# Mirabolic group

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In mathematics, a **mirabolic subgroup** of the [general linear group](/source/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 (Gelfand & Kajdan 1975). The image of a mirabolic subgroup in the [projective general linear group](/source/Projective_general_linear_group) is a [parabolic subgroup](/source/Borel_subgroup) consisting of all elements fixing a given point of [projective space](/source/Projective_space). The word "mirabolic" is a [portmanteau](/source/Portmanteau) of "miraculous parabolic". As an algebraic group, a mirabolic subgroup is the [semidirect product](/source/Semidirect_product) of a vector space with its group of automorphisms, and such groups are called **mirabolic groups**. The mirabolic subgroup is used to define the [Kirillov model](/source/Kirillov_model) of a representation of the general linear group.

As an example, the group of all matrices of the form \begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix} where a is a nonzero element of the [field](/source/Field_(mathematics)) k and b is any element of k is a mirabolic subgroup of the 2-dimensional general linear group.

## References

- Bernstein, Joseph N. (1984), "P-invariant distributions on GL(N) and the classification of unitary representations of GL(N) (non-Archimedean case)", "Lie group representations, II (College Park, Md., 1982/1983)", Vol. 1041, Lecture Notes in Math., Berlin, New York: [Springer-Verlag](/source/Springer-Verlag), pp. 50–102, [doi:10.1007/BFb0073145](https://doi.org/10.1007/BFb0073145). MR 748505
- Bushnell, Colin J. & Henniart, Guy (2006), *The local Langlands conjecture for GL(2)*, Vol. 335, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Berlin, New York: [Springer-Verlag](/source/Springer-Verlag), [doi:10.1007/3-540-31511-X](https://doi.org/10.1007/3-540-31511-X). ISBN 978-3-540-31486-8. MR 2234120
- Finkelberg, Michael & Ginzburg, Victor (2010), "On mirabolic D-modules", *International Mathematics Research Notices*. '***(15): 2947–2986, [doi:10.1093/imrn/rnp216](https://doi.org/10.1093/imrn/rnp216). [ISSN 1073-7928](https://www.worldcat.org/issn/1073-7928). MR 2673716***
- Gelfand, I. M. & Kajdan, D. A. (1975 [1971]), ["Representations of the group GL(n,K) where K is a local field"](https://books.google.com/books?id=HyrvAAAAMAAJ), *Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971)*, Halsted, New York, pp. 95–118, ISBN 978-0-470-29600-4. MR 0404534

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