Mirabolic Schur–Weyl decomposition for $\mathrm{MT}_\mathbf{v}(2,d)$

Let $R_d$ be the mirabolic Hecke algebra, and let $\mathrm{MT}_\mathbf{v}(2,d)$ be the corresponding mirabolic tensor space. For a bipartition $\boldsymbol\lambda=(\lambda,1^s)$ of $d$ with $\lambda=(\lambda_1,\lambda_2)$, $|\lambda|+s=d$, and $0\leq s\leq 2$, write $M^{\boldsymbol\lambda}$ for the irreducible representation of $R_d$ associated with $\boldsymbol\lambda$. Define

$$L^+_{\boldsymbol\lambda}=\begin{cases} L^+(\lambda_1-\lambda_2,1) & \text{if }\boldsymbol\lambda=(\lambda,\emptyset),\\ L^+(\lambda_1-\lambda_2+1,01) & \text{if }\boldsymbol\lambda=(\lambda,1),\\ L^+(\lambda_1-\lambda_2,0) & \text{if }\boldsymbol\lambda=(\lambda,11). \end{cases}$$

Mirabolic Schur–Weyl decomposition. The decomposition of $\mathrm{MT}_\mathbf{v}(2,d)$ as a bimodule for $MU_\mathbf{v}(2,d)$ and $R_d$ should be

$$\mathrm{MT}_\mathbf{v}(2,d)\simeq\bigoplus_{\boldsymbol\lambda\in\Lambda}L^+_{\boldsymbol\lambda}\otimes M^{\boldsymbol\lambda},$$

where $\Lambda$ is the set of such bipartitions. This is the proposed mirabolic analogue of quantum Schur–Weyl duality; the preceding double-commutant decomposition explains why such a parametrization is expected, but the conjecture's resolution is not supplied here.