A positivity property of the Satake isomorphism
type: Item
instance of: scholarly article
title: A positivity property of the Satake isomorphism
zbMATH Open document ID: 0941.22006
published in: Manuscripta Mathematica
publication date: 2000-08-01T00:00:00Z
review text: Let \(G\) be a connected reductive group over a local field \(F\) which is quasisplit and which is split over an unramified extension. In this paper the author considers the matrix of the Satake isomorphism in terms of the natural bases of the source and of the target, and proves that all coefficients of this matrix that are not zero are actually positive numbers. The theorem is proved by induction on the semi-simple rank of \(G\). In the case \(G=GL_n\) one also shows how this positivity result can be deduced from the theory of symmetric functions. The (positivity) result is then applied to an existence problem of \(F\)-crystals which is a partial converse to Mazur's theorem relating the Hodge polygon and the Newton polygon.
Mathematics Subject Classification ID: 22E35
Mathematics Subject Classification ID: 14F30
zbMATH DE Number: 1423142
zbMATH Keywords: reductive groups
zbMATH Keywords: local fields
zbMATH Keywords: Satake isomorphism
zbMATH Keywords: positivity result
zbMATH Keywords: symmetric functions
zbMATH Keywords: \(F\)-crystals
zbMATH Keywords: Hodge polygon
zbMATH Keywords: Newton polygon
author: Michael Rapoport
reviewed by: Lucian Bădescu
MaRDI profile type: Publication
full work available at URL: s002290050010
OpenAlex ID: W2080910340
DOI: 10.1007/S002290050010
description: scientific article; zbMATH DE number 1423142