On linear Hodge-Newton decomposition for reductive monoids.
type: Item
instance of: scholarly article
title: On linear Hodge-Newton decomposition for reductive monoids.
zbMATH Open document ID: 1277.20077
author: Sandeep Varma
published in: Semigroup Forum
publication date: 2013-03-04T00:00:00Z
review text: 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 \…
Mathematics Subject Classification ID: 20M32
Mathematics Subject Classification ID: 20G25
Mathematics Subject Classification ID: 22E35
zbMATH DE Number: 6141004
zbMATH Keywords: Bruhat-Renner decomposition
zbMATH Keywords: Hodge-Newton decomposition
zbMATH Keywords: reductive groups
zbMATH Keywords: reductive monoids
zbMATH Keywords: \(p\)-adic fields
zbMATH Keywords: monoids of matrices
reviewed by: Huang Wenxue
MaRDI profile type: Publication
full work available at URL: s00233-011-9366-y
OpenAlex ID: W4237407714
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DOI: 10.1007/S00233-011-9366-Y
description: scientific article; zbMATH DE number 6141004