We present J-band spectroscopy of passive galaxies focusing on the Na I doublet at 1.14 {mu}m. Like the Na I 0.82 {mu}m doublet, this feature is strong in low-mass stars and hence may provide a useful probe of the initial mass function (IMF). From high signal-to-noise composite spectra, we find that Na I 1.14 {mu}m increases steeply with increasing velocity dispersion, {sigma}, and for the most massive galaxies (sigma > 300 km/s) is much stronger than predicted from synthetic spectra with Milky-Way-like IMFs and solar abundances. Reproducing Na I 1.14 {mu}m at high {sigma} likely requires either a very high [Na/H], or a bottom-heavy IMF, or a combination of both. Using the Na D line to break the degeneracy between IMF and abundance, we infer [Na/H] $approx$ +0.5 and a steep IMF (single-slope-equivalent x $approx$ 3.2, where x = 2.35 for Salpeter), for the high-sigma galaxies. At lower mass ({sigma} = 50-100 km/s), the line strengths are compatible with MW-like IMFs and near-solar [Na/H]. We highlight two galaxies in our sample where strong gravitational lensing masses favour MW-like IMFs. Like the high-{sigma} sample on average, these galaxies have strong Na I 1.14 mu m; taken in isolation their sodium indices imply bottom-heavy IMFs which are hard to reconcile with the lensing masses. An alternative full-spectrum-fitting approach, applied to the high-sigma sample, recovers an IMF less heavy than Salpeter, but under-predicts the Na I 1.14 mu m line at the 5{sigma} level. We conclude that current models struggle to reproduce this feature in the most massive galaxies without breaking other constraints, and caution against over-reliance on the sodium lines in spectroscopic IMF studies.
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