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Complexity and rank of homogeneous spaces. (English) Zbl 0706.14032
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.
Reviewer: I.Dotti Miatello

##### MSC:
 14M17 Homogeneous spaces and generalizations 22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings 43A85 Harmonic analysis on homogeneous spaces 14L30 Group actions on varieties or schemes (quotients)
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