Small automorphic representations and degenerate Whittaker vectors
Journal article, 2016

We investigate Fourier coefficients of automorphic forms on split simply-laced Lie groups G. We show that for automorphic representations of small Gelfand-Kirillov dimension the Fourier coefficients are completely determined by certain degenerate Whittaker vectors on G. Although we expect our results to hold for arbitrary simply-laced groups, we give complete proofs only for G=SL(3) and G=SL(4). This is based on a method of Ginzburg that associates Fourier coefficients of automorphic forms with nilpotent orbits of G. Our results complement and extend recent results of Miller and Sahi. We also use our formalism to calculate various local (real and p-adic) spherical vectors of minimal representations of the exceptional groups E_6, E_7, E_8 using global (adelic) degenerate Whittaker vectors, correctly reproducing existing results for such spherical vectors obtained by very different methods.


Henrik Gustafsson

Chalmers, Physics, Theoretical Physics

Axel Kleinschmidt

International Solvay Institute for Physics and Chemistry

Max Planck Society

Daniel Persson

Chalmers, Physics, Theoretical Physics

Journal of Number Theory

0022-314X (ISSN) 1096-1658 (eISSN)

Vol. 166 344-399

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