In the mathematical theory of
automorphic representation
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 ...
s, a multiplicity-one theorem is a result about the
representation theory of an
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
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value i ...
s, given in a concrete way.
A multiplicity one theorem may also refer to a result about the
restriction
Restriction, restrict or restrictor may refer to:
Science and technology
* restrict, a keyword in the C programming language used in pointer declarations
* Restriction enzyme, a type of enzyme that cleaves genetic material
Mathematics and log ...
of a
representation
Representation may refer to:
Law and politics
*Representation (politics), political activities undertaken by elected representatives, as well as other theories
** Representative democracy, type of democracy in which elected officials represent a ...
of a
group ''G'' to a
subgroup ''H''. In that context, the pair (''G'', ''H'') is called a strong
Gelfand pair.
Definition
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 be a
continuous unitary character In mathematics, a character is (most commonly) a special kind of function from a group to a field (such as the complex numbers). There are at least two distinct, but overlapping meanings. Other uses of the word "character" are almost always qualifie ...
from ''Z''(''K'')\Z(A)
× to C
×. Let ''L''
20(''G''(''K'')/''G''(A), ) denote the
space of cusp forms with central character ω on ''G''(A). This space decomposes into a
direct sum of Hilbert spaces
:
where the sum is over
irreducible subrepresentations and ''m''
are non-negative
integers.
The group of adelic points of ''G'', ''G''(A), is said to satisfy the multiplicity-one property if any
smooth irreducible
admissible representation of ''G''(A) occurs with multiplicity at most one in the space of
cusp forms of central character , i.e. ''m''
is 0 or 1 for all such .
Results
The fact that the
general linear group, ''GL''(''n''), has the multiplicity-one property was proved by for ''n'' = 2 and independently by and for ''n'' > 2 using the uniqueness of the
Whittaker model. Multiplicity-one also holds for
''SL''(2), but not for ''SL''(''n'') for ''n'' > 2 .
Strong multiplicity one theorem
The strong multiplicity one theorem of and 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.
See also
*
Gan-Gross-Prasad conjecture
References
*
*
*
*
*
*
*{{Citation , last1=Shalika , first1=J. A. , title=The multiplicity one theorem for GL
''n'' , jstor=1971071 , mr=0348047 , year=1974 , journal=
Annals of Mathematics , series=Second Series , issn=0003-486X , volume=100 , pages=171–193 , doi=10.2307/1971071
Representation theory of groups
Automorphic forms
Theorems in number theory
Theorems in representation theory