In mathematics, parabolic induction is a method of constructing representation

Group representation

In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...

s of a reductive group

Reductive group

In mathematics, a reductive group is an algebraic group G over an algebraically closed field such that the unipotent radical of G is trivial . Any semisimple algebraic group is reductive, as is any algebraic torus and any general linear group...

 from representations of its parabolic subgroups.

If G is a reductive algebraic group and is the Langlands decomposition

Langlands decomposition

In mathematics, the Langlands decomposition writes a parabolic subgroup P of a semisimple Lie group as a product P=MAN of a reductive subgroup M, an abelian subgroup A, and a nilpotent subgroup N.- Applications :...

 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

Induced representation

In mathematics, and in particular group representation theory, the induced representation is one of the major general operations for passing from a representation of a subgroup H to a representation of the group G itself. It was initially defined as a construction by Frobenius, for linear...

 the result from P to G.

There are some generalizations of parabolic induction using cohomology

Cohomology

In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...

, such as cohomological parabolic induction and Deligne–Lusztig theory

Deligne–Lusztig theory

In mathematics, Deligne–Lusztig theory is a way of constructing linear representations of finite groups of Lie type using ℓ-adic cohomology with compact support, introduced by ....

.

Philosophy of cusp forms

The philosophy of cusp forms was a slogan of Harish-Chandra

Harish-Chandra

Harish-Chandra was an Indian mathematician, who did fundamental work in representation theory, especially Harmonic analysis on semisimple Lie groups. -Life:...

, expressing his idea of a kind of reverse engineering of automorphic form

Automorphic form

In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms...

 theory, from the point of view of representation theory

Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...

. The discrete group

Discrete group

In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one...

 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 representation

Cuspidal representation

In number theory, cuspidal representations are certain representations of algebraic groups that occur discretely in L^2 spaces. The term cuspidal is derived, at a certain distance, from the cusp forms of classical modular form theory...

s. A similar philosophy was enunciated by Israel Gelfand

Israel Gelfand

Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand was a Soviet mathematician who made major contributions to many branches of mathematics, including group theory, representation theory and functional analysis...

, and the philosophy is a precursor of the Langlands program

Langlands program

The Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. It was proposed by ....

. A consequence for thinking about representation theory is that cuspidal representation

Cuspidal representation

In number theory, cuspidal representations are certain representations of algebraic groups that occur discretely in L^2 spaces. The term cuspidal is derived, at a certain distance, from the cusp forms of classical modular form theory...

s are the fundamental class of objects, from which other representations may be constructed by procedures of induction.

According to Nolan Wallach

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. This is almost possible and leads to a description of all automorphic forms in terms of these constructs and cusp forms. The construction that does this is the Eisenstein series

Eisenstein series

Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly...

.