I am wondering about the equivalence of two notions of a "nilpotent orbit".

The first notion, which I am familiar with, is as follows: given a lie group $G$ and a lie algebra $\frak{g}$, the orbit of a nilpotent element $n \in \frak{g}$ under the adjoint action of $G$ on $\frak{g}$ is called a $\textit{nilpotent orbit}$.

I encountered a seemingly different notion in Example 4.2 of the following survey on variations of Hodge structures. http://people.math.umass.edu/~cattani/ICTP/cattani_vhs.pdf In this setting, a nilpotent orbit is a certain map from a product of copies of the upper half plane to a certain space of filtrations of a complex vector space.

These two notions don't seem to obviously coincide, but since they share the same name, I would hope that they are related.

Is this the case?

  • 2
    $\begingroup$ I don't think there is any serious relationship apart from the fact that they both involve nilpotent elements of a Lie algebra. In fact, 'nilpotent' plays different roles: In the first, one is looking a the orbit of a nilpotent element under conjugation by the group, while in the second one is looking at the orbit of a Hodge structure under the action of a group generated by nilpotent elements. In other words, in the former, 'nilpotent' is qualifying the object, while in the latter it is qualifying the subject. $\endgroup$ – Keerthi Madapusi Pera Oct 2 '18 at 1:58

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