Gelfand-Kirillov dimension of representations of ${\mathrm{G}\mathrm{L}}_n$ over a non-archimedean local field

We calculate the asymptotic behavior of the dimension of the fixed vectors of $\pi$ with respect to compact open subgroups $1+M_n({\mathfrak{p}}^N)\subset {\mathrm{G}\mathrm{L}}_n(F)$ for $\pi$ an admissible representation of ${\mathrm{G}\mathrm{L}}_n(F)$, and $F$ a nonarchimedean local field. Such dimensions can be calculated by germs of the character of $\pi$. We also make some observations on how those dimensions behave under instances of Langlands functoriality, such as the Jacquet-Langlands correspondence and cyclic base change, where relations between characters are known.