Caught in the act: diffuse stellar and Ly$\alpha$ emission around a halo-less quasar in the assembling cluster CARLA J0800+4029 at z=2

Extended Ly$\alpha$ emission is commonly detected around luminous quasars at $z\sim 2$-3 and traces cold gas in the circumgalactic environment of the AGN. Together with the spatial distribution of galaxies, Ly$\alpha$ probes gas and galaxy assembly in high-redshift overdensities. We use deep KCWI wide-field spectroscopy and HST/F140W imaging to investigate the assembling cluster CARLA J0800+4029 at $z=1.986\pm 0.014$. No robust extended Ly$\alpha$ emission is detected at the radio-loud quasar position. Instead, the dominant Ly$\alpha$ emission is offset from the AGN and coincides with a disturbed structure composed of galaxies, tidal features, and diffuse intra-halo light (IHL). The detected structure has $L_{\mathrm{L}\mathrm{y}\alpha }=1.42\times {10}^{43}$ erg/s and reaches surface-brightness levels of $\sim {10}^{-17}$ erg/s/cm${}^2$/arcsec${}^2$. At the quasar position, a matched $3\sigma$ limit gives $L_{\mathrm{L}\mathrm{y}\alpha }<4.20\times {10}^{42}$ erg/s, a factor of $\sim 3.4$ below the detected offset emission and among the most stringent constraints at similar redshift. The Ly$\alpha$ emission is dominated by one extended region near systemic velocity, with FWHM $\sim 300$-500 km/s. The associated tidal structure has an IHL fraction of 18.8%, and contains a spatially coherent absorption-like decrement at $\sim +1000$ km/s. The offset and alignment between Ly$\alpha$ emission and IHL suggest that gas and stars have been redistributed together during assembly. The lack of a comparable Ly$\alpha$ halo around the quasar further suggests that gas in the immediate AGN environment is ionized, depleted, or suppressed, while the offset region remains favorable for Ly$\alpha$ emission. CARLA J0800+4029 reveals baryon redistribution during cluster assembly, with the dominant observable Ly$\alpha$ reservoir decoupled from the central AGN and linked to ongoing tidal assembly.