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Invariant differential operators and an homomorphism of Harish-Chandra. (English) Zbl 0837.22011
Let \({\mathfrak g}\) be a complex reductive Lie algebra with adjoint group \(G\), Cartan subalgebra \({\mathfrak h}\), and Weyl group \(W\). Then \(G\) acts naturally on the algebra \({\mathcal O}({\mathfrak g})\) of polynomial functions and the ring \({\mathcal D}({\mathfrak g})\) of polynomial coefficient differential operators on \({\mathfrak g}\). Similarly \(W\) acts on \({\mathcal O}({\mathfrak h})\) and \({\mathcal D}({\mathfrak h})\). Harish-Chandra defined an algebra homomorphism \(\delta\) between the invariant subalgebras \({\mathcal D}({\mathfrak g})^G\) and \({\mathcal D}({\mathfrak h})^W\) [Am. J. Math. 86, 534-564 (1964; Zbl 0115.108)]. N. R. Wallach has recently shown that this map is surjective if \({\mathfrak g}\) has no simple factor of type \(E\) [J. Am. Math. Soc. 6, 779-816 (1993; Zbl 0804.22004)]. This enables him to give an easy proof of an important theorem of Harish-Chandra on invariant eigendistributions and an elegant new approach to the Springer correspondence.
Here the authors use an abstract result on fixed rings under a finite group action of rings of differential operators to give an elementary proof of Wallach’s surjectivity result that works for all \({\mathfrak g}\). They are able to simplify Wallach’s arguments at several crucial points.

MSC:
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
14L30 Group actions on varieties or schemes (quotients)
16S32 Rings of differential operators (associative algebraic aspects)
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