Pompeiu sets in compact Heisenberg manifolds (Q1976487)

Let \(G\) be any unimodular locally compact group and \(\Gamma\) be a discrete subgroup such that \(\Gamma\setminus G= \{\Gamma g\mid g\in G\}\) is compact. A subset \(E\) of \(\Gamma\setminus G\) is called a Pompeiu set (or a \(P\)-set) provided that, for every \(f\in L^2(\Gamma\setminus G)\), \(\int_{Eg}f= 0\) for all \(g\in G\) implies that \(f=0\). In this paper, a necessary and sufficient condition is presented for a set to be a Pompeiu subset which is expressed in terms of the intertwining operators of each primary summand of the quasiregular representation. The condition to be a Pompeiu set is then interpreted in the case of the compact Heisenberg manifolds. Examples are presented demonstrating that the condition to be a Pompeiu set in these manifolds is quite different from the corresponding condition for a torus of the same dimension. This provides a contrast with the existing comparison between the Heisenberg group itself and the Euclidean space in terms of Pompeiu sets.