Let $N$ be a connected, simply connected, nilpotent Lie group and let $K$ be a compact subgroup of the automorphism group of $N$. We study the density of Schwartz functions in the kernels of $K$-orbits and characterize $K$-prime ideals. For this purpose a retract theory for $K$-actions has to be established.
"Compact actions, retract theory and prime ideals." Illinois J. Math. 55 (3) 1235 - 1266, Fall 2011. https://doi.org/10.1215/ijm/1369841804