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 groupGL2 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 ƒ on G(F) satisfying
Let be the
general linear group, a smooth complex valued non-trivial additive character of and the subgroup of consisting of unipotent upper triangular matrices. A non-degenerate character on is of the form
for ∈ and non-zero ∈ . If is a smooth representation of , a Whittaker functional is a continuous linear functional on such that for all ∈ , ∈ . Multiplicity one states that, for 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 IndG U(χ), where χ is a non-degenerate character of U, such as the sum of the characters corresponding to simple roots.
Jacquet, Hervé (1966), "Une interprétation géométrique et une généralisation P-adique des fonctions de Whittaker en théorie des groupes semi-simples", Comptes Rendus de l'Académie des Sciences, Série A et B, 262: A943–A945,
ISSN0151-0509,
MR0200390