Chapter 1. Introduction ......................................... 1
Chapter 2. Roots and Coxeter Groups ............................. 7
2.1. Split tori over F ........................................ 7
2.2. Roots, coroots and the Langlands' dual ................... 8
2.3. Coxeter groups ........................................... 9
Chapter 3. The First Three Algebra Problems and the Parameter
Spaces ∑ for K\G/K .................................. 12
3.1. The generalized eigenvalues of a sum problem Q1 and
the parameter space ∑ K-double cosets ................... 13
3.2. The generalized singular values of a product and
the parameter space ∑ of K-double cosets ................ 13
3.3. The generalized invariant factor problem and
the parameter space ∑ of K-double cosets ................ 14
3.4. Comparison of the parameter spaces for the four algebra
problems ................................................ 16
3.5. Linear algebra problems ................................. 16
Chapter 4. The existence of polygonal linkages and solutions
to the algebra problems ............................. 19
4.1. Setting up the general geometry problem ................. 19
4.2. Geometries modeled on Coxeter complexes ................. 21
4.3. Bruhat-Tits buildings associated with nonarchimedean
reductive Lie groups .................................... 24
4.4. Geodesic polygons ....................................... 25
Chapter 5. Weighted Configurations, Stability and the Relation
to Polygons ......................................... 29
5.1. Gauss maps and associated dynamical systems ............. 30
5.2. The polyhedron Dn(X) .................................... 33
5.3. The polyhedron for the root system B2 ................... 35
Chapter 6. Polygons in Euclidean Buildings and the Generalized
Invariant Factor Problem ............................ 37
6.1. Folding polygons into apartments ........................ 37
6.2. A solution of Problem Q2 is not necessarily a solution
of Problem Q3 ........................................... 40
Chapter 7. The Existence of Fixed Vertices in Buildings and
Computation of the Saturation Factors for
Reductive Groups .................................... 45
7.1. The saturation factors associated to a root system ...... 45
7.2. The existence of fixed vertices ......................... 50
7.3. Saturation factors for reductive groups ................. 56
Chapter 8. The Comparison of Problems Q3 and Q4 ................ 60
8.1. The Hecke ring .......................................... 60
8.2. A geometric interpretation of mαβγ ....................... 62
8.3. The Satake transform .................................... 64
8.4. A solution of Problem Q4 is a solution of Problem Q3 .... 67
8.5. A solution of Problem Q3 is not necessarily a solution
of Problem Q4 ........................................... 71
8.6. The saturation theorem for GL(l) ........................ 73
8.7. Computations for the root systems B2 and G2 .............. 75
Appendix A. Decomposition of Tensor Products and Mumford
Quotients of Products of Coadjoint orbits .......... 77
A.1. The existence of semistable triples and nonzero
invariant vectorsin triple tensor products .............. 77
A.2. The semigroups of solutions to Problems Q1 and Q4 ....... 80
Bibliography ................................................... 82
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