In mathematics, the Kirillov model, studied by , is a realization of a representation of GL<sub>2</sub> over a local field on a space of functions on the local field.
If G is the algebraic group GL<sub>2</sub> and F is a non-Archimedean local field, and ÃÂ is a fixed nontrivial character of the additive group of F and ÃÂ is an irreducible representation of G(F), then the Kirillov model for ÃÂ is a representation ÃÂ on a space of locally constant functions f on F<sup>*</sup> 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<sup>*</sup> 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<sub>þ</sub> of a vector þ of the Kirillov model by
where ÃÂ(g) is the image of g in the Kirillov model.
defined the Kirillov model for the general linear group GL<sub>n</sub> 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.