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

Abstract

A strict monoidal category referred to as affine Brauer category $\mathcal{A}\mathcal{B}$ is introduced over a commutative ring $\kappa$ containing multiplicative identity $1$ and invertible element $2$. We prove that morphism spaces in $\mathcal{A}\mathcal{B}$ are free over $\kappa$. The cyclotomic (or level $k$) Brauer category ${\mathcal{CB}}^f(\omega )$ is a quotient category of $\mathcal{A}\mathcal{B}$. 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 $\mathcal{A}\mathcal{B}$ 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].

36 pages