I recently came across to the following action of $\text{SL}(2,\Bbb Z)$ on the space $\Bbb R^n\times\Bbb R^n$ defined as $$ \begin{pmatrix} a & b \\ c & d \end{pmatrix}\cdot \big(v,\,w\big)\mapsto \big(dv-cw,\,-bv+aw\big)=\big(v,\,w\big) \begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} $$ where $v=(v_1,\dots,v_n),\, w=(w_1,\dots,w_n)\in \Bbb R^n$.

I would like to determine the $\text{SL}(2,\Bbb Z)$-invariant closed subsets.

Some examples

Of course the trivial subspaces $\{0\}$ and $\Bbb R^n\times\Bbb R^n$ are among these. One can easily check that every linear subspace of the form $V\times V$, where $V<\Bbb R^n$ is invariant under these action. On the other hand, if $V,W<\Bbb R^n$ are different subspaces, then $V\times W$ is not invariant. Similarly, if $\Lambda$ is any lattice in $\Bbb R^n$, then $\Lambda\times\Lambda$ is also invariant.

Not only, considering $\Bbb R^n\times\Bbb R^n$ as $\Bbb R^2\times\cdots\times\Bbb R^2$, where $\big(v,\,w\big)$ is identified with $\big(\big(v_1,\,w_1\big),\dots,\big(v_n,\,w_n\big)\big)$, if $\Lambda$ is a lattice in $\Bbb R^2$, the $n$-times product $\Lambda\times\cdots\times\Lambda$ is $\text{SL}(2,\Bbb Z)$-invariant.

These are some examples of invariant closed subspaces I have found so far. I would like to determine all of them. Any idea, hint or reference is very appreciated!

I recently came across to the following action of $\text{SL}(2,\Bbb Z)$ on the space $\Bbb R^n\times\Bbb R^n$ defined as $$ \begin{pmatrix} a & b \\ c & d \end{pmatrix}\cdot \big(v,\,w\big)\mapsto \big(dv-cw,\,-bv+aw\big)=\big(v,\,w\big) \begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} $$ where $v=(v_1,\dots,v_n),\, w=(w_1,\dots,w_n)\in \Bbb R^n$.

I would like to determine the $\text{SL}(2,\Bbb Z)$-invariant closed subsets.

Some examples

Of course the trivial subspaces $\{0\}$ and $\Bbb R^n\times\Bbb R^n$ are among these. One can easily check that every linear subspace of the form $V\times V$, where $V<\Bbb R^n$ is invariant under these action. On the other hand, if $V,W<\Bbb R^n$ are different subspaces, then $V\times W$ is not invariant. Similarly, if $\Lambda$ is any lattice in $\Bbb R^n$, then $\Lambda\times\Lambda$ is also invariant.

Not only, considering $\Bbb R^n\times\Bbb R^n$ as $\Bbb R^2\times\cdots\times\Bbb R^2$, where $\big(v,\,w\big)$ is identified with $\big(\big(v_1,\,w_1\big),\dots,\big(v_n,\,w_n\big)\big)$, if $\Lambda$ is a lattice in $\Bbb R^2$, the $n$-times product $\Lambda\times\cdots\times\Lambda$ is $\text{SL}(2,\Bbb Z)$-invariant.

These are some examples of invariant closed subspaces I have found so far. I would like to determine all of them. Any idea, hint or reference is very appreciated!

Edit: I was thinking this problem from a generic point of view. However, I am mainly interested on invariant closed subset of $\Bbb R^n\times\Bbb R^n$ which are also subgroups.

I recently came across to the following action of $\text{SL}(2,\Bbb Z)$ on the space $\Bbb R^n\times\Bbb R^n$ defined as $$ \begin{pmatrix} a & b \\ c & d \end{pmatrix}\cdot \big(v,\,w\big)\mapsto \big(dv-cw,\,-bv+aw\big)=\big(v,\,w\big) \begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} $$ where $v=(v_1,\dots,v_n),\, w=(w_1,\dots,w_n)\in \Bbb R^n$.

I would like to determine the $\text{SL}(2,\Bbb Z)$-invariant closed subspaces.

Some examples

Of course the trivial subspaces $\{0\}$ and $\Bbb R^n\times\Bbb R^n$ are among these. One can easily check that every linear subspace of the form $V\times V$, where $V<\Bbb R^n$ is invariant under these action. On the other hand, if $V,W<\Bbb R^n$ are different subspaces, then $V\times W$ is not invariant. Similarly, if $\Lambda$ is any lattice in $\Bbb R^n$, then $\Lambda\times\Lambda$ is also invariant.

Not only, considering $\Bbb R^n\times\Bbb R^n$ as $\Bbb R^2\times\cdots\times\Bbb R^2$, where $\big(v,\,w\big)$ is identified with $\big(\big(v_1,\,w_1\big),\dots,\big(v_n,\,w_n\big)\big)$, if $\Lambda$ is a lattice in $\Bbb R^2$, the $n$-times product $\Lambda\times\cdots\times\Lambda$ is $\text{SL}(2,\Bbb Z)$-invariant.

