Affine Brauer category and parabolic category \({\mathcal {O}}\) in types BCD

  • Hebing RuiEmail author
  • Linliang Song


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).


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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical Science, Tongji UniversityShanghaiChina

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