Automorphic functionals for the minimal representations of groups of type $D_n$ and $E_n$

Let $G$ be a split simply-connected group of type $D$ or $E$. The minimal automorphic representation $\mathrm{\Pi }$ of $G(\mathbb{A})$ admits a realization on a space of functions $\mathcal{S}(X(\mathbb{A}))$ for a variety $X$. In this paper we write explicitly an automorphic, i.e. $G(F)$-invariant, functional on $\mathcal{S}(X(\mathbb{A}))\mathrm{.}$