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 ''GL''₂ 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 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 ''θ''₁₀ of the symplectic group Sp₄ is the simplest example of a degenerate representation.

Whittaker models for GL₂

If ''G'' is the algebraic group ''GL''₂ and F is a local field, and is a fixed non-trivial character of the additive group of F and is an irreducible representation of a general linear group ''G''(F), then the Whittaker model for is a representation on a space of functions ''ƒ'' on ''G''(F) satisfying

:$f\left(\begin1 & b \\ 0 & 1\endg\right) = \tau(b)f(g).$

used Whittaker models to assign L-functions to admissible representations of ''GL''₂.

Whittaker models for GL_''n''

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

:$\chi(u)=\psi(\alpha_1 x_+\alpha_2 x_+\cdots+\alpha_x_),$

for $u=(x_)$ ∈ $U$ and non-zero $\alpha_1, \ldots, \alpha_$ ∈ $F$. 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 Ind(), where 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 $GL_n$'', The Annals of Mathematics, 2nd. Ser., Vol. 100, No. 2 (1974), 171-193.

Further reading

* {{Cite journal, last1=Jacquet, first1=Hervé, last2=Shalika, first2=Joseph, date=1983, title=The Whittaker models of induced representations., url=https://projecteuclid.org/euclid.pjm/1102720206, journal=Pacific Journal of Mathematics, language=en, volume=109, issue=1, pages=107–120, doi=10.2140/pjm.1983.109.107, issn=0030-8730, doi-access=free

Representation theory
Automorphic forms
Langlands program
E. T. Whittaker