Steinberg-Chevalley Groups

Last updates: 18 February 2012

Steinberg-Chevalley Groups

This section gives a brief treatment of the theory of Chevalley groups. The primary reference is [St] and the extensions to the Kac-Moody case are found in [Ti].

Let $A$ be a Cartan matrix and let ${R}_{\mathrm{re}}$ be the real roots of the corresponding Borcherds-Kac-Moody Lie algebra $𝔤$. Let $U$ be the enveloping algebra of $𝔤$. For each $\mathrm{alpha}\in {R}_{\mathrm{re}}$ fix a choice of ${e}_{\alpha }$ in (2.18) (a choice of $\stackrel{˜}{w}$). Use the notation

Then $xα(t) xα(u) =xα (t+u) in U[[t,u]] .$

Following [Ti, 3.2], a prenilpotent pair is a pair of roots $\alpha ,\beta \in {R}_{\mathrm{re}}$ such that there exists $w,w\prime \in W$ with

 $wα,wβ ∈Rre+ and w′α,w′β ∈-Rre+.$
This condition guarantees that the Lie subalgebra of $𝔤$ generated by ${𝔤}_{\alpha }$ and ${𝔤}_{\beta }$ is nilpotent. Let $\alpha ,\beta$ be a prenilpotent pair and let ${e}_{\alpha }\in {𝔤}_{\alpha }$ and ${e}_{\beta }\in {𝔤}_{\beta }$ be as in (2.18). By [St, Lemma 15] there exist unique integers ${C}_{\alpha ,\beta }^{i,j}$ such that $xα(t) xβ(u) = xβ(u) xα(t) xα+β (Cα,β 1,1 tu) x2α+β (Cα,β 2,1 t2u) xα+2β (Cα,β 1,2 tu2)$

Let $𝔽$ be a commutative ring. The Steinberg group $St is given by generators xα(f) forα∈Rre, and$

 $xα(f1) xβ(f2) = xβ(f2) xα(f1) xα+β (Cα,β 1,1 f1f2) x2α+β (Cα,β 2,1 f12 f2) xα+2β (Cα,β 1,2 f1 f22)$
for prenilpotent pairs $\alpha ,\beta$. In St define
 $nα(g) =xα(g) x-α (-g-1) xα(g), nα =nα(1), and hα∨(g) =nα(g) nα-1,$
for $\alpha \in {R}_{\mathrm{re}}$ and $g\in {𝔽}^{×}$.

Let ${𝔥}_{ℤ}$ be a $ℤ$-lattice in $𝔥$ which is stable under the $W$-action and such that $𝔥ℤ⊇Q∨ , where Q∨ =ℤ-span{ h1,…, hn}$ with ${h}_{1},\dots ,{h}_{n}$ as in (2.2). With

 $T given by generators hλ∨(g) for λ∨∈ 𝔥ℤ, g∈𝔽×,$ $and relations hλ∨ (g1) hλ∨ (g2) =hλ∨ (g1g2) and hλ∨(g) hμ∨(g) =hλ∨ +μ∨ (g) ,$
the Tits group $Gis the group generated by St and T$ with the relations coming from the third equation of (1.3) and the additional relations
 $hλ∨(g) xα(f) hλ∨ (g)-1 = xα(g ⟨λ∨ ,α⟩ f) and ni hλ∨(g) ni-1 =hsi λ∨(g).$

For $\alpha ,\beta \in {R}_{\mathrm{re}}$ let ${\epsilon }_{\alpha \beta }=±1$ be given by $s˜α (eβ) =εαβ esαβ, where s˜α =exp( adeα) exp(-ad fα) exp(adeα)$ (see [CC, p48] and [Ti, (3.3)]). By [St, Lemma 37] (see also [Ti, 3.7a])

 $nα(g) xβ(f) nα(g) -1 = xsαβ (εαβ g-⟨β, α∨⟩f) , hλ∨(g) xβ(f) hλ∨ (g)-1 = xβ(g ⟨β, λ∨⟩ f),$
 $and nα(g) hλ∨ (g′) nα (g)-1 =hsα λ∨ (g′).$

