## The group $\mathrm{Sp}\left(n\right)\simeq {U}_{n}\left(ℍ\right)$

The 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 fundamental representation $\theta :{U}_{1}\left(ℍ\right)\stackrel{\sim }{⟶}{\mathrm{SU}}_{2}\left(ℂ\right).$

The action of ${ℍ}^{*}$ on the 2-dimensional $ℂ$-vector space $ℍ$ by right multiplication provides

 $\begin{array}{cccccc}\theta :& {ℍ}^{*}& ⟶& {\mathrm{GL}}_{2}\left(ℂ\right)& \stackrel{\text{transpose}}{⟶}& {\mathrm{GL}}_{2}\left(ℂ\right)\\ & {x}_{0}+{x}_{1}i+{x}_{2}j+{x}_{3}k& ⟼& \left(\begin{array}{cc}{x}_{0}+{x}_{1}i& -{x}_{2}+{x}_{3}i\\ {x}_{2}+{x}_{3}i& {x}_{0}-{x}_{1}i\end{array}\right)& ⟼& \left(\begin{array}{cc}{x}_{0}+{x}_{1}i& -{x}_{2}+{x}_{3}i\\ {x}_{2}+{x}_{3}i& {x}_{0}-{x}_{1}i\end{array}\right)\end{array}$
This gives a group homomorphism
 $\begin{array}{cccc}\theta :& {ℍ}^{*}& ⟶& {\mathrm{GL}}_{2}\left(ℂ\right)\\ & a+cj& ⟼& \left(\begin{array}{cc}a& c\\ -\stackrel{‾}{c}& \stackrel{‾}{a}\end{array}\right)\end{array}\phantom{\rule{2em}{0ex}}\text{for}\phantom{\rule{0.5em}{0ex}}a={x}_{0}+{x}_{1}i\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}c={x}_{2}+{x}_{3}i\phantom{\rule{0.5em}{0ex}}\text{in}\phantom{\rule{0.5em}{0ex}}ℂ.$
The Pauli matrices are
 $\theta \left(i\right)=\left(\begin{array}{cc}i& 0\\ 0& -i\end{array}\right),\phantom{\rule{2em}{0ex}}\theta \left(j\right)=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right),\phantom{\rule{2em}{0ex}}\theta \left(k\right)=\left(\begin{array}{cc}0& i\\ i& 0\end{array}\right),\phantom{\rule{2em}{0ex}}$
and
 $\theta :{U}_{1}\left(ℍ\right)\stackrel{\sim }{⟶}{\mathrm{SU}}_{2}\left(ℂ\right).$

More generally, for $n\in {ℤ}_{>0}$, the function

 $\begin{array}{cccc}\theta :& {\mathrm{GL}}_{n}\left(ℍ\right)& ⟶& {\mathrm{GL}}_{2n}\left(ℂ\right)\\ & \left({g}_{ij}\right)& ⟼& \left(\theta \left({g}_{ij}\right)\right)\end{array}\phantom{\rule{2em}{0ex}}\text{satisfies}\phantom{\rule{2em}{0ex}}\theta \left({\stackrel{‾}{g}}^{t}\right)={\stackrel{‾}{\theta \left(g\right)}}^{t}$
and is a group homomorphism. Restriction to ${U}_{n}\left(ℍ\right)$ gives an isomorphism,
 $\theta :{U}_{n}\left(ℍ\right)\stackrel{\sim }{⟶}\mathrm{Sp}\left(n\right)={U}_{2n}\left(ℂ\right)\cap {\mathrm{Sp}}_{2n}\left(ℂ\right)$
where
 ${U}_{n}\left(ℍ\right)=\left\{g\in {\mathrm{GL}}_{n}\left(ℍ\right)\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}g{\stackrel{‾}{g}}^{t}=1\right\},\phantom{\rule{2em}{0ex}}{U}_{2n}\left(ℂ\right)=\left\{g\in {\mathrm{GL}}_{2n}\left(ℂ\right)\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}g{\stackrel{‾}{g}}^{t}=1\right\}$
and
 ${\mathrm{Sp}}_{2n}\left(ℂ\right)=\left\{g\in {\mathrm{GL}}_{2n}\left(ℂ\right)\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}gJ{g}^{t}=J\right\},\phantom{\rule{2em}{0ex}}\text{where}\phantom{\rule{1em}{0ex}}J=\left(\begin{array}{cc}0& \begin{array}{ccc}1& & 0\\ & \ddots & \\ 0& & 1\end{array}\\ \begin{array}{ccc}-1& & 0\\ & \ddots & \\ 0& & -1\end{array}& 0\end{array}\right)$

${\mathrm{Sp}}_{2}\left(ℂ\right)={\mathrm{SL}}_{2}\left(ℂ\right)$.

## 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.