Consider the projective symplectic group $\mathrm{PSp}(n+1)$ over $\mathbb{C}$. This parametrizes $(n+1)\times (n+1)$ symplectic matrices modulo scalar multiplication.

Is $\mathrm{PSp}(n+1)$ irreducible?

Consider $4\times 4$ symplectic matrices. A matrix $A$ has a symplectic representative (modulo scalar) if and only if $A^{t}\Omega A = \lambda\Omega$ for some $\lambda\in\mathbb{C}^{*}$, where $\Omega$ is the standard symplectic form. Set $N = A^{t}\Omega A$. The condition $N = \lambda\Omega$ translates into the following equations in the entries of $A$: $$N_{12} = a_{00}a_{21}-a_{01}a_{20}+a_{10}a_{31}-a_{11}a_{30}=0$$ $$N_{14} = a_{00}a_{23}-a_{03}a_{20}+a_{10}a_{33}-a_{13}a_{30}=0$$ $$N_{23} = a_{01}a_{22}-a_{02}a_{21}+a_{11}a_{32}-a_{12}a_{31}=0$$ $$N_{34} = a_{02}a_{23}-a_{03}a_{22}+a_{12}a_{33}-a_{13}a_{32}=0$$ $$N_{13}-N_{24} = a_{00}a_{22}-a_{01}a_{23}-a_{02}a_{20}+a_{03}a_{21}+a_{10}a_{32}-a_{11}a_{33}-a_{12}a_{30}+a_{13}a_{31}=0$$

Consider the variety $X$ defined by these equations in the $\mathbb{P}^{15}$ of $4\times 4$ matrices modulo scalar. MacAulay2 tells me that $X = X_1\cup X_2$ has two irreducible components both of dimension $10$ and of degree $12$ and $20$ respectively.

This is where the confusion comes from. What am I missing here?

connectedsemisimple group is generated by unipotent subgroups. Maybe we should understand the question as follows: Is the group ${\rm PSp}(2n,{\Bbb C})$ connected? $\endgroup$ – Mikhail Borovoi Mar 6 at 18:08