We report new HST COS and STIS spectroscopy of a star-forming region (~100 solar masses/year) in the center of the X-ray cluster RXJ1532.9+3021 (z=0.362), to follow-up the CLASH team discovery of luminous UV filaments and knots in the central massive galaxy. We detect broad (~500 km/s) Lyman alpha emission lines with extraordinarily high equivalent width (EQW~200 Angstroms) and somewhat less broadened H-alpha (~220 km/s). Emission lines of N V and O VI are not detected, which constrains the rate at which gas cools through temperatures of 10^6 K to be less than about 10 solar masses/year. The COS spectra also show a flat rest-frame UV continuum with weak stellar photospheric features, consistent with the presence of recently-formed hot stars forming at a rate of ~10 solar masses/year, uncorrected for dust extinction. The slope and absorption lines in these UV spectra are similar to those of Lyman Break Galaxies at z approximately 3, albeit those with the highest Lyman-alpha equivalent widths and star-formation rates. This high-EQW Lyman-alpha source is a high-metallicity galaxy rapidly forming stars in structures that look nothing like disks. This mode of star formation could significantly contribute to the spheroidal population of galaxies. The constraint on the luminosity of any O VI line emission is stringent enough to rule out steady and simultaneous gas cooling and star formation, unlike similar systems in the Phoenix Cluster and Abell 1795. The fact that the current star formation rate differs from the local mass cooling rate is consistent with recent simulations of episodic AGN feedback and star formation in a cluster atmosphere.
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