Affine Hecke algebras of type D and generalisations of quiver Hecke algebras

TL;DR

This paper establishes an isomorphism between cyclotomic quotients of affine Hecke algebras of type D and generalized quiver Hecke algebras, extending the classical type A correspondence to type D.

Contribution

It introduces a new isomorphism linking type D affine Hecke algebras with generalized quiver Hecke algebras, completing the classical types correspondence.

Findings

Established isomorphism between type D cyclotomic quotients and generalized quiver Hecke algebras

Related type D fixed point subalgebras to type B analogues

Extended the classical Brundan-Kleshchev result to type D

Abstract

We define and study cyclotomic quotients of affine Hecke algebras of type D. We establish an isomorphism between (direct sums of blocks of) these cyclotomic quotients and a generalisation of cyclotomic quiver Hecke algebras which are a family of Z-graded algebras closely related to algebras introduced by Shan, Varagnolo and Vasserot. To achieve this, we first complete the study of cyclotomic quotients of affine Hecke algebras of type B by considering the situation when a deformation parameter p squares to 1. We then relate the two generalisations of quiver Hecke algebras showing that the one for type D can be seen as fixed point subalgebras of their analogues for type B, and we carefully study how far this relation remains valid for cyclotomic quotients. This allows us to obtain the desired isomorphism. This isomorphism completes the family of isomorphisms relating affine Hecke algebras of classical types to (generalisations of) quiver Hecke algebras, originating in the famous result of Brundan and Kleshchev for the type A.