The Plancherel Formula for the Universal Covering Group of SL(2,R) Revisited
Abstract
Description
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 ($s\mathrm{Re}\tanhπ(s+\frac{iτ}{2}), 0\leqτ<1)$ which is different from the Pukanszky-Herb-Wolf measure ($s\mathrm{Re}\tanhπ(s+iτ), 0\leqτ<1)$.
Since a substitution changes f(z) to a multi-valued function we will remove the argument given in the introduction which was never used in the text. Everything comes out naturally from the integral kernel of the group ring
Since a substitution changes f(z) to a multi-valued function we will remove the argument given in the introduction which was never used in the text. Everything comes out naturally from the integral kernel of the group ring