On linear Hodge-Newton decomposition for reductive monoids.
R. Kottwitz and E. Viehmann, in course of their study of certain affine Springer fibers, developed linear variants of Katz's Hodge-Newton decomposition -- for an arbitrary reductive group over a \(p\)-adic field, as well as for the monoid of \(n\times n\) matrices over a \(p\)-adic field. The proof in the general linear monoid case involved perturbing a matrix by a scalar one so as to make it invertible and using the result for the group case. It is thus natural to ask if this phenomenon of linear Hodge-Newton decomposition is monoid-theoretic, for instance whether there is a monoid-theoretic proof. Accordingly, the author considers a reductive linear algebraic monoid \(\widetilde G\) over a \(p\)-adic field \(F\) of characteristic zero, whose group of units is defined and split over the ring of integers of \(F\). We define the notions of ``Newton points'' and ``Hodge points'' associated to \(F\)-rational points of \(\widetilde G\), and prove a relationship between them, that constitutes part of the linear Hodge-Newton decomposition. Doing so requires one to develop some structure theory for reductive monoids over \(p\)-adic fields. For instance, one needs an Iwasawa decomposition and a Cartan decomposition for these objects, and a ``Bruhat-Tits inequality'' that relates these. A crucial ingredient in proving the Iwasawa and Cartan decompositions is an adaptation of the Bruhat-Renner decomposition to non-algebraically closed fields. Other, related, structure theoretic results such as the affine Bruhat decomposition are also discussed. Let \(\widetilde G\) be an irreducible linear reductive monoid over a characteristic zero field \(F\) of fractions of a complete discrete valuation ring \(\sigma\), such that its group \(G\) of units is split over \(\sigma\). This paper concerns a relation between the Hodge point and the Newton point associated to an element \(r\in\widetilde G(F)\), proved by Kottwitz and Viehmann when \(\widetilde G\) is either a connected reductive \(\sigma\)-split linear algebraic group over \(F\) or the monoid of \(n\times n\) matrices over \(F\). On the way to proving this relation, we apply the Putcha-Renner theory of linear algebraic monoids over algebraically closed fields to study \(\widetilde G(F)\) by generalizing various results for linear algebraic groups over \(F\) such as the Iwasawa, Cartan and affine Bruhat decompositions.
scientific article; zbMATH DE number 6141004