Fourier Transform and the minimal representation of $E_7$

TL;DR

This paper studies the minimal representation of the split group E_7 over p-adic fields, describing its Fourier transform and geometric action on a 17-dimensional cone, connecting it to generalized Fourier transforms.

Contribution

It provides explicit formulas for the Fourier transform and the action of involutions on the minimal representation of E_7, linking geometric and algebraic structures.

Findings

Explicit formula for the Fourier transform on the cone 

Description of the action of involutive elements on the representation

Connection to Braverman and Kazhdan's generalized Fourier transform

Abstract

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 $\Omega$, 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 $\Omega$, is related to generalized Fourier transform, defined by Braverman and Kazhdan.