Hypergeometry, integrability and lie theory : Virtual Conference on Hypergeometry, Integrability and Lie Theory, December 7-11, 2020, Lorentz Center, Leiden, Netherlands

This volume contains the proceedings of the virtual conference on Hypergeometry, Integrability and Lie Theory, held from December 7-11, 2020, which was dedicated to the 50th birthday of Jasper Stokman. The papers represent recent developments in the areas of representation theory, quantum integrable systems and special functions of hypergeometric type. Cover Title page Contents Preface 1. Sectionformat {Background}{1} 2. Sectionformat {Structure of workshop}{1} 3. Sectionformat {A special occasion}{1} Characteristic functions of p-adic integral operators 1. Introduction 2. Zeta-functions 3. Realization of A_{P,s,χ} on analytic functions 4. q-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 K-matrix, revisited 4. The bar involution for quantum symmetric pairs, revisited References Charting the q-Askey scheme 1. Introduction 2. Askey–Wilson polynomials and Verde-Star’s theorem 3. The q-Verde-Star scheme 4. The q-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 g 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 k 3.2. Generalized Satake diagrams 3.3. Basic properties of k 3.4. Iwasawa decomposition for pseudo-symmetric pairs 3.5. A combinatorial description of k’ 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 V^{σ} and V^{-σ}. 4.5. The Weyl group of the restricted root system 4.6. The group W^{σ} and the restricted Weyl group \overlineW 4.7. A combinatorial prescription of the simple restricted reflections: the group ̃W 4.8. The group W(\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. n-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: SU(p,r). Acknowledgment References Elementary symmetric polynomials and martingales for Heckman-Opdam processes 1. Introduction 2. Heckman-Opdam theory 3. The compact case of type A_{N-1} 4. The non-compact case of type A_{N-1} 5. The non-compact case of type BC_{N} 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 F_{p}-hypergeometric solutions under ample reduction 1. Introduction 2. KZ equations 3. Coefficients of polynomials 4. F_{p}-Beta integral and KZ equations for n=2 5. Leading term of a polynomial solution 6. Leading term of an F_{p}-hypergeometric solution 7. Determinant of F_{p}-hypergeometric solutions 8. Properties of F_{p}-hypergeometric solutions Acknowledgment References Notes on solutions of KZ equations modulo p^{s} and p-adic limit s→∞ 1. Introduction 2. KZ equations 3. Complex solutions 4. Solutions modulo p^{s} 5. Independence of modules from the choice of M 6. Filtrations and homomorphisms 7. Coefficients of solutions 8. Multiplication by p and Cartier-Manin matrix 9. Change of variables 10. p-Adic convergence Appendix A. The case n=3 and Dwork’s theory Acknowledgments References Back Cover This volume contains the proceedings of the virtual conference on Hypergeometry, Integrability and Lie Theory, held in December 2020. The papers represent recent developments in the areas of representation theory, quantum integrable systems and special functions of hypergeometric type.