| Summary: | A generalization of a theorem of N.R. 0'Brian, zeroes of holomorphic vector fields and the Grothendieck residue, Bull. London Math. Soc, 7 (1975) is given.
The theorem of Riemann-Roch and Hirzebruch for V-equivariant holomorphic vector bundles is obtained, via holomorphic vector fields, in the case all zeroes of the holomorphic vector field V are isolated.
The Bruhat decomposition of G/B is obtained from the G -action on G/B .
It is shown that a theorem of A. Bialynicki-Birula, Some
theorems on actions of algebraic groups, Ann. of Math. 98, 480-497 (1973)
is the generalization of the Bruhat decomposition on G/B , which was
a conjecture of B. Iversen.
The existence of a G -action on G/P with only one fixed
a
point is proved, where G is a connected linear algebraic group defined over an algebraically closed field k of characteristic zero and P is a parabolic subgroup of G .
The following is obtained
P = N[sub G](Pu) = {geG: Adg(Pu) = Pu}
where G is a connected linear algebraic group, P is a parabolic subgroup of G and P^ is the tangent space of the set of unipotent elements of P at the identity.
An elementary proof of P = N[sub G](P) = {geG: gPg ⁻¹=P} is given,
where G is a connected linear algebraic group and P is a parabolic subgroup of G . === Science, Faculty of === Mathematics, Department of === Graduate
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