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 (Jacquet 1966, 1967) 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 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 f on G(F) satisfying
Whittaker models for GLn
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.
- Gelfand–Graev representation, roughly the sum of Whittaker models over a finite field.
- Kirillov model
- Jacquet, Hervé (1966), "Une interprétation géométrique et une généralisation P-adique des fonctions de Whittaker en théorie des groupes semi-simples", Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Séries a et B 262: A943––A945, ISSN 0151-0509, MR 0200390
- Jacquet, Hervé (1967), "Fonctions de Whittaker associées aux groupes de Chevalley", Bulletin de la Société Mathématique de France 95: 243–309, ISSN 0037-9484, MR 0271275
- Jacquet, H.; Langlands, Robert P. (1970), Automorphic forms on GL(2), Lecture Notes in Mathematics, Vol. 114 114, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0058988, ISBN 978-3-540-04903-6, MR 0401654
- J. A. Shalika, The multiplicity one theorem for , The Annals of Mathematics, 2nd. Ser., Vol. 100, No. 2 (1974), 171-193.