On the general faces of the moment polytope. (English) Zbl 0946.14025

Brion, Michel

Int. Math. Res. Not. 1999, No. 4, 185-201 (1999).

In the paper under review the author considers the following situation. Let \(X\) be a projective algebraic variety endowed with an action of a connected reductive group \(G\), both defined over \({\mathbb C}\). Let \({\mathcal L}\) be an ample line bundle over \(X\). Assume that \({\mathcal L}\) is \(G\)-linearized; that is, \(G\) acts on the total space of \({\mathcal L}\) compatibly with its action on \(X\), and acts linearly on fibers. Then the space \(\Gamma(X,{\mathcal L})\) of global sections of \({\mathcal L}\) is a rational \(G\)-module. Its decomposition into simple modules is described by its highest weight vectors, that is, the set \(\Gamma(X,{\mathcal L})^{B}\) of eigenvectors of a fixed Borel subgroup \(B\) of \(G\). To study simultaneously the \(G\)-modules \(\Gamma(X,{\mathcal L}^{\otimes n})\) for all positive integers \(n\) the author introduces a set \(P_{G}(X,{\mathcal L})\) as follows. Let \({\mathcal X}\) be the character group of \(B\) and \({\mathcal X}_{\mathbb Q} := {\mathcal X}{\otimes}_{{\mathbb Z}}{\mathbb Q}\). For each dominant weight \(\chi \in {\mathcal X}\), let \(V(\chi)\) be a simple \(G\)-module with highest weight \(\chi\). Then the moment polytope is \[ P_{G}(X,{\mathcal L}) = \{p \in {\mathcal X}_{{\mathbb Q}} \mid V(np)\text{ occurs in }\Gamma(X,{\mathcal L}^{\otimes n}) \text{ for some positive integer }n\}. \] Then \(P_{G}(X,{\mathcal L})\) is a convex polytope in \({\mathcal X}_{\mathbb Q}\). By definition \(P_{G}(X,{\mathcal L})\) is contained in the Weyl chamber \(\rho\) of dominant rational weights. The moment polytope is indeed closely related to the image of the moment map of symplectic geometry.
Let \(\rho_X\) be the smallest face Weyl chamber which contains \(P_{G}(X,{\mathcal L})\) (it turns out that \(\rho_X\) is independent of \({\mathcal L}\)). A face of \(P_{G}(X,{\mathcal L})\) which meets the relative interior of \(\rho_X\) is called general. The author describes the general faces of the moment polytope in terms of fixed points of certain subtori of \(G\). When \(X\) is normal he describes the corresponding sets of highest weight vectors as well and as application he determines the general vertices of polytopes associated with tensor products or Schur powers of simple modules. In the former case, the author obtains an inductive description of the faces of the polytope.

Reviewer: Nikolai I.Osetinski (Moskva)

MSC:

14L30	Group actions on varieties or schemes (quotients)
22E99	Lie groups
14F05	Sheaves, derived categories of sheaves, etc. (MSC2010)

Keywords:

moment polytope; action of a connected reductive group; weight vectors; Borel subgroup; global section; Weyl chamber; general faces