Steinberg-Chevalley Groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

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 $\tilde{w}$). Use the notation $$x_{\alpha }\left(t\right)=\exp \left(t{e}_{\alpha }\right)=1+e_{\alpha }+\frac{1}{2!}{t}^2{e}_{\alpha }^2+\frac{1}{3!}{t}^3{e}_{\alpha }^3+\dots ,\text{in }U\left[\right[t\left]\right].$$

Then $$x_{\alpha }\left(t\right)x_{\alpha }\left(u\right)=x_{\alpha }\left(t+u\right)\text{in}U\left[\right[t,u\left]\right].$$

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

$$w\alpha ,w\beta \in R_{\mathrm{re}}^{+}\text{and}w\prime \alpha ,w\prime \beta \in -R_{\mathrm{re}}^{+}.$$

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 $$x_{\alpha }\left(t\right)x_{\beta }\left(u\right)=x_{\beta }\left(u\right)x_{\alpha }\left(t\right)x_{\alpha +\beta }\left(C_{\alpha ,\beta }^{1,1}tu\right)x_{2\alpha +\beta }\left(C_{\alpha ,\beta }^{2,1}{t}^{2}u\right)x_{\alpha +2\beta }\left(C_{\alpha ,\beta }^{1,2}t{u}^2\right)$$

Let $𝔽$ be a commutative ring. The Steinberg group $$\text{St is given by generators}{x}_{\alpha }\left(f\right)\text{for}\alpha \in R_{\mathrm{re}},\text{and}$$

$$x_{\alpha }\left(f_1\right)x_{\beta }\left(f_2\right)=x_{\beta }\left(f_2\right)x_{\alpha }\left(f_1\right)x_{\alpha +\beta }\left(C_{\alpha ,\beta }^{1,1}{f}_1{f}_2\right)x_{2\alpha +\beta }\left(C_{\alpha ,\beta }^{2,1}{f_1}^2{f}_2\right)x_{\alpha +2\beta }\left(C_{\alpha ,\beta }^{1,2}{f}_1{f_2}^2\right)$$

for prenilpotent pairs $\alpha ,\beta$. In St define

$$n_{\alpha }\left(g\right)=x_{\alpha }\left(g\right)x_{-\alpha }\left(-g^{-1}\right)x_{\alpha }\left(g\right),n_{\alpha }=n_{\alpha }\left(1\right),\text{and}{h}_{{\alpha }^{\vee }}\left(g\right)=n_{\alpha }\left(g\right)n_{\alpha }^{-1},$$

for $\alpha \in R_{\mathrm{re}}$ and $g\in {𝔽}^{\times }$.

Let ${𝔥}_{\mathbb{Z}}$ be a $\mathbb{Z}$-lattice in $𝔥$ which is stable under the $W$-action and such that $${𝔥}_{\mathbb{Z}}\supseteq Q^{\vee },\text{where}{Q}^{\vee }=\mathbb{Z}\text{-span}\{h_1,\dots ,h_n\}$$ with $h_1,\dots ,h_n$ as in (2.2). With

$$T\text{given by generators}{h}_{{\lambda }^{\vee }}\left(g\right)\text{for}{\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}},g\in {𝔽}^{\times },$$ $$\text{and relations}{h}_{{\lambda }^{\vee }}\left(g_1\right)h_{{\lambda }^{\vee }}\left(g_2\right)=h_{{\lambda }^{\vee }}\left(g_1{g}_2\right)\text{and}{h}_{{\lambda }^{\vee }}\left(g\right)h_{{\mu }^{\vee }}\left(g\right)=h_{{\lambda }^{\vee }+{\mu }^{\vee }}\left(g\right),$$

the Tits group $$G\text{is the group generated by St and}T$$ with the relations coming from the third equation of (1.3) and the additional relations

$$h_{{\lambda }^{\vee }}\left(g\right)x_{\alpha }\left(f\right)h_{{\lambda }^{\vee }}{\left(g\right)}^{-1}=x_{\alpha }\left(g^{⟨{\lambda }^{\vee },\alpha ⟩}f\right)\text{and}{n}_i{h}_{{\lambda }^{\vee }}\left(g\right)n_i^{-1}=h_{s_i{\lambda }^{\vee }}\left(g\right).$$

