A correction to Conducteur des Représentations du groupe linéaire

Nadir Matringe has indicated to me that the paper Conducteur des Représentations du groupe linéaire ([JPSS81a], citeError2) contains an error. I correct the error in this note. The correct proof is actually simpler than the erroneous proof. Separately, Matringe has given a different, interesting proof of the result in question ([Mat11]). We recall the result in question. Let F be a local field. We denote by α the absolute value, by q the cardinality of the residual field and finally by v the valuation of F . Thus α(x) = |x| = q−v(x). Let ψ be an additive character of F whose conductor is the ring of integers OF . Let Gr be the group GL(r) regarded as an algebraic group. We denote by dg the Haar measure of Gr(F ) for which the compact group Gr(OF ) has volume 1. Let Nr be the subgroup of upper triangular matrices with unit diagonal. We define a character θr,ψ : Nr(F )→ C×