In mathematics, parabolic induction is a method of constructing representations of a

reductive group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct ...

from representations of its

parabolic subgroup In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgro ...

s. If ''G'' is a reductive algebraic group and

P=MAN

is the

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 A key applicat ...

of a parabolic subgroup ''P'', then parabolic induction consists of taking a representation of

MA

, 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 In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...

, such as

cohomological parabolic induction In mathematics, a Zuckerman functor is used to construct representations of real reductive Lie groups from representations of Levi subgroups. They were introduced by Gregg Zuckerman (1978). The Bernstein functor is closely related. Notation an ...

and

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 . used these representations to find all representations of all ...

Philosophy of cusp forms

The ''philosophy of

cusp form In number theory, a branch of mathematics, a cusp form is a particular kind of modular form with a zero constant coefficient in the Fourier series expansion. Introduction A cusp form is distinguished in the case of modular forms for the modular gro ...

s'' was a slogan of

Harish-Chandra Harish-Chandra FRS (11 October 1923 – 16 October 1983) was an Indian American mathematician and physicist who did fundamental work in representation theory, especially harmonic analysis on semisimple Lie groups. Early life Harish-Chandra ...

, expressing his idea of a kind of reverse engineering of automorphic form theory, from the point of view of

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 studies modules over these abstract algebraic structures. In essen ...

. The

discrete group In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and o ...

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 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. In the c ...

s. A similar philosophy was enunciated by

Israel Gelfand Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand ( yi, ישראל געלפֿאַנד, russian: Изра́иль Моисе́евич Гельфа́нд, uk, Ізраїль Мойсейович Гел� ...

, and the philosophy is a precursor of the

Langlands program In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic num ...

. A consequence for thinking about representation theory is that

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

Nolan Wallach Nolan Russell Wallach (born August 3, 1940) is a mathematician known for work in the representation theory of reductive algebraic groups. He is the author of the 2-volume treatise ''Real Reductive Groups''. Education and career Wallach did his un ...

Nolan Wallac
Introductory lectures on automorphic forms
p.80.

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 n!-- source says "and" --> 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, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series can be generaliz ...
.

Notes

References

*A. W. Knapp, ''Representation Theory of Semisimple Groups: An Overview Based on Examples'', Princeton Landmarks in Mathematics, Princeton University Press, 2001. . *{{Citation, first=Daniel, last=Bump, title=Lie Groups, series=Graduate Texts in Mathematics, volume=225, publisher=Springer-Verlag, location=New York, year=2004, isbn=0-387-21154-3 Representation theory