Poisson-Plancherel Formula for a Real Semialgebraic Group .1. Fourier-Transform of Orbital Integrals

AbstractIn this part of our work, a (real) semialgebraic group G being given, first we associate, to any linear form on the Lie algebra of G, a so-called canonical acceptable subgroup and we study the conjugacy classes of the reductive factors of these subgroups. Then we define orbital integrals on the Lie algebra of G, generalizing those introduced by Harish-Chandra [Han] and Duflo [Du4], and we express in a rather simple way their Fourier transforms in terms of the Fourier transforms of Hanish-Chandra′s orbital integrals of some reductive factors of the generic canonical acceptable subgroups of G