Whittaker models

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.

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

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

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

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