We investigate the Demazure product in a double affine setting. A recent preprint by Muthiah and Puskás 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 Puskás using the Kac-Moody affine Hecke algebra.
