Large sieve inequalities for GL(n)-forms in the conductor aspect

Duke and Kowalski in [A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations, Invent. Math. 139(1) (2000) 1–39 (with an appendix by Dinakar Ramakrishnan)] derive a large sieve inequality for automorphic forms on GL(n) via the Rankin–Selberg method. We give here a partial complement to this result: using some explicit geometry of fundamental regions, we prove a large sieve inequality yielding sharp results in a region distinct to that in [Duke and Kowalski, A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations, Invent. Math. 139(1) (2000) 1–39 (with an appendix by Dinakar Ramakrishnan)]. As an application, we give a generalization to GL(n) of Duke's multiplicity theorem from [Duke, The dimension of the space of cusp forms of weight one, Internat. Math. Res. Notices (2) (1995) 99–109 (electronic)]; we also establish basic estimates on Fourier coefficients of GL(n) forms by computing the ramified factors for GL(n)×GL(n) Rankin–Selberg integrals.