Correction to “Harish-Chandra’s Plancherel theorem for $\mathfrak {p}$-adic groups”

Abstract

K. Hiraga [1] has informed the author that the proof of Theorem 5 given in [2] is incorrect, and he has also supplied a correct proof for this result. In what follows the author will present, with insignificant modifications, the example and the argument of Hiraga's letter. First, as Hiraga points out, the numbers C(n) of [2, 3.28] are not in general finite, as the following example makes clear. We employ the notation of [2]. Let 7T denote a prime element of the p-field Q. For the moment take G = GL3, P the upper triangular subgroup, and N the lower triangular unipotent subgroup of G. The set F of [2, ?2], a fundamental set for the Cartan decomposition of G /Z7 consists of the set of matrices El-#2 ~0 0 m=O IT-F2 0 [ 0 0 1