We present the discovery of a giant $gtrsim$100~kpc Ly$alpha$ nebula detected in the core of the X-ray emitting cluster CL~J1449+0856 at $z=1.99$ through Keck/LRIS narrow-band imaging. This detection extends the known relation between Ly$alpha$ nebulae and overdense regions of the Universe to the dense core of a $5-7times10^{13}$ M$_{odot}$ cluster. The most plausible candidates to power the nebula are two Chandra-detected AGN host cluster members, while cooling from the X-ray phase and cosmological cold flows are disfavored primarily because of the high Ly$alpha$ to X-ray luminosity ratio ($L_{mathrm{Lyalpha}}/L_{mathrm{X}} approx0.3$, $gtrsim10-1000times$ higher than in local cool-core clusters) and by current modeling. Given the physical conditions of the Ly$alpha$-emitting gas and the possible interplay with the X-ray phase, we argue that the Ly$alpha$ nebula would be short-lived ($lesssim10$ Myr) if not continuously replenished with cold gas at a rate of $gtrsim1000$ M$_{odot}$ yr$^{-1}$. We investigate the possibility that cluster galaxies supply the required gas through outflows and we show that their total mass outflow rate matches the replenishment necessary to sustain the nebula. This scenario directly implies the extraction of energy from galaxies and its deposition in the surrounding intracluster medium, as required to explain the thermodynamic properties of local clusters. We estimate an energy injection of the order of $thickapprox2$ keV per particle in the intracluster medium over a $2$ Gyr interval. In our baseline calculation AGN provide up to $85$% of the injected energy and 2/3 of the mass, while the rest is supplied by supernovae-driven winds.
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