In mathematics, a mirabolic subgroup of the general linear group GL<sub>n</sub>(k) is a subgroup consisting of automorphisms fixing a given non-zero vector in k<sup>n</sup>. Mirabolic subgroups were introduced by . 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 where is a nonzero element of the field and is any element of is a mirabolic subgroup of the 2-dimensional general linear group.