Affine and cyclotomic Brauer categorification via $ı$-Kac--Moody $2$-categories: the half-integral type $\mathrm{AIII}$ case

Abstract

Cyclotomic Brauer or cyclotomic Nazarov--Wenzl algebras arise in higher Schur--Weyl dualities involving parabolic categories $\mathcal{O}$ for Lie algebras of types $B,C$ and $D$. Their connection with Kac--Moody-type categorification is substantially less developed than the corresponding type $A$ theory for cyclotomic Hecke algebras. We construct a categorical bridge between affine Brauer-type representation theory and the half-integral quasi-split type $\mathrm{AIII}$ $ı$-Kac--Moody $2$-category of Bao--Shan--Wang--Webster. More precisely, an action of the affine Brauer category on a locally Schurian category, with dot spectrum exactly $\frac{1}{2}+\mathbb{Z}$, determines a generalized nilpotent $2$-representation of the even component ${\mathfrak{U}}_{+}^{ı}$. Conversely, every nilpotent $2$-subrepresentation of an ambient locally Schurian $2$-representation of ${\mathfrak{U}}_{+}^{ı}$ carries a compatible affine Brauer action whose dot spectrum is contained in $\frac{1}{2}+\mathbb{Z}$. Applying these constructions to cyclotomic quotients, we prove that the locally unital algebra attached to a $\mathbf{u}$-admissible cyclotomic Brauer category is isomorphic to the locally unital algebra attached to the corresponding cyclotomic quotient of the principal $2$-representation of ${\mathfrak{U}}_{+}^{ı}$. Consequently, the associated cyclotomic Brauer (or cyclotomic Nazarov--Wenzl) algebras acquire natural $\mathbb{Z}$-gradings. To our knowledge, this is the first such categorical realization of $\mathbf{u}$-admissible half-integral cyclotomic Brauer algebras by means of an $ı$-Kac--Moody $2$-category. It provides the categorical and graded framework toward a Brauer-type extension of the Brundan--Kleshchev--Ariki theory in which coideal algebras and $ı$-canonical bases are expected to replace ordinary quantum groups and canonical bases.

76 pages