Is the representation of $GL_n(\mathcal{O})$ in functions on Grassmannian multiplicity free?

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.)

• Perhaps it would help to give some examples, e.g. what happens when $i=n-1$? Or for small $n$ such as $n=2$? A classification of irreducible representations of $\operatorname{GL}_2(\mathcal O)$ seems to be known. Jan 12, 2020 at 7:42
• It is indeed known, see sciencedirect.com/science/article/pii/S0001870808002260 Jan 12, 2020 at 9:35

Yes, this is due to Hill:

Hill, Gregory, On the nilpotent representations of (GL_ n({\mathcal O})), Manuscr. Math. 82, No. 3-4, 293-311 (1994). See especially Corollary 3.2.

This was generalised and extended by Bader and Onn in

Bader, Uri; Onn, Uri, Geometric representations of (\text{GL}(n,R)), cellular Hecke algebras and the embedding problem., J. Pure Appl. Algebra 208, No. 3, 905-922 (2007).

and

Bader, Uri; Onn, Uri, On some geometric representations of (GL_N(\mathfrak{o})), Commun. Algebra 40, No. 9, 3169-3191 (2012).

This may be redundant as a complete answer with references has already been posted. Just in case this is still useful: I think multiplicity one can be proved by the usual Gefland trick: we need to check that endomorphisms of the induced representation is a commutative ring, this follows once we check that the identity automorphism of that algebra is an anti-involution. The endomorphism algebra coincides with the Hecke algebra $${\mathbb{C}}[P(O)\backslash G(O)/P(O)]$$ where $$P$$ is the group of block upper triangular matrices with blocks of sizes $$i, \, n-i$$. This can also be realized as the space of $$G(O)$$-invariant functions on pairs of rank $$i$$ summands in $$O^n$$. Switching the two elements in the pair induces an anti-involution. Looking at relative positions of two direct summands in $$O^n$$ one sees that the relative position of a pair $$(N,M)$$ is the same as the relative position of $$(M,N)$$, which yields the statement.