Bott towers, complete integrability, and the extended character of representations
type: Item
instance of: scholarly article
title: Bott towers, complete integrability, and the extended character of representations
zbMATH Open document ID: 0826.22018
DOI: 10.1215/S0012-7094-94-07602-3
author: Yael Karshon
author: Michael D. Grossberg
published in: Duke Mathematical Journal
publication date: 1995-11-22T00:00:00Z
review text: A Bott tower is an iterated family of \({\mathbf C} {\mathbf P}^1\) bundles over a point with distinguished sections of the fiber maps; if \(M_0\) (the zeroth stage is a point), \(M_1 = {\mathbf C} {\mathbf P}^1\), then \(M_2\) is defined as the projectivization of \(\mathbf{1} \oplus {\mathbf L}_1\), where \({\mathbf L}_1\) is a holomorphic line bundle over \(M_1\), \(M_3\) is the projectivization of \(\mathbf{1} \oplus {\mathbf L}_2\), where \({\mathbf L}_2\) is a holomorphic line bundle over \(M_2\), and so on. The distinguished sections are defined by the zero sections of \textbf{1} and of \({\mathbf L}_j\). The \(n\)th step in this construction can also be obtained from any collection of \(n(n-1)/2\) integers \(\{c_{ij}\}_{1 \leq i < j \leq n}\) by \(M_n = ({\mathbf C}_2 \setminus \mathbf{0})^n / ({\mathbf C}^*)^n\), where the \(i\)-th factor \(a_i\) of \(({\mathbf C}^*)^n\) acts on the right by \((z_1, w_1, \dots, z_n, w_n) \cdot a_i = (z_1,w_1, \dots, z_i a_i, w_i a_i, \dots, z_j, a_i^{c_{ij}} w_j, \dots)\). All Bott towers arise in this way and the authors prove that the maps \(\mathbb{Z}^{n(n-1)/2} \to\) \{Isomorphism classes of \(n\)-step Bott towers\} and, for a given \(M_n\) in the image of this map, \(\mathbb{Z}^n \to\) \{Isomorphism classes of holomorphic line bundles over \(M_n\}\) are bijections. This notion is related to representation theory in a natural way using ``twisted cubes''. A twisted cube \(C\) in an affine space \(V\) with lattice \(l_V\) is a subset of \(V\), called the support of \(C\) and a density function \(\rho : V \to \mathbb{R}\) which takes the values 1 and \(-1\) on the support. These data determine a signed measure \(m_C\) in \(V\). The authors prove: If \(M_n\) is a Bott tower, \(T\) acts on it by a complete torus action. \(\omega\) is a closed \(T\)-invariant 2-form, and \(\Phi\): \(M_n \to {\mathfrak t}^*\) is a moment map, then the corresponding Duistermaat-Heckman measure \(\Phi_* \omega^n/n!\) coincides with the measure \…
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 32M10
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 53D50
Mathematics Subject Classification ID: 22E45
zbMATH DE Number: 727818
zbMATH Keywords: holomorphic line bundle
zbMATH Keywords: projectivization
zbMATH Keywords: Bott towers
zbMATH Keywords: torus action
zbMATH Keywords: irreducible representation
zbMATH Keywords: highest weight
zbMATH Keywords: moment map
zbMATH Keywords: complex structure
zbMATH Keywords: flag variety
zbMATH Keywords: Bott-Samelson manifold
zbMATH Keywords: connected compact Lie group
zbMATH Keywords: virtual character
zbMATH Keywords: multiplicities
zbMATH Keywords: Demazure's character formulas
zbMATH Keywords: Schubert varieties
MaRDI profile type: Publication
cites work: Convexity and Commuting Hamiltonians
cites work: The topology of torus actions on symplectic manifolds. Transl. from the French by the author
cites work: The index of elliptic operators. I
cites work: Q3993373
cites work: Homogeneous vector bundles
cites work: Applications of the Theory of Morse to Symmetric Spaces
cites work: Q3961493
cites work: Désingularisation des variétés de Schubert généralisées
cites work: On the variation in the cohomology of the symplectic form of the reduced phase space
cites work: Q4195061
cites work: Equivariant index and the moment map for completely integrable torus actions
cites work: Q5690465
cites work: On the Kostant multiplicity formula
cites work: Convexity properties of the moment mapping
cites work: Geometric quantization and multiplicities of group representations
cites work: Convexity properties of the moment mapping. II
cites work: Q4089001
cites work: Q3800267
cites work: The moment map and line bundles over presymplectic toric manifolds
cites work: On crystal bases of the \(q\)-analogue of universal enveloping algebras
cites work: The crystal base and Littelmann's refined Demazure character formula
cites work: Q3741813
cites work: On convexity, the Weyl group and the Iwasawa decomposition
cites work: A Formula For the Multiplicity of a Weight
cites work: Q3995493
cites work: Crystal graphs and Young tableaux
cites work: Canonical Bases Arising from Quantized Enveloping Algebras
cites work: Q3874497
description: scientific article; zbMATH DE number 727818