Cuspidal Automorphic Representation

A cuspidal automorphic representation of a reductive group over a global field is an irreducible representation occurring in its cuspidal automorphic spectrum. Cuspidality means that every constant term along a proper parabolic subgroup vanishes. If is the unipotent radical of such a subgroup, a vector satisfies

for every , where is the global field and its adele ring. For a quasi-split group, a representation is globally generic when some vector has a nonzero Whittaker coefficient. This coefficient is the integral over against the inverse of a nondegenerate character trivial on , where is the unipotent radical of a Borel subgroup.