Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Rahul Dalal, Mathilde Gerbelli-Gauthier, Simon Marshall

Abstract

We prove two results on the growth of dimensions of fixed vectors of representations $π$ 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 $π$, which we extend to a uniform bound on the Harish-Chandra--Howe coefficients. Second, for $π$ unitary, a quantitative relationship between the GK-dimension of $π$ 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.

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 of -adic under principal congruence subgroups: First, a uniform bound on the growth of fixed vectors in terms of the GK-dimension , which we extend to a uniform bound on the Harish-Chandra--Howe coefficients. Second, for unitary, a quantitative relationship between the GK-dimension of 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.

Paper Structure

This paper contains 39 sections, 38 theorems, 158 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Analytic invariants
  3. Fixed-Vector Growth
  4. Local Character Expansion
  5. Matrix Coefficient Decay
  6. Results
  7. Generalizations
  8. Background
  9. Nilpotent orbits in $\mathrm{GL}_N$
  10. Bernstein-Zelevinsky Classifications
  11. Multisegments
  12. Langlands Classification
  13. Zelevinsky Classification
  14. Duality
  15. Degenerate Whittaker Models
  16. ...and 24 more sections

Key Result

Theorem 1.1.1

For a suitable normalization of the $\mu_O$, there are constants $c_O(\pi)$ for $O \in \mathcal{N}$ such that for small enough $X \in \mathfrak{g}$, the trace character of $\pi$ admits the expansion

Figures (1)

  • Figure 1: Tightness of Bounds in Theorem \ref{['thm:atypemctogk']} with $N = 50$.

Theorems & Definitions (85)

  • Theorem 1.1.1: Local Character Expansion, HC99admissibleHowe74Fourier
  • Definition 1.1.2
  • Definition 1.1.3
  • Theorem 1.2.1: Corollary \ref{['cor:bodyfixedvector']}
  • Corollary 1.2.2: Corollary \ref{['cor:maincoefficientsalt']}
  • Theorem 1.2.3: Theorem \ref{['thm:atypemctogk']}
  • Remark 1.2.4
  • Conjecture 1.3.1
  • Definition 2.2.1
  • Definition 2.2.2
  • ...and 75 more