  We study the natural labeling of the one dimensional representations for
Ariki-Koike algebras at roots of unity. For Hecke algebras of types A and B,
some of these representations can be identified with the socle of the Steinberg
representation of a finite reductive group. We here give closed formulas for
them. This uses, in particular, several results concerning crystal isomorphisms
and the Mullineux involution.
