# Whittaker model

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In [representation theory](/source/Representation_theory), a branch of mathematics, the **Whittaker model** is a realization of a [representation](/source/Group_representation) of a [reductive algebraic group](/source/Reductive_algebraic_group) such as *GL*2 over a [finite](/source/Finite_field) or [local](/source/Local_field) or [global field](/source/Global_field) on a space of functions on the group. It is named after [E. T. Whittaker](/source/E._T._Whittaker) even though he never worked in this area, because pointed out that for the group SL2(**R**) some of the functions involved in the representation are [Whittaker functions](/source/Whittaker_function).

[Irreducible representations](/source/Irreducible_representation) without a Whittaker model are sometimes called "degenerate", and those with a Whittaker model are sometimes called "generic". The representation [*θ*10](/source/%CE%9810) of the [symplectic group](/source/Symplectic_group) Sp4 is the simplest example of a degenerate representation.

## Whittaker models for GL2

If *G* is the [algebraic group](/source/Algebraic_group) *GL*2 and **F** is a local field, and *τ* is a fixed non-trivial [character](/source/Character_theory) of the additive group of **F** and π is an irreducible representation of a general linear group *G*(**F**), then the Whittaker model for π is a representation π on a space of functions *ƒ* on *G*(**F**) satisfying

- f\left(\begin{pmatrix}1 & b \\ 0 & 1\end{pmatrix}g\right) = \tau(b)f(g).

Jacquet & Langlands (1970) used Whittaker models to assign L-functions to [admissible representations](/source/Admissible_representation) of *GL*2.

## Whittaker models for GL*n*

Let G be the [general linear group](/source/General_linear_group) \operatorname{GL}_n, \psi a smooth complex valued non-trivial additive character of F and U the subgroup of \operatorname{GL}_n consisting of unipotent upper triangular matrices. A non-degenerate character on U is of the form

- \chi(u)=\psi(\alpha_1 x_{12}+\alpha_2 x_{23}+\cdots+\alpha_{n-1}x_{n-1n}),

for u=(x_{ij}) ∈ U and non-zero \alpha_1, \ldots, \alpha_{n-1} ∈ F. If (\pi,V) is a smooth representation of G(F), a **Whittaker functional** \lambda is a continuous linear functional on V such that \lambda(\pi(u)v)=\chi(u)\lambda(v) for all u ∈ U, v ∈ V. **Multiplicity one** states that, for \pi unitary irreducible, the space of Whittaker functionals has dimension at most equal to one.

## Whittaker models for reductive groups

If *G* is a split reductive group and *U* is the unipotent radical of a Borel subgroup *B*, then a Whittaker model for a representation is an embedding of it into the induced ([Gelfand–Graev](/source/Gelfand%E2%80%93Graev_representation)) representation Ind*G**U*(*χ*), where *χ* is a non-degenerate character of *U*, such as the sum of the characters corresponding to simple roots.

## See also

- [Gelfand–Graev representation](/source/Gelfand%E2%80%93Graev_representation), roughly the sum of Whittaker models over a finite field.
- [Kirillov model](/source/Kirillov_model)

## References

- Jacquet, Hervé (1966), "Une interprétation géométrique et une généralisation P-adique des fonctions de Whittaker en théorie des groupes semi-simples", *Comptes Rendus de l'Académie des Sciences, Série A et B*. **262**: A943–A945, [ISSN 0151-0509](https://www.worldcat.org/issn/0151-0509). MR 0200390
- Jacquet, Hervé (1967), ["Fonctions de Whittaker associées aux groupes de Chevalley"](http://www.numdam.org/item?id=BSMF_1967__95__243_0), *Bulletin de la Société Mathématique de France*. **95**: 243–309, [doi:10.24033/bsmf.1654](https://doi.org/10.24033/bsmf.1654). [ISSN 0037-9484](https://www.worldcat.org/issn/0037-9484). MR 0271275
- Jacquet, H. & Langlands, Robert P. (1970), [*Automorphic forms on GL(2)*](http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/JL.html#book), Vol. 114, Lecture Notes in Mathematics, Vol. 114, Berlin, New York: [Springer-Verlag](/source/Springer-Verlag), [doi:10.1007/BFb0058988](https://doi.org/10.1007/BFb0058988). ISBN 978-3-540-04903-6. MR 0401654. [S2CID 122773458](https://api.semanticscholar.org/CorpusID:122773458)
- J. A. Shalika, *The multiplicity one theorem for GL_n*, The Annals of Mathematics, 2nd. Ser., Vol. 100, No. 2 (1974), 171–193.

## Further reading

- Jacquet, Hervé & Shalika, Joseph (1983). ["The Whittaker models of induced representations."](https://projecteuclid.org/euclid.pjm/1102720206). *Pacific Journal of Mathematics*. **109** (1): 107–120. [doi:10.2140/pjm.1983.109.107](https://doi.org/10.2140/pjm.1983.109.107). [ISSN 0030-8730](https://www.worldcat.org/issn/0030-8730)

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