Minimal representations, spherical vectors, and exceptional theta series I

41 pages, uses JHEP.cls, form and mathematica files at http://www.lpthe.jussieu.fr/~pioline/minrep/; v2: discussion of p-adic spherical vector and adelic formulation of theta series in sec 2.3, discussion of standard minimal rep improved in sec 3.1, complex spherical vector obtained in sec 4.5, plus various cosmetic changes. Final version to appear in CMPTheta 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\\ödinger 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