Minimal representations, spherical vectors, and exceptional theta series I

Details Minimal representations, spherical vectors, and exceptional theta series I Minimal representations, spherical vectors, and exceptional theta series I

2001-07-25

arxiv.org/abs/hep-th/0107222v2

Commun.Math.Phys.226:1-40,2002

Physics

High Energy Physics

High Energy Physics - Theory

41 pages, uses JHEP.cls, form and mathematica files at http://www.lpthe.jussieu.fr/~pioline/minrep/; v2: discussion of p-adic

Scientific paper

10.1007/s002200200601

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.

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