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

Abstract:

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