Hallo,
I am trying to read the paper "Calibrated embeddings in the special Lagrangian and coassociative cases" by Robert Bryant and I have a question. I hope that maybe one of you could give me some answers. The paper can be found here: http://arxiv.org/abs/math/9912246 . Here is my question: On page 12, in the middle, there are defined the spaces $\mathfrak{h}_{k}$ = { $x\in \mathbb{R}^{n\times n}$ | $\iota^{*}_{k}(x.\alpha)=0$, $\forall \alpha \in $ $\Lambda^{*}(\mathbb{R}^{n})^{G}$ }. I am not very familiar with the notation. I think $x.\alpha$ means the Lie-derivative of $\alpha$ in the direction $x$. Am I right? If so, using the formula for the Lie-derivative $\mathcal{L}_{X}= \iota d + d\iota$. I do not see how to plug in a matrix $x \in \mathbb{R}^{n\times n}$ in a form defined on $\mathbb{R}^{n}$. How can this be understood? I think, by using the $\mathbb{R}^{n}$-valued form $\nu$ defined by $\nu (v) = u(\pi ' (v))$ , for all $v \in T _{u} F$ (also on page 12 in the middle). But still, for $x \in \mathbb{R}^{n\times n} = Lie(GL(n,\mathbb{R}))=ker(\pi ')$ one gets $ \nu(x)=0 $. How is this to be understood? I would be very tankfull if somebody could help me with this.
greetings hapchiu