These are some examples of invariant closed subspaces I have found so far. I would like to determine all of them. Any idea, hint or reference is very appreciated!

I recently came across to the following action of $\text{SL}(2,\Bbb Z)$ on the space $\Bbb R^n\times\Bbb R^n$ defined as $$ \begin{pmatrix} a & b \\ c & d \end{pmatrix}\cdot \big(v,\,w\big)\mapsto \big(dv-cw,\,-bv+aw\big)=\big(v,\,w\big) \begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} $$ where $v=(v_1,\dots,v_n),\, w=(w_1,\dots,w_n)\in \Bbb R^n$.

I would like to determine the $\text{SL}(2,\Bbb Z)$-invariant closed subsets.

Some examples

Of course the trivial subspaces $\{0\}$ and $\Bbb R^n\times\Bbb R^n$ are among these. One can easily check that every linear subspace of the form $V\times V$, where $V<\Bbb R^n$ is invariant under these action. On the other hand, if $V,W<\Bbb R^n$ are different subspaces, then $V\times W$ is not invariant. Similarly, if $\Lambda$ is any lattice in $\Bbb R^n$, then $\Lambda\times\Lambda$ is also invariant.

Not only, considering $\Bbb R^n\times\Bbb R^n$ as $\Bbb R^2\times\cdots\times\Bbb R^2$, where $\big(v,\,w\big)$ is identified with $\big(\big(v_1,\,w_1\big),\dots,\big(v_n,\,w_n\big)\big)$, if $\Lambda$ is a lattice in $\Bbb R^2$, the $n$-times product $\Lambda\times\cdots\times\Lambda$ is $\text{SL}(2,\Bbb Z)$-invariant.

These are some examples of invariant closed subspaces I have found so far. I would like to determine all of them. Any idea, hint or reference is very appreciated!

I recently came across to the following action of $\text{SL}(2,\Bbb Z)$ on the space $\Bbb R^n\times\Bbb R^n$ defined as $$ \begin{pmatrix} a & b \\ c & d \end{pmatrix}\cdot \big(v,\,w\big)\mapsto \big(dv-cw,\,-bv+aw\big)=\big(v,\,w\big) \begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} $$ where $v=(v_1,\dots,v_n),\, w=(w_1,\dots,w_n)\in \Bbb R^n$.

I would like to determine the $\text{SL}(2,\Bbb Z)$-invariant subspaces.

Some examples

Of course the trivial subspaces $\{0\}$ and $\Bbb R^n\times\Bbb R^n$ are among these. One can easily check that every linear subspace of the form $V\times V$, where $V<\Bbb R^n$ is invariant under these action. On the other hand, if $V,W<\Bbb R^n$ are different subspaces, then $V\times W$ is not invariant. Similarly, if $\Lambda$ is any lattice in $\Bbb R^n$, then $\Lambda\times\Lambda$ is also invariant.

Not only, considering $\Bbb R^n\times\Bbb R^n$ as $\Bbb R^2\times\cdots\times\Bbb R^2$, where $\big(v,\,w\big)$ is identified with $\big(\big(v_1,\,w_1\big),\dots,\big(v_n,\,w_n\big)\big)$, if $\Lambda$ is a lattice in $\Bbb R^2$, the $n$-times product $\Lambda\times\cdots\times\Lambda$ is $\text{SL}(2,\Bbb Z)$-invariant.

These are some examples of invariant subspaces I have found so far. I would like to determine all of them. Any idea, hint or reference is very appreciated!

I recently came across to the following action of $\text{SL}(2,\Bbb Z)$ on the space $\Bbb R^n\times\Bbb R^n$ defined as $$ \begin{pmatrix} a & b \\ c & d \end{pmatrix}\cdot \big(v,\,w\big)\mapsto \big(dv-cw,\,-bv+aw\big)=\big(v,\,w\big) \begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} $$ where $v=(v_1,\dots,v_n),\, w=(w_1,\dots,w_n)\in \Bbb R^n$.

I would like to determine the $\text{SL}(2,\Bbb Z)$-invariant closed subspaces.

Some examples

Of course the trivial subspaces $\{0\}$ and $\Bbb R^n\times\Bbb R^n$ are among these. One can easily check that every linear subspace of the form $V\times V$, where $V<\Bbb R^n$ is invariant under these action. On the other hand, if $V,W<\Bbb R^n$ are different subspaces, then $V\times W$ is not invariant. Similarly, if $\Lambda$ is any lattice in $\Bbb R^n$, then $\Lambda\times\Lambda$ is also invariant.

Not only, considering $\Bbb R^n\times\Bbb R^n$ as $\Bbb R^2\times\cdots\times\Bbb R^2$, where $\big(v,\,w\big)$ is identified with $\big(\big(v_1,\,w_1\big),\dots,\big(v_n,\,w_n\big)\big)$, if $\Lambda$ is a lattice in $\Bbb R^2$, the $n$-times product $\Lambda\times\cdots\times\Lambda$ is $\text{SL}(2,\Bbb Z)$-invariant.

These are some examples of invariant closed subspaces I have found so far. I would like to determine all of them. Any idea, hint or reference is very appreciated!