q-Hypergeometric Functions and Representation Theory

Abstract

• Introduction

• 1. One-dimensional differential example

• 2. One-dimensional difference example

• 3. The hypergeometric Riemann identity

  • 3.1 Basic notations

  • 3.2 The hypergeometric integral

  • 3.3 The hypergeometric spaces and the hypergeometric pairing

  • 3.4 The Shapovalov pairings

  • 3.5 The hypergeometric Riemann identity

• 4. Tensor coordinates on the hypergeometric spaces and the hypergeometric maps

  • 4.1 Bases of the hypergeometric spaces

  • 4.2 Tensor coordinates and the hypergeometric maps

  • 4.3 Difference equations for the hypergeometric maps

  • 4.4 Asymptotics of the hypergeometric maps

  • 4.5 Proof of the hypergeometric Riemann identity