Harmonic Analysis on the Heisenberg nilpotent Lie group.

The real Heisenberg group A(R) is :a connected and simply_connected, two-step nilpotent, analytic group having 0·1e-dimensional centre C. Therefore A(R) fonns the simplest possible non-contnutative, non-compact Lie group. The name and the quantum mechanical meaning of the real Heisenberg nilpotent Lie group ... - A(R) stem from the fact that the Lie algebra n of A(R )over R is defined by the Heisenberg canonical co1T111uta ti ·:>n re 1 at ions. Thus , according to the philosophy of Niels Bohr, the geometric intuition necessarily fails to describe the action of A(R). It i '; the purpose of these notes to study nilpotent harmonic anal-ysis in a unified manner and specifically to determine the unitary dual of A(R) by an application of the Mackey machinery as well as by the Kirillov orbit picture.