Hi,

Let $W$ be a real algebraic group, and $G$ the associated complex group. Then $W$ is of Hodge type if there is a $\mathbb{C}^*$ action on $G$ such that $U(1)$ preserves $W$ and the action of $-1$ on $G$ is a Cartan involution.

I have trouble understanding the definition, I guess because I don't understand very well the definition of Cartan involution : it should be something like the complex conjugation relative to a compact real form of $G$.

Examples: $SU(p,q)$, $SO(2n)$ ($n\geq 3$), $Sp(n)$, $Sp(p,q)$, $SO(p,2q)$ ($q\geq 2$) and some other classical Lie groups are apparently of Hodge type. But I don't see the action of $U(1)$, the compact real form etc.

Also, I was told that a group is of Hodge type if the Lie algebra has a Hodge structure. Is it easy to see the Hodge structure in the examples given by Simpson ?

Thanks for your help.