Harmonic analysis for spinors on real hyperbolic spaces (Q2717698)

The \(L^2\) harmonic analysis for spinors on the real hyperbolic space \(H^n(R)\) is developed. The notions introduced and the properties investigated in the paper are: the Poisson transform, the spherical function theory, the spherical Fourier transform and the Fourier transform. It is shown that the spherical Fourier analysis of radial functions on the spinor bundle \(\Sigma H^n(R)\) can be reduced to the Jacobi analysis of functions defined on \(R\). The basic structure of the real hyperbolic spaces \(H^n(R)\) is recalled. Then, an additional structure for the semisimple Lie group \(G\) \(=\) \(Spin_e(n,1)\) is introduced and its representation theory is used to state the Plancherel theorem for spinors on \(H^n(R)\). The eigenfunctions (eigenspinors) for the generators of the algebra of differential operators acting on smooth sections of \(\Sigma H^n(R)\) are constructed and the results are used to obtain the \(L^2\) spectrum of the Dirac operator. All the statements and properties are established by reduction to the Jacobi analysis on \(L^2(R)\). The Abel transform is studied and explicit expressions for the heat kernel associated with the spinor Laplacian are derived. The results of the paper are compared with those of other authors, especially of \textit{U. Bunke} [Math. Nachr. 153, 179-190 (1991; Zbl 0803.58056)]. An important error contained in the Bunke's paper in the case of \(n\) even is corrected.