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

TL;DR

This paper introduces a new affine Brauer algebra of type B/C that extends the classical algebra, establishing a duality with orthogonal and symplectic groups and linking their modules to graded Hecke algebra representations.

Contribution

It defines the type B/C affine Brauer algebra and constructs functors connecting modules of orthogonal and symplectic groups to this algebra and graded Hecke algebras, providing a new duality framework.

Findings

Defined the type B/C affine Brauer algebra.

Established functors linking group modules to the algebra.

Connected principal series modules to graded Hecke algebra modules.

Abstract

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 $F_{\mu,k}$ from the category of admissible $O(p,q)$ or $Sp_{2n}(\mathbb{R})$ modules to representations of the type B/C affine Brauer algebra $\mathfrak{B}_k^\theta$. Thus providing a Akawaka-Suzuki-esque link between $O(p,q)$ (or $Sp_{2n}(\mathbb{R})$) and $\mathfrak{B}_k^\theta$. 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.