A Cartan decomposition for \(p\)-adic loop groups
Let \(G\) be a simply connected Chevalley group, \({\mathcal L}_k\) the field of formal Laurent series with coefficients in a \(p\)-adic field \(k\), \({\mathcal O}\) the ring of integers in \(k\), \({\mathcal L}_{{\mathcal O}}\) the ring of elements in \({\mathcal L}_k\) with coefficients in \({\mathcal O}\), \(\overline G= G_{{\mathcal L}_k}\) the group of \({\mathcal L}_k\)-rational points of \(G\), \(\overline K= G_{{\mathcal L}_{\mathcal O}}\) the group of \({\mathcal L}_{{\mathcal O}}\)-rational points of \(G\), and \(\overline A\) the \(k\)-rational points of a maximal split torus in \(G\). The author's purpose is to prove the existence of a Cartan decomposition of \(\overline G\) with respect to \(\overline A\) and \(\overline K\). For technical simplicity the author restricts the considerations to the case when \(G\) corresponds to a simply-laced root system (however, the proof of another result, the existence of maxima, has no such restriction). Actually, the author does not prove the existence of Cartan decompositions for all of \(\overline G\), but only for a subgroup of \(\overline G\) whose elements are bounded.
scientific article; zbMATH DE number 769473
scientific article