Editing Cusp form (section)

==Related concepts==
In the larger picture of [[automorphic form]]s, the cusp forms are complementary to [[Eisenstein series]], in a ''discrete spectrum''/''continuous spectrum'', or ''discrete series representation''/''induced representation'' distinction typical in different parts of [[spectral theory]]. That is, Eisenstein series can be 'designed' to take on given values at cusps. There is a large general theory, depending though on the quite intricate theory of parabolic subgroups, and corresponding [[cuspidal representation]]s.

Consider <math>P=MU</math> a standard parabolic subgroup of some reductive group <math>G</math> (over <math>\mathbb{A}</math>, the [[adele ring]]), an automorphic form <math>\phi</math> on <math>U(\mathbb{A})M(k)\backslash G</math> is called cuspidal if for all parabolic subgroups <math>P'</math> such that <math>P_0\subset P'\subsetneq P</math> we have <math>\phi_{P'}=0</math>, where <math>P_0</math> is the standard minimal parabolic subgroup. The notation <math>\phi_{P}</math> for <math>P=MU</math> is defined as <math>\phi_P (g) =\int_{U(k)\backslash U(\mathbb{A})} \phi(ug) du</math>.