This article surveys the use of harmonic analysis in the study of rigidity properties of discrete subgroups of Lie groups, actions of such on manifolds and related phenomena in geometry and dynamics. Let me call this circle of ideas rigidity theory for short. Harmonic analysis has most often come into play in the guise of the representation theory of a group, such as the automorphism group of a system. I will concentrate on this avenue in this survey. There are certainly other ways in which harmonic analysis enters the subject. For example, harmonic functions on manifolds of nonpositive curvature, the harmonic measures on boundaries of such spaces and the theory of harmonic maps play an important role in rigidity theory. I will mention some of these developments.

Rigidity theory became established as an important field of research during the last three decades. The first rigidity results date back to about 1960 when A. Selberg, E. Calabi and A. Vesentini and later A. Weil discovered various deformation, infinitesimal and perturbation rigidity theorems for certain discrete subgroups of Lie groups. At about the same time, M. Berger proved his purely geometric 1/4-pinching rigidity theorem for posi- tively curved manifolds [184, 28, 27, 205, 206]. But the most important and influential early result was achieved by G. D. Mostow in 1968. In proving his celebrated Strong Rigidity Theorem, Mostow not only provided a global version of the earlier local results, but also introduced a battery of novel ideas and tools from topology, differential and conformal geometry, group theory, ergodic theory, and harmonic analysis. Mostow's results were the catalyst for a host of diverse developments in the ensuing years.