Spherical harmonic analysis for spinors on \({\mathbf H}^n(\mathbb{C})\)
In this article \textit{R. Camporesi} reviews his recent results on harmonic analysis for spinors on the complex hyperbolic space \(H^n({\mathbb{C}})\) obtained in [Adv. Math. 154, 367-442 (2000; Zbl 0978.43004)]. The general theory of \(\tau\)-spherical harmonic analysis on homogeneous vector bundles on noncompact Riemannian symmetric spaces \(G/K\) is well understood, however, the concrete and explicit formulas are difficult, for example, the explicit form of the Arthur-Campoli condition in the Paley-Wiener theorem. Let \(G=SU(n,1)\) and \(G=KAK\) be the Cartan decomposition of \(G\). Let \(M\) be the centralizer of \(A\) in \(K\). For \((\tau, V)\in\hat K\) the author considers \(\tau\)-radical functions \(f\) on \(G\) satisfying \(f(k_1xk_2)=\tau(k_1)f(x)\tau(k_2)\), \(k_1,k_2\in K, x\in G\). In this case \(f\) is completely determined by its restriction on \(A\) and \(f(a)\), \(a\in A\), is a scalar valued function \(f_\sigma\) on each \(M\)-isotypic subspace \(V_\sigma\) of \(V\). Hence, for each scalar component \(f_\sigma\) the spherical transform coincides with the scalar spherical function of rank one, that is, the Jacobi transform. The author reduces harmonic analysis on \(\tau\)-radical functions to the one on Jacobi functions, and obtains the inversion formula, the Plancherel formula and the Paley-Wiener theorem.
scientific article; zbMATH DE number 1879581