A finite linear group W acting on a complex vector space V is a reflection group if and only if the algebra of invariant polynomials is again a polynomial algebra; then, the algebra of polynomials on V is a free module of rank |W| over the invariants (Chevalley-Shepphard-Todd theorem). Moreover, for Coxeter groups, Demazure shows that the inclusion of the invariant subalgebra is a Frobenius extension, whose trace can be expressed using divided difference operators (known as Demazure operators). This Frobenius property is an important preliminary for the construction of the singular Hecke 2-category, which gives a presentation by generators and relations of singular Soergel bimodules. I will talk about a joint work with Ben Elias and Ben Young, where we start from the affine type A Weyl group, but with a q-deformation of its Cartan matrix (due to Elias); when q is specialized to a 2m-th root of unity, the reflection representation factors through the finite quotient G(m,m,n). We show that, in this case, we have again a Frobenius extension, and we study the exotic nilCoxeter algebra generated by the Demazure operators associated to the simple roots of the affine Weyl group. This algebra has interesting new features, including some roundabout relations, but it remains mysterious. For n = 3, we determine which words of length 3m (in the Demazure operators) give a Frobenius trace (rather than 0). I will give some experimental results at the end.