To any reductive group G (such as SL_n) one can associate its affine flag variety X, whose geometry is related to the representation theory of G and of its loop group G[t,t^{-1}]. Affine Kazhdan-Lusztig R-polynomials relate some of the structure of X to the combinatorics of a Coxeter group associated to G, namely its affine Weyl group. These polynomials are a cornerstone in the famous affine Kazhdan-Lusztig theory. If we try to replace G by a Kac-Moody, non-reductive group, X can still be defined, but has no reasonable topology, and some of its structure is lost. In particular there is an analog W^+ of the affine Weyl group, but it is only a semigroup, and it has no proper Coxeter structure. However, in 2016 Braverman-Kazhdan-Patnaik have introduced a partial order on W^+ which could play the role of the Bruhat order, and Muthiah-Orr have defined a Z-valued length associated to it. Since then, some key combinatorial properties of this semi-group have been obtained, making the definition of affine Kazhdan-Lusztig R-polynomials in this context reasonable.

In my talk I will introduce these polynomials in the reductive setting, and explain a possible construction to define them in the Kac-Moody setting. This construction was schemed in a prepublication by Muthiah in 2019, and relies on later work by Bardy-Panse, Hébert and Rousseau. This is a joint work in progress with A. Hébert.