A Harish-Chandra homomorphism for reductive group actions
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 algebras \({\mathcal D} (X)^G\), \(({\mathcal D} (X)^G)'\), \({\mathfrak z} (X)'\) and \({\mathcal D} (X)\) are all free \({\mathfrak z} (X)\)-modules. (5) Suppose a Borel subgroup of \(G\) has a dense orbit in \(X\), then \({\mathcal D} (X)^G = {\mathfrak z} (X)\), so the former is a polynomial ring. -- For the proof one considers the natural filtrations by degree of all the above algebras. The corresponding graded objects have been considered in an earlier paper by the author [Invent. Math. 99, 1-23 (1990; Zbl 0726.20031)]. The central problem which is solved in the present paper is to lift the results from the graded to the filtered case.
scientific article; zbMATH DE number 742782