Kirillov Model

In mathematics, the Kirillov model, studied by , is a realization of a representation of ''GL''₂ over a local field on a space of functions on the local field.

If ''G'' is the algebraic group ''GL''₂ 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
:$\pi\left(\begina & b \\ 0 & 1\end\right)f(x) = \tau(bx)f(ax).$

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.

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 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.

References

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*{{Citation , author2-link=Robert Langlands , last1=Jacquet , first1=H. , last2=Langlands , first2=Robert P. , title=Automorphic forms on GL(2) , url=http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/JL.html#book , publisher=Springer-Verlag , location=Berlin, New York , series=Lecture Notes in Mathematics, Vol. 114 , doi=10.1007/BFb0058988 , mr=0401654 , year=1970, volume=114 , isbn=978-3-540-04903-6

Representation theory
Automorphic forms
Langlands program