Algebraic Geometry IV: Linear Algebraic Groups Invariant by T. A. Springer (auth.), A. N. Parshin, I. R. Shafarevich

By T. A. Springer (auth.), A. N. Parshin, I. R. Shafarevich (eds.)

The difficulties being solved by means of invariant thought are far-reaching generalizations and extensions of difficulties at the "reduction to canonical shape" of varied is sort of an analogous factor, projective geometry. items of linear algebra or, what Invariant conception has a ISO-year background, which has obvious alternating classes of progress and stagnation, and adjustments within the formula of difficulties, equipment of resolution, and fields of program. within the final twenty years invariant idea has skilled a interval of development, influenced through a prior improvement of the speculation of algebraic teams and commutative algebra. it's now seen as a department of the idea of algebraic transformation teams (and less than a broader interpretation will be pointed out with this theory). we are going to freely use the speculation of algebraic teams, an exposition of which might be chanced on, for instance, within the first article of the current quantity. we'll additionally imagine the reader knows the fundamental suggestions and easiest theorems of commutative algebra and algebraic geometry; whilst deeper effects are wanted, we are going to cite them within the textual content or offer compatible references.

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32]. Example. Let fl' ... nlfl(x) = ... ';;d has rank d J 1 ~J~n for all x E U. Then X is defined on F. d with cp(x) = (/1 (x), ... ,fAx)). 3. 1. Let X be an F -variety and E an extension of F contained in k. An E-form of X is an F -variety Y which is isomorphic to X over E. The set of F -isomorphism classes of E-forms of X is denoted by (/>(E/F, X). It should be noted that if Y is any F -variety isomorphic to X there is a finite algebraic extension E as above such that Y is an E-form of X. So (/>(k/F, X) = (/>(F/F, X).

Here the important result is Kempf's vanishing theorem. Theorem. If p. » = 0 for i "# O. If char(k) = 0 this is a particular case of an earlier theorem of Bott. This states that for arbitrary p. » "# O. This I. Linear Algebraic Groups 45 i can be described precisely. Bott's theorem is not true in positive characteristics. For a discussion of these matters see [J, II. 5]. 4. Weyl's Character Formula. We introduce the group ring Z[X] of the character group X. The basis element of that ring defined by A E X is denoted by exp()_).

2), (b) Let R be a root system and R+ a system of positive roots. If rJ. and /3 are linearly independent roots there is an element w of the Weyl group such that WrJ. and w/3 both lie in R+. Corollary 1. If S is a subtorus of G then Ru(ZG(S» c Ru(G). 5, proposition. Corollary 2. If G is reductive then ZG(T) = T. Hence Cartan subgroups and maximal tori coincide. 1 we now deduce the following properties. Proposition. Let G be reductive, let R be the root system of (G, T). (i) The roots of R are the non-zero weights of T in the Lie algebra of G; (ii) For any rJ.

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