Steinberg-Chevalley Groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

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 Rre be the real roots of the corresponding Borcherds-Kac-Moody Lie algebra 𝔤. Let U be the enveloping algebra of 𝔤. For each alphaRre fix a choice of eα in (2.18) (a choice of w˜). Use the notation xαt = exp(teα) =1+ eα+ 12! t2eα2 +13! t3eα3 +, in   U[[t]].

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

Following [Ti, 3.2], a prenilpotent pair is a pair of roots α,β Rre such that there exists w,wW with

wα,wβ Rre+ and wα,wβ -Rre+.
This condition guarantees that the Lie subalgebra of 𝔤 generated by 𝔤α and 𝔤β is nilpotent. Let α,β be a prenilpotent pair and let eα 𝔤α and eβ 𝔤β be as in (2.18). By [St, Lemma 15] there exist unique integers Cα,β 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) forαRre, 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 α,β. In St define
nα(g) =xα(g) x-α (-g-1) xα(g), nα =nα(1), and hα(g) =nα(g) nα-1,
for αRre and g𝔽×.

Let 𝔥 be a -lattice in 𝔥 which is stable under the W-action and such that 𝔥Q , where Q =-span{ h1,, hn} with h1,, hn as in (2.2). With

T given by generators hλ(g) for λ 𝔥, g𝔽×, 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 α,β Rre let εαβ =±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, αf) , 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 αRre, g𝔽×.

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 =Q and the first relation of (1.5) holds in St then there is a surjective homomorphism ψ:StG. 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 xβ(f) so that ker(ψ) Z(St). In many cases St is the universal central extension of G (see [Ti, 3.7c] and [St, Theorems 10-12]).

Remark 2. The algebra 𝔤=[𝔤,𝔤] in (2.12) is generated by eα,α Rre. A 𝔤-module V is integrable if eα, αRre, act locally nilpotently so that

xα(c), αRre, cC, with relations xα(c1) = exp(ceα) for α Rre, c ,
are well defined operators on V. The Chevalley group GV is the subgroup of GL(V) generated by the operators in (1.10). To do this integrally use a Kostant -form and choose a lattice in the module V (see [Ti, 4.3-4] and [St, Ch1]). The Kac-Moody group is the group GKM generated by the symbols xα(c), αRre, 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-module. This is essentially the Chevalley group GV for the case when V is the adjoint representation and so GKM Aut(𝔤). There are surjective homomorphisms St() 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 hα g for α R re ,g 𝔽 × . Then h α 1 g 1 h α n g n =1   if and only if   g 1 μ, α 1 g n μ, α n =1  for all weights  μ  of  V, Z G V = h α 1 g 1 h α n g n | g 1 β, α 1 g n β, α n =1   for all   βR , and if 𝔽 is big enough T V = h ω 1 g 1 h ω n g n | g 1 ,, g n 𝔹 × , where ω 1 ,, ω n is a -basis of the -span of the weights of V [St, Lemma 35].

Notes and References

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


[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

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