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In mathematics, the '''Kirillov model''', studied by {{harvs|txt|authorlink=Alexandre Kirillov|last=Kirillov|year=1963}}, is a realization of a representation of ''GL''<sub>2</sub> over a local field on a space of functions on the local field.
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If ''G'' is the algebraic group ''GL''<sub>2</sub> 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'''<sup>*</sup> with compact support in '''F''' such that
:<math> \pi({ab\choose 01})f(x) = \tau(bx)f(ax). </math>
 
{{harvtxt|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'''<sup>*</sup> 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''<sub>ξ</sub> of a vector ξ of the Kirillov model by
:''W''<sub>ξ</sub>(''g'') = π(g)ξ(1)
where π(''g'') is the image of ''g'' in the Kirillov model.
 
{{harvtxt|Bernstein|1984}} defined the Kirillov model  for the general linear group GL<sub>''n''</sub> 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==
 
*{{Citation | last1=Bernstein | first1=Joseph N. | title=Lie group representations, II (College Park, Md., 1982/1983) | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Math. | doi=10.1007/BFb0073145 | mr=748505 | year=1984 | volume=1041 | chapter=P-invariant distributions on GL(N) and the classification of unitary representations of GL(N) (non-Archimedean case) | pages=50–102}}
*{{Citation | authorlink=Alexandre Kirillov | last1=Kirillov | first1=A. A. | title=Infinite-dimensional unitary representations of a second-order matrix group with elements in a locally compact field | mr=0151552 | year=1963 | journal=Doklady Akademii Nauk SSSR | issn=0002-3264 | volume=150 | pages=740–743}}
*{{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}}
 
[[Category:Representation theory]]
[[Category:Automorphic forms]]
[[Category:Langlands program]]

Revision as of 11:23, 27 February 2014

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