Agnes Scott College

Doris Schattschneider

Restricted Roots of a Semi-Simple Algebraic Group
Yale University, 1966


The ideas of root system and Weyl group of a semi-simple algebraic group are basic to the theory of algebraic groups. Recently I. Satske and J. Tits have relativized these ideas, i.e., they have considered the case of a connected reductive algebraic group G defined over a ground field k, and developed the notions of "restricted roots of G" and "small Weyl group of G" relative to a maximal k-trivial torus of G. One of the main results of their work concerning a maximal k-trivial torus Q of G is the "the set of restricted roots τ of G relative to Q is a root system, and N(Q)/Z(Q) (normalizer of Q in G mod the centralizer of Q in G) is isomorphic to the Weyl group of the root system τ (the group generated by reflections in X(Q)Q with respect to elements of τ, X(Q) = character group of Q). A key factor in the proofs of the theorems they obtained was the Galois group G(K/k) (K a splitting field for a maximal torus of G). One of the motivations for this paper was to examine Satake's arguments in particular, to find out just how dependent they were on the use of G(K/k).

The main purpose of this paper is to consider a much more general relativization of the classic idea of root system of a connected semi-simple algebraic group G. The problem is this: develop the theory of "restricted roots of G" relative to an arbitrary subtorus Q of G, and then find necessary and sufficient conditions on Q to insure that the following theorem is true, "the set of restricted roots τ of G with respect to Q is a root system with Weyl group isomorphic to N(Q)/Z(Q)," By obtaining necessary and sufficient conditions for an arbitrary subtorus of G to satisfy this theorem, we automatically obtain necessary conditions for a torus of G to be a maximal k-trivial torus (where G is defined over k). These conditions are helpful in the classification of maximal k-trivial tori of the classical groups, and a partial list of possible maximal k-trivial tori of the classical groups appears in this paper at the end of section IV. Also, by considering the case of an arbitrary subtorus of a semi-simple algebraic group G, without any reference to the ground field of definition we obtain theorems which have as corollaries some of the results of Satake and Tits.