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Derivatives of invariant polynomials on a semisimple Lie algebra. (English) Zbl 0637.17004
Harmonic analysis and operator algebras, Miniconf. Canberra/Aust. 1987, Proc. Cent. Math. Anal. Aust. Natl. Univ. 15, 228-241 (1987).
[For the entire collection see Zbl 0628.00008.]
Let $${\mathfrak g}$$ be a semisimple complex Lie algebra and let G be the adjoint group of $${\mathfrak g}$$. It is known that the algebra $$I({\mathfrak g})=C[{\mathfrak g}]^ G$$ of G-invariant polynomial functions on $${\mathfrak g}$$ is generated by $$n=rank({\mathfrak g})$$ algebraically independent homogeneous polynomials $$P_ 1,...,P_ n$$. Let $$\pi: {\mathfrak g}\to {\mathbb{C}}^ n$$ be the polynomial map given by $$\pi (x)=(P_ 1(x),...,P_ n(x))$$. Let $$d\pi_ x$$ be the differential at a point x of $${\mathfrak g}$$. It is known that rank$$(d\pi_ x)=n$$ if and only if x is regular [B. Kostant, Am. J. Math. 85, 327-404 (1963; Zbl 0124.268)].
In this paper, the author gives rank$$(d\pi_ x)$$ for every $$x\in {\mathfrak g}$$ except for the case of one nilpotent class in type $$E_ 8$$. For the classical simple Lie algebras, one can give a reasonably straightforward computation of the ranks. For the exceptional Lie algebras, one need to use the classification of nilpotent conjugacy classes and detailed information in the closure of nilpotent classes. The author shows that the rank$$(d\pi_ x)$$ is equal to the dimension of the vector subspace of $${\mathfrak g}$$ spanned by $$\{\phi_ 1(x),...,\phi_ n(x)\}$$, where $$\phi_ i=\text{grad}(P_ i)$$, and also to the multiplicity of the adjoint representation of G in the G-module C[$$\overline{Gx}]$$ of the regular functions on $$\overline{Gx}$$ the closure of the orbit Gx.
Using this fact together with a theorem of W. Borho and H. Kraft [Comment. Math. Helv. 54, 61-104 (1979; Zbl 0395.14013)], the author gives a number of examples of non-normal nilpotent orbit closures in the exceptional Lie algebras of types $$E_ 6, E_ 7, E_ 8$$. This paper contains only a statement of the results, with outlines of proofs.
Reviewer: E.Abe

##### MSC:
 17B20 Simple, semisimple, reductive (super)algebras 22E46 Semisimple Lie groups and their representations