Mirabolic group

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In mathematics, a mirabolic subgroup of the general linear group GLn(k), studied by Gel'fand & Kazhdan (1975), is a subgroup consisting of automorphisms fixing a given non-zero vector in kn. Its image 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.

  • Example: The group of all matrices of the form (**
    ) is a mirabolic subgroup of the 2-dimensional general linear group.