  Let G be the three dimensional connected real semisimple Lie group and let
KAN be the Iwasawa decomposition of G.Let J be the Jacobi group, which is the
semidirect product of the two groups Heisenberg group with G. The Jacobi group
plays an important role in Quantum Mechanics. The purpose of this paper is to
define the Fourier transform in order to obtain the Plancherel theorem for the
group J. To this end a classification of all left ideals of the group algebra
for the Jacobi group
