The symplectic group Sp2n (𝔽)

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

Last updates: 5 November 2011

The symplectic group Sp2n (𝔽)

Let V be a vector space oover 𝔽 and let ,: VV𝔽 be a skew symmetric bilinear form. The symplectic group is

Sp(,) = {gGL(V) | gv1, gv2 =v1, v2 }.
The Lie algebra of Sp(,) is
𝔰𝔭(,) = {g𝔤𝔩(V) | gv1, v2 +v1, gv2=0 }.

By Gram-Schmidt there exists a basis {e1,, en,e1*, ,en*} of V such that

ei, ej* =δij and ei,ej =0, so that J= ( 0 10 01 -10 0-1 0 )
is the matrix of , with respect to the basis {e1,, en,e1*, ,en* }. Using the basis {e1,, en,e1*, ,en* } to identify GL(V) with GL2n(𝔽),
identifies Sp(,) with Sp2n(𝔽) ={gGL2n (𝔽) | gJgt=J}
identifies 𝔰𝔭(,) with 𝔰𝔭2n (𝔽) ={g𝔤𝔩 2n(𝔽) | gJ+Jgt=0} .

A maximal compact subgroup of Sp2n() is

Sp(n) =U2n() Sp2n() .
The group Sp(n) is a compact, connected, and simply connected real Lie group.

HW: Show that Sp2() =SL2().

Notes and References

These notes were influenced by the Wikipedia articles ????. They were prepared for lectures and working seminars in Representation Theory at University of Melbourne in 2008-2011.


[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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