  As a sequel to [14], in this article we first introduce a so-called duplex
Hecke algebras of type B which is a Q(q)-algebra associated with the Weyl group
W (B) of type B, and symmetric groups S_l for l = 0, 1, . . . ,m, satisfying
some Hecke relations. This notion originates from the degenerate duplex Hecke
algebra arising from the course of study of a kind of Schur-Weyl duality of
Levi-type, extending the duplex Hecke algebra of type A arising from the
related q-Schur-Weyl duality of Levi-type. A duplex Hecke algebra of type B
admits natural representations on certain tensor spaces. We then establish a
Levi-type q-Schur-Weyl duality of type B, which reveals the double centralizer
property between such duplex Hecke algebras and {\i}quantum groups studied by
Bao-Wang in [1].
