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If $G$ is a complex Chevalley group and $H\leq G(\mathbb Z)$ dense in $G(\mathbb C)$, can I find $g\in H$ conjugated in $G(\mathbb Z)$ to an element in the Bruhat cell $BwB$ where $w$ represent a Coxeter element in the Weyl group?
I know that the Bruhat cells that are not the big cell are closed so they are not containing any open sets, but maybe the union of all the conjugation in $G(\mathbb Z)$ of Bruhat-Coxeter cells is Zariski dense in $G(\mathbb C)$?

Edit:$H$ is also of finite index.

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    $\begingroup$ "I know that the Bruhat cells that are not the big cell are closed"? Only the small cell (corresponding to $e \in W$) is closed, unless you mean the closed Bruhat cells (in which case, the big cell is the entire variety). The rest are the intersection of a closed set (the closed Bruhat cell) with an open set (the complement of the union of Bruhat cells with smaller $w$ in the Bruhat order). The statement after is still true (they contain no open sets), though. $\endgroup$
    – user44191
    Commented Aug 23, 2019 at 1:28
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    $\begingroup$ What exactly do you mean by "Chevalley group"? This could be the adjoint-type group, as in Chevalley's 1955 paper, based on the adjoint representation, or could be the more general version treated in Steinberg's 1967-68 Yale lectures, based on a more general faithful representation for groups such as SL$_n(\mathbb{C})$. In either setting, the group over $\mathbb{Z}$ has a Tits system (BN-pair), with "Weyl group" the full $W$ involved. This ensures nonrtrivial intersection with each Bruhat cell. $\endgroup$
    – Jim Humphreys
    Commented Aug 23, 2019 at 19:55
  • $\begingroup$ I meant the more general version by Steinberg, so you are saying that the Tits system of $G(\mathbb Z)$ is the Tits system of $G(\mathbb C)$ intersected with $G(\mathbb Z)$ and have the same presentation for the Weyl group in $G(\mathbb C)$? so $G(\mathbb Z)$ contains nontrivial elements from every cell, but I don't see how it shows $H$ contain a conjugation to an element in a Bruhat-Coexeter cell? Also, I forgot to mention that $H$ has a finite index. $\endgroup$
    – Ami
    Commented Aug 23, 2019 at 23:22
  • $\begingroup$ @JimHumphreys is it also true that for any algebraic extension of $\mathbb Z$, $\overline {\mathbb Z}$, that $G(\overline {\mathbb Z})$ has a Tits system, with the same presentation for the Weyl group in $G(\mathbb C)$? Is there a good source for that? $\endgroup$
    – Ami
    Commented Aug 27, 2019 at 20:33
  • $\begingroup$ @Ami: I don't have a suitable reference in mind. But in any case, I don't see why a Coxeter element of $W$ should plaly a special role in the Bruhat decomposition here, or why other elements of $W$ should automatically be involved in a Chevalley group over $\mathbb{Z}$. (I did study arithmetic groups at an early stage but got more attracted to modular representation theory of algebraic groups.) $\endgroup$
    – Jim Humphreys
    Commented Aug 28, 2019 at 19:24

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