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

Abstract

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 * which does not factor through the norm map from K to F , then χ ∈ K * with (χ.θ −1 )| F * = ω K/F occurs in r θ + if and only if ǫ(θχ −1 , ψ 0 ) = ǫ(θχ −1 , ψ 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.