Whittaker model

Template:Short description 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 Template:Harvs 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 Template:Math is a fixed non-trivial character of the additive group of F and Template:Pi is an irreducible representation of a general linear group G(F), then the Whittaker model for Template:Pi is a representation Template:Pi on a space of functions ƒ on G(F) satisfying

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

Template:Harvtxt used Whittaker models to assign L-functions to admissible representations of GL₂.

Whittaker models for GL_n

Let $G$ be the general linear group ${\mathrm{GL}}_n$, $\psi$ a smooth complex valued non-trivial additive character of $F$ and $U$ the subgroup of ${\mathrm{GL}}_n$ 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 ${\alpha }_1,\dots ,{\alpha }_{n-1}$ ∈ $F$. 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 IndTemplate:Su(Template:Math), where Template:Math is a non-degenerate character of U, such as the sum of the characters corresponding to simple roots.

See also

• Gelfand–Graev representation, roughly the sum of Whittaker models over a finite field.
• Kirillov model

References

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• J. A. Shalika, The multiplicity one theorem for $G{L}_n$, The Annals of Mathematics, 2nd. Ser., Vol. 100, No. 2 (1974), 171–193.

Further reading

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