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

Abstract

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

18 pages, 1 figure; v2, removed Section 7 since there was an error in our proof of Theorem 7.4, added isotropic imaginary roots case