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In [[number theory]], '''cuspidal representations''' are certain [[group representation|representations]] of [[algebraic groups]] that occur discretely in <math>L^2</math> 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 representation]]s, representations take the place of holomorphic functions; these representations may be of [[adelic algebraic group]]s. | |||
When the group is the [[general linear group]] <math>\operatorname{GL}_2</math>, the cuspidal representations are directly related to cusp forms and [[Maass form]]s. For the case of cusp forms, each [[Hecke eigenform]] ([[newform]]) corresponds to a cuspidal representation. | |||
==Formulation== | |||
Let ''G'' be a [[reductive group|reductive]] algebraic group over a [[number field]] ''K'' and let '''A''' denote the [[adele group|adele]]s of ''K''. Let ''Z'' denote the [[center of a group|centre]] of ''G'' and let ω be a [[continuous (mathematics)|continuous]] [[character (mathematics)|unitary character]] from ''Z''(''K'')\Z('''A''')<sup>×</sup> to '''C'''<sup>×</sup>. Fix a [[Haar measure]] on ''G''('''A''') and let ''L''<sup>2</sup><sub>0</sub>(''G''(''K'')\''G''('''A'''), ω) denote the [[Hilbert space]] of [[measurable function|measurable]] complex-valued functions, ''f'', on ''G''('''A''') satisfying | |||
#''f''(γ''g'') = ''f''(''g'') for all γ ∈ ''G''(''K'') | |||
#''f''(''gz'') = ''f''(''g'')ω(''z'') for all ''z'' ∈ ''Z''('''A''') | |||
#<math>\int_{Z(\mathbf{A})G(K)\backslash G(\mathbf{A})}|f(g)|^2\,dg < \infty</math> | |||
#<math>\int_{U(K)\backslash U(\mathbf{A})}f(ug)\,du=0</math> for all [[unipotent radical]]s, ''U'', of all proper [[parabolic subgroup]]s 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 a [[unitary representation]] of the group ''G''('''A''') where the [[group action|action]] of ''g'' ∈ ''G''('''A''') on a cuspidal function ''f'' is given by | |||
:<math>(g\cdot f)(x)=f(xg).</math> | |||
The space of cusp forms with central character ω decomposes into a [[direct sum of Hilbert spaces]] | |||
:<math>L^2_0(G(K)\backslash G(\mathbf{A}),\omega)=\hat{\bigoplus}_{(\pi,V_\pi)}m_\pi V_\pi</math> | |||
where the sum is over [[irreducible representation|irreducible]] [[subrepresentation]]s of ''L''<sup>2</sup><sub>0</sub>(''G''(''K'')\''G''('''A'''), ω) and ''m''<sub>π</sup> 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 ''m''<sub>π</sub> all equal one are said to have the [[multiplicity-one property]]. | |||
==References== | |||
*James W. Cogdell, Henry Hyeongsin Kim, Maruti Ram Murty. ''Lectures on Automorphic L-functions'' (2004), Chapter 5. | |||
[[Category:Representation theory of algebraic groups]] | |||
Latest revision as of 16:55, 17 July 2013
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 forms 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 groups.
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.
Formulation
Let G be a reductive algebraic group over a number field K and let A denote the adeles 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
- 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 a 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 integers (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.
References
- James W. Cogdell, Henry Hyeongsin Kim, Maruti Ram Murty. Lectures on Automorphic L-functions (2004), Chapter 5.