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Let $H\subset \operatorname{GL}(n,\mathbb{C})$ be a connected, semisimple algebraic group defined over $\mathbb{Q}$. Fix a number field $K$ with $[K:\mathbb{Q}]=3$ that is not totally real. Denote its ring of integers by $\mathcal{O}_{K}$. The Borel-Harish Chandra theorem says that $H(\mathcal{O}_{K})$ embeds as a lattice in $H(\mathbb{R})\times H(\mathbb{C})$. Is it true that the projection of $H(\mathcal{O}_{K})$ into the first factor $H(\mathbb{R})$ is dense?

Edit: I am asking because I am trying to understand the proof of Theorem 3 in "Dense subgroups with property (T) in Lie groups" by de Cornulier https://ems.press/journals/cmh/articles/1565. In the last paragraph of the first step, this Borel-Harish-Chandra construction shows up. It is somehow essential that the projection into $H(\mathbb{R})$ is dense, since the aim of the paper is to construct dense subgroups with property (T).

Also, I got quite confused reading about these things. Theorem 3.7 of http://constantinkogler.com/Files/Topics_on_Linear_Groups.pdf claims that the projections of $H(\mathcal{O}_{K})$ into both factors are dense. But if $H(\mathbb{R})$ is compact, then the projection of $H(\mathcal{O}_{K})$ in the second factor $H(\mathbb{C})$ should again be a lattice hence discrete. Am I missing something?

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    $\begingroup$ Yes, if $H(\mathbf{R})$ is compact then the projection is discrete (and hence not dense unless $H=1$). Indeed the theorem in the linked paper is not precise enough in the definition of "irreducible lattice" (it is even false or ambiguous in the case without compact factors, since $\mathrm{SL}_2(\mathbf{Z}[\sqrt{2}])^2$ is a non-irreducible lattice in $\mathrm{SL}_2(\mathbf{R})^4$ but all projections to simple factors are dense). $\endgroup$ Commented Dec 4, 2024 at 18:07
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    $\begingroup$ Keep in mind that these are notes written by a student. $\endgroup$ Commented Dec 5, 2024 at 0:26
  • $\begingroup$ @MoisheKohan as written, it's authored by a professor. It's unclear anyway from the formulation whether it's been checked by the professor. $\endgroup$ Commented Dec 5, 2024 at 0:59
  • $\begingroup$ @YCor: Yes, but written by a student. To which extent the student understood the material is also far from clear. $\endgroup$ Commented Dec 5, 2024 at 2:17
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    $\begingroup$ OK. I'm not sure we should argue much more about it, but it means it would be better to find a reference with a more precise result. Have you checked Dave Witte Morris' book? $\endgroup$ Commented Dec 5, 2024 at 7:42

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