The Plancherel Formula for the Universal Covering Group of SL(2,R) Revisited

The Plancherel formula for the universal covering group of $SL(2,R)$ derived earlier by Pukanszky on which Herb and Wolf build their Plancherel theorem for general semisimple groups is reconsidered. It is shown that a set of unitarily equivalent representations is treated by these authors as distinct. Identification of this equivalence results in a Plancherel measure ($s\mathrm{R}\mathrm{e}\tanh \pi (s+\frac{i\tau }{2}),0\le \tau <1)$ which is different from the Pukanszky-Herb-Wolf measure ($s$\mathrm{R}\mathrm{e}$\tanh\pi(s+i\tau), 0\leq\tau<1)$.