**Journal of Lie Theory
**

Vol. 9, No. 1, pp. 113-123 (1999)

#
** Z**-Gradations of Lie algebras and infinitesimal generators

##
D. Richter

School of Mathematics

University of Minnesota

206 Church St SE

Minneapolis MN 55455

drichter@math.umn.edu

**Abstract:** In this paper we arrive at explicit formulae for the infinitesimal generators of the action of a complex simple Lie group $G$ on the manifold $M=G/P$ where $P$ is a maximal parabolic subgroup. These formulae are obtained by assuming that local coordinates on $M$ are furnished by the nilpotent subalgebra ${\frak n}$ complementary to the maximal parabolic subalgebra ${\frak p}$ corresponding to $P$. For the classical isogeny classes $A_r$, $B_r$, $C_r$, and $D_r$, the components of the infinitesimal generators are never worse than quartic polynomials in the coordinate functions, but for the exceptional cases, $G_2$, $F_4$, and $E_r$, higher-degree polynomials frequently occur.

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