  Theta series for exceptional groups have been suggested as a possible
description of the eleven-dimensional quantum supermembrane. We present
explicit formulae for these automorphic forms whenever the underlying Lie group
$G$ is split (or complex) and simply laced. Specifically, we review and
construct explicitly the minimal representation of $G$, generalizing the
Schr\"odinger representation of symplectic groups. We compute the spherical
vector in this representation, i.e. the wave function invariant under the
maximal compact subgroup, which plays the role of the summand in the
automorphic theta series. We also determine the spherical vector over the
complex field. We outline how the spherical vector over the $p$-adic number
fields provides the summation measure in the theta series, postponing its
determination to a sequel of this work. The simplicity of our result is
suggestive of a new Born-Infeld-like description of the membrane where
U-duality is realized non-linearly. Our results may also be used in
constructing quantum mechanical systems with spectrum generating symmetries.
