Hypergeometry, Integrability and Lie Theory

Cover
Title page
Contents
Preface
1. Sectionformat {Background}{1}
2. Sectionformat {Structure of workshop}{1}
3. Sectionformat {A special occasion}{1}
Characteristic functions of ?-adic integral operators
1. Introduction
2. Zeta-functions
3. Realization of ?_{?,?,?} on analytic functions
4. ?-hypergeometric functions and proof of Theorem 1.1
5. Examples
6. The non-homogeneous case
Acknowledgments
References
Shuffle algebras, lattice paths and the commuting scheme
1. Introduction
2. Hecke algebra and lattice paths
3. The shuffle algebra
4. Matching the partition functions with shuffle elements
5. Application to the commuting scheme
Acknowledgments
References
The bar involution for quantum symmetric pairs –hidden in plain sight
1. Introduction
2. Preliminaries
3. The quasi ?-matrix, revisited
4. The bar involution for quantum symmetric pairs, revisited
References
Charting the ?-Askey scheme
1. Introduction
2. Askey–Wilson polynomials and Verde-Star’s theorem
3. The ?-Verde-Star scheme
4. The ?-Verde-Star scheme as a four-manifold
5. Further perspectives
Appendix A. Explicit data for the families in Figure 1
Appendix B. Some explicit limit transitions
Acknowledgement
References
Filtered deformations of elliptic algebras
1. Introduction
2. Filtered deformations
3. Resolutions of elliptic algebras
4. Elliptic noncommutative del Pezzo surfaces
5. Filtered deformations from del Pezzo surfaces
6. Classifications
Acknowledgments
References
Pseudo-symmetric pairs for Kac-Moody algebras
1. Introduction
1.1. Pseudo-involutions and pseudo-fixed-point subalgebras
1.2. Applications in the quantum deformed setting
1.3. Outline
2. Pseudo-involutions in terms of compatible decorations
2.1. Generalized Cartan matrices and Dynkin diagrams
2.2. Braid group and Weyl group
2.3. Minimal realization and bilinear forms
2.4. Kac-Moody algebra and roots
2.5. Kac-Moody group and triple exponentials
2.6. Subdiagrams of finite type
2.7. Automorphisms of ?
2.8. Twisted involutions and compatible decorations
2.9. Classification of pseudo-involutions of the second kind
3. Pseudo-fixed-point subalgebras in terms of generalized Satake diagrams
3.1. The subalgebra ?
3.2. Generalized Satake diagrams
3.3. Basic properties of ?
3.4. Iwasawa decomposition for pseudo-symmetric pairs
3.5. A combinatorial description of ?’
4. The restricted Weyl group and restricted root system
4.1. The ℚ-span of the root system
4.2. Root system involutions and the corresponding orthogonal decompositions
4.3. The restricted root system
4.4. Combinatorial bases for ?^{?} and ?^{-?}.
4.5. The Weyl group of the restricted root system
4.6. The group ?^{?} and the restricted Weyl group \overline?
4.7. A combinatorial prescription of the simple restricted reflections: the group ̃?
4.8. The group ?(\overlineΦ) revisited
4.9. The restricted Weyl group as a Coxeter group
4.10. Non-reduced and non-crystallographic root systems
Appendix A. Classification of generalized Satake diagrams
A.1. Notation
A.2. Low-rank coincidences
A.3. Finite type
A.4. Affine type
Acknowledgments
References
Asymptotic boundary KZB operators and quantum Calogero-Moser spin chains
1. Introduction
2. ?-Point spherical functions
3. Structure theory of real semisimple Lie groups
4. Generalised radial component maps
5. The quantum Calogero-Moser spin chain
6. The asymptotic boundary KZB operators
7. Example: ??(?,?).
Acknowledgment
References
Elementary symmetric polynomials and martingales for Heckman-Opdam processes
1. Introduction
2. Heckman-Opdam theory
3. The compact case of type ?_{?-1}
4. The non-compact case of type ?_{?-1}
5. The non-compact case of type ??_{?}
References
Conformal hypergeometry and integrability
1. Introduction
2. Conformal field theory and partial waves
3. Conformal partial waves and hypergeometry
4. Integrability of multipoint conformal partial waves
5. Concluding comments
Acknowledgment
References
Determinant of ?_{?}-hypergeometric solutions under ample reduction
1. Introduction
2. KZ equations
3. Coefficients of polynomials
4. ?_{?}-Beta integral and KZ equations for ?=2
5. Leading term of a polynomial solution
6. Leading term of an ?_{?}-hypergeometric solution
7. Determinant of ?_{?}-hypergeometric solutions
8. Properties of ?_{?}-hypergeometric solutions
Acknowledgment
References
Notes on solutions of KZ equations modulo ?^{?} and ?-adic limit ?→∞
1. Introduction
2. KZ equations
3. Complex solutions
4. Solutions modulo ?^{?}
5. Independence of modules from the choice of ?
6. Filtrations and homomorphisms
7. Coefficients of solutions
8. Multiplication by ? and Cartier-Manin matrix
9. Change of variables
10. ?-Adic convergence
Appendix A. The case ?=3 and Dwork’s theory
Acknowledgments
References
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