Let $F$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers. Let $Gr\_{i,n}$ denote the Grassmannian of $i$-dimensional linear subspaces in $F^n$. 



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> Can one describe explicitly the $GL(n,\mathcal{O})$-orbits on $Gr\_{i,n}\times Gr\_{i,n}$, i.e. on pairs of subspaces?
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Remark. $GL(n,\mathcal{O})$ acts transitively on $Gr\_{i,n}$. 



Orbits of $GL(n, \mathcal{O})$ on pairs of linear subspaces over non-Archimedean local fields
The $\text{GL}(n,\mathcal{O})$-orbits refine the $\text{GL}(n,F)$-orbits, i.e., the Bruhat cells. Up to replacing $i$ by $n-i$, assume that $2i\leq n.$ For every integer $m$ with $0\leq m\leq i,$ the **Bruhat cell** $U\_m$ in $\text{Gr}\_{i,n}\times \text{Gr}\_{i,n}$ is the set of pairs $([V],[W])$ of $F$-vector subspaces of $F^{\oplus n},$ $V\cong F^{\oplus i}$ and $W\cong F^{\oplus i},$ such that $V\cap W$ is an $F$-vector subspace of codimension $m$ in both $V$ and in $W$. The Bruhat cell $U\_m$ is an affine space over $F$ of dimenion $i(n-i)+m(n-m-i).$


For every finite, free $\mathcal{O}$-module, say $\mathcal{O}^{\oplus n}$, with its associated $F$-vector space, $\mathcal{O}^{\oplus n} \otimes\_{\mathcal{O}} F = F^{\oplus n},$ for every $F$-vector subspace $V\subset F^{\oplus n},$ denote by $V\_\mathcal{O}$ the intersection $V\cap \mathcal{O}^{\oplus n}.$ 


**Lemma 1.** The $\mathcal{O}$-submodule $V\_{\mathcal{O}}$ of $\mathcal{O}^{\oplus n}$ is a finite, free $\mathcal{O}$-module that is a direct summand of $\mathcal{O}^{\oplus n}.$


**Proof.**
Since $\mathcal{O}$ is a Noetherian ring, the $\mathcal{O}$-submodule $V\_{\mathcal{O}}$ of $\mathcal{O}^{\oplus n}$ is a finitely generated $\mathcal{O}$-module. Since $V\_\mathcal{O}$ is an $\mathcal{O}$-submodule of $\mathcal{O}^{\oplus n},$ it is also torsion-free. Since $\mathcal{O}$ is a DVR, also $V\_{\mathcal{O}}$ is a finite, free $\mathcal{O}$-module. Finally, since $V\_{\mathcal{O}}$ is a saturated $\mathcal{O}$-submodule of $\mathcal{O}^{\oplus n}$, also the cokernel $\mathcal{O}^{\oplus n}/V\_{\mathcal{O}}$ is also a finitely generated and torsion-free $\mathcal{O}$-module. Thus, also $\mathcal{O}^{\oplus n}/V\_{\mathcal{O}}$ is a finite, free $\mathcal{O}$-module. Altogether, this implies that $V\_{\mathcal{O}}$ is a direct summand of $\mathcal{O}^{\oplus n}.$ **QED**


Denote by $\pi$ a uniformizing element of $\mathcal{O}.$


**Lemma 2.** For every pair of $F$-linear subspaces of $F^{\oplus n},$ $V\cong F^{\oplus \ell}$ and $W\cong F^{\oplus m},$ giving a direct sum decomposition, there exists a free basis $(b\_1,\dots,b\_m)$ of $W\_{\mathcal{O}},$ and an ordered sequence of integers $0\leq e\_1\leq \dots \leq e\_m$ such that for every $i=1,\dots,m,$ the image of $b\_i$ in $\mathcal{O}^{\oplus n}/V\_{\mathcal{O}}$ equals $\pi^{e\_i}\cdot c\_i$ for some $c\_i,$ and $(c\_1,\dots,c\_m)$ is a free basis for $\mathcal{O}^{\oplus n}/V\_{\mathcal{O}}.$ The ordered sequence of integers is independent of the choice of basis.


