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

edit

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 ƒ on G(F) satisfying

 

Jacquet & Langlands (1970) used Whittaker models to assign L-functions to admissible representations of GL2.

Whittaker models for GLn

edit

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

edit

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.

See also

edit

References

edit
  • 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, 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, doi:10.24033/bsmf.1654, ISSN 0037-9484, MR 0271275
  • Jacquet, H.; Langlands, Robert P. (1970), Automorphic forms on GL(2), Lecture Notes in Mathematics, Vol. 114, vol. 114, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0058988, ISBN 978-3-540-04903-6, MR 0401654, S2CID 122773458
  • J. A. Shalika, The multiplicity one theorem for  , The Annals of Mathematics, 2nd. Ser., Vol. 100, No. 2 (1974), 171-193.

Further reading

edit