Groups of Hodge type, hodge structure on Lie algebra

Question

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.

Answer 1

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.