Representation theory of mirabolic quantum ${\mathfrak{s}\mathfrak{l}}_n$

We show that the mirabolic quantum group $MU(n)$ is a comodule algebra over the quantized enveloping algebra $U_v({\mathfrak{s}\mathfrak{l}}_n)$, and use this structure to give a complete classification of its finite dimensional representations. In particular, we explicitly describe the construction of all irreducible finite dimensional representations of $MU(n)$ and show that the category of finite dimensional representations is semisimple. A crucial step involves constructing and analyzing Verma-type universal representations of $MU(n)$.