We report new HST NICMOS and WFPC2 imaging of emission-line nebulae in the central galaxies of 3 clusters of galaxies purported to host massive cooling flows, NGC1275, A2597, and PKS0745. The spectral signature of vibrationally- excited molecular hydrogen (VEMH) has been seen in every galaxy searched so far in a cluster cooling flow with an optical emission line nebula. We have discovered that the VEMH gas extends several kpc from the centers of A2597 and PKS0745, while the

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... molecular hydrogen in NGC1275 appears to be mostly confined to its nucleus, with some extended emission <1 kpc from the center. The VEMH in A2597 and PKS0745-191 seems to be nearly co-spatial with the optical emission-line filaments in those systems. Candidates for heating the nebulae are X-ray irradiation by the ICM, UV fluorescence by young stars, and shocks. UV heating by young stars provides the most satisfactory explanation for the H2 emission in A2597 and PKS0745; X-ray irradiation is energetically unlikely and strong shocks (v>40 km/s) are ruled out by the high H2/H-alpha ratios. If UV heating is the main energy input, a few billion solar masses of molecular gas is present in A2597 and PKS0745. UV irradiation models predict a significant amount of 1-2 micron emission from higher excitation H2 transitions and moderate far infrared luminosities (~1e44/h^2 erg/s) for A2597 and PKS0745. Even in the context of UV fluorescence models, the total amount of H2 gas and star formation inferred from these observations is too small to account for the cooling flow rates and longevities inferred from the X-rays. We note an interesting new constraint on cooling flow models: the radio sources do not provide a significant amount of shock heating, and therefore cannot counterbalance the cooling of the X-ray gas in the cluster cores.