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 B_n^(f) be the two-sided ideal of the Brauer algebra B_n(-2m) over K generated by e₁e₃ ... _2f-1 where 0 ≤ f ≤ [n/2]. Let HT_f^⊗n be the subspace of partial-harmonic tensors of valence f in V^⊗n. In this paper we prove that dimHT_f^⊗nand dim End_KSp(V)(V^⊗n/V^⊗nB_n^(f) are both independent of K, and the natural homomorphism from B_n(-2m)/B_n^(f) to End_KSp(V)(V^⊗n/V^⊗nB_n^(f) is always surjective. We show that has a Weyl filtration and is isomorphic to the dual of V^⊗nB_n^(f)/V^⊗nB_n^(f+1) as an Sp(V)-(B_n(-2m)/B_n^(f+1))-bimodule. We obtain an Sp(V)-B_n-bimodules filtration of V^⊗n such that each successive quotient is isomorphic to some ∇(λ) ⊗ Z_g,λ B_n with ⊢ n 2g, ℓ(λ)≤m and 0 ≤ g ≤ [n/2], where ∇(λ) is the co-Weyl module associated to λ and z_g,λ is an explicitly constructed maximal vector of weight λ. As a byproduct, we show that each right B_n-module is integrally defined and stable under base change.