# Multiplicity-one theorem

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In the mathematical theory of [automorphic representations](/source/Automorphic_representation), a **multiplicity-one theorem** is a result about the [representation theory](/source/Representation_theory) of an [adelic](/source/Adelic_algebraic_group) [reductive algebraic group](/source/Reductive_algebraic_group). The multiplicity in question is the number of times a given abstract [group representation](/source/Group_representation) is realised in a certain space, of [square-integrable functions](/source/Square-integrable_function), given in a concrete way.

A multiplicity one theorem may also refer to a result about the [restriction](/source/Restricted_representation) of a [representation](/source/Group_representation) of a [group](/source/Group_(mathematics)) *G* to a [subgroup](/source/Subgroup) *H*. In that context, the pair (*G*, *H*) is called a strong [Gelfand pair](/source/Gelfand_pair).

## Definition

Let *G* be a reductive algebraic group over a [number field](/source/Number_field) *K* and let **A** denote the [adeles](/source/Adele_ring) of *K*. Let *Z* denote the [centre](/source/Center_of_a_group) of *G* and let *ω* be a [continuous](/source/Continuous_(mathematics)) [unitary character](/source/Character_(mathematics)) from *Z*(*K*)\Z(**A**)× to **C**×. Let *L*20(*G*(*K*)/*G*(**A**), *ω*) denote the [space of cusp forms with central character ω](/source/Cuspidal_representation) on *G*(**A**). This space decomposes into a [direct sum of Hilbert spaces](/source/Direct_sum_of_Hilbert_spaces)

- L^2_0(G(K)\backslash G(\mathbf{A}),\omega)=\widehat{\bigoplus}_{(\pi,V_\pi)}m_\pi V_\pi

where the sum is over [irreducible](/source/Irreducible_representation) [subrepresentations](/source/Subrepresentation) and *m*π are non-negative [integers](/source/Integer).

The group of adelic points of *G*, *G*(**A**), is said to satisfy the **multiplicity-one property** if any [smooth](/source/Smooth_representation) irreducible [admissible representation](/source/Admissible_representation) of *G*(**A**) occurs with multiplicity at most one in the space of [cusp forms](/source/Cusp_form) of central character *ω*, i.e. *m*π is 0 or 1 for all such π.

## Results

The fact that the [general linear group](/source/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 txt for *n* > 2 using the uniqueness of the [Whittaker model](/source/Whittaker_model). Multiplicity-one also holds for [*SL*(2)](/source/Special_linear_group), but not for *SL*(*n*) for *n* > 2 (Blasius 1994).

## Strong multiplicity one theorem

The strong multiplicity one theorem of Piatetski-Shapiro (1979) and txt 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](/source/Gan%E2%80%93Gross%E2%80%93Prasad_conjecture)

## References

- Blasius, Don (1994), "On multiplicities for SL(*n*)", *[Israel Journal of Mathematics](/source/Israel_Journal_of_Mathematics)*. **88** (1): 237–251, [doi:10.1007/BF02937513](https://doi.org/10.1007/BF02937513). [ISSN 0021-2172](https://www.worldcat.org/issn/0021-2172). MR 1303497
- Cogdell, James W. (2004), ["Lectures on L-functions, converse theorems, and functoriality for GL*n*"](https://books.google.com/books?id=jb3ZCp0-MQsC), *Lectures on automorphic L-functions*, Vol. 20, Fields Inst. Monogr., Providence, R.I.: [American Mathematical Society](/source/American_Mathematical_Society), pp. 1–96, ISBN 978-0-8218-3516-6. MR 2071506
- Jacquet, Hervé & Langlands, Robert (1970), "Automorphic forms on GL(2)", Vol. 114, Lecture Notes in Mathematics, Springer-Verlag
- Jacquet, H. & Shalika, J. A. (1981a), ["On Euler products and the classification of automorphic representations. I"](http://www.math.columbia.edu/~hj/On%20Euler%20products%20I.pdf), *[American Journal of Mathematics](/source/American_Journal_of_Mathematics)*. **103** (3): 499–558, [doi:10.2307/2374103](https://doi.org/10.2307/2374103). [ISSN 0002-9327](https://www.worldcat.org/issn/0002-9327). [JSTOR 2374103](https://www.jstor.org/stable/2374103). MR 618323, retrieved 2021-08-06
- Jacquet, H. & Shalika, J. A. (1981b), ["On Euler products and the classification of automorphic forms. II"](http://www.math.columbia.edu/~hj/On%20Euler%20products%20II.pdf), *[American Journal of Mathematics](/source/American_Journal_of_Mathematics)*. **103** (4): 777–815, [doi:10.2307/2374050](https://doi.org/10.2307/2374050). [ISSN 0002-9327](https://www.worldcat.org/issn/0002-9327). [JSTOR 2374050](https://www.jstor.org/stable/2374050). MR 618323, retrieved 2021-08-06
- Piatetski-Shapiro, I. I. (1979), ["Multiplicity one theorems"](https://www.ams.org/publications/online-books/pspum331-index), *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](/source/American_Mathematical_Society), pp. 209–212, ISBN 978-0-8218-1435-2. MR 546599
- Shalika, J. A. (1974), "The multiplicity one theorem for GL*n*", *[Annals of Mathematics](/source/Annals_of_Mathematics)*. **100** (2): 171–193, Second Series, [doi:10.2307/1971071](https://doi.org/10.2307/1971071). [ISSN 0003-486X](https://www.worldcat.org/issn/0003-486X). [JSTOR 1971071](https://www.jstor.org/stable/1971071). MR 0348047

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