# Do these Zariski-dense subgroups of complex Chevalley group have non-empty intersection with this Bruhat cell?

Let $$G$$ be a complex Chevalley group (not necessarily adjoint type) with $$\operatorname{\mathbb{C}-rank}\geq2$$ and let $$H$$ be a normal subgroup of $$G(\mathbb Z)$$ with a finite index (so $$H$$ is Zariski dense in $$G$$).

Let $$T$$ a maximal torus in $$G$$ and $$B$$ a Borel subgroup containing $$T$$, let $$w_{\alpha_1},...,w_{\alpha_n}$$ represent the simple reflections of the Weyl group in $$G$$. From Steinberg's 1967-68 Yale lectures $$w_α(t)=x_α(t)x_{−α}(−t^{−1})x_α(t),w_α:=w_α(1)$$ where $$x_α(t)$$ is the root element in $$G$$.

Denote $$w:=w_{\alpha_1}\cdots w_{\alpha_n}$$ now

is it true that $$H∩BwB$$ is necessarily non-empty?

The answer can be positive if the answer to this is positive:

does $$\bigcup_{A\in G(\mathbb{Z})} A^{-1}BwBA$$ Zariski-dense in $$G$$?

When $$G=\operatorname{SL}_{n}(\mathbb C)$$ we get that $$w$$ is the permutation matrix of $$(123..n)$$ when $$B$$ is the upper triangular matrices and $$T$$ the diagonal matrices. Then since $$J=\{A\in \operatorname{SL}_{n}(\mathbb C)|\operatorname{disc}(A)\neq 0\}$$ is Zariski open there is a $$g\in J\cap H$$ and since every element of $$J(\mathbb Z)$$ is similar in $$\operatorname{SL}_{n}(\mathbb Z)$$ to some $$b_1wb_2$$ ,$$b_i\in B$$ and $$H$$ is normal, we get $$g\in H\cap BwB$$.

Here $$disc(A)$$ is the discriminant of the characteristic polynomial of A, when $$disc(A)\neq0$$ that mean that $$A$$ is similar to a companion matrix since all the roots of the characteristic polynomial of A are distinct.

When $$G=\operatorname{SO}_{2n}(\mathbb C)=\{A\in\operatorname{SL}_{2n}(\mathbb C)\mid A^TJ_{2n}A=J_{2n}\}$$ where $$J_{2n}$$ is the identity matrix flipped 90 degrees and the index order is $$1,2...n,-n,...,-1$$. We get that $$w$$ is the permutation matrix of $$((n-1)(n-2)...1(1-n)(2-n)...(-1))(n(-n))$$ when $$B$$ is the upper triangular matrices and $$T$$ the diagonal matrices. I'm thinking of looking for an open set inside $$\{A\in \operatorname{SO}_{2n}(\mathbb C)|A$$ is conjugate to $$C(p_1)\oplus C(p_2)$$ in $$\operatorname{SL}_{2n}(\mathbb C),p_1|p_2,p_i\in\mathbb{C}[x]$$ monic and $$deg(p_1)=2 \}$$.

$$C(p)$$ is the companion matrix of the monic polynomial $$p$$.

In general $$G$$ where the Coxeter number $$h$$ is even there is a Coxeter element $$w_c$$ such that $$w_c^{h/2}=w_0$$ the longest element. So the open set $$Bw_0B=\prod_{i=1}^{h/2}Bw_iB$$ may say that $$\bigcup_{A\in G(\mathbb{Z})} A^{-1}Bw_cBA$$ is open?