Whittaker model

In representation theory, a branch of mathematics, the Whittaker model is a realization of a representation of a reductive algebraic group such as GL₂ over a finite or local or global field on a space of functions on the group. It is named after E. T. Whittaker even though he never worked in this area, because (Jacquet 1966, 1967) pointed out that for the group SL₂(R) some of the functions involved in the representation are Whittaker functions.

Irreducible representations without a Whittaker model are sometimes called "degenerate", and those with a Whittaker model are sometimes called "generic". The representation θ₁₀ of the symplectic group Sp₄ is the simplest example of a degenerate representation.

Whittaker models for GL₂

If G is the algebraic group GL₂ and F is a local field, and τ is a fixed non-trivial character 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 f 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 of GL₂.

Whittaker models for GL_n

Let $G$ be the general linear group ${\rm GL}_n$ , $\psi$ a smooth complex valued non-trivial additive character of $F$ and $U$ the subgroup of ${\rm GL}_n$ consisting of unipotent upper triangular matrices. A non-degenerate character on $U$ is of the form

\chi(u)=\psi(\alpha_1x_{12}+\alpha_2x_{23}+\cdots+\alpha_{n-1}x_{n-1n}),

for $u=(x_{ij})$ ∈ $U$ and non-zero $\alpha_1$ , ..., $\alpha_{n-1}$ ∈ $F$ . 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) representation IndG
U(χ), where χ is a non-degenerate character of U, such as the sum of the characters corresponding to simple roots.

References

Jacquet, Hervé (1966), "Une interprétation géométrique et une généralisation P-adique des fonctions de Whittaker en théorie des groupes semi-simples", Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Séries a et B 262: A943––A945, ISSN 0151-0509, MR 0200390
Jacquet, Hervé (1967), "Fonctions de Whittaker associées aux groupes de Chevalley", Bulletin de la Société Mathématique de France 95: 243–309, ISSN 0037-9484, MR 0271275
Jacquet, H.; Langlands, Robert P. (1970), Automorphic forms on GL(2), Lecture Notes in Mathematics, Vol. 114 114, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0058988, ISBN 978-3-540-04903-6, MR 0401654

J. A. Shalika, The multiplicity one theorem for $GL_n$ , The Annals of Mathematics, 2nd. Ser., Vol. 100, No. 2 (1974), 171-193.

Whittaker model

Whittaker models for GL2

Whittaker models for GLn

Whittaker models for reductive groups

See also

References

Whittaker models for GL₂

Whittaker models for GL_n