In mathematics, the Kirillov model, studied by Kirillov (1963), 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, 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 Jacquet & Langlands (1970) 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.
Bernstein (1984) defined the Kirillov model for the general linear group GLn 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.