The formal affine Demazure algebra and real finite reflection groups

Abstract: In this paper, we generalize the formal affine Demazure algebra of Hoffnung-Malag\'on-L\'opez-Savage-Zainoulline to all real finite reflection groups. We begin by generalizing the formal group ring of Calm`es-Petrov-Zainoulline to all real finite reflection groups. We then define and study the formal Demazure operators that act on the formal group ring. Using these results and constructions, we define and study the formal affine Demazure algebra for all real finite reflection groups. Finally, we compute several structure coefficients that appear in a braid relation among the formal Demazure elements, and we conclude this paper by computing all structure coefficients for the reflection groups $I_2(5)$, $I_2(7)$, $H_3$, and $H_4$.