Double Affine Bruhat Order

Abstract

We classify cocovers and covers of a given element of the double affine Weyl semigroup W with respect to the Bruhat order, specifically when W is associated to a finite root system that is irreducible and simply laced. We show two approaches: one extending the work of Lam and Shimozono [LS], and its strengthening by Milićević [Mi], where cocovers are characterized in the affine case using the quantum Bruhat graph of W fin , and another, which takes a more geometrical approach by using the length difference set defined by Muthiah and Orr [MO].