# Derivative/Chain Rule (for MANLYfolds) Computation

Embarrasingly, I can't compute the following derivative.

$dh(X)=\left.\frac{d}{dt}h(e^{XT}]\right|_{t=0}$, where $X$ resides in the lie algebra of $\rm SL(3,\Bbb C)$ [ie $\mathfrak{sl}(3,\Bbb C)$] and h is the action $h(g)p(z)=p(g^{-1}z)$ where $g\in\rm{SL}(3,\Bbb C)$, $p$ is from the vector space of polynomials of degree $\le2$ in three variables, and $z\in \Bbb C^3$.

This is a small piece to a larger problem I'm struggling with. But I think diverging in this way is reasonable. If those with the power think otherwise, you may close the question without hesitation!

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Uh, what kind of folds? –  Nate Eldredge Apr 26 '12 at 5:13
@Nate: Think Charles Bronson in "Once Upon A Time In The West" sort of manifolds. –  Asaf Karagila Apr 26 '12 at 6:29
@AsafKaragila: Then I suppose the study of their harmonic functions must be of great interest... –  Nate Eldredge Apr 26 '12 at 12:39
@NateEldredge Actually their genera are of greater interest. As well as their respective...coverings... –  AsinglePANCAKE Apr 26 '12 at 13:06