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

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

The difficulties being solved via invariant concept are far-reaching generalizations and extensions of difficulties at the "reduction to canonical shape" of assorted is nearly a similar factor, projective geometry. items of linear algebra or, what Invariant conception has a ISO-year historical past, which has noticeable alternating sessions of progress and stagnation, and adjustments within the formula of difficulties, tools of resolution, and fields of software. within the final 20 years invariant idea has skilled a interval of development, influenced via a prior improvement of the speculation of algebraic teams and commutative algebra. it really is now seen as a department of the idea of algebraic transformation teams (and below a broader interpretation could be pointed out with this theory). we are going to freely use the speculation of algebraic teams, an exposition of which are came upon, for instance, within the first article of the current quantity. we are going to additionally think the reader knows the elemental options and easiest theorems of commutative algebra and algebraic geometry; whilst deeper effects are wanted, we'll cite them within the textual content or offer compatible references.

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Then G has fixed points in X. 1 one reduces to the case that G is either Ga or Gm • If x E X is not a fixed point the orbit G. x is an affine variety, being the image of G under a finite morphism. Then G. x can not be closed in X and the finite complement of G . x in its closure must consist of fixed points. Example. Let G be a (connected, solvable) closed subgroup of GL n. e. of all sequences V1 c V2 C ... c v" = V of subspaces of V, with dim V; = i. This is a projective, hence complete, variety, on which GL n acts.

1 we see that » L(B n Ga/Ru(Ga = L(ZG(T)/ZG(T) n Ru(G» EfjkXp, where /3 is either rJ. or -rJ. (the left hand side being viewed as a subspace of /3, rJ. running through the L( Gal Ru( Ga}}. We denote by R + (B) the set of these roots of (G, T). Proposition. R+(B) is a system of positive roots in R. The proof uses information about SL 2 ([Sp3, p. 197J). 6. Characterization of the Unipotent Radical, Applications. The next result, due to Chevalley, is a basic one. Theorem. The unipotent radical of G is the identity component of the intersection of the Borel subgroups containing T.

Every semi-simple element of G lies in a maximal torus. 2 (ii)). Corollary 2. The centralizer of a subtorus of G is connected. In particular: Cartan subgroups are connected. 3 to S and the closed subvariety (B being a Borel group) {yBly-1xy E B} of GIB shows that there is a Borel group containing both Sand x. 2 (iii). Proposition. Let S be a subtorus of G and B a Borel group containing S. Then ZG(S) n B is a Borel subgroup of ZG(S). 3). Theorem 2. A parabolic subgroup is connected. Recall that G is assumed to be connected.

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