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 $$e_n’=\frac{1}{q_n}\Sigma _{g\in \Sigma _n}sgn(g)g \Sigma _{g\in B _n}g \in Z_{(2)}[Gl_n(Z/2)]$$ where $q_n$ is an appropriate odd number, $sgn$ is the signature of the permutation. Now, let’s … Read more

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 C–rank≥2 and let H be a normal subgroup of G(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 wα1,…,wαn represent the simple reflections of the … Read more

Steinberg relations for elementary subgroup of a Chevalley group over an arbitrary ring

Given a semisimple Lie algebra g of type Φ with a Lie algebra representation ρ:g→gl(v) and an arbitrary commutative ring one can associate the following gadgets: A simply connected split reductive group scheme G(Φ) over R. An abstract group G(Φ,R) which is the R-points of the reductive group scheme G(Φ) which comes with an action … Read more

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’s lecture notes, every simple untwisted Chevalley group G can be obtained as follows. Consdider a finite-dimensional, complex simple Lie algebra L together with … Read more

Chevalley groups GF2G_{\mathbb{F}_2} in GQG_\mathbb{Q}

Is every (adjoint) Chevalley group over the field with two elements GF2 isomorphic to a subgroup of its counterpart over the rationals GQ? Answer No. Here’s a cheap argument. Let G=PGLn for n>8 even. Inside GF2, we have the group of upper-triangular matrices which differ from the identity matrix only in the upper-right quadrant of … Read more

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 xb:Ga→Ub of the corresponding root subgroup, and for each element w of the Weyl group (of the root system) there are lifts ˙w, such that ˙wxb(c)˙w−1 = xw(b)(±c). Are there any … Read more

What meanings does “Chevalley group” have?

It appears to me that there are at least two working definitions of the term “Chevalley group” operative in the literature. For example, one can consider Steinberg’s notes on the subject. Starting from a complex Lie algebra g and a finite dimensional representation V of it, he constructs a lattice M in V and a … Read more

Is a presentation of the hyperbolic orthogonal group of rank 2 over the integers known?

The hyperbolic orthogonal group Og,g(Z) often appears in the study of high-dimensional manifolds, see e.g. work of Kreck or Galatius and Randal-Williams. Let H denote the lattice Z{e,f} with symmetric bilinear form λ determined by λ(e,e)=λ(f,f)=0 and λ(e,f)=1. Then Og,g(Z) is defined to be the group of automorphisms of the orthogonal direct sum H⊕g. This … Read more

Finite index subgroup of GLn(C)\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(Z) is Zariski dense in G(C). (G(Z) is the Chevalley group over Z and similarly for G(C)) But I’m struggling to understand some basic stuff, It seems to me that there can’t be finite index subgroup in G(C) … Read more