Trilinear Forms and Triple Product Epsilon Factors

Abstract

We give a short and simple proof of a theorem of Dipendra Prasad on the existence or nonexistence of invariant trilinear forms on a triple of irreducible representations of GL 2 (F) or D × , where $F$ is a nonarchimedean local field of zero or odd characteristic and $D$ is the unique quaternion division F-algebra. The answer is controlled by the central value of the triple product epsilon factor. Our proof works uniformly for all representations and without restriction on residual characteristic. It also gives an analogous theorem for any separable cubic F-algebra.