Philosophy of cusp forms
In mathematics, parabolic induction is a method of constructing representations of a reductive group from representations of its parabolic subgroups.
If G is a reductive algebraic group and is the Langlands decomposition of a parabolic subgroup P, then parabolic induction consists of taking a representation of , 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. The discrete group Γ fundamental to the classical theory disappears, superficially. What remains is the basic idea that representations in general are to be constructed by parabolic induction of cuspidal representations. A similar philosophy was enunciated by Israel Gelfand, and the philosophy is a precursor of the Langlands program. A consequence for thinking about representation theory is that cuspidal representations are the fundamental class of objects, from which other representations may be constructed by procedures of induction.
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