On counting twists of a character appearing in its associated Weil representation

Abstract.

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