Kirillov model

In mathematics, the Kirillov model, studied by , is a realization of a representation of "GL"2 over a local field on a space of functions on the local field.
If "G" is the algebraic group "GL"2 and F is a non-Archimedean local field,
and τ is a fixed nontrivial character of the additive group of F
and π is an irreducible representation of "G"(F), then the Kirillov model for π is 
a representation π on a space of locally constant functions "f" on F* with compact support in F such that 
 showed that an irreducible representation of dimension greater than 1 has an essentially unique Kirillov model. 
Over a local field, the space of functions with compact support in F* has codimension 0, 1, or 2 in the Kirillov model, depending on whether the irreducible representation is cuspidal, special, or principal. 
The Whittaker model can be constructed from the Kirillov model, by defining the image "W"ξ of a vector ξ of the Kirillov model by
where π("g") is the image of "g" in the Kirillov model.
 defined the Kirillov model for the general linear group GL"n" using the mirabolic subgroup. More precisely, a Kirillov model for a representation of the general linear group is an embedding of it in the representation of the mirabolic group induced from a non-degenerate character of the group of upper triangular matrices.