Verma Howe duality and LKB representations

Abstract

We establish a version of quantum Howe duality with two general linear quantum enveloping algebras that involves a tensor product of Verma modules. We prove that the (colored higher) LKB representations arise from this duality and use this description to show that they are simple as modules for various subgroups of the braid group, including the pure braid group.1A. Schur-Weyl(-Brauer) and Howe dualities. Three main themes in Weyl's seminal book "The classical groups" [Wey97] are the study of polynomial invariants for actions of the eponymous classical groups, and, more or less equivalent, decomposition of the tensor algebra for such an action, and, again more or less equivalent, the description of the invariants in the tensor algebra.The two most prominent examples that fit into Weyl's setting are the celebrated Schur-Weyl duality [Sch01] for tensor invariants of GL m (C) and Brauer duality [Bra37] for tensor invariants of O m (C) and SP m (C) (for the symplectic group m is even). Both of these were studied by using commuting actions of GL m (C), and O m (C), SP m (C) on one side and the symmetric group S n and the Brauer algebra, respectively, on the other side, both acting on a tensor product of the defining representation of the classical groups. In this commuting-action-approach, Schur-Weyl duality for example essentially reads:(A) There are commuting actions of GL m (C) and S n on (C m ) ⊗n .(B) The two actions generate each others centralizer.(C) The GL m (C)-S n bimodule (C m ) ⊗n can be explicitly decomposed into a direct sum of simple GL m (C) modules tensored with simple S n modules. A statement of this form is what we call a double centralizer (a.k.a. double commutant) approach.Howe [How89], [How95] studied polynomial invariants, e.g. via symmetric powers, of classical groups using a double centralizer approach, and the resulting dualities are called Howe dualities