Whittaker model
Encyclopedia
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 pointed out that for the group SL2(R) some of the functions involved in the representation are Whittaker function
s.
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.
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 G(F), then the Whittaker model for π is a representation π on a space of functions f on G(F) satisfying
used Whittaker models to assign L-functions to admissible representation
s of GL2.
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.
) representation Ind(χ), where χ is a non-degenerate character of U.
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
, a branch of mathematics, the Whittaker model is a realization of a representation
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
of a reductive algebraic group such as GL2 over a finite
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
or local
Local field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...
or global field
Global field
In mathematics, the term global field refers to either of the following:*an algebraic number field, i.e., a finite extension of Q, or*a global function field, i.e., the function field of an algebraic curve over a finite field, equivalently, a finite extension of Fq, the field of rational functions...
on a space of functions on the group. It is named after E. T. Whittaker
E. T. Whittaker
Edmund Taylor Whittaker FRS FRSE was an English mathematician who contributed widely to applied mathematics, mathematical physics and the theory of special functions. He had a particular interest in numerical analysis, but also worked on celestial mechanics and the history of physics...
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 function
Whittaker function
In mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced by to make the formulas involving the solutions more symmetric...
s.
Irreducible representations without a Whittaker model are sometimes called "degenerate", and those with a Whittaker model are sometimes called "generic". The representation θ10
Θ10
In representation theory, a branch of mathematics, θ10 is a cuspidal unipotent complex irreducible representation of the symplectic group Sp4 over a finite, local, or global field....
of the symplectic group Sp4 is the simplest example of a degenerate representation.
Whittaker models for GL2
If G is the algebraic groupAlgebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
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 G(F), then the Whittaker model for π is a representation π on a space of functions f on G(F) satisfying
used Whittaker models to assign L-functions to admissible representation
Admissible representation
In mathematics, admissible representations are a well-behaved class of representations used in the representation theory of reductive Lie groups and locally compact totally disconnected groups. They were introduced by Harish-Chandra....
s of GL2.
Whittaker models for GLn
Let be the general linear groupGeneral linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
, 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 formfor
∈ 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–GraevGelfand–Graev representation
In representation theory, a branch of mathematics, the Gelfand–Graev representation is a representation of a reductive group over a finite field introduced by , induced from a non-degenerate character of a Sylow subgroup....
) representation Ind(χ), where χ is a non-degenerate character of U.
See also
- Gelfand–Graev representationGelfand–Graev representationIn representation theory, a branch of mathematics, the Gelfand–Graev representation is a representation of a reductive group over a finite field introduced by , induced from a non-degenerate character of a Sylow subgroup....
, roughly the sum of Whittaker models over a finite field. - Kirillov modelKirillov modelIn mathematics, the Kirillov model, studied by , is a realization of a representation of GL2 over a local field on a space of functions on the local field.If G is the algebraic group GL2 and F is a non-Archimedean local field,...