Complexity and rank of homogeneous spaces (Q919052)

Let G be a connected reductive Lie group acting on an algebraic variety X and let \(B\supset T\) be a fixed Borel group and a maximal torus of G. The complexity of X, c(X), is defined to be the codimension of a generic B- orbit in X. Let \({\mathcal P}=\{f\in k(X)^*:\;bf=\lambda_ f(b),\quad b\in B\}\) where \(\lambda_ f\in {\mathfrak X}(B)\), the character group of B and let \(\Gamma\) (X) be the image of \(\lambda_ f\). \(\Gamma\) (X) is a free, finitely generated abelian group and the rank of \(\Gamma\) (X) is said to be the rank of X, r(X). The purpose of this paper is to study relations between c(X), r(X) and stabilizers of some actions of G and B. When X is a homogeneous space of G, the author obtains explicit formulas for the rank and complexity of quasiaffine G/H in terms of the co-isotropy representation of H.