On Counting Twists of a Character Appearing in its Associated Weil Representation

TL;DR

This paper investigates the number of characters with specific epsilon factor properties that appear in the Weil representation associated with a character of a quadratic extension, providing explicit counts at each conductor level.

Contribution

It offers a precise calculation of how many characters with given epsilon factors occur in the Weil representation for each conductor, advancing understanding of character distributions.

Findings

Explicit counts of characters at each conductor level.

Characterization of epsilon factor conditions for representation components.

Enhanced understanding of Weil representation decompositions.

Abstract

Consider an irreducible, admissible representation $\pi$ of GL(2,$F$) whose restriction to GL(2,$F)^+$ breaks up as a sum of two irreducible representations $\pi_+ + \pi_-$. If $\pi=r_{\theta}$, the Weil representation of GL(2,$F$) attached to a character $\theta$ of $K^*$ which does not factor through the norm map from $K$ to $F$, then $\chi\in \hat{K^*}$ with $(\chi >. \theta ^{-1})|_{F^{*}}=\omega_{{K/F}}$ occurs in ${r_{\theta}}_+$ if and only if $\epsilon(\theta\chi^{-1},\psi_0)=\epsilon(\bar \theta\chi^{-1},\psi_0)=1$ and in ${r_{\theta}}_-$ if and only if both the epsilon factors are -1. But given a conductor $n$, can we say precisely how many such $\chi$ will appear in $\pi$? We calculate the number of such characters at each given conductor $n$ in this work.