 Multiplicityone theorem

In the mathematical theory of automorphic representations, a multiplicityone 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 functions, given in a concrete way.
Contents
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 ω be a continuous unitary character from Z(K)\Z(A)^{×} to C^{×}. Let L^{2}_{0}(G(K)/G(A), ω) 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_{π} are nonnegative integers.
The group of adelic points of G, G(A), is said to satisfy the multiplicityone 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 ω, i.e. m_{π} is 0 or 1 for all such π.
Results
The fact that the general linear group, GL(n), has the multiplicityone property was proved by Jacquet & Langlands (1970) for n = 2 and independently by PiatetskiShapiro (1979) and Shalika (1974) for n > 2 using the uniqueness of the Whittaker model. Multiplicityone also holds for SL(2), but not for SL(n) for n > 2 (Blasius 1994).
Strong multiplicity one theorem
The strong multiplicity one theorem of PiatetskiShapiro (1979) and 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
 Blasius, Don (1994), "On multiplicities for SL(n)", Israel Journal of Mathematics 88 (1): 237–251, doi:10.1007/BF02937513, ISSN 00212172, MR1303497
 Cogdell, James W. (2004), "Lectures on Lfunctions, converse theorems, and functoriality for GL_{n}", in Cogdell, James W.; Kim, Henry H.; Murty, Maruti Ram, Lectures on automorphic Lfunctions, Fields Inst. Monogr., 20, Providence, R.I.: American Mathematical Society, pp. 1–96, ISBN 9780821835166, MR2071506, http://www.math.osu.edu/~cogdell/
 Jacquet, Hervé; Langlands, Robert (1970), Automorphic forms on GL(2), Lecture Notes in Mathematics, 114, SpringerVerlag
 Jacquet, H.; Shalika, J. A. (1981), "On Euler products and the classification of automorphic representations. I", American Journal of Mathematics 103 (3): 499–558, doi:10.2307/2374103, ISSN 00029327, MR618323 Jacquet, H.; Shalika, J. A. (1981), "On Euler products and the classification of automorphic representations. II", American Journal of Mathematics 103 (4): 777–815, ISSN 00029327, JSTOR 2374050, MR618323
 PiatetskiShapiro, I. I. (1979), "Multiplicity one theorems", in Borel, Armand; Casselman., W., Automorphic forms, representations and Lfunctions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, Providence, R.I.: American Mathematical Society, pp. 209–212, ISBN 9780821814352, MR546599, http://www.ams.org/publications/onlinebooks/pspum331index
 Shalika, J. A. (1974), "The multiplicity one theorem for GL_{n}", Annals of Mathematics. Second Series 100: 171–193, ISSN 0003486X, JSTOR 1971071, MR0348047
Categories: Representation theory of groups
 Automorphic forms
 Theorems in number theory
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