Fourier Transform and the minimal representation of $E_7$

We consider the minimal representation of the adjoint split group $E_7$ over a p-adic field. The representation has a model in a space of functions on a 17 dimensional cone $\mathrm{\Omega }$, and elements of the unique parabolic subgroup Q with abelian radical act by simple geometric formulas. We write a formula for the action of an involutive element $s$, conjugating $Q$ to the opposite parabolic $\bar{Q}$. The resulting integral operator, called a Fourier transform on $\mathrm{\Omega }$, is related to generalized Fourier transform, defined by Braverman and Kazhdan.