Minimal representations, spherical vectors, and exceptional theta series I

TL;DR

This paper provides explicit formulas for automorphic theta series associated with exceptional groups, focusing on minimal representations and spherical vectors, with implications for quantum supermembranes and U-duality.

Contribution

It explicitly constructs minimal representations and spherical vectors for split and complex exceptional groups, extending previous work and suggesting new physical models.

Findings

Explicit formulas for automorphic theta series for exceptional groups

Construction of minimal representations and spherical vectors

Potential applications in quantum supermembranes and symmetry models

Abstract

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.