Hello all

As is probably well-known to most, in the upper halfplane we have a natural action of $SL_2(\mathbb{R})$ through linear fractional transformations, and a $2$-form $\frac{dx dy}{y^2}$ which is invariant under this action.

In the case of the upper half-space $H_3 = \{ (z,t) \ | \ z\in \mathbb{C} \ , \ t>0 \ \}$ we have an analogous situation, only now we have a natural action of all of $SL_2(\mathbb{C})$. The fastest way of defining this action is by identifying a point $(z,t)$ in $H_3$ with the quaternion $q:= z+t j$, and then setting $$ \begin{pmatrix} a&b\cr c& d\cr \end{pmatrix}.(z,t) := (aq+b)(cq+d)^{-1}$$ Now for my question; I want to try to construct an $SL_2(\mathbb{Z}[i])$-invariant two-form on $H_3$ which when restricted to the plane $Im(z)=0$ takes the form $\frac{dxdt}{t^2}$. Do such a thing exist ? How would I go about constructing it? Any thoughts/references would be greatly appreciated.