We study cool neutral gas traced by NaD absorption in 140 local ($rm z<0.1)$ early-type ``red geyser galaxies. These galaxies show unique signatures in spatially-resolved strong-line emission maps that have been interpreted as large-scale active galactic nuclei driven ionized winds. To investigate the possible fuel source for these winds, we examine the abundance and kinematics of cool gas ($rm T sim 100-1000 K$) inferred from Na I D absorption in red geysers and matched control samples drawn from SDSS-IV MaNGA. We find that red geysers host greater amounts of NaD-associated material. Substantial cool gas components are detected in more than $rm 50 %$ of red geysers (compared to 25% of the control sample) going up to 78$%$ for radio-detected red geysers. Our key result is that cool gas in red geysers is predominantly infalling. Among our 30 radio-detected red geysers, 86$%$ show receding NaD absorption velocities (with respect to the systemic velocity) between $rm 40 - 50~km~s^{-1}$. We verify this result by stacking NaD profiles across each sample which confirms the presence of infalling NaD velocities within red geysers ( $simrm 40~km~s^{-1}$) with no velocity offsets detected in the control samples. Interpreting our observations as signatures of inflowing cool neutral clouds, we derive an approximate mass inflow rate of $rm dot{M}_{in} sim 0.1 M_{odot} yr^{-1}$, similar to that expected from minor merging and internal recycling. Some red geysers show much higher rates ($rm dot{M}_{in} sim 5 M_{odot} yr^{-1}$) that may indicate an ongoing accretion event.
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