I am starting to read up on Hilbert Modular Forms and I am a bit confused with the definition. I am looking at multiple references and I am seeing some definitions that define Hilbert Modular Forms to be functions invariant under the action of $$GL(\mathcal{O}_L \oplus\mathfrak{U})^+ = \{\begin{pmatrix}a&b\\c&d \end{pmatrix}:a,d\in \mathcal{O}_L, b\in \mathfrak{U} , c\in \mathfrak{U}^{-1}, ad-bc\in {\mathcal{O}_L^\times}^+ \}$$ or maybe of a subgroup of finite index.
I have also seen several definitions of it as being functions invariant under a group $\Gamma$ which is a subgroup of $SL_2(L)$ and is commensurable with $SL_2(\mathcal{O}_L)$.
I am having a hard time reconciling both definitions. How are these two definitions related? I imagine that they're not entirely equivalent but there's perhaps very few differences in working with either?