The Hausdorff-Young theorem for the matricial groups Gnm = ax + b

Abstract

Let G nm = {ax + b} be the matricial group of a local field. The Hausdorff-Young theorem for G 11 was proved by Eymard-Terp [3] in 1978. We will establish here the HausdorffYoung theorem for G nm for all n m ∈ N.Introduction. The Hausdorff-Young theorem was generalized by several authors as they are passing over from a locally compact Abelian group to a locally compact group which is not Abelian but which is unimodular. In this paper we will tackle G nm = ax + b, the matricial group of a local field, to be defined below, which is not unimodular, and the theory will involve unbounded operators, as the case was before if m = n = 1.Let K be a local field, n, m ∈ N, m n 1. Let M nm be the space of all n × m-matrices with elements from K and GL n be the multiplicative group of all n × n-invertible matrices with elements from K. G nm denotes the group of pairs (b, a), where b ∈ M nm and a ∈ GL n , with multiplication given by (b, a)(b , a ) = (b + ab , aa ). It is the semi-direct product M nm GL n .Let H be the Hilbert space L 2 (GL n , du | det(u)| n ). For all λ in M mn , the formula [π λ (b, a)ξ ](u) = τ (T r(bλu))ξ(ua),