Let $\mathbb{F}$ be a local non-Archimedean field. Let $\mathcal{O}\subset \mathbb{F}$ be its ring of integers. Let $GL_n(\mathcal{O})$ be the (compact) group of $n\times n$ invertible matrices with entries in $\mathcal{O}$ such the inverse matrix also has entries in $\mathcal{O}$. Let $Gr_{i,n}$ be the Grassmannnian of linear $i$-dimensional subspaces in $\mathbb{F}^n$.

Is it true that the natural representation of $GL_n(\mathcal{O})$ in the space of locally constant functions on $Gr_{i,n}$ is multiplicity free? In other words is it true that any irreducible representation of $GL_n(\mathcal{O})$ appears with multiplicity at most 1?

A reference would be helpful.

**Remark.** Archimedean analogue of the above statement, i.e. $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$, is true; here $GL_n(\mathcal{O})$ should be replaced by $O(n)$ or $U(n)$ respectively. (That follows from the fact that the Grassmannian is a symmetric space.)