A major trend in non-commutative harmonic analysis is to investigate function spaces related to Fourier analysis (and representation theory) of non-abelian groups.The Fourier algebra and the Fourier-Steiltjes algebra, which are associated with the regular representation and the universal representation of the ambient group respectively, are important examples of such function spaces. These function algebras encode the properties of the group in various ways; for instance the non-existence of derivations on such algebras indicates their lack of analytic properties, which in turn translates into forms of either commutativity or discreteness for the group itself.

In this talk, we present explicit constructions of continuous derivations on the Fourier algebras of two important matrix groups, namely the group of ${\mathbb R}$-affine transformations and the Heisenberg group. We then use techniques of non-commutative harmonic analysis, and in particular the theory of square-integrable representations and the fusion rules for tensor products of Schr\"{o}dinger representations, to prove that such derivations are indeed continuous.

Using the structure theory of Lie groups, we extend our results to semisimple Lie groups and nilpotent Lie groups. If time permits, we will discuss weighted versions of the Fourier algebra, called the Beurling-Fourier algebra, and some of their properties.

This talk is mainly based on joint articles with Y. Choi.