In
mathematics, parabolic induction is a method of constructing
representation
Representation may refer to:
Law and politics
*Representation (politics), political activities undertaken by elected representatives, as well as other theories
** Representative democracy, type of democracy in which elected officials represent a ...
s 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 direc ...
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 subgroup ...
s.
If ''G'' is a reductive algebraic group and
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
Abelian may refer to:
Mathematics Group theory
* Abelian group, a group in wh ...
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
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 and ...
and
Deligne–Lusztig theory.
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 g ...
s'' was a slogan of
Harish-Chandra, expressing his idea of a kind of reverse engineering of
automorphic form
In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...
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 representations. 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 n ...
. 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.
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 u ...
[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 general ...
.
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