Kirillov model

In mathematics, the Kirillov model is a certain realization of representations of algebraic groups over local fields.

If G is the algebraic group GL₂ and F is a non-archimedean local field, and τ is a fixed non-tric`vial 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

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

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