Philosophy Of Cusp Forms
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 that has a finite kernel and is a ...
from representations of its parabolic subgroups. 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 applica ...
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 viewed ...
, such as cohomological parabolic induction 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 gr ...
s'' was a slogan of
Harish-Chandra Harish-Chandra (né Harishchandra) 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 ...
, 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 o ...
theory, from the point of view of
representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
. 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 ...
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 (, , ; – 5 October 2009) was a prominent Soviet and American mathematician, one of the greatest mathematicians of the 20th century, biologist, teache ...
, and the philosophy is a precursor of the
Langlands program In mathematics, the Langlands program is a set of conjectures about connections between number theory, the theory of automorphic forms, and geometry. It was proposed by . It seeks to relate the structure of Galois groups in algebraic number t ...
. A consequence for thinking about representation theory is that
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 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 two-volume treatise ''Real Reductive Groups''. Education and career Wallach did his ...
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 generalize ...
.


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