Differential operators on quotients of simple groups
type: Item
instance of: scholarly article
title: Differential operators on quotients of simple groups
zbMATH Open document ID: 0835.14019
published in: Journal of Algebra
publication date: 1996-04-15T00:00:00Z
review text: Let \(X\) be an affine complex algebraic variety, and let \(\Delta (X)\) denote the (non-commutative) algebra of algebraic differential operators on \(X\). Then \(\Delta (X)\) has a filtration \(\{\Delta^n (X)\}\) by order of differentiation, and the associated graded algebra \(\text{gr} \Delta (X)\) is commutative. Let \(X\) be smooth and a \(G\)-variety \((G\) is a reductive complex algebraic group). Let \(\pi_X : X \to X/G\) be the quotient morphism. Then one has a natural map \((\pi_X)_* : (\Delta^n (X))^G \to \Delta^n (X/G)\). The main aim of the paper is the following: Theorem. Let \(G\) be simple and connected, and let \(V\) be an irreducible nontrivial \(G\)-module such that \(\dim V/G > 0\). Then the following are equivalent (1) \(V\) is 2-modular; (2) \(V\) is 2-large; (3) \((\pi_V)_*\) is graded surjective; (4) \((\pi_V)_*\) is surjective; (5) \(V/G\) has no codimension one strata; (6) \(V\) is not coregular.
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 16S32
Mathematics Subject Classification ID: 13N10
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 665866
zbMATH Keywords: algebraic group acting on a complex algebraic variety
zbMATH Keywords: algebra of algebraic differential operators
zbMATH Keywords: quotient morphism
author: Gerald W. Schwarz
reviewed by: Rodolfo Salvi
MaRDI profile type: Publication
full work available at URL: jabr.1994.1282
OpenAlex ID: W2015119263
DOI: 10.1006/JABR.1994.1282
description: scientific article; zbMATH DE number 665866