Whittaker model

In representation theory, a branch of mathematics, the Whittaker model is a realization of a representation of a reductive algebraic group such as "GL"2 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 "θ"10 of the symplectic group Sp4 is the simplest example of a degenerate representation.
Whittaker models for GL2.
If "G" is the algebraic group "GL"2 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
 used Whittaker models to assign L-functions to admissible representations of "GL"2.
Whittaker models for GL"n".
Let formula_2 be the general linear group formula_3, formula_4 a smooth complex valued non-trivial additive character of formula_5 and formula_6 the subgroup of formula_3 consisting of unipotent upper triangular matrices. A non-degenerate character on formula_6 is of the form
for formula_10 ∈ formula_6 and non-zero formula_12 ∈ formula_5. If formula_14 is a smooth representation of formula_15, a Whittaker functional formula_16 is a continuous linear functional on formula_17 such that formula_18 for all formula_19 ∈ formula_6, formula_21 ∈ formula_17. Multiplicity one states that, for formula_23 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.