Frobenius extensions and the exotic nilCoxeter algebra for 𝐺(𝑚,𝑚,3)

In a previous paper of the first author, the type A n − 1 A_{n-1} affine Cartan matrix was q q -deformed to produce a deformation of the reflection representation of the affine Weyl group W a f f W_{aff} . This deformation plays a role in the quantum geometric Satake equivalence. In this paper we introduce the study of q q -deformed divided difference operators. When q q is specialized to a primitive 2 m 2m -th root of unity, this reflection representation of W a f f W_{aff} factors through a quotient, the complex reflection group G ( m , m , n ) G(m,m,n) . The divided difference operators now generate a finite-dimensional algebra we call the exotic nilCoxeter algebra. This algebra is new and has surprising features. In addition to the usual braid relations, we prove a new relation called the roundabout relation. A classic result of Demazure, for Weyl groups, states that the polynomial ring of the reflection representation is a Frobenius extension over its subring of invariant polynomials, and describes how the Frobenius trace can be constructed within the nilCoxeter algebra. We study the analogous Frobenius extension for G ( m , m , n ) G(m,m,n) , and identify the Frobenius trace within the exotic nilCoxeter algebra for G ( m , m , 3 ) G(m,m,3) .