Given an analytic group action of $SL(n, \mathbb{R})$ on $\mathbb{R}^m$ fixing the origin, in this article the author then proceeds to "complexify the analytic $SL(n, \mathbb{R})$ action to obtain a local holomorphic action of $SL(n, \mathbb{C})$ on a neighbourhood of the origin in $\mathbb{C}^m$" (pg 139, 1st paragraph of theorem 2.6).
That sounds quite reasonable, given that the action is analytic. I believe I see how one obtains $SL(n, \mathbb{C})$ as the complexification of $SL(n, \mathbb{R})$, but how may I describe the action of an element in $SL(n, \mathbb{C})$ on $\mathbb{C}^m$ given that the group action is the result of a complexified action of $SL(n, \mathbb{R})$?
I've the following for the complexification of $SL(n, \mathbb{R})$: we think of this group as sitting in $SL(n, \mathbb{C})$. The lie sub-algebra $\mathfrak{sl}(n, \mathbb{R})$ is the set of n-by-n matrices over $\mathbb{R}$ with vanishing trace. Looking at $\mathfrak{sl}(n, \mathbb{R})\oplus i \mathfrak{sl}(n, \mathbb{R})$ it is clear that we obtain all n-by-n matrices over $\mathbb{C}$ with vanishing trace, that is $\mathfrak{sl}(n, \mathbb{C})$. The complexification of $SL(n, \mathbb{R})$ is then the image of $\mathfrak{sl}(n, \mathbb{C})$ under the exponential map, which translates to the connected component of $SL(n, \mathbb{C})$ containing the identity. Since $SL(n, \mathbb{C})$ is connected the image is all of $SL(n, \mathbb{C})$.
So taking an element $g\in SL(n, \mathbb{C})$ we can write it as $g=e^{\alpha+i\beta}$ for $\alpha, \beta \in \mathfrak{sl}(n, \mathbb{R})$, but I'm not sure how to put these pieces together to give me the action of $g$ on $\mathbb{C}^m$. Can this reasoning be completed, or is there perhaps a better approach?
The following is only a partial answer, because I'm working only locally in both the group and the space on which it acts. For a complete answer, one would have to show that the resulting action is (or can be extended to be) well-defined on the whole group.
The given action of $SL(n,\mathbb R)$ on $\mathbb R^n$ is given, in a neighborhood of the identity in $SL(n,\mathbb R)$ and the origin in $\mathbb R^n$, by convergent power series in the entries of matrices in $SL(n,\mathbb R)$ and the coordinates of points in $\mathbb R^n$. Use the same power series with complex matrix entries and complex coordinates to define a local action of $SL(n,\mathbb C)$ on $\mathbb C^n$.