The quantum Bruhat graph for $$\widehat{SL}_2$$ and double affine Demazure products

Abstract 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 $$q=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> 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 $$\widehat{SL}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mrow> <mml:mi>SL</mml:mi> </mml:mrow> <mml:mo>^</mml:mo> </mml:mover> <mml:mn>2</mml:mn> </mml:msub> </mml:math> , 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 $$W_\mathcal {T}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>T</mml:mi> </mml:msub> </mml:math> (for level greater than one). We give results regarding the Demazure product and Muthiah and Orr’s length function for $$W_\mathcal {T}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>T</mml:mi> </mml:msub> </mml:math> , and we verify that our proposal matches specific examples computed by Muthiah and Puskás using the Kac-Moody affine Hecke algebra.