Reductive algebraic groups

A linear algebraic group $G$ over an algebraically closed field $K$ is a Zariski closed subgroup of the group $GL(n,K)$. In other words, as a subset in $GL(n,K)$, the group $G$ is defined by polynomial equations in the matrix variables $X_{i,j}$ ( $1\leq i,j\leq n$). We call such a group simple if it has no nontrivial proper closed connected normal subgroups. A reductive algebraic group is an almost direct product of a finite number of simple algebraic groups and a finite number of copies of $K^*$, the multiplicative group of the field $K$. The group $GL(n,K)$ itself is the basic example of a reductive linear algebraic group. It is the product of the normal subgroups $Z$ (the scalar matrices) and $SL(n,K)$. Here $Z$ is isomorphic to $K^*$, and $SL(n,K)$ is a simple algebraic group. (The product is called almost direct because $Z\cap SL(n,K)$ is finite.)

Let $F$ be a subfield of $K$, and assume that $K/F$ is separable. We say that a linear algebraic group $G$ is defined over $F$ when its defining polynomial equations can be chosen with coefficients in $F$. We can form the group $G(F):= G\cap GL(n,F)$ of $F$-rational points of $G$. In other words, $G(F)$ is the set of fixed points in $G$ for the standard action of the Galois group $Gal(K/F)$ on $GL(n,K)$. The group $G(F)$ is called an $F$-form of $G$. More generally, one defines $F$-forms $G(F)$ of $G$ by ``twisting'' the standard action of the Galois group on $G$ by automorphisms of $G$.

Plain examples of $F$-forms of reductive groups are $GL(n,F)$, $SL(n,F)$, the symplectic groups $Sp(2n,F)$, and the orthogonal groups $O(n,F)$. Depending on the properties of the field $F$, there may exist many other forms of such ``classical'' matrix groups. For instance, $SU(p,q,\mathbb C)$ and $SO(p,q,{\mathbb R})$ are real forms of the simple groups $SL(p+q,\mathbb C)$ and $SO(p+q,\mathbb C)$ respectively.

In addition, there are certain exceptional simple groups that occur over any field $F$.

A local field $F$ is a field equipped with a locally compact non-discrete Hausdorff topology. The local fields of characteristic $0$ are $\mathbb C$, $\mathbb R$, and finite extensions of the p-adic field ${\mathbb Q}_p$. The finite extensions of ${\mathbb Q}_p$ are called p-adic fields (of characteristic $0$). Given a local field $F$, a form $G(F)$ of $F$-rational points of an algebraic group $G$ can be given the topology induced by the topology of $F$. This gives $G(F)$ the structure of a locally compact group. There is an important result saying that forms $G(F)$ of reductive algebraic groups over local fields are always of type I. This statement provides a large class of examples of type I groups for which the questions raised in the previous section are meaningful.