The adjoint representation L-function for GL(n)

Ideas underlying the proof of the "simple" trace formula are used to show the following. Let F be a global field, and A its ring of adeles. Let π be a cuspidal representation of GL(n, A) which has a supercuspidal component, and ω a unitary character of A^x/F^x. Let S₀ be a complex number such that for every separable extension E of F of degree n, the L-function L(s, ω o Norm_E/F) over E vanishes at s = s₀ to the order m ≥ 0.; Then the product L-function L(s, π ⊗ ω × π) vanishes at s = So to the order m. This result is a reflection of the fact that the tensor product of a finite dimensional representation with its contragredient contains a copy of the trivial representation.