  Let $V$ be a finite dimensional vector space over a local field $F$. Let
$\chi: F^\times \rightarrow \mathbb C^\times$ be an arbitrary character of
$F^\times$. We determine the structure of the natural representation of
$\mathrm{GL}(V)$ on the space $\mathcal{S}^*(V)^\chi$ of $\chi$-invariant
distributions on $V$.
