  In this article, by studying the Bernstein degrees and Goldie rank
polynomials, we establish a comparison between the irreducible representations
of $G=\textrm{GL}_n(\mathbb{C})$ possessing the minimal Gelfand-Kirillov
dimension and those induced from finite-dimensional representations of the
maximal parabolic subgroup of $G$ of type $(n-1,1)$. We give the transition
matrix between the two bases for the corresponding coherent families.
