Selberg's orthogonality conjecture for automorphic L -functions

Abstract

Let π and π' be automorphic irreducible unitary cuspidal representations of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] and [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /], respectively. Assume that either π or π' is self contragredient. Under the Ramanujan conjecture on π and π', we deduce a prime number theorem for L ( s , π × [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /]), which can be used to asymptotically describe whether π' ≅ π, or π' ≅ π ⊗ |det(·)| i τ0 for some nonzero τ0 ∈ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /], or π' ≅ π ⊗|det(·)| it for any t ∈ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="05i" /]. As a consequence, we prove the Selberg orthogonality conjecture, in a more precise form, for automorphic L -functions L ( s , π) and L ( s , π'), under the Ramanujan conjecture. When m = m ' = 2 and π and π' are representations corresponding to holomorphic cusp forms, our results are unconditional.