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In the mathematical theory of [[automorphic representation]]s, a '''multiplicity-one theorem''' is a result about the [[representation theory]] of an [[adelic algebraic group|adelic]] [[reductive algebraic group]]. The multiplicity in question is the number of times a given abstract [[group representation]] is realised in a certain space, of [[square integrable function]]s, given in a concrete way.


==Definition==
Let ''G'' be a reductive algebraic group over a [[number field]] ''K'' and let '''A''' denote the [[adele ring|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>&times;</sup> to '''C'''<sup>&times;</sup>. Let ''L''<sup>2</sup><sub>0</sub>(''G''(''K'')/''G''('''A'''), ω) denote the [[cuspidal representation|space of cusp forms with central character &omega;]] on ''G''('''A'''). This space 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 and ''m''<sub>π</sup> are non-negative [[integer]]s.


The group of adelic points of ''G'', ''G''('''A'''), is said to satisfy the '''multiplicity-one property''' if any [[smooth representation|smooth]] irreducible [[admissible representation]] of ''G''('''A''') occurs with multiplicity at most one in the space of [[cusp form]]s of central character ω, i.e. ''m''<sub>π</sub> is 0 or 1 for all such π.
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==Results==
The fact that the [[general linear group]], ''GL''(''n''), has the multiplicity-one property was proved by {{harvtxt|Jacquet|Langlands|1970}} for ''n''&nbsp;=&nbsp;2 and independently by {{harvtxt|Piatetski-Shapiro|1979}} and {{harvs|txt|authorlink=Joseph Shalika|last=Shalika|year=1974}} for ''n''&nbsp;>&nbsp;2 using the uniqueness of the [[Whittaker model]]. Multiplicity-one also holds for [[Special linear group|''SL''(2)]], but not for ''SL''(''n'') for ''n''&nbsp;>&nbsp;2 {{harv|Blasius|1994}}.
 
==Strong multiplicity one theorem==
 
The strong multiplicity one theorem of {{harvtxt|Piatetski-Shapiro|1979}} and {{harvtxt|Jacquet|Shalika|1981}} states that two cuspidal automorphic representations of the general linear group are isomorphic if their local components are isomorphic for all but a finite number of places.
 
==References==
 
*{{Citation | last1=Blasius | first1=Don | title=On multiplicities for  SL(n) | doi=10.1007/BF02937513 | mr=1303497 | year=1994 | journal=Israel Journal of Mathematics | issn=0021-2172 | volume=88 | issue=1 | pages=237–251}}
*{{Citation | last1=Cogdell | first1=James W. | editor1-last=Cogdell | editor1-first=James W. | editor2-last=Kim | editor2-first=Henry H. | editor3-last=Murty | editor3-first=Maruti Ram | title=Lectures on automorphic L-functions | url=http://books.google.com/books?id=jb3ZCp0-MQsC | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Fields Inst. Monogr. | isbn=978-0-8218-3516-6  | mr=2071506 | year=2004 | volume=20 | chapter=Lectures on L-functions, converse theorems, and functoriality for  GL<sub>n</sub> | chapterurl=http://www.math.osu.edu/~cogdell/ | pages=1–96}}
*{{Citation
| last=Jacquet
| first=Hervé
| last2=Langlands
| first2=Robert
| title=Automorphic forms on GL(2)
| series=Lecture Notes in Mathematics
| publisher=Springer-Verlag
| volume=114
| year=1970
}}
*{{Citation | last1=Jacquet | first1=H. | last2=Shalika | first2=J. A. | title=On Euler products and the classification of automorphic representations. I | doi=10.2307/2374103 | mr=618323 | year=1981 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=103 | issue=3 | pages=499–558}} {{Citation | last1=Jacquet | first1=H. | last2=Shalika | first2=J. A. | title=On Euler products and the classification of automorphic representations. II | jstor=2374050 | mr=618323 | year=1981 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=103 | issue=4 | pages=777–815}}
*{{Citation | last1=Piatetski-Shapiro | first1=I. I. | editor1-last=Borel | editor1-first=Armand | editor1-link=Armand Borel | editor2-last=Casselman. | editor2-first=W. | title=Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1 | url=http://www.ams.org/publications/online-books/pspum331-index | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Proc. Sympos. Pure Math., XXXIII | isbn=978-0-8218-1435-2  | mr=546599 | year=1979 | chapter=Multiplicity one theorems | pages=209–212}}
*{{Citation | last1=Shalika | first1=J. A. | title=The multiplicity one theorem for GL<sub>n</sub> | jstor=1971071 | mr=0348047 | year=1974 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=100 | pages=171–193}}
 
[[Category:Representation theory of groups]]
[[Category:Automorphic forms]]
[[Category:Theorems in number theory]]
[[Category:Theorems in representation theory]]

Latest revision as of 14:36, 14 April 2014


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