The Kapteyn moving group is not tidal debris from $\omega$ Centauri

The Kapteyn moving group has been postulated as tidal debris from $\omega$ Centauri. If true, members of the group should show some of the chemical abundance patterns known for stars in the cluster. We present an optical and near-infrared high-resolution, high-S/N spectroscopic study of 14 stars of the Kapteyn group, plus 10 additional stars (the $\omega$ Cen-group) that, while not listed as members of the Kapteyn group as originally defined, have been nevertheless associated dynamically with $\omega$ Centauri. Abundances for Na, O, Mg, Al, Ca and Ba were derived from the optical spectra, while the strength of the chromospheric He I 10830 {\AA} line is studied as a possible helium abundance indicator. The resulting Na-O and Mg-Al patterns for stars of the combined Kapteyn and $\omega$ Cen-group samples do not resemble those of $\omega$ Centauri, and are not different from those of field stars of the Galactic halo. The distribution of equivalent widths of the He I 10830 {\AA} line is consistent with that found among non-active field stars. Therefore, no evidence is found for second-generation stars within our samples, which most likely rules out a globular-cluster origin. Moreover, no hint of the unique Ba-overabundance at the metal-rich end, well-established for $\omega$ Centauri stars, is seen among stars of the combined samples. Because this specific Ba pattern is present in $\omega$ Centauri irrespective of stellar generation, this would rule out the possibility that our entire sample might be composed of only first generation stars from the cluster. Finally, for the stars of the Kapteyn group, the possibility of an origin in the hypothetical $\omega$ Centauri's parent galaxy is disfavored by the different run of $\alpha$-elements with metallicity between our targets and stars from present-day dwarf galaxies.