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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.

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  • $\begingroup$ It's not an unusual action. First since this is the restriction of an algebraic action of $G=\mathrm{SL}_2(\mathbf{R})$ in which it's Zariski dense, its invariant subspaces are the same as for the $G$-action and since $G$ is semisimple this behaves well. Also this action, call it $W_n$, obviously splits coordinate-wise as $W_1^{\otimes n}$, and $W_1$ is absolutely irreducible. Describing invariant subspaces is then an exercise. $\endgroup$
    – YCor
    Commented Apr 19, 2020 at 7:02
  • $\begingroup$ Hi @YCor, thanks for the answer first. I have a couple of doubts. Why do you say that the invariant subspaces for the $\text{SL}(2\Bbb Z)$ action are the same as for the $G$-action? The $n$-times product of lattice is not $G$-invariant. Am I missing something? The second doubt is about the splitting, the action is diagonal on $\Bbb R^2\times\cdots\times \Bbb R^2$. So the splitting may contain factors greater than $W_1$. $\endgroup$
    – InsideOut
    Commented Apr 19, 2020 at 7:11
  • $\begingroup$ When you say "subspace", what exactly do you mean? Presumably not vector subspace (because you mention lattices) - that was probably the context of the answer of @YCor . So what is it? Subset ? Algebraic subvariety? $\endgroup$
    – Vladimir Dotsenko
    Commented Apr 19, 2020 at 7:26
  • $\begingroup$ Yes, maybe the term "subspace" has been misleading. I actually mean topological subspaces, where $\Bbb R^n\times \Bbb R^n$ is endowed with the product of the standard euclidean topology. The point Is that I have examples with linear subspaces and lattices. $\endgroup$
    – InsideOut
    Commented Apr 19, 2020 at 7:29
  • $\begingroup$ Topological subspace just means any subset (with the endowed topology, but this is irrelevant to the question). You really want no further assumption such as "closed"? $\endgroup$
    – YCor
    Commented Apr 19, 2020 at 7:30

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