Ben Elias, University of Oregon

Abstract: The reflection representation of the affine Weyl group in type A admits a q-deformation, which plays a role in quantum geometric Satake. Specializing q to a primitive 2m-th root of unity, one obtains a reflection representation of the complex reflection group G(m,m,n). In this way, a category of "singular Soergel bimodules" for G(m,m,n) is related to representations of quantum groups at roots of unity. To properly study these bimodules, additional algebraic background is needed.

Associated to a reflection representation one has an action of a reflection group W on a polynomial ring R. A classic result of Demazure for Coxeter groups states that the inclusion of invariant polynomials into all polynomials is a Frobenius extension. A key tool in Demazure's proof is the nilCoxeter algebra, which acts on R by certain operators now called Demazure operators. Demazure proves that the Frobenius trace agrees with the Demazure operator associated to the longest element. These facts are what make the theory of Soergel bimodules tick.

In joint work with Juteau and Young, we study the affine reflection representation of G(m,m,n). We prove the corresponding Frobenius extension result. We investigate an exotic nilCoxeter algebra with Demazure operators associated to elements of the affine Weyl group, acting on the ring R. When n=3, we determine which elements of the affine Weyl group are "longest elements" for G(m,m,n).