Affine Brauer category and parabolic category $\mathcal O$ in types $B, C, D$

TL;DR

This paper introduces the affine Brauer category and its quotients, establishing their algebraic properties and connecting them to parabolic category O in types B, C, D, with implications for duality and decomposition matrices.

Contribution

It constructs the affine Brauer category and cyclotomic variants, proves their morphism spaces are free, and links these categories to Nazarov-Wenzl algebras and parabolic category O.

Findings

Morphisms in affine Brauer category are free over the base ring.

Maximal rank of morphism spaces in cyclotomic Brauer category depends on admissibility.

Higher Schur-Weyl duality established between Nazarov-Wenzl algebras and parabolic category O.

Abstract

A strict monoidal category referred to as affine Brauer category $\mathcal{AB}$ is introduced over a commutative ring $\kappa$ containing multiplicative identity $1$ and invertible element $2$. We prove that morphism spaces in $\mathcal{AB}$ are free over $\kappa$. The cyclotomic (or level $k$) Brauer category $\mathcal{CB}^f(\omega)$ is a quotient category of $\mathcal{AB}$. We prove that any morphism space in $\mathcal{CB}^f(\omega)$ is free over $\kappa$ with maximal rank if and only if the $\mathbf u$-admissible condition holds in the sense of (1.30). Affine Nazarov-Wenzl algebras and cyclotomic Nazarov-Wenzl algebras will be realized as certain endomorphism algebras in $\mathcal{AB}$ and $\mathcal{CB}^f(\omega)$, respectively. We will establish higher Schur-Weyl duality between cyclotomic Nazarov-Wenzl algebras and parabolic BGG categories $\mathcal O$ associated to symplectic and orthogonal Lie algebras over the complex field $\mathbb C$. This enables us to use standard arguments in [1,26,27] to compute decomposition matrices of cyclotomic Nazarov-Wenzl algebras. The level two case was considered by Ehrig and Stroppel in [14].