The symplectic group ${\mathrm{Sp}}_{2n}\left(𝔽\right)$

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

Last updates: 5 November 2011

The symplectic group ${\mathrm{Sp}}_{2n}\left(𝔽\right)$

Let $V$ be a vector space oover $𝔽$ and let $⟨,⟩:V\otimes V\to 𝔽$ be a skew symmetric bilinear form. The symplectic group is

$\mathrm{Sp}(⟨,⟩)=\{g\in \mathrm{GL}(V\left)\right|⟨g{v}_1,g{v}_2⟩=⟨v_1,v_2⟩\}.$

The Lie algebra of $\mathrm{Sp}(⟨,⟩)$ is

By Gram-Schmidt there exists a basis $\{e_1,\dots ,e_n,e_1^{*},\dots ,e_n^{*}\}$ of $V$ such that

$⟨e_i,e_j^{*}⟩={\delta }_{ij}\text{and}⟨e_i,e_j⟩=0,\text{so that}J=\left(\begin{array}{cc}0& \begin{array}{ccc}1& & 0\\ & \ddots & \\ 0& & 1\end{array}\\ \begin{array}{ccc}-1& & 0\\ & \ddots & \\ 0& & -1\end{array}& 0\end{array}\right)$

is the matrix of $⟨,⟩$ with respect to the basis $\{e_1,\dots ,e_n,e_1^{*},\dots ,e_n^{*}\}$. Using the basis $\{e_1,\dots ,e_n,e_1^{*},\dots ,e_n^{*}\}$ to identify $\mathrm{GL}\left(V\right)$ with ${\mathrm{GL}}_{2n}\left(𝔽\right)$,

$\text{identifies}\mathrm{Sp}(⟨,⟩)\text{with}{\mathrm{Sp}}_{2n}\left(𝔽\right)=\{g\in {\mathrm{GL}}_{2n}(𝔽\left)\right|gJ{g}^t=J\}$

and

A maximal compact subgroup of ${\mathrm{Sp}}_{2n}\left(ℂ\right)$ is

$\mathrm{Sp}\left(n\right)=U_{2n}\left(ℂ\right)\cap {\mathrm{Sp}}_{2n}\left(ℂ\right).$

The group $\mathrm{Sp}\left(n\right)is\; a\; compact,\; connected,\; and\; simply\; connected\; real\; Lie\; group.$

HW: Show that ${\mathrm{Sp}}_2\left(ℂ\right)={\mathrm{SL}}_2\left(ℂ\right)$.

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.

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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