The group ${\mathrm{PGL}}_2$

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

Last updates: 17 July 2012

The group ${\mathrm{PGL}}_2$

The group ${\mathrm{PGL}}_2$ is given by $${\mathrm{PGL}}_2\left(𝔽\right)={\mathrm{GL}}_2\left(𝔽\right)/Z\left({\mathrm{GL}}_2\right(𝔽\left)\right)={\mathrm{GL}}_2\left(𝔽\right)/{𝔽}^{\times },$$ since the center of ${\mathrm{GL}}_2\left(𝔽\right)$ is $Z\left({\mathrm{GL}}_2\right(𝔽\left)\right)={𝔽}^{\times }$, the nonzero constant multiples of the identity matrix.

Let $𝔤={\mathrm{𝔰𝔩}}_2=\text{span}\{X_{\alpha },H_{{\alpha }^{\vee }},X_{-\alpha }\}$ and let $V=𝔤$ be the adjoint representation, so that $$X_{\alpha }=\left(\begin{array}{ccc}0& -2& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right),X_{-\alpha }=\left(\begin{array}{ccc}0& 0& 0\\ -1& 0& 0\\ 0& 2& 0\end{array}\right),\text{and}{H}_{{\alpha }^{\vee }}=\left(\begin{array}{ccc}2& 0& 0\\ 0& 0& 0\\ 0& 0& -2\end{array}\right).$$ Then $$x_{\alpha }\left(f\right)=\left(\begin{array}{ccc}1& -2f& -f^2\\ 0& 1& f\\ 0& 0& 1\end{array}\right),\text{and}{x}_{-\alpha }\left(f\right)=\left(\begin{array}{ccc}1& 0& 0\\ -f& 1& 0\\ -f^2& 2f& 1\end{array}\right),$$ and we compute $$n_{\alpha }\left(g\right)=\left(\begin{array}{ccc}0& 0& -g^2\\ 0& -1& 0\\ -g^{-2}& 0& 0\end{array}\right),\text{and}{h}_{{\alpha }^{\vee }}\left(g\right)=\left(\begin{array}{ccc}{g}^2& 0& 0\\ 0& 1& 0\\ 0& 0& g^{-2}\end{array}\right),$$ so that $h_{{\alpha }^{\vee }}\left(1\right)=h_{{\alpha }^{\vee }}(-1)=1$. Then $G\cong {\mathrm{PGL}}_2\left(𝔽\right),$ $$h_{{\alpha }^{\vee }}\left(g\right)=1\text{if and only if}{g}^{⟨\alpha ,{\alpha }^{\vee }⟩}=1\text{if and only if}{g}^2=1,$$ and $$Z\left(G\right)=\left\{h_{{\alpha }^{\vee }}\right(g\left)\right|g^{⟨\alpha ,{\alpha }^{\vee }⟩}=1\}=\left\{1\right\}.$$ Note that $G$ contains $$h_{{\alpha }^{\vee }}\left(g\right)=h_{2{\omega }^{\vee }}\left(g\right)=h_{{\omega }^{\vee }}\left(g^2\right),$$ and so if $𝔽$ is closed under square roots then $h_{{\omega }^{\vee }}\left(g\right)\in ⟨x_{\alpha }\left(f\right),x_{-\alpha }\left(f\right)|f\in 𝔽⟩.$????

Notes and References

These notes follow Steinberg [St, ????].

References

[St] R. Steinberg, Lectures on Chevalley groups, Notes prepared by John Faulkner and Robert Wilson, Yale University, New Haven, Conn., 1968. iii+277 pp. MR0466335.

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