A Harish-Chandra homomorphism for reductive group actions
type: Item
instance of: scholarly article
title: A Harish-Chandra homomorphism for reductive group actions
zbMATH Open document ID: 0828.22017
DOI: 10.2307/2118600
author: Friedrich Knop
published in: Annals of Mathematics. Second Series
publication date: 1995-07-16T00:00:00Z
review text: For a semisimple complex Lie algebra \(\mathfrak g\), Harish-Chandra constructed an isomorphism of the center \(\mathfrak z\) of the universal enveloping algebra with the algebra of invariant polynomials \(\mathbb{C} [{\mathfrak t}^*]^W\), where \({\mathfrak t} \subset {\mathfrak g}\) is a Cartan subalgebra and \(W\) is the Weyl group of \(({\mathfrak g}, {\mathfrak t})\). Suppose \(\mathfrak g\) is the complexification of the real Lie algebra of a semisimple Lie group \(G\) then the admissible spectrum of \(G\) is a finite cover of the spectrum of the commutative algebra \(\mathfrak z\). Therefore the above isomorphism is one of the most basic tools in representation theory of semisimple groups. Later on this construction was extended to the algebra of invariant differential operators on a symmetric space \(G/K\). Surprisingly these results can be generalized to the case of an arbitrary smooth, affine \(G\)-variety \(X\), where we consider \(G\) as defined over some algebraically closed field \(k\) of characteristic zero now. Let \({\mathcal D} (X)\) denote the algebra of all differential operators on \(X\) and \({\mathcal D} (X)^G\) the subalgebra of all \(G\)-invariant operators. Further write \({\mathfrak z} (X)\) for the center of \({\mathcal D} (X)^G\). In the paper under consideration the following results are proven: (1) There is a subspace \({\mathfrak a}^*_X \subset {\mathfrak t}^*\) and a subgroup \(W_X \subset W\) stabilizing \({\mathfrak a}^*_X\) such that there is an isomorphism \({\mathfrak z} (X) @>\sim>> k[\rho + {\mathfrak a}^*_X]^{W_X}\). (2) The ``little Weyl group'' \(W_X\) acts on \({\mathfrak a}^*_X\) as a reflection group. In particular, \({\mathfrak z}(X)\) is a polynomial ring. (3) For any subalgebra \(A \subset {\mathcal D} (X)\) write \(A'\) for the commutant of \(A\) in \({\mathcal D} (X)\) then there is a canonical isomorphism \({\mathcal D} (X)^G \otimes_{{\mathfrak z} (X)} ({\mathcal D} (X)^G)' @>\sim>> {\mathfrak z} (X)'\). (4) The a…
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 742782
zbMATH Keywords: Harish-Chandra homomorphism
zbMATH Keywords: invariant differential operators
reviewed by: Anton Deitmar
MaRDI profile type: Publication
OpenAlex ID: W2318894543
description: scientific article; zbMATH DE number 742782