In representation theory, a branch of mathematics, the Whittaker model is a realization of a representation of a reductive algebraic group such as GL<sub>2</sub> 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 SL<sub>2</sub>(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 ø<sub>10</sub> of the symplectic group Sp<sub>4</sub> is the simplest example of a degenerate representation.
If G is the algebraic group GL<sub>2</sub> 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 ÃÂ on G(F) satisfying
used Whittaker models to assign L-functions to admissible representations of GL<sub>2</sub>.
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 â and non-zero â . If is a smooth representation of , a Whittaker functional is a continuous linear functional on such that for all â , â . Multiplicity one states that, for unitary irreducible, the space of Whittaker functionals has dimension at most equal to one.
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 (GelfandâÂÂGraev) representation Ind(), where is a non-degenerate character of U, such as the sum of the characters corresponding to simple roots.