Levi-type Schur-Sergeev duality for general linear super groups

In this note, we investigate a kind of double centralizer property for general linear supergroups. For the super space $V=\mathbb{K}^{m\mid n}$ over an algebraically closed field $\mathbb{K}$ whose characteristic is not equal to $2$, we consider its $\mathbb{Z}_2$-homogeneous one-dimensional extension $\underline V=V\oplus\mathbb{K}v$, and the natural action of the supergroup $\tilde G:=\text{GL}(V)\times \textbf{G}_m$ on $\underline V$. Then we have the tensor product supermodule ($\underline{V}^{\otimes r}$, $\rho_r$) of $\tilde G$. We present a kind of generalized Schur-Sergeev duality which is said that the Schur superalgebras $S'(m|n,r)$ of $\tilde G$ and a so-called weak degenerate double Hecke algebra $\underline{\mathcal{H}}_r$ are double centralizers. The weak degenerate double Hecke algebra is an infinite dimensional algebra, which has a natural representation on the tensor product space. This notion comes from \cite{B-Y-Y2020}, with a little modification.