector space. A representation on a Hilbert space is unitary if it preserves the inner product.
Unitary representations of Lie groups are ubiquitous in mathematics and science. They are basic objects in quantum mechanics, geometry, and harmonic analysis, and elsewhere. They play a key role in the proof of Fermat's Last Theorem.
The major problem in representation is that of the Unitary Dual: classifying the irreducible unitary representations of a Lie group. This is known in only a few cases.
It has been proposed to attack this problem by computer.
I these lectures I will discuss a very special case of this program. I will give a precise algorithm which may be done by computer and would give new results on the unitary dual.