For $\alpha ,\beta \in R_{\mathrm{re}}$ let ${\epsilon }_{\alpha \beta }=\pm 1$ be given by $${\tilde{s}}_{\alpha }\left(e_{\beta }\right)={\epsilon }_{\alpha \beta }{e}_{s_{\alpha }\beta },\text{where}{\tilde{s}}_{\alpha }=\exp \left(\mathrm{ad}{e}_{\alpha }\right)\exp \left(-\mathrm{ad}{f}_{\alpha }\right)\exp \left(\mathrm{ad}{e}_{\alpha }\right)$$ (see [CC, p48] and [Ti, (3.3)]). By [St, Lemma 37] (see also [Ti, 3.7a])

$$n_{\alpha }\left(g\right)x_{\beta }\left(f\right)n_{\alpha }{\left(g\right)}^{-1}=x_{s_{\alpha }\beta }\left({\epsilon }_{\alpha \beta }{g}^{-⟨\beta ,{\alpha }^{\vee }⟩}f\right),h_{{\lambda }^{\vee }}\left(g\right)x_{\beta }\left(f\right)h_{{\lambda }^{\vee }}{\left(g\right)}^{-1}=x_{\beta }\left(g^{⟨\beta ,{\lambda }^{\vee }⟩}f\right),$$

$$\text{and}{n}_{\alpha }\left(g\right)h_{{\lambda }^{\vee }}\left(g\prime \right)n_{\alpha }{\left(g\right)}^{-1}=h_{s_{\alpha }{\lambda }^{\vee }}\left(g\prime \right).$$

Thus $G$ has a symmetry under the subgroup $$N\text{generated by}T\text{and the}{n}_{\alpha }\left(g\right)\text{for}\alpha \in R_{\mathrm{re}},g\in {𝔽}^{\times }.$$

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

$$\begin{array}{rcl}N& \to & W\\ n_{\alpha }\left(g\right)& \mapsto & s_{\alpha }\end{array}\text{is surjective with kernel}T.$$

Remark 1. [Ti, 3.7b] If ${𝔥}_{\mathbb{Z}}=Q^{\vee }$ and the first relation of (1.5) holds in $\mathrm{St}$ then there is a surjective homomorphism $\psi :\mathrm{St}\twoheadrightarrow G$. By [St, Lemma 22], the elements $$n_{\alpha }{h}_{{\lambda }^{\vee }}\left(g\right)n_{\alpha }^{-1}{h}_{s_{\alpha }{\lambda }^{\vee }}{\left(g\right)}^{-1}\text{and}{n}_{\alpha }\left(g\right)n_{\alpha }^{-1}{h}_{{\alpha }^{\vee }}{\left(g\right)}^{-1}$$ automatically commute with each $x_{\beta }\left(f\right)$ so that $\ker \left(\psi \right)\subseteq Z\left(\mathrm{St}\right)$. 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 $𝔤\prime =[𝔤,𝔤]$ 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

$$x_{\alpha }\left(c\right),\alpha \in R_{\mathrm{re}},c\in C,\text{with relations}{x}_{\alpha }\left(c_1\right)=\exp \left(c{e}_{\alpha }\right)\text{for}\alpha \in R_{\mathrm{re}},c\in ℂ,$$

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 $\mathbb{Z}$-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 $$x_{\alpha }\left(c\right),\alpha \in R_{\mathrm{re}},c\in ℂ,\text{with relations}{x}_{\alpha }\left(c_1\right)x_{\alpha }\left(c_2\right)=x_{\alpha }\left(c_1+c_2\right)$$ 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 $$\mathrm{St}(ℂ)\twoheadrightarrow G_{KM}\twoheadrightarrow G_V.$$ 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 $\alpha \in R_{\mathrm{re}},g\in {𝔽}^{\times }.$ Then $$h_{{{\alpha }_1}^{\vee }}\left(g_1\right)\dots h_{{{\alpha }_n}^{\vee }}\left(g_n\right)=1\text{ if and only if }{g}_1^{⟨\mu ,{\alpha }_1^{\vee }⟩}\dots g_n^{⟨\mu ,{\alpha }_n^{\vee }⟩}=1\text{ for all weights }\mu \text{ of }V,$$ $$Z\left(G_V\right)=\left\{h_{{{\alpha }_1}^{\vee }}\left(g_1\right)\dots h_{{{\alpha }_n}^{\vee }}\left(g_n\right)|g_1^{⟨\beta ,{\alpha }_1^{\vee }⟩}\dots g_n^{⟨\beta ,{\alpha }_n^{\vee }⟩}=1\text{ for all }\beta \in R\right\},$$ and if $𝔽$ is big enough $$T_V=\left\{h_{{{\omega }_1}^{\vee }}\left(g_1\right)\dots h_{{{\omega }_n}^{\vee }}\left(g_n\right)|g_1,\dots ,g_n\in {𝔹}^{\times }\right\},$$ where ${\omega }_1^{\vee },\dots ,{\omega }_n^{\vee }$ is a $\mathbb{Z}$-basis of the $\mathbb{Z}$-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

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