Uniform bounds and uncertainty for asymptotics of representations of $p$-adic ${\rm GL}_N$

Abstract

We prove two results on the growth of dimensions of fixed vectors of representations $\pi$ of $p$-adic ${\rm GL}_N$ under principal congruence subgroups: First, a uniform bound on the growth of fixed vectors in terms of the GK-dimension $\pi$, which we extend to a uniform bound on the Harish-Chandra--Howe coefficients. Second, for $\pi$ unitary, a quantitative relationship between the GK-dimension of $\pi$ and the rate of decay of its matrix coefficients. These results are independent of one another and proved in the framework of the Langlands and Zelevinsky classifications.