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

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

Jacquet & Langlands (1970) used Whittaker models to assign L-functions to admissible representations of GL2.

Whittaker models for GLn

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

${\displaystyle \chi (u)=\psi (\alpha _{1}x_{12}+\alpha _{2}x_{23}+\cdots +\alpha _{n-1}x_{n-1n}),}$

for ${\displaystyle u=(x_{ij})}$${\displaystyle U}$ and non-zero ${\displaystyle \alpha _{1}}$, ..., ${\displaystyle \alpha _{n-1}}$${\displaystyle F}$. If ${\displaystyle (\pi ,V)}$ is a smooth representation of ${\displaystyle G(F)}$, a Whittaker functional ${\displaystyle \lambda }$ is a continuous linear functional on ${\displaystyle V}$ such that ${\displaystyle \lambda (\pi (u)v)=\chi (u)\lambda (v)}$ for all ${\displaystyle u}$${\displaystyle U}$, ${\displaystyle v}$${\displaystyle V}$. Multiplicity one states that, for ${\displaystyle \pi }$ 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.