zbMATH — the first resource for mathematics

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.

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)
Full Text: DOI
[1] Harish-Chandra, Differential operators on a semisimple Lie algebra, Amer. J. Math. 79 (1957), 87 – 120. · Zbl 0072.01901 · doi:10.2307/2372387 · doi.org
[2] Harish-Chandra, Invariant differential operators and distributions on a semisimple Lie algebra, Amer. J. Math. 86 (1964), 534 – 564. , https://doi.org/10.2307/2373023 Harish-Chandra, Some results on an invariant integral on a semisimple Lie algebra, Ann. of Math. (2) 80 (1964), 551 – 593. , https://doi.org/10.2307/1970664 Harish-Chandra, Invariant eigendistributions on a semisimple Lie algebra, Inst. Hautes Études Sci. Publ. Math. 27 (1965), 5 – 54. Harish-Chandra, Invariant eigendistributions on a semisimple Lie group, Trans. Amer. Math. Soc. 119 (1965), 457 – 508. · Zbl 0161.33804
[3] -, Invariant eigendistributions on a semi-simple Lie algebra, Inst. Hautes Études Sci. Publ. Math. 27 (1965), 5-54. · Zbl 0199.46401
[4] Günter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, Revised edition, Graduate Studies in Mathematics, vol. 22, American Mathematical Society, Providence, RI, 2000. · Zbl 0957.16001
[5] T. Levasseur and J. T. Stafford, Rings of differential operators on classical rings of invariants, Mem. Amer. Math. Soc. 81 (1989), no. 412, vi+117. · Zbl 0691.16019 · doi:10.1090/memo/0412 · doi.org
[6] J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1987. With the cooperation of L. W. Small; A Wiley-Interscience Publication. · Zbl 0644.16008
[7] Susan Montgomery, Fixed rings of finite automorphism groups of associative rings, Lecture Notes in Mathematics, vol. 818, Springer, Berlin, 1980. · Zbl 0449.16001
[8] Nolan R. Wallach, Invariant differential operators on a reductive Lie algebra and Weyl group representations, J. Amer. Math. Soc. 6 (1993), no. 4, 779 – 816. · Zbl 0804.22004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.