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 ($s\mathrm{Re}\tanh\pi(s+\frac{i\tau}{2}), 0\leq\tau<1)$ which is different from the Pukanszky-Herb-Wolf measure ($s\mathrm{Re}\tanh\pi(s+i\tau), 0\leq\tau<1)$.