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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.)

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    $\begingroup$ 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. $\endgroup$
    – WhatsUp
    Jan 12, 2020 at 7:42
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    $\begingroup$ It is indeed known, see sciencedirect.com/science/article/pii/S0001870808002260 $\endgroup$
    – kneidell
    Jan 12, 2020 at 9:35

2 Answers 2

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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).

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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.

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