Let $S_g$ be a Riemann surface of genus $g\ge 2$ and $P=\{p_1,\dots,p_n\}\subset S_g$ a set of $n\ge 1$ points. Denote by $Diff_P^0(S_g)$ the set of diffeomorphisms of $S_g$ homotopic to the identity and which fix the points of $P$.
For any $\alpha=(\alpha_1,\dots,\alpha_n)\in \mathbb{N}^n$ such that $\sum_{i=1}^n\alpha_i=4g-4$, and $l\in \{1,-1\}$ one can define the set $Q(\alpha,l)$ of half translation surfaces with zeroes on $P$ prescribed by $\alpha$ (this means that the half-translation surfaces have a zero of order $\alpha_i$ on $p_i$, $i=1,\dots,n$) and $l$ (if $l=1$ then the half-translation surfaces have trivial holonomy, if $l=-1$ then the holonomy is $-Id$). The Teichmuller space $TQ(\alpha,l)$ of half-translation surfaces with zeroes on $P$ prescribed by $\alpha$ and $l$ is the quotient of $Q(\alpha,l)$ by the action of $Diff_P^0(S_g)$ (which acts by pullback).
I have read in many articles that there is an action of $SL(2,\mathbb{R})$ on $TQ(\alpha,l)$, which for every $A\in SL(2,\mathbb{R})$ is realized simply composing the charts of the half-translation surface with $A$.
My question is: is there a similar action for $GL(2,\mathbb{R})$? Yoccoz in this article this article defines it for translation surfaces, so the answer is "yes" for translation surfaces. Is there a problem extending the action of $GL(2,\mathbb{R})$ on $TQ(\alpha,l)$? Maybe the holonomy can be changed? I am asking because I can not figure it out.