Abstract Harmonic Analysis and Group Algebra on the Group AN

Consider the Iwasawa decomposition of the real semisimple Lie group. The purpose of this paper is to define the Fourier transform in order to obtain the Plancherel theorem on its maxima solvable Lie group. Besides, we prove the existence theorems for the invariant differential operators on this solvable. To this end, we give a classification of all left ideals of its group algebra