Schwartz space for the hypergeometric Fourier transform. (With an appendix by Mustapha Tinfou)
The author determines the hypergeometric Fourier transform of the related Schwartz space of functions invariant under the Weyl group \((W\)-invariant functions) on a Euclidean vector space. The result is quite similar to that of Harish-Chandra. It is shown that the simple difference operators play the role of some elements of the unipotent radical of a parabolic subalgebra. This requires a theory of constant term for \(W\)-invariant functions satisfying the hypergeometric system. It is emphasized that the parameter of cuspidal hypergeometric functions can be singular and in such cases new difficulties appear. Finally, it is suggested that the techniques used for real reductive symmetric spaces can be applied to study truncation and its consequences.
scientific article; zbMATH DE number 1374986
scientific article