Codimensions of Newton strata for \(\text{SL}_3(F)\) in the Iwahori case.
type: Item
instance of: scholarly article
title: Codimensions of Newton strata for \(\text{SL}_3(F)\) in the Iwahori case.
zbMATH Open document ID: 1185.20051
published in: Mathematische Zeitschrift
publication date: 2009-10-23T00:00:00Z
review text: Let \(F\) denote the field of Laurent power series over the algebraic closure \(\overline k\) of a finite field. One has the skew polynomial ring over \(F\) generated by the Frobenius automorphism \(\sigma\). An isocrystal is a tuple \((V,\Phi)\) where \(V\) is a finite-dimensional vector space over \(F\) and \(\Phi\) is a \(\sigma\)-linear automorphism of \(V\). As \(V\) is cyclic as a module over the afore-mentioned skew polynomial ring \(F[\sigma]\), there is a polynomial \(f\) over it such that \(V\cong F[\sigma]/F[\sigma]f\). One has the Newton polygon associated to \(f\); this is the convex hull of the points \((0,0)\) and \((i,-v(a_i))\) where \(a_i\)'s are the coefficients of \(f\) (this notation differs slightly from the usual one where one considers \((i,v(a_i))\)). The author considers the group \(\text{SL}_3(F)\) and looks at a stratification of it which is associated to the slope sequence of a Newton polygon. Indeed, if \(A\in\text{SL}_3(F)\), then \(\Phi=A\sigma\) gives an isocrystal on \(F^3\) and the Newton polygon is considered for the associated characteristic polynomial. Thus, \(\text{SL}_3(F)\) is the disjoint union of the so-called Newton strata \(G_\lambda\), parametrized by different slope sequences \(\lambda\) arising as \(A\) varies in \(\text{SL}_3(F)\). The set \(\mathcal N\) of slope sequences is in bijective correspondence with \(\sigma\)-conjugacy classes in the group. The author provides two versions of a formula for the codimensions of Newton strata inside each component of the affine Bruhat decomposition on \(I\widetilde WI\) where \(I\) is the Iwahori subgroup of \(\text{SL}_3(F)\) and \(\widetilde W\) is the affine Weyl group. For a particular double coset \(IxI\), there is a corresponding subset \(\mathcal N(x)\) of the finitely many possible slope sequences \(\lambda\) such that \(G_\lambda\cap IxI\neq\emptyset\). The author calls certain subsets of \(IxI\) to be admissible; one can define their codimension. For slopes \(\mu\leq\…
Mathematics Subject Classification ID: 20G25
Mathematics Subject Classification ID: 14L05
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 5620112
zbMATH Keywords: Newton polygons
zbMATH Keywords: Iwahori subgroups
zbMATH Keywords: isocrystals
zbMATH Keywords: Deligne-Lusztig varieties
author: Elizabeth Beazley
reviewed by: Balasubramanian Sury
MaRDI profile type: Publication
full work available at URL: s00209-008-0429-z
OpenAlex ID: W2024254638
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DOI: 10.1007/S00209-008-0429-Z
description: scientific article; zbMATH DE number 5620112