On the Plancherel Formulas for Some Types of Simple Lie Groups

The problem of finding the explicit Plancherel formulas for semisimple Lie groups has been solved completely in the case of complex semisimple Lie groups (see [3 (b)]). Moreover Harish-Chandra showed [3(f)] that the problem is solved also for a real semisimple Lie group having only one conjugate class of Cartan subgroups. In the case of real semisimple Lie groups with several conjugate classes of Cartan subgroups, the problem is very difficult to attack. As far as the auther knows, the problem was taken up and solved for SL(2y R) by V. Bargman, [1], Harish-Chandra [3 (a)], R. Takahashi [9 (a)] and L. Pukanszky [7] also for the universal covering group of SL(2, R) by L. Pukanszky. In the previous note [6], we gave a method of finding the Plancherel formula for the universal covering group of De Sitter group. The purpose of this paper is to generalize this method and to obtain the explicit Plancherel formulas for simple Lie groups G which satisfy the following conditions (A. 1)~(A. 5).