Minimal representations, spherical vectors, and exceptional theta series I

TL;DR

This work develops explicit formulas for minimal representations and spherical vectors of split (or complex) simply-laced groups, providing concrete real and complex spherical vectors for $A$, $D$, and $E$ types and detailing how these enter exceptional theta series. By exploiting a 5-graded nilpotent orbit, Weyl generators $S$ and $A$, and both real and complex polarizations, the authors construct automorphic forms whose summation measures and invariants under discrete subgroups underpin membrane-inspired partition functions. They connect the resulting spherical vectors to Bessel-kernel structures and invariant polynomials $I_2,I_3,I_4$, with explicit examples such as $D_4$ and the exceptional groups $E_6,E_7,E_8$, and they discuss p-adic and complex-analytic generalizations. The results point to potential Born-Infeld-like membrane descriptions with nonlinear realizations of U-duality and offer a framework for quantum systems with spectrum-generating exceptional symmetries, while laying groundwork for future intructions to p-adic measures and non-simply-laced cases.

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 Schrödinger 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.