The adjoint representation L-function for GL(n)

Yuval Zvi Flicker

Vol. 154 (1992), No. 2, 231–244

DOI: 10.2140/pjm.1992.154.231

Abstract

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^×∕F^×. Let s₀ be a complex number such that for every separable extension E of F of degree n, the L-function L(s,ω ∘ 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 = s₀ 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.

Mathematical Subject Classification 2000

Primary: 11F70

Secondary: 22E55

Milestones

Received: 26 December 1990

Published: 1 June 1992