What does this equation mean?

The hole’s evaporation is modeled, as it is throughout this line of work, as an isometry V from the interior-plus-infalling-system Hilbert space to a radiation-plus-remaining-interior Hilbert space. What makes the conservation law bite is that V must commute with the group action on both sides: $V\,U_{\mathrm{in}}(g) \;=\; \big[U_{\mathrm{rad}}(g)\otimes U_{\mathrm{rem}}(g)\big]\,V$. Every symbol here is a unitary representation of the same rotation on a different Hilbert space — dimensionless linear operators, none of them observables in their own right. The equation says the hole’s internal dynamics cannot tell which way “up” was chosen before the compass fell in: rotate the input, and the output rotates the same way, split between radiation and remainder. This is not an assumption invented for this article. It is simply what “the hole conserves angular momentum” means once evaporation is written as a unitary process, and it is the exact setting Nakata, Wakakuwa, and Koashi analyzed when they asked how a global symmetry constraint changes Hayden-Preskill recovery [ 8 ] .