Harmonic analysis for differential forms on complex hyperbolic spaces

Emmanuel Pedon (1)

1 IECN - Institut Élie Cartan de Nancy

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 Hodge--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 $H_t(e)$

Mots clés: Complex hyperbolic space; Differential forms; Plancherel theorem; Hodge-de Rham Laplacian spectrum; Spherical functions; Fourier transform; Heat kernel

Domaines: Théorie des représentations [math.RT]

Emmanuel Pedon. Harmonic analysis for differential forms on complex hyperbolic spaces. Journal of Geometry and Physics, 1999, 32 (2), pp.102-130. ⟨hal-00160424⟩