# Questions tagged [chevalley-groups]

The Chevalley group is a way, uniform over all fields (and commutative rings), to define a split simple algebraic group of a given type.

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### Twisted root subgroups in twisted Chevalley groups (reference request)

I am trying to find a standard reference for the natural analogue of root subgroups (and their properties) in twisted Chevalley groups. Let me first recall the classical set-up. According to Steinberg'...
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### Conjugation of root subgroups by the Weyl group

Fix a field $k$ of characteristic zero, and let $G$ be a connected reductive algebraic $k$-group of isotropic rank $\ge 1$. Fix a maximal $k$-split torus $S$, and let $\Phi_k$ be the relative root ...
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### Chevalley groups $G_{\mathbb{F}_2}$ in $G_\mathbb{Q}$

Is every (adjoint) Chevalley group over the field with two elements $G_{\mathbb{F}_2}$ isomorphic to a subgroup of its counterpart over the rationals $G_\mathbb{Q}$?
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### Signs in Chevalley systems for reductive groups

Let $G$ be a pinned split reductive group. There exists a Chevalley system: For each root $b$ in its root system there are parametrisations $x_b: \mathbb{G}_a \rightarrow U_b$ of the corresponding ...
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### If $\Lambda \cap U$ is Zariski-dense in $U$, then $\Lambda$ contains $U(k\mathbb Z)$ for some $k ≥ 1$?

If $G$ is a $\mathbb Q$-defined subgroup of $\operatorname{GL}_n(\mathbb C)$, $\Lambda$ is a subgroup of $G(\mathbb Z)$, and $U$ is a unipotent subgroup of $G(\mathbb C)$ such that $\Lambda \cap U$ is ...
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### For $G$ an adjoint Chevalley group, are all of $G(\mathbb Z)$'s finite-index subgroups congruence subgroups?

Let $G$ be an adjoint Chevalley group. Are all of $G(\mathbb Z)$'s finite-index subgroups congruence subgroups? I read a theorem that states: When $G$ is the universal Chevalley group and it's not of ...
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### How to prove that Chevalley groups over $\mathbb R$ have no compact factors

I am trying to see why the Chevalley groups (not limited to the adjoint group) over $\mathbb R$ are without compact factors in order to use the Borel density theorem. I've been told in another thread ...
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### Character of a semisimple connected Lie groups [closed]

I'm trying to see why the Chevalley groups over $\mathbb C$ have no nontrivial character? I know that a compact connected semisimple Lie group has no nontrivial character but is the compactness ...
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### Finite index subgroup of $\mathrm{GL}_n(\Bbb C)$ and Chevalley groups

I'm trying to show that if $G$ is a Chevalley group, then every finite index subgroup of $G(\Bbb Z)$ is Zariski dense in $G(\Bbb C)$. ($G(\Bbb Z)$ is the Chevalley group over $\Bbb Z$ and similarly ...
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### Representations of GL(n,2) over a field of characteristic 2

I would appreciate very much if you can point to me some references on the following: 1) Representations of the linear group $GL(n,2)$ over $F_2$. 2) Representations of $GL(n,2)$ over an algebraic ...
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### Exotic 2-adic lifts of mod $2$ Steinberg idempotent
Denote $B_n$ the Borel subgroup of $Gl_n(Z/2)$, i.e., the subgroup of upper triangular matrices, $\Sigma _n$ the subgroup of permutation matrices. The (conjugate) Steinberg idempotent is defined to be ...