Schur–Sergeev duality for Ariki–Koike algebras

Let $U_q(\mathfrak g)$ be the quantized enveloping superalgebra of $\mathfrak g=\mathfrak{gl}(k_1|\ell_1)\oplus \cdots \oplus \mathfrak{gl}(k_m|\ell_m)$ and let $\mathcal H_{m,n}$ be the Ariki–Koike algebra. We define a right $\mathcal H$-action on the $n$-fold tensor (super) space of the vector representation of $U_q(\mathfrak g)$ and prove Schur–Sergeev duality between $U_q(\mathfrak g)$ and $\mathcal H_{m,n}$.