# The Generalized Triangle Inequalities in Symmetric Spaces and Buildings with Applications to Algebra

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*Michael Kapovich; Bernhard Leeb; John J. Millson*

In this paper the authors apply their results on the geometry of polygons in infinitesimal symmetric spaces and symmetric spaces and buildings to four problems in algebraic group theory. Two of these problems are generalizations of the problems of finding the constraints on the eigenvalues (resp. singular values) of a sum (resp. product) when the eigenvalues (singular values) of each summand (factor) are fixed. The other two problems are related to the nonvanishing of the structure constants of the (spherical) Hecke and representation rings associated with a split reductive algebraic group over \(\mathbb{Q}\) and its complex Langlands' dual. The authors give a new proof of the “Saturation Conjecture” for \(GL(\ell)\) as a consequence of their solution of the corresponding “saturation problem” for the Hecke structure constants for all split reductive algebraic groups over \(\mathbb{Q}\).

#### Table of Contents

# Table of Contents

## The Generalized Triangle Inequalities in Symmetric Spaces and Buildings with Applications to Algebra

- Contents v6 free
- Chapter 1. Introduction 110 free
- Chapter 2. Roots and Coxeter Groups 716 free
- Chapter 3. The First Three Algebra Problems and the Parameter Spaces ∑ for K\G/K 1221
- 3.1. The generalized eigenvalues of a sum problem Q1 and the parameter space ∑ of k-double cosets 1322
- 3.2. The generalized singular values of a product and the parameter space ∑ of K-double cosets 1322
- 3.3. The generalized invariant factor problem and the parameter space ∑ of K-double cosets 1423
- 3.4. Comparison of the parameter spaces for the four algebra problems 1625
- 3.5. Linear algebra problems 1625

- Chapter 4. The existence of polygonal linkages and solutions to the algebra problems 1928
- Chapter 5. Weighted Configurations, Stability and the Relation to Polygons 2938
- Chapter 6. Polygons in Euclidean Buildings and the Generalized Invariant Factor Problem 3746
- Chapter 7. The Existence of Fixed Vertices in Buildings and Computation of the Saturation Factors for Reductive Groups 4554
- Chapter 8. The Comparison of Problems Q3 and Q4 6069
- 8.1. The Hecke ring 6069
- 8.2. A geometric interpretation of m[sub(α,β,γ)](0) 6271
- 8.3. The Satake transform 6473
- 8.4. A solution of Problem Q4 is a solution of Problem Q3 6776
- 8.5. A Solution of Problem Q3 is not necessarily a solution of Problem Q4 7180
- 8.6. The saturation theorem for GL(l) 7382
- 8.7. Computations for the root systems B[sub(2)] and G[sub(2)] 7584

- Appendix A. Decomposition of Tensor Products and Mumford Quotients of Products of Coadjoint orbits 7786
- Bibliography 8291