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In the mathematical theory of automorphic representations, 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 functions, given in a concrete way.

A multiplicity one theorem may also refer to a result about the restriction of a representation 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 ω be a continuous unitary character from Z(K)\Z(A)× to C×. Let L20(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

$${\displaystyle L_{0}^{2}(G(K)\backslash G(\mathbf {A} ),\omega )={\widehat {\bigoplus }}_{(\pi ,V_{\pi })}m_{\pi }V_{\pi }}$$

where the sum is over irreducible subrepresentations and mπ 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 ω, i.e. mπ is 0 or 1 for all such π.
Results

The fact that the general linear group, GL(n), has the multiplicity-one property was proved by Jacquet & Langlands (1970) for n = 2 and independently by Piatetski-Shapiro (1979) and Shalika (1974) 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 (Blasius 1994).
Strong multiplicity one theorem

The strong multiplicity one theorem of Piatetski-Shapiro (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 0021-2172, MR 1303497
Cogdell, James W. (2004), "Lectures on L-functions, converse theorems, and functoriality for GLn", in Cogdell, James W.; Kim, Henry H.; Murty, Maruti Ram (eds.), Lectures on automorphic L-functions, Fields Inst. Monogr., 20, Providence, R.I.: American Mathematical Society, pp. 1–96, ISBN 978-0-8218-3516-6, MR 2071506
Jacquet, Hervé; Langlands, Robert (1970), Automorphic forms on GL(2), Lecture Notes in Mathematics, 114, Springer-Verlag
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 0002-9327, MR 0618323 Jacquet, H.; Shalika, J. A. (1981), "On Euler products and the classification of automorphic representations. II" (PDF), American Journal of Mathematics, 103 (4): 777–815, doi:10.2307/2374050, ISSN 0002-9327, JSTOR 2374050, MR 0618323
Piatetski-Shapiro, I. I. (1979), "Multiplicity one theorems", in Borel, Armand; Casselman., W. (eds.), Automorphic forms, representations and L-functions (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 978-0-8218-1435-2, MR 0546599
Shalika, J. A. (1974), "The multiplicity one theorem for GLn", Annals of Mathematics, Second Series, 100: 171–193, doi:10.2307/1971071, ISSN 0003-486X, JSTOR 1971071, MR 0348047