In mathematics, parabolic induction is a method of constructing representations of a reductive group from representations of its parabolic subgroups.
, extending it to P by letting N act trivially, and inducing the result from P to G. There are some generalizations of parabolic induction using cohomology, such as cohomological parabolic induction and Deligne–Lusztig theory.
The philosophy of cusp forms was a slogan of Harish-Chandra, expressing his idea of a kind of reverse engineering of automorphic form theory, from the point of view of representation theory.
[1] The discrete group Γ fundamental to the classical theory disappears, superficially.
According to Nolan Wallach[4] Put in the simplest terms the "philosophy of cusp forms" says that for each Γ-conjugacy classes of Q-rational parabolic subgroups one should construct automorphic functions (from objects from spaces of lower dimensions) whose constant terms are zero for other conjugacy classes and the constant terms for [an] element of the given class give all constant terms for this parabolic subgroup.