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In [[representation theory]], a branch of mathematics, the '''Whittaker model''' is a realization of a [[Group representation|representation]] of a [[reductive algebraic group]] such as ''GL''<sub>2</sub> over a [[finite field|finite]] or [[local field|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 {{harvs|last=Jacquet|year1=1966|year2=1967}} pointed out that for the group SL<sub>2</sub>('''R''')  some of the functions involved in the representation are [[Whittaker function]]s.


[[Irreducible representation]]s without a Whittaker model are sometimes called "degenerate", and those with a Whittaker model are sometimes called "generic". The representation [[θ10|θ<sub>10</sub>]] of the [[symplectic group]] Sp<sub>4</sub> is the simplest example of a degenerate representation.


==Whittaker models for GL<sub>2</sub>==
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If ''G'' is the [[algebraic group]] ''GL''<sub>2</sub> 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 ''G''('''F'''), then the Whittaker model for π is a representation π on a space of functions ''f'' on ''G''('''F''') satisfying
 
:<math>f\left(\begin{pmatrix}1 & b \\ 0 & 1\end{pmatrix}g\right) = \tau(b)f(g). </math>
 
{{harvtxt|Jacquet|Langlands|1970}} used Whittaker models to assign L-functions to [[admissible representation]]s of ''GL''<sub>2</sub>.
 
==Whittaker models for GL<sub>''n''</sub>==
 
Let <math>G</math> be the [[general linear group]] <math>{\rm GL}_n</math>, <math>\psi</math> a smooth complex valued non-trivial additive character of <math>F</math> and <math>U</math> the subgroup of <math>{\rm GL}_n</math> consisting of unipotent upper triangular matrices. A non-degenerate character on <math>U</math> is of the form
 
:<math>\chi(u)=\psi(\alpha_1x_{12}+\alpha_2x_{23}+\cdots+\alpha_{n-1}x_{n-1n}),</math>
 
for <math>u=(x_{ij})</math> ∈ <math>U</math> and non-zero <math>\alpha_1</math>, ..., <math>\alpha_{n-1}</math> ∈ <math>F</math>. If <math>(\pi,V)</math> is a smooth representation of <math>G(F)</math>, a '''Whittaker functional''' <math>\lambda</math> is a continuous linear functional on <math>V</math> such that <math>\lambda(\pi(u)v)=\chi(u)\lambda(v)</math> for all <math>u</math> ∈ <math>U</math>, <math>v</math> ∈ <math>V</math>. '''Multiplicity one''' states that, for <math>\pi</math> 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|Gelfand–Graev]]) representation Ind{{su|b=''U''|p=''G''}}(χ), where χ is a non-degenerate character of ''U''.
 
==See also==
 
*[[Gelfand–Graev representation]], roughly the sum of Whittaker models over a finite field.
*[[Kirillov model]]
 
==References==
 
*{{Citation | last1=Jacquet | first1=Hervé | title=Une interprétation géométrique et une généralisation P-adique des fonctions de Whittaker en théorie des groupes semi-simples | mr=0200390 | year=1966 | journal=Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Séries a et B | issn=0151-0509 | volume=262 | pages=A943––A945}}
*{{Citation | last1=Jacquet | first1=Hervé | title=Fonctions de Whittaker associées aux groupes de Chevalley | url=http://www.numdam.org/item?id=BSMF_1967__95__243_0 | mr=0271275 | year=1967 | journal=Bulletin de la Société Mathématique de France | issn=0037-9484 | volume=95 | pages=243–309}}
*{{Citation | author2-link=Robert Langlands | last1=Jacquet | first1=H. | last2=Langlands | first2=Robert P. | title=Automorphic forms on  GL(2) | url=http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/JL.html#book | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics, Vol. 114 | doi=10.1007/BFb0058988 | mr=0401654 | year=1970 | volume=114 | isbn=978-3-540-04903-6}}
 
*J. A. Shalika, ''The multiplicity one theorem for <math>GL_n</math>'', The Annals of Mathematics, 2nd. Ser., Vol. 100, No. 2 (1974), 171-193.
 
[[Category:Representation theory]]
[[Category:Automorphic forms]]

Revision as of 23:46, 24 February 2014


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