**Proof.** This is part of the theory of finitely generated modules over a Principal Ideal Domain. The sequence of powers $(\pi^{e\_1},\dots,\pi^{e\_m})$ are the **elementary divisors**. **QED**


Denote $i-m$ by $r$. By Lemma 1, the quotient $\mathcal{O}$-module $(V+W)\_{\mathcal{O}}/(V\cap W)\_{\mathcal{O}}$ is a finite, free $\mathcal{O}$-module with associated $F$-vector space $(V+W)/(V\cap W).$ The $F$-vector subspaces, $V/(V\cap W)\cong F^{\oplus m}$ and $W/(V\cap W)\cong F^{\oplus m}$ give a direct sum decomposition of $(V+W)/(V\cap W).$ For every sequence $\underline{e}=(e\_1,\dots,e\_m)$ of nonnegative integers, denote by $U\_{\underline{e},m}\subset U\_m$ the set of pairs such that the elementary divisors of the $\mathcal{O}$-module homomorphism, $$(W/(V\cap W))\_{\mathcal{O}} \to (V+W)\_{\mathcal{O}}/V\_{\mathcal{O}},$$ equal $(\pi^{e\_1},\dots,\pi^{e\_m}).$


**Proposition.** The group $\text{GL}(n,\mathcal{O})$ acts transitively on $U\_{\underline{e},m}.$


**Proof.** By Lemma 1, $(V\cap W)\_{\mathcal{O}}$ is a finite, free direct summand in $V\_{\mathcal{O}},$ in $W\_{\mathcal{O}},$ and in $\mathcal{O}^{\oplus n}.$ Similarly, $(V+W)\_{\mathcal{O}}$ is a finite, free direct summand in $\mathcal{O}^{\oplus n}.$ Thus, there exists a free basis $(d\_1,\dots,d\_r)$ for $(V\cap W)\_{\mathcal{O}},$ and there exists an extension $(d\_1,\dots,d\_r,a\_1,\dots,a\_m)$ to a free basis of $V\_{\mathcal{O}}.$ By Lemma 2, there exists an extension $(d\_1,\dots,d\_r,a\_1,\dots,a\_m,c\_1,\dots,c\_m)$ to a free basis of $(V+W)\_{\mathcal{O}}$ and there exists a free basis $(d\_1,\dots,d\_r,b\_1,\dots,b\_m)$ of $W\_{\mathcal{O}}$ such that for the images in $(V+W)\_{\mathcal{O}}/V\_{\mathcal{O}}$, each $\overline{b}\_i$ equals $\pi^{e\_i}\overline{c}\_i.$ Finally, there exists an extension $(d\_1,\dots,d\_r,a\_1,\dots,a\_m,c\_1,\dots,c\_m,g\_1,\dots,g\_{n-(2m+r)})$ to a free basis for $\mathcal{O}^{\oplus n}.$ 


For any other pair $(V',W')$ with associated bases $(d'\_i),$ etc., there exists a unique $\mathcal{O}$-module isomorphism $V\_{\mathcal{O}}\to V'\_{\mathcal{O}}$ sending the free basis $(d\_1,\dots,d\_r,a\_1,\dots,a\_m)$ to $(d'\_1,\dots,d'\_r,a'\_1,\dots,a'\_m).$ There exists a unique $\mathcal{O}$-module isomorphism $(V+W)\_{\mathcal{O}}/V\_{\mathcal{O}}\to (V'+W')\_{\mathcal{O}}/W'\_{\mathcal{O}}$ sending $(\overline{c}\_1,\dots,\overline{c}\_m)$ to $(\overline{c}'\_1,\dots,\overline{c}'\_m).$ There is a unique lift of these moduli isomorphisms to an $F$-vector space isomorphism $V+W\to V'+W'$ that maps $(b\_1,\dots,b\_m)$ to $(b'\_1,\dots,b'\_m).$ Because the elementary divisors are equal, this $F$-vector space isomorphism maps the $\mathcal{O}$-module saturation $(V+W)\_{\mathcal{O}}$ of $V\_{\mathcal{O}}+W\_{\mathcal{O}}$ to the $\mathcal{O}$-module saturation $(V'+W')\_{\mathcal{O}}$ of $V'\_{\mathcal{O}}+W'\_{\mathcal{O}}.$ Finally, there is a unique extension to a $\mathcal{O}$-module isomorphism of $\mathcal{O}^{\oplus n}$ to $\mathcal{O}^{\oplus n}$ sending $(d\_1,\dots,d\_r,a\_1,\dots,a\_m,b\_1,\dots,b\_m,g\_1,\dots,g\_{n-(r+2m)})$ to $(d'\_1,\dots,d'\_r,a'\_1,\dots,a'\_m,b'\_1,\dots,b'\_m,g'\_1,\dots,g'\_{n-(r+2m)}).$ **QED**


**Edit.** As Uri Bader points out, the data of an integer $m$ satisfying $0\leq m \leq i$ together with the sequence of elementary divisors is equivalent to the data of the $\mathcal{O}$-module isomorphism class of the finitely generated $\mathcal{O}$-module, $$\mathcal{O}^{\oplus n}/(V\_{\mathcal{O}}+W\_{\mathcal{O}}).$$ So, as in Uri Bader's answer, a more intrinsic index for the $\text{GL}(n,\mathcal{O})$-orbits of $\text{Gr}\_{i,n}\times \text{Gr}\_{i,n}$ is an isomorphism class of an $\mathcal{O}$-module that is generated by $\leq n-i$ generators and whose free rank is at least $n-2i$.


