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 $\mathbb{F}^n$.



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> 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? 
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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.)



Is the representation of $GL_n(\mathcal{O})$ in functions on Grassmannian multiplicity free?
Yes, this is due to Hill:


*Hill, Gregory*, [**On the nilpotent representations of (GL\_ n({\mathcal O}))**](http://dx.doi.org/10.1007/BF02567703), 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.**](http://dx.doi.org/10.1016/j.jpaa.2006.02.002), J. Pure Appl. Algebra 208, No. 3, 905-922 (2007).


and


*Bader, Uri; Onn, Uri*, [**On some geometric representations of (GL\_N(\mathfrak{o}))**](http://dx.doi.org/10.1080/00927872.2011.561508), Commun. Algebra 40, No. 9, 3169-3191 (2012).


