Invariant differential operators on a real semisimple Lie algebra and their radial components

Author:
Mohsen Pazirandeh

Journal:
Trans. Amer. Math. Soc. **182** (1973), 119-131

MSC:
Primary 22E45; Secondary 17B20

MathSciNet review:
0335689

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Abstract: Let be the symmetric algebra over the complexification of the real semisimple Lie algebra . For is the corresponding differential operator on . denotes the algebra generated by and multiplication by polynomials on . For any open set is the algebra of differential operators with -coefficients on *U*. Let be a Cartan subalgebra of the set of its regular points and , *P* some positive system of roots. Let , *G* the connected adjoint group of .

Harish-Chandra showed that, for each , there is a unique differential operator on such that for all *G*-invariant , and that if , then for some . In particular and invariant.

We prove these results by different, yet simpler methods. We reduce evaluation of , invariant) via Weyl's unitarian trick, to the case of compact *G*. This case is proved using an evaluation of a family of *G*-invariant eigenfunctions on:

For *G*-invariant , we prove using properties of derivations of induced by and of the algebra of polynomials on invariant under the Weyl group.

**[1]**Harish-Chandra,*Differential operators on a semisimple Lie algebra*, Amer. J. Math.**79**(1957), 87–120. MR**0084104****[2]**Harish-Chandra,*Invariant differential operators and distributions on a semisimple Lie algebra*, Amer. J. Math.**86**(1964), 534–564. MR**0180628****[3]**Harish-Chandra,*The characters of semisimple Lie groups*, Trans. Amer. Math. Soc.**83**(1956), 98–163. MR**0080875**, 10.1090/S0002-9947-1956-0080875-7**[4]**Harish-Chandra,*On some applications of the universal enveloping algebra of a semisimple Lie algebra*, Trans. Amer. Math. Soc.**70**(1951), 28–96. MR**0044515**, 10.1090/S0002-9947-1951-0044515-0**[5]**G. C. Shephard and J. A. Todd,*Finite unitary reflection groups*, Canadian J. Math.**6**(1954), 274–304. MR**0059914****[6]***Selecta Hermann Weyl*, Birkhäuser Verlag, Basel und Stuttgart, 1956 ( German). MR**0075883**

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DOI:
https://doi.org/10.1090/S0002-9947-1973-0335689-7

Keywords:
Radial components,
invariant differential operators

Article copyright:
© Copyright 1973
American Mathematical Society