Irreducible lattices in $G=G_1\times G_2$

Question

First, we shall recall the definition of an irreducible lattice.

Let $G$ be a Lie group which admits a direct product decomposition into simple non-compact factors $G_1\times\dots\times G_k$. A lattice $\Gamma$ in $G$ is irreducible if the intersection of $\Gamma$ with any factor $G_{i_1}\times\dots\times G_{i_j}$ is not a lattice [Wikipedia].

Now, let $G$ be a Lie group which admits a direct product decomposition into simple factors $G_1\times G_2$. Let $\Gamma$ be a lattice in $G$ and assume that $\Gamma\cap G_1$ is not a lattice in $G_1$. What conditions are sufficient to say $\Gamma$ is irreducible?

Obviously we can simply satisfy the above definition, but in the case of two factors of $G$ are there other equivalent conditions which could hold that would not in the case of $3$ or more factors?

Answer 1

Let $G_1,G_2$ be simple, noncompact connected Lie groups with finite center.

For a lattice $\Gamma$ in $G=G_1\times G_2$, the following are equivalent:

1. $\Gamma\cap G_1$ is infinite
2. $\Gamma\cap G_2$ is infinite
3. $\Gamma\cap G_1$ is a lattice in $G_1$
4. $\Gamma\cap G_2$ is a lattice in $G_2$
5. the projection of $\Gamma$ on $G_1$ is dense
6. the projection of $\Gamma$ on $G_2$ is dense
7. There exist lattices $\Lambda_1,\Lambda_2$ in $G_1$ and $G_2$ such that $\Lambda_1\times\Lambda_2$ is contained in $\Gamma$ as a finite index subgroup.

(The negation of either of these conditions is called being "irreducible".)

With more factors, the definition is a little more tricky. Let $I$ be a finite set, $G_i$ be simple noncompact connected Lie groups with finite center. For $J\subset I$, write $G_J=\prod_{j\in J}G_j$, and $G=G_I$. Let $\Gamma$ be a lattice in $G$. The following are equivalent

1. there exists a subset $J\notin\{\emptyset,I\}$ of $I$ such that $\Gamma\cap G_J$ is a lattice in $G_J$
2. for every $i$, the projection of $\Gamma$ on $G_{I\smallsetminus\{i\}}$ is dense
3. for every proper subset $J$ of $I$, the projection of $\Gamma$ on $G_J$ is dense
4. there exists a partition $I=J\sqcup K$ with $J,K$ both nonempty, and lattices $\Lambda_J,\Lambda_K$ in $G_J$ and $G_K$ such that $\Lambda_J\cap\Lambda_K$ is contained in $\Gamma$ as a finite index subgroup.

Note that the condition "for every $i$, the projection of $\Gamma$ on $G_i$ is dense", sometimes abusively called "irreducible" by confusion with the case $|I|=2$, is not equivalent: it is false for $|I|=1$, and strictly weaker if $|I|\ge 4$.

I think all this material is more or less covered in Raghunathan's book.