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## Reductive algebraic groups

A linear algebraic group over an algebraically closed field is a Zariski closed subgroup of the group . In other words, as a subset in , the group is defined by polynomial equations in the matrix variables (). 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 , the multiplicative group of the field . The group itself is the basic example of a reductive linear algebraic group. It is the product of the normal subgroups (the scalar matrices) and . Here is isomorphic to , and is a simple algebraic group. (The product is called almost direct because is finite.)

Let be a subfield of , and assume that is separable. We say that a linear algebraic group is defined over when its defining polynomial equations can be chosen with coefficients in . We can form the group of -rational points of . In other words, is the set of fixed points in for the standard action of the Galois group on . The group is called an -form of . More generally, one defines -forms of by twisting'' the standard action of the Galois group on by automorphisms of .

Plain examples of -forms of reductive groups are , , the symplectic groups , and the orthogonal groups . Depending on the properties of the field , there may exist many other forms of such classical'' matrix groups. For instance, and are real forms of the simple groups and respectively.

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

A local field is a field equipped with a locally compact non-discrete Hausdorff topology. The local fields of characteristic are , , and finite extensions of the p-adic field . The finite extensions of are called p-adic fields (of characteristic ). Given a local field , a form of -rational points of an algebraic group can be given the topology induced by the topology of . This gives the structure of a locally compact group. There is an important result saying that forms 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.

Next: Harish-Chandra's Plancherel formula Up: Representation Theory of Algebraic Previous: Locally compact groups