\varphi({\mathbf x}_0,{\mathbf y}) \varphi_{{\mathrm{KM}}} \left( \sqrt{v_0} x_0+ \frac{y \cdot v'_{00} }{ \sqrt{v_0}} \right)\wedge { \mathfrak{g}}(ya_0^*)    \prod_{i=0}^d \varphi_{\infty_i}( \sigma_i ({\mathbf x}_0, {\mathbf y}) a_i)  \ e( T \mathbf u )$$ converges absolutely to a smooth form on ${\mathbb H}