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

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 (1.32). Affine Nazarov–Wenzl algebras (Nazarov in J Algebra 182(3):664–693, 1996) and cyclotomic Nazarov–Wenzl algebras (Ariki et al. in Nagoya Math J 182:47–134, 2006) 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 (Anderson et al. in Pac J Math 292(1):21–59, 2018; Rui and Song in Math Zeit 280(3–4):669–689, 2015; Rui and Song in J Algebra 444:246–271, 2015), to compute decomposition matrices of cyclotomic Nazarov–Wenzl algebras. The level two case was considered by Ehrig and Stroppel in (Adv. Math. 331:58–142, 2018).