Complement to the appendix of: “On the Howe duality conjecture”

Abstract

Let F {\mathbb F} be a local field, nonarchimedean and of characteristic not 2. Let ( V , Q ) (V,Q) be a nondegenerate quadratic space over F {\mathbb F} , of dimension n n . Let M r M_r be the direct sum of r r copies of V V . We prove that, for r > n r>n there is no nonzero distribution on M r M_r which under the action of the orthogonal group transforms according to the character determinant.