### Restricted Roots of a Semi-Simple Algebraic Group

Yale University, 1966

Introduction

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.