Cuspidal representation
Encyclopedia
In number theory
, cuspidal representations are certain representations
of algebraic groups that occur discretely in
spaces. The term cuspidal is derived, at a certain distance, from the cusp form
s of classical modular form
theory. In the contemporary formulation of automorphic representations, representations take the place of holomorphic functions; these representations may be of adelic algebraic group
s.
When the group is the general linear group
, the cuspidal representations are directly related to cusp forms and Maass forms. For the case of cusp forms, each Hecke eigenform (newform) corresponds to a cuspidal representation.
algebraic group over a number field K and let A denote the adele
s of K. Let Z denote the centre of G and let ω be a continuous unitary character
from Z(K)\Z(A)× to C×. Fix a Haar measure
on G(A) and let L20(G(K)\G(A), ω) denote the Hilbert space
of measurable
complex-valued functions, f, on G(A) satisfying
This is called the space of cusp forms with central character ω on G(A). A function occurring in such a space is called a cuspidal function. This space is unitary representation
of the group G(A) where the action
of g ∈ G(A) on a cuspidal function f is given by
The space of cusp forms with central character ω decomposes into a direct sum of Hilbert spaces
where the sum is over irreducible subrepresentations of L20(G(K)\G(A), ω) and mπ are positive integer
s (i.e. each irreducible subrepresentation occurs with finite multiplicity). A cuspidal representation of G(A) is such a subrepresentation (π, V) for some ω.
The groups for which the multiplicities mπ all equal one are said to have the multiplicity-one property.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, cuspidal representations are certain representations
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...
of algebraic groups that occur discretely in
spaces. The term cuspidal is derived, at a certain distance, from the cusp formCusp form
In number theory, a branch of mathematics, a cusp form is a particular kind of modular form, distinguished in the case of modular forms for the modular group by the vanishing in the Fourier series expansion \Sigma a_n q^n...
s of classical modular form
Modular form
In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...
theory. In the contemporary formulation of automorphic representations, representations take the place of holomorphic functions; these representations may be of adelic algebraic group
Adelic algebraic group
In abstract algebra, an adelic algebraic group is a topological group defined by an algebraic group G over a number field K, and the adele ring A = A of K. It consists of the points of G having values in A; the definition of the appropriate topology is straightforward only in case G is a linear...
s.
When the group is the general linear group
General linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
, the cuspidal representations are directly related to cusp forms and Maass forms. For the case of cusp forms, each Hecke eigenform (newform) corresponds to a cuspidal representation.Formulation
Let G be a reductiveReductive 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...
algebraic group over a number field K and let A denote the adele
Adele
Adele is a female given name of European origin used in English, French, German and Italian with a meaning of noble, kind, and tender...
s of K. Let Z denote the centre of G and let ω be a continuous unitary character
Character (mathematics)
In mathematics, a character is a special kind of function from a group to a field . There are at least two distinct, but overlapping meanings...
from Z(K)\Z(A)× to C×. Fix a Haar measure
Haar measure
In mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of locally compact topological groups and subsequently define an integral for functions on those groups....
on G(A) and let L20(G(K)\G(A), ω) denote the Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
of measurable
Measurable function
In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...
complex-valued functions, f, on G(A) satisfying
- f(γg) = f(g) for all γ ∈ G(K)
- f(gz) = f(g)ω(z) for all z ∈ Z(A)
for all unipotent radicals, U, of all proper parabolic subgroups of G(A).
This is called the space of cusp forms with central character ω on G(A). A function occurring in such a space is called a cuspidal function. This space is unitary representation
Unitary representation
In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π is a unitary operator for every g ∈ G...
of the group G(A) where the action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
of g ∈ G(A) on a cuspidal function f is given by
The space of cusp forms with central character ω decomposes into a direct sum of Hilbert spaces
where the sum is over irreducible subrepresentations of L20(G(K)\G(A), ω) and mπ are positive integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s (i.e. each irreducible subrepresentation occurs with finite multiplicity). A cuspidal representation of G(A) is such a subrepresentation (π, V) for some ω.
The groups for which the multiplicities mπ all equal one are said to have the multiplicity-one property.