  The quantum enveloping algebra of $\mathfrak{sl}_n$ (and the quantum Schur
algebras) was constructed by Beilinson-Lusztig-MacPherson as the convolution
algebra of $GL_d$-invariant functions over the space of pairs of partial
$n$-step flags over a finite field. In this paper we expand the construction to
the mirabolic setting of triples of two partial flags and a vector, and examine
the resulting convolution algebra. In the case of $n=2$, we classify the finite
dimensional irreducible representations of the mirabolic quantum algebra and we
prove that the category of such representations is semisimple. Finally, we
describe a mirabolic version of the quantum Schur-Weyl duality, which involves
the mirabolic Hecke algebra.
