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We present J-band spectroscopy of passive galaxies focusing on the Na Idoublet at 1.14 {\mu}m. Like the Na I 0.82 {\mu}m doublet, this feature isstrong in low-mass stars and hence may provide a useful probe of the initialmass function (IMF). From high signal-to-noise composite spectra, we find thatNa I 1.14 {\mu}m increases steeply with increasing velocity dispersion,{\sigma}, and for the most massive galaxies (\sigma > 300 km/s) is muchstronger than predicted from synthetic spectra with Milky-Way-like IMFs andsolar abundances. Reproducing Na I 1.14 {\mu}m at high {\sigma} likely requireseither 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 IMFsand near-solar [Na/H]. We highlight two galaxies in our sample where stronggravitational lensing masses favour MW-like IMFs. Like the high-{\sigma} sampleon average, these galaxies have strong Na I 1.14 \mu m; taken in isolationtheir sodium indices imply bottom-heavy IMFs which are hard to reconcile withthe lensing masses. An alternative full-spectrum-fitting approach, applied tothe high-\sigma sample, recovers an IMF less heavy than Salpeter, butunder-predicts the Na I 1.14 \mu m line at the 5{\sigma} level. We concludethat current models struggle to reproduce this feature in the most massivegalaxies without breaking other constraints, and caution against over-relianceon the sodium lines in spectroscopic IMF studies.

The IMF-sensitive 1.14-micron Na I doublet in early-type galaxies