The group PGL2

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 PGL2

The group PGL2 is given by PGL2(𝔽) =GL2(𝔽) /Z(GL2(𝔽)) =GL2(𝔽)/ 𝔽×, since the center of GL2(𝔽) is Z(GL2(𝔽)) =𝔽×, the nonzero constant multiples of the identity matrix.

Let 𝔤=𝔰𝔩2 =span{ Xα, Hα, X-α} 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 hα(1) =hα (-1)=1 . Then G PGL2(𝔽), hα(g)=1 if and only if g α,α =1 if and only if g2=1, and Z(G)= {hα(g) | gα, α =1} ={1}. Note that G contains hα(g) =h2ω (g) =hω (g2), and so if 𝔽 is closed under square roots then hω (g) xα(f), x-α(f) | f𝔽. ????

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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