Let $\Lambda = \{T \in \operatorname{Her}_2(\mathcal{O}) ; T \ge 0\})$ and $\mathcal{O}$ the maximal order of some quadratic imaginary number field. I write $T[U] := U^* \cdot T \cdot U$ where $U$ is some $2\times2$ matrix and $U^*$ is its conjugated transpose.
For $T_1, T_2 \in \Lambda$, define the equivalence relation $$T_1 \sim T_2 \quad \Leftrightarrow \quad \exists\ U \in \operatorname{GL}_2(\mathcal{O}) \colon\ T_1[U] = T_2.$$
Now, we say that $a \in \mathbb{Q}^{\Lambda}$ is $k$-invariant under $\operatorname{GL}_2(\mathcal{O})$ iff $$ \operatorname{det}(U)^k a(T[U]) = a(T) \quad U \in \operatorname{GL}_2(\mathcal{O}), T \in \Lambda.$$
Thus, for the set $$\mathcal{F} := \{a \in \mathbb{Q}^\Lambda; \text{$a$ is $k$-invariant under $\operatorname{GL}_2(\mathcal{O})$}\} ,$$ we can identify a basis by $$ \Lambda / {\sim} . $$
I wonder about common notations for $\mathcal{F}$. I have seen $\mathcal{F} = \left(\mathbb{Q}^\Lambda \right)^{\operatorname{GL}_2(\mathcal{O})}$ but I got confused myself a bit at the beginning with this (confused it with the function notation $A^B$).
Also, is there a common notation for $\Lambda / {\sim}$? Maybe $\Lambda /{\sim} = \Lambda / [\operatorname{GL}_2(\mathcal{O})]$?