The Quantum Bruhat Graph for $\widehat{SL}_2$ and Double Affine Demazure Products

We investigate the Demazure product in a double affine setting. Work by Muthiah and Pusk\'as gives a conjectural way to define this in terms of the $q=0$ specialisation of these Hecke algebras. We instead take a different approach generalising work by Felix Schremmer, who gave an equivalent formula for the (single) affine Demazure product in terms of the quantum Bruhat graph. We focus on type $\widehat{SL}_2$, where we prove that the quantum Bruhat graph of this type satisfies some nice properties, which allows us to construct a well-defined associative Demazure product for the double affine Weyl semigroup $W_{\mathcal{T}}$ (for level greater than one). We give results regarding the Demazure product and Muthiah and Orr's length function for $W_{\mathcal{T}}$, and we verify that our proposal matches specific examples computed by Muthiah and Pusk\'as using the Kac-Moody affine Hecke algebra