Compact Schur-Weyl duality and the affine Type B/C Brauer algebra

We define an extension of the affine Brauer algebra, the type B/C affine Brauer algebra. This new algebra contains the hyperoctahedral group and it naturally acts on $END_K(X \otimes V^{\otimes k})$ for Orthogonal and Symplectic groups. Thus we obtain a compact analogue of Schur-Weyl duality. We study functors $F_{μ,k}$ from the category of admissible $O(p,q)$ or $Sp_{2n}(\mathbb{R})$ modules to representations of the type B/C affine Brauer algebra $\mathfrak{B}_k^θ$. Thus providing a Akawaka-Suzuki-esque link between $O(p,q)$ (or $Sp_{2n}(\mathbb{R})$) and $\mathfrak{B}_k^θ$. Furthermore these functors take non spherical principal series modules to principal series modules for the graded Hecke algebra of type $D_k$, $C_{n-k}$ or $B_{n-k}$. With this we get a functorial correspondence between admissible simple $O(p,q)$ (or $Sp_{2n}(\mathbb{R})$) modules and graded Hecke algebra modules.

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