A constraint for twist equivalence of cusp forms on GL$(n)$

Abstract

This Note answers, and generalizes, a question of Kaisa Matomäki. We show that give two cuspidal automorphic representations ${\pi }_1$ and ${\pi }_2$ of $G{L}_n$ over a number field $F$ of respective conductors $N_1,$ $N_2,$ every character $\chi$ such that ${\pi }_1\otimes \chi \simeq {\pi }_2$ of conductor $Q,$ satisfies the bound: $Q^n\mid N_1{N}_2.$ If at every finite place $v,$ ${\pi }_{1,v}$ is a discrete series whenever it is ramified, then $Q^n$ divides the least common multiple $[N_1,N_2].$

The main result of the earlier version remains unchanged in this version. The main change is that the proof has been modified to "not use" anything about Galois representations or the local Langlands conjecture. Instead this new proof works totally on the "automorphic side", using only the representation theory of GL(n) and division algebras