Rigged configurations and the $\ast$-involution for generalized Kac--Moody algebras

We construct a uniform model for highest weight crystals and $B(\mathrm{\infty })$ for generalized Kac–Moody algebras using rigged configurations. We also show an explicit description of the $\ast$-involution on rigged configurations for $B(\mathrm{\infty })$: that the $\ast$-involution interchanges the rigging and the corigging. We do this by giving a recognition theorem for $B(\mathrm{\infty })$ using the $\ast$-involution. As a consequence, we also characterize $B(\lambda )$ as a subcrystal of $B(\mathrm{\infty })$ using the $\ast$-involution.