Residual automorphic forms and spherical unitary representations of exceptional groups


Arthur has conjectured that the unitarity of a number of representations can be shown by finding appropriate automorphic realizations. This has been verified for classical groups by M\oe glin and for the exceptional Chevalley group $G_2$ by Kim. In this paper we extend their results on spherical representations to the remaining exceptional groups $E_6$, $E_7$, $E_8$, and $F_4$. In particular, we prove Arthur’s conjecture that the spherical constituent of an unramified principal series of a Chevalley group over any local field of characteristic zero is unitarizable if its Langlands parameter coincides with half the weighted marking of a coadjoint nilpotent orbit of the Langlands dual Lie algebra.

Computer code referenced in this paper is available for download at the following location:


Stephen D. Miller

Department of Mathematics, Hill Center-Busch Campus, Rutgers, The State University of New Jersey, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019