We develop the L² harmonic analysis for (Dirac) spinors on the real hyperbolic space Hⁿ(ℝ) and give the analogue of the classical notions and results known for functions and differential forms: we investigate the Poisson transform, spherical function theory, spherical Fourier transform and Fourier transform. Very explicit expressions and statements are obtained by reduction to Jacobi analysis on L²(ℝ). As applications, we describe the exact spectrum of the Dirac operator, study the Abel transform...

Denote by $L^1(K\setminus G/K)$ the algebra of spherical integrable functions on $SU(1,1)$, with convolution as multiplication. This is a commutative semi-simple algebra, and we use its Gelfand transform to study the ideals in $L^1(K\setminus G/K)$. In particular, we are interested in conditions on an ideal that ensure that it is all of $L^1(K\setminus G/K)$, or that it is $L_0^1(K\setminus G/K)$. Spherical functions on $SU(1,1)$ are naturally represented as radial functions on the unit disk $D$ in the complex plane. Using this representation, these results are applied to characterize harmonic...