Whittaker model explained

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 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 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} + … + \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 λ is a continuous linear functional on V such that λ (\pi(u)v) = \chi(u) λ (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

• • • • 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

• Jacquet. Hervé. Shalika. Joseph. 1983. The Whittaker models of induced representations.. Pacific Journal of Mathematics. en. 109. 1. 107–120. 10.2140/pjm.1983.109.107. 0030-8730. free.