Dual partially harmonic tensors and Brauer-Schur-Weyl duality

Let $V$ be a $2m$-dimensional symplectic vector space over an algebraically closed field $K$. Let ${\text{mbb}}_n^{(f)}$ be the two-sided ideal of the Brauer algebra ${\text{mbb}}_n(-2m)$ over $K$ generated by $e_1{e}_3\mathrm{.}\mathrm{.}\mathrm{.}{e}_{2f-1}$, where $0\leq f\leq [n/2]$\mathrm{.}Let$$\mathcal{H}\mathcal{T}${f}^{\otimes n}$ be the subspace of partially harmonic tensors of valence $f$ in $V^{\otimes n}$. In this paper, we prove that $\dim {\mathcal{H}\mathcal{T}}_f^{\otimes n}$ and $\dim\End{KSp(V)}\Bigl(V^{\otimes n}/V^{\otimes n}\mbb_n^{(f)}\Bigr)$arebothindependentof$K$, and the natural homomorphism from ${\text{mbb}}_n(-2m)\mathrm{/}{\text{mbb}}_n^{(f)}$ to ${\text{End}}_{KSp(V)}\left(V^{\otimes n}\mathrm{/}{V}^{\otimes n}{\text{mbb}}_n^{(f)}\right)$ is always surjective. We show that ${\mathcal{H}\mathcal{T}}_f^{\otimes n}$ has a Weyl filtration and is isomorphic to the dual of $V^{\otimes n}\mbb_n^{(f)}/V^{\otimes n}\mbb_n^{(f+1)}$asa$Sp(V)$-$(\mbb_n(-2m)/\mbb_n^{(f+1)})$-bimodule. We obtain a $Sp(V)$-${\text{mbb}}_n$-bimodules filtration of $V^{\otimes n}$ such that each successive quotient is isomorphic to some $\nabla(\lam)\otimes z_{g,\lam}\mbb_n$with$\lam\vdash n-2g$,$\ell(\lam)\leq m$and$0\leq g\leq [n/2]$,where$\nabla(\lam)$istheco-Weylmoduleassociatedto$\lam$and$z_{g,\lam}$isanexplicitlyconstructedmaximalvectorofweight$\lam$. As a byproduct, we show that each right ${\text{mbb}}_n$-module $z_{g,\text{lam}}{\text{mbb}}_n$ is integrally defined and stable under base change.