the Action of $SL_2(\mathbb{R})$ on a fundamental domain of $\Gamma$ Let $F=\{z\in H: |z|\geq1, |Re(z)|\leq 1/2\}$. It is a fundamental domain of the modular group $\Gamma$ acting on the upper half plane $H$. (Strictly speaking, one should take part of the boundary from $F$ to make it a fundamental domain. For example, one can use $F_1=F-\{z\in F:Re(z)=1/2\}-\{z\in F:Re(z)>0 \text{ and }|z|=1\}$.
For a matrix $A\in SL_2(\mathbb{R})$, it is said that $A(F)=\{A(x):x\in F\}$ gives a fundamental domain for $\Gamma$. My question is how to prove this.
To prove it, one thing we should show is: for every $s\in H$, there is a $g\in \Gamma$ such that $g(s)\in A(F)$.
This claim is used in the proof that the Hecke operators are Hermitian with respect the Petersson inner product. It seems like a trivial result, but I couldn't prove this. I think I must have missed something. Thus any comment would be appreciated.
 A: As @SamNead notes, this is simply not true. It would be convenient if it were true, because then there'd be an easy proof of the self-adjointness of Hecke operators, yes. Indeed, this error is one of the standard, popular unfortunate-choices to try to prove that self-adjointness. The self-adjointness certainly does hold, and for reasons not so distant from this literally-incorrect argument.
It seems to me that arguments depending on rearrangements of fundamental domains, not to mention false "principles" about such, are to be mistrusted. Also, there is the clumsiness that the completely classical presentation of Hecke operators as sums has summands that are not modular forms of the same level... and it's only by summing up, that is, averaging, that the outcome is again of the same type. This awkwardness (which cannot be trivially overcome), and the very simple general fact that a unimodular topological group $G$ acts on the right on $L^2(\Gamma\backslash G)$ unitarily (by translation), suggest that a rewrite of the whole situation will make things clearer, and give a more persuasive self-adjointness argument.
Indeed, via some sort of Strong Approximation argument and other standard devices, automorphic forms on domains and Lie groups are converted to automorphic forms on adele quotients $G_k\backslash G_{\mathbb A}$, and Heck operators are converted to integral operators (often misleading called "convolution operators") of the form $Tf(g)=\int_H f(gh)\cdot \varphi(h)\;dh$ for a closed subgroup $H$ of $G$ and continuous compactly-supported function $\varphi$ on $H$. One directly determines the condition for such operators to be self-adjoint on $L^2(\Gamma\backslash G)$, and also sees that the Hecke operators meet that general criterion.
(Further, the integration on the quotient can be completely characterized without choice or specification of any fundamental domain.)
In particular, the translations involved in Hecke operators are converted to right translations, which obviously have no interaction with left quotients.
A: As Misha says, this is false as stated.  To see this: Apply the dilation $A \colon z \mapsto z/2$ to $F$.  Note $i$ lies in the interior of $A(F)$.  Since $i$ is fixed by a torsion element of $SL(2,\mathbb{Z})$, deduce that $A(F)$ is not a fundamental domain.  
There are several different things you might be trying to ask here.  I suggest you have a bit of a think, and then ask a new question. 
