Sp(2n,C) - Knowlpedia

For every integer $n\ge1$ , $\mathrm{Sp}(2n,\mathbb C)$ is the symplectic group

\mathrm{Sp}(2n,\mathbb C)=\{A\in\mathrm{GL}(2n,\mathbb C):A^{\mathsf T}JA=J\},\qquad J=\begin{pmatrix}0&I_{n}\\-I_{n}&0\end{pmatrix}.

It preserves the displayed nondegenerate alternating bilinear form.

Dimensions and structure

This group has real dimension $2(n(2n+1))$ , complex dimension $n(2n+1)$ . The parameter $n$ is the number of symplectic pairs, while the matrix size is $2n$ . Its tangent matrices have block form $\left(\begin{smallmatrix}P&Q\\R&-P^{\mathsf T}\end{smallmatrix}\right)$ with $Q=Q^{\mathsf T}$ and $R=R^{\mathsf T}$ . Counting these blocks gives the dimension. This is a connected simply connected complex Lie group.

Its complex Lie group structure uses holomorphic multiplication and inversion. Forgetting the complex structure gives the underlying real Lie group; real and holomorphic homomorphisms belong to different categories.

References

Pavel Etingof, Lie Groups and Lie Algebras (MIT lecture notes), §6.1, pp. 38–39; Proposition 6.3 and Exercise 6.5.