The Plancherel Formula

Abstract

We shall put together the facts we have learned about traces in order to prove the Plancherel formula, giving an expansion of a function in terms of its characters. The proof is due to Harish-Chandra [H-C 6]. It consists in expanding out the Fourier series of the Harish transform H ^Kψ(θ) using the relation between the trace of the discrete series and the Harish transform on A given in Theorem 6 of the preceding chapter, and then performing a Fourier transform on some of the terms to get the final formula. It turns out that H ^Kψ(k_θ) is not continuous, the discontinuity occurring at those elements of k which also lie in A, i.e. at ± 1. The first calculus lemma serves to determine the jumps, and also shows that the derivative (H ^Kψ)’(k_θ) is continuous at those points, and that the Fourier series converges for the derivative at those points. This gives us the value ψ(1), in terms of a series involving traces in the discrete and principal series.