Twin masures associated with Kac-Moody groups over Laurent polynomials

Abstract

Let $\mathfrak{G}$ be a split reductive group, $\mathbb{k}$ be a field and $\varpi$ be an indeterminate. In order to study $\mathfrak{G}(\mathbb{k}[\varpi ,{\varpi }^{-1}])$ and $\mathfrak{G}(\mathbb{k}(\varpi ))$, one can make them act on their twin building $\mathcal{I}={\mathcal{I}}_{\oplus }\times {\mathcal{I}}_{\ominus }$, where ${\mathcal{I}}_{\oplus }$ and ${\mathcal{I}}_{\ominus }$ are related via a ''codistance''. Masures are generalizations of Bruhat-Tits buildings adapted to the study of Kac-Moody groups over valued fields. Motivated by the work of Dinakar Muthiah on Kazhdan-Lusztig polynomials associated with Kac-Moody groups, we study the action of $\mathfrak{G}(\mathbb{k}[\varpi ,{\varpi }^{-1}])$ and $\mathfrak{G}(\mathbb{k}(\varpi ,{\varpi }^{-1}))$ on their ''twin masure'', when $\mathfrak{G}$ is a split Kac-Moody group instead of a reductive group.