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

TL;DR

This paper revisits the Plancherel formula for the universal cover of SL(2,R), clarifying representation equivalences and deriving a corrected Plancherel measure differing from previous results.

Contribution

It identifies unitarily equivalent representations previously treated as distinct, leading to a revised Plancherel measure for the universal covering group of SL(2,R).

Findings

Corrected the set of unitarily equivalent representations.

Derived a new Plancherel measure different from prior formulas.

Clarified the relationship between different formulations of the Plancherel formula.

Abstract

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 ($s\mathrm{Re}\tanh\pi(s+i\tau), 0\leq\tau<1)$.