## The group ${\mathrm{PGL}}_{2}$

The group ${\mathrm{PGL}}_{2}$ is given by $PGL2(𝔽) =GL2(𝔽) /Z(GL2(𝔽)) =GL2(𝔽)/ 𝔽×,$ since the center of ${\mathrm{GL}}_{2}\left(𝔽\right)$ is $Z\left({\mathrm{GL}}_{2}\left(𝔽\right)\right)={𝔽}^{×}$, the nonzero constant multiples of the identity matrix.

Let $𝔤={\mathrm{𝔰𝔩}}_{2}=\text{span}\left\{{X}_{\alpha },{H}_{{\alpha }^{\vee }},{X}_{-\alpha }\right\}$ and let $V=𝔤$ be the adjoint representation, so that $Xα= ( 0-20 001 000 ), X-α= ( 000 -100 020 ) , and Hα∨= ( 200 000 00-2 ) .$ Then $xα(f)= ( 1-2f-f2 01f 001 ) , and x-α(f)= ( 100 -f10 -f22f1 ),$ and we compute $nα(g)= ( 00-g2 0-10 -g-200 ), and hα∨(g)= ( g200 010 00g-2 ),$ so that ${h}_{{\alpha }^{\vee }}\left(1\right)={h}_{{\alpha }^{\vee }}\left(-1\right)=1$. Then $G\cong {\mathrm{PGL}}_{2}\left(𝔽\right),$ $hα∨(g)=1 if and only if g ⟨α,α∨⟩ =1 if and only if g2=1,$ and $Z(G)= {hα∨(g) | g⟨α, α∨⟩ =1} ={1}.$ Note that $G$ contains $hα∨(g) =h2ω∨ (g) =hω∨ (g2),$ and so if $𝔽$ is closed under square roots then ${h}_{{\omega }^{\vee }}\left(g\right)\in ⟨{x}_{\alpha }\left(f\right),{x}_{-\alpha }\left(f\right)\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}f\in 𝔽⟩.\phantom{\rule{2em}{0ex}}$????

## 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.