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

RESUMO

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 $\operatorname{AIII}$ $\imath$-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 $\mathfrak U^{\imath}_{+}$. Conversely, every nilpotent $2$-subrepresentation of an ambient locally Schurian $2$-representation of $\mathfrak U^{\imath}_{+}$ 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 $\mathfrak U^{\imath}_{+}$. 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 $\imath$-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 $\imath$-canonical bases are expected to replace ordinary quantum groups and canonical bases.