Let $\alpha\in\mathfrak{gl}(n,\mathbb{R})$ and $F\colon GL(n,\mathbb{R})\to \mathbb{C}$ be smooth. We have the following differential operator $D_{\alpha}$ acting on $F$ by $$(D_{\alpha}F)(g)=\frac{\partial}{\partial t}F(g\cdot \exp(t\alpha)) \Big{|}_{t=0} $$ One can think of $D_{\alpha}$ as the left invariant vector field $X_{\alpha}$ on $GL(n,\mathbb{R})$ associated to the matrix $\alpha$, where $$X_{\alpha}(g)=\sum_{i,j}(g\cdot\alpha)_{i j}\frac{\partial}{\partial g_{ij}}$$ where $(g\cdot\alpha)_{i j}$ denoted the $(i,j)$th entry of the matrix $g\cdot\alpha$. These differential operators $D_{\alpha}$ generate an associative algebra ${\mathcal{D}} ^n$ over $\mathbb{R}$ where the operation is the composition of the operators. I want to show that if $D$ is in the centre of $\mathcal{D}^n$, then $D$ is well defined on the space of smooth functions $$f\colon GL(n,\mathbb{Z})\big\backslash GL(n,\mathbb{R})\big/ (O(n,\mathbb{R})Z_n)\to \mathbb{C} $$ i.e., $$(Df)(\gamma\cdot g\cdot k\cdot \delta)=(Df)(g) $$ for all $g\in GL(n,\mathbb{R}), \gamma\in GL(n,\mathbb{Z}), \delta\in Z_n$ and $k\in O(n,\mathbb{R})$. ($Z_n$ denotes the diagonal matrices with all diagonal entries equal).

I am not able to show the right invariance under an element of $O(n,\mathbb{R})$. Need some hints.


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