  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 $5-7\times10^{13}$ 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
($L_{\mathrm{Ly\alpha}}/L_{\mathrm{X}} \approx0.3$, $\gtrsim10-1000\times$
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.
