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

TL;DR

This paper investigates the asymptotic growth of fixed vector dimensions in admissible representations of  over non-archimedean local fields, linking character germs to Langlands functoriality.

Contribution

It provides explicit calculations of fixed vector dimensions and explores their behavior under Langlands functoriality, connecting representation theory and number theory.

Findings

Dimensions grow asymptotically with N

Character germs determine fixed vector dimensions

Behavior under Langlands transfers observed

Abstract

We calculate the asymptotic behavior of the dimension of the fixed vectors of $\pi$ with respect to compact open subgroups $1+ M_n(\mathfrak{p}^N)\subset\mathrm{GL}_n(F)$ for $\pi$ an admissible representation of $\mathrm{GL}_n(F)$, 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.