In mathematics, a mirabolic subgroup of the general linear group GLn(k) is a subgroup consisting of automorphisms fixing a given non-zero vector in kn. Mirabolic subgroups were introduced by (Gel'fand & Kazhdan 1975). The image of a mirabolic subgroup in the projective general linear group is a parabolic subgroup consisting of all elements fixing a given point of projective space. The word "mirabolic" is a portmanteau of "miraculous parabolic". As an algebraic group, a mirabolic subgroup is the 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 of a representation of the general linear group.
As an example, the group of all matrices of the form \( {\displaystyle {\begin{pmatrix}a&b\\0&1\end{pmatrix}}} \)where a is a nonzero element of the field 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), Lecture Notes in Math., 1041, Berlin, New York: Springer-Verlag, pp. 50–102, doi:10.1007/BFb0073145, MR 0748505
Bushnell, Colin J.; Henniart, Guy (2006), The local Langlands conjecture for GL(2), Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 335, Berlin, New York: Springer-Verlag, doi:10.1007/3-540-31511-X, ISBN 978-3-540-31486-8, MR 2234120
Finkelberg, Michael; Ginzburg, Victor (2010), "On mirabolic D-modules" (PDF), International Mathematics Research Notices (15): 2947–2986, doi:10.1093/imrn/rnp216, 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", in Gelfand, I. M. (ed.), 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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