Affine Kazhdan-Lusztig R-polynomials for Kac-Moody groups

by Paul Philippe (Universität Münster)

Monday 27 Apr 2026, 14:00 → 15:00 Europe/Brussels

B/2nd floor-B.203 - Seminar room (Marc de Hemptinne (chemin du Cyclotron, 2, Louvain-la-Neuve))

To any reductive group G (such as SLn) one can associate an affine flag variety X, whose geometry is related to the representation theory of G and of its loop group G[t,t−1]. 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: there is an analog W+ of the affine Weyl group, but which is only a semi-group, and has no proper Coxeter structure. However in 2016 Braverman Kazhdan and Patnaik have introduced a partial order on W+ which could play the role of the Bruhat order. 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, after introducing these polynomials in the reductive setting, I will present a path model construction for affine Kazhdan-Lusztig R polynomials associated to Kac-Moody groups. This path model was schemed in a prepublication of Muthiah in 2019, and relies on later work of Bardy-Panse, Hébert and Rousseau on twin masures, which we will use as a black box. This is a joint work with A. Hébert.