Some results on \({\mathcal D}\)-modules on Borel varieties in characteristic \(p>0\)
type: Item
instance of: scholarly article
title: Some results on \({\mathcal D}\)-modules on Borel varieties in characteristic \(p>0\)
zbMATH Open document ID: 0845.14014
author: Rikard Bøgvad
published in: Journal of Algebra
publication date: 1995-06-14T00:00:00Z
review text: Let \(G\) be a semisimple connected and simply connected group over an algebraically closed field \(k\) of characteristic \(p > 0\), let \(B = TU\) be a Borel subgroup of \(G\), and let \(X = G/B\) be the Borel variety. Let \({\mathcal D}_X\) denote the sheaf of differential operators on the structure sheaf \({\mathcal O}_X\). \({\mathcal D}_X\) has distinct right and left \({\mathcal O}_X\)-module structures and is not Noetherian. For an invertible sheaf \(L\) on \(X\), \({\mathcal D}_L \simeq L \otimes {\mathcal D}_X \otimes L^{-1}\) denotes the sheaf of differential operators on \(L \). In particular, one can take the invertible sheaf \(L = {\mathcal O} (\lambda)\) on \(X\) associated to a dominant weight \(\lambda\) of \(T\); the corresponding sheaf of differential operators is denoted by \({\mathcal D}_\lambda\) and its global sections by \(\Gamma_\lambda\). The paper under review studies certain modules over \(\Gamma_\lambda\) and \({\mathcal D}_\lambda\). In particular, it is shown that the Weyl module with highest weight \(\lambda\) is irreducible over \(\Gamma_\lambda\). Furthermore, up to isomorphism, \({\mathcal O}(\lambda)\) is the only indecomposable coherent \({\mathcal D}_\lambda\)-module (theorem 2.2). Another result of the paper is that there exists a unique \(B^-\)-equivariant \({\mathcal D}_\lambda\)-module with support on a given Schubert variety.
Mathematics Subject Classification ID: 14F10
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 14G15
zbMATH DE Number: 763965
zbMATH Keywords: characteristic \(p\)
zbMATH Keywords: Borel subgroup
zbMATH Keywords: Borel variety
zbMATH Keywords: differential operators
reviewed by: F. L. Zak
MaRDI profile type: Publication
full work available at URL: jabr.1995.1107
OpenAlex ID: W2090265957
DOI: 10.1006/JABR.1995.1107
description: scientific article; zbMATH DE number 763965