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 $2$, consider its $\mathbb{Z}_2$-homogeneous one-dimensional extension $\underline V=V\oplus\mathbb{K}v$, and natural action supergroup $\tilde G:=\text{GL}(V)\times \textbf{G}_m$ on V$. Then have tensor product supermodule ($\underline{V}^{\otimes r}$, $\rho_r$) G$. We present generalized Schur-Sergeev duality which said that Schur superalgebras $S'(m|n,r)$ G$ so-called weak degenerate Hecke algebra $\underline{\mathcal{H}}_r$ are centralizers. The infinite dimensional algebra, has representation space. This notion comes from \cite{B-Y-Y2020}, with little modification.