The Pompeiu problem and spherical spectral analysis

Let $K$ be a compact subgroup of a locally compact group $G$. We investigate a Pompeiu type problem for homogeneous spaces $G\mathrm{/}K$. Suppose $E$ is a compact subset of $G\mathrm{/}K$. Using recent work of László Székelyhidi on $K$-spectral analysis \cite{Szekelyhidi17} we are able to give necessary and sufficient conditions for $E$ to have the Pompeiu property when $(G,K)$ is a Gelfand pair.