mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Kirillov model, studied by , is a realization of a representation of ''GL''₂ over a

local field In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact t ...

on a space of functions on the local field. If ''G'' is the

algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...

''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 In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W,W ...

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 In representation theory, a branch of mathematics, the Whittaker model is a realization of a representation of a reductive algebraic group such as ''GL''2 over a finite or local or global field on a space of functions on the group. It is named aft ...

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 In mathematics, a mirabolic subgroup of the general linear group GL''n''(''k'') is a subgroup consisting of automorphisms fixing a given non-zero vector in ''k'n''. Mirabolic subgroups were introduced by . The image of a mirabolic subgroup in th ...

. 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

* * *{{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 Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...

, 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