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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.

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Not exactly an answer, but you might like Gross's article "A Remark on Tube Domains". Although it doesn't cover all cases you're interested in, it presents some details you're interested in, in some very illustrative cases. Since it seems like you're looking for more background/examples, perhaps this reading will help. – Marty Sep 30 '10 at 17:55

That $-1$ is a Cartan involution is the same as saying that the subgroup $K$ of points in G fixed under the action of the adjoint of this copy of $-1$ is a maximal compact subgroup of $G$.

For example, in $W=U(p,q)$ the subgroup $K=U(p)\times U(q)$ is the centraliser of the diagonal matrix $k$ whose first $p$ entries are $-1$ and whose last $q$ entries are $+1$. This element $k$ may be viewed as an element of the diagonal ${\mathbb C}^*$ embedded in $GL_{p+q}({\mathbb C})$ (the latter is the complexification of $W$), where $z\in {\mathbb C}^*$ is embedded $G$ as the diagonal matrix whose first $p$ entries are $z$ and the last $q$ entries are $1$. It is clear that $K$ is a maximal compact of $W$.

Similar constructions exist for $Sp(2n)$ with $K=U(n)$. In general, you may like to replace your semi-simple group by a reuctive group, to see the ${\mathbb C}^*$ action more clearly; for example, $U(p,q)$ was more convenient that $SU(p,q)$ in the above example.

You can look up Milne's articles on "Shimura varieties" for detailed descriptions of these Hodge groups.

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