Harmonic analysis for differential forms on complex hyperbolic spaces

We use representation theory for the semisimple Lie group G = SU (n, 1) to develop the L-2 harmonic analysis for differential forms on the complex hyperbolic space H-n (C). In this setting, most of the basic notions and results known for functions are generalized: the abstract Plancherel theorem, the spectrum of the Hedge-de Rham Laplacian, the spherical function theory, the spherical Fourier transform and the Fourier transform. In addition, we calculate explicitly the Plancherel measure and estimate the decay at infinity of the heat kernel H-t (e). (C) 1999 Elsevier Science B.V. All rights reserved. Subj. Class.: Differential geometry 1991 MSG: Primary: 22E30, 32M15, 43A85, 58A10, 58C40; Secondary: 22E46, 43A90, 58G05, 58G11.