Thus $G$ has a symmetry under the subgroup $Ngenerated by Tand the nα(g) for α∈Rre, g∈𝔽×.$

If $𝔽$ is big enough then $N$ is the normaliser of $T$ in $G$ [St, Ex(b) p36] and, by [St, Lemma 27], the homomorphism

 $N → W nα(g) ↦ sα is surjective with kernel T.$

Remark 1. [Ti, 3.7b] If ${𝔥}_{ℤ}={Q}^{\vee }$ and the first relation of (1.5) holds in $\mathrm{St}$ then there is a surjective homomorphism $\psi :\mathrm{St}↠G$. By [St, Lemma 22], the elements $nα hλ∨(g) nα-1 hsα λ∨ (g)-1 and nα(g) nα-1 hα∨ (g)-1$ automatically commute with each ${x}_{\beta }\left(f\right)$ so that $\mathrm{ker}\left(\psi \right)\subseteq Z\left(\mathrm{St}\right)$. In many cases $\mathrm{St}$ is the universal central extension of $G$ (see [Ti, 3.7c] and [St, Theorems 10-12]).

Remark 2. The algebra $𝔤\prime =\left[𝔤,𝔤\right]$ in (2.12) is generated by ${e}_{\alpha },\alpha \in {R}_{\mathrm{re}}$. A ${𝔤}^{\prime }$-module $V$ is integrable if ${e}_{\alpha }$, $\alpha \in {R}_{\mathrm{re}}$, act locally nilpotently so that

 $xα(c), α∈Rre, c∈C, with relations xα(c1) = exp(ceα) for α∈ Rre, c ∈ℂ,$
are well defined operators on $V$. The Chevalley group ${G}_{V}$ is the subgroup of $\mathrm{GL}\left(V\right)$ generated by the operators in (1.10). To do this integrally use a Kostant $ℤ$-form and choose a lattice in the module $V$ (see [Ti, 4.3-4] and [St, Ch1]). The Kac-Moody group is the group ${G}_{KM}$ generated by the symbols $xα(c), α∈Rre, c∈ℂ, with relations xα(c1) xα(c2) =xα(c1 +c2)$ and the additional relations coming from forcing an element to be 1 if it acts by 1 on every integrable $𝔤\prime$-module. This is essentially the Chevalley group ${G}_{V}$ for the case when $V$ is the adjoint representation and so ${G}_{KM}\subseteq \mathrm{Aut}$(𝔤). There are surjective homomorphisms $St(ℂ)↠ GKM↠ GV.$ See [Kac, Exercises 3.16-19] and [Ti, Proposition 1].

Remark 3. [St, Lemma 28] In the setting of Remark 2 let ${T}_{V}$ be the subgroup of ${G}_{V}$ generated by ${h}_{{\alpha }^{\vee }}\left(g\right)$ for $\alpha \in {R}_{\mathrm{re}},g\in {𝔽}^{×}.$ Then and if $𝔽$ is big enough $T V = h ω 1 ∨ g 1 … h ω n ∨ g n | g 1 ,…, g n ∈ 𝔹 × ,$ where ${\omega }_{1}^{\vee },\dots ,{\omega }_{n}^{\vee }$ is a $ℤ$-basis of the $ℤ$-span of the weights of $V$ [St, Lemma 35].

Notes and References

These notes are a retyped version of Section 3 of [PRS].

References

[CC] R. Carter and Y. Chen, Automorphisms of affine Kac-Moody groups and related Chevalley groups over rings, J. Algebra 155 (1) (1993) 44-94. MR1206622

[PRS] J. Parkinson, A. Ram and C. Schwer, Combinatorics in affine flag varieties, J. Algebra 321 (2009) 3469-3493.

[St] R. Steinberg, Lecture Notes on Chevalley groups, Yale University, 1967.

[Ti] J. Tits, Uniqueness and presentation of Kac-Moody groups over fields, J. Algebra 105 (2) (1987) 542-573, MR0873684