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A complex algebraic group G is in this note a subgroup of GL(n, C), the elements of which are all invertible matrices whose coefficients annihilate some set of polynomials {P M [Xn, • • • , X nn ]} in n 2 indeterminates. It is said to be defined over a field KQC if the polynomials can be chosen so as to have coefficients in K. Given a subring B of C, we denote by GB the subgroup of elements of G which have coefficients in JB, and whose determinant is a unit of B. Assume in particular G to be defined over Q. Then Gz is an "arithmetically defined discrete subgroup" of G Rl or, more briefly, an arithmetic subgroup of GR. A typical example is the group of units of a nondegenerate integral quadratic form, and as a matter of fact, the main results stated below generalize facts known in this case from reduction theory. The proofs will be published elsewhere. Reductive groups.A complex algebraic group G is an algebraic torus (a torus in the terminology of [l]) if it is connected and can be diagonalized or, equivalently, if it is birationally isomorphic to a product of groups C* [l, Chapter II]. The group G is reductive if its identity component G° may be written as G°=T-G', where T is a central algebraic torus, and G' is an invariant connected semi-simple group, or, equivalently, if all rational representations of G are fully reducible.LEMMA 1. Let GO • • • DG m be reductive algebraic subgroups of GL(n, C), defined over R. Then there exists aÇzSL(n, R) such that the groups a-Gi R -a~l are stable under x-**x (i=l, • • • , m).

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