Complexity and rank of homogeneous spaces. (English) Zbl 0706.14032

Panyushev, D. I.

Geom. Dedicata 34, No. 3, 249-269 (1990).

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)

Keywords:

homogeneous spaces; reductive Lie group; algebraic variety; complexity; rank