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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 ''L''20(''G''(''K'')\''G''(A), ω) denote the Hilbert space of 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) # # 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 ''L''20(''G''(''K'')\''G''(A), ω) and ''m''π 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 ''m''π all equal one are said to have the multiplicity-one property. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「cuspidal representation」の詳細全文を読む スポンサード リンク
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