Whittaker model

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

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

Whittaker models for GL_''n''

Let G be the general linear group \operatorname{GL}_n, \psi a smooth complex valued non-trivial additive character of F and U the subgroup of \operatorname{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, \ldots, \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 Ind(χ), where χ is a non-degenerate character of U, such as the sum of the characters corresponding to simple roots.

References

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