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 Kirillov model can also be defined for the general linear group GL_n using the mirabolic subgroup.

See also

• Whittaker model

References

• 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, MR0151552
• 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, MR0401654