The functional equation of zeta distributions associated with formally real Jordan algebras (Q1059099)

Let V be a formally real simple Jordan algebra over \({\mathbb{R}}\) and let N be the reduced norm of V. G denotes the structure group of V, \(G^ 0\) is the connected component of the identity of G. The set of real invertible elements from V is decomposed into the union of \(G^ 0\)-orbits \(\prod_{i}\Omega_ i\). The zeta distribution on V is a map \[ f\to \Phi_ i(f,s)=\int_{\Omega_ i}f(u) | N(u)|^ s du \] where f is from the Schwartz space of V. The authors give some explicit expression for the Fourier transform of the zeta distributions on a certain class of prehomogeneous spaces defined by Jordan algebras. The method is based on the theory of Jordan algebras [see \textit{H. Braun} and \textit{M. Koecher}, Jordan-Algebren (1966; Zbl 0145.260); \textit{M. Sato} and \textit{T. Shintani}, Ann. Math., II. Ser. 100, 131-170 (1974; Zbl 0309.10014)].