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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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##### References:
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