Harmonic analysis on vector bundles over \(\text{Sp}(1,n)/\text{Sp}(1)\times\text{Sp}(n)\)
A natural generalization of the harmonic analysis over Riemannian symmetric spaces is to consider (sections of) equivariant vector bundles over such spaces. Some methods of the function case carry over, others have to be modified, some reinvented. In the present paper the authors consider bundles over \(Sp(1,n)/Sp(1)\times Sp(n)\) given by irreducible representations of \(Sp(1)\). They use Godement's approach of trace spherical functions which they write in terms of Jacobi functions. They give an explicit inversion formula for the Abel transform, the Paley-Wiener theorem, and the Plancherel theorem for the \(\tau\)-spherical transform. The methods are rooted in the work of Godement and Harish-Chandra, but are kept explicit and down to earth here to make the work accessible to a broader audience.
scientific article; zbMATH DE number 1558743