Kirillov model

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

$\pi({ab\choose 01})f(x) = \tau(bx)f(ax).$

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.

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

W_ξ(g) = π(g)ξ(1)

where π(g) is the image of g in the Kirillov model.

Bernstein (1984) 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 mirabolic 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.

References

Bernstein, Joseph N. (1984), "P-invariant distributions on GL(N) and the classification of unitary representations of GL(N) (non-Archimedean case)", Lie group representations, II (College Park, Md., 1982/1983), Lecture Notes in Math., 1041, Berlin, New York: Springer-Verlag, pp. 50–102, doi:10.1007/BFb0073145, MR 748505
Kirillov, A. A. (1963), "Infinite-dimensional unitary representations of a second-order matrix group with elements in a locally compact field", Doklady Akademii Nauk SSSR 150: 740–743, ISSN 0002-3264, MR 0151552
Jacquet, H.; Langlands, Robert P. (1970), Automorphic forms on GL(2), Lecture Notes in Mathematics, Vol. 114, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0058988, MR 0401654, http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/JL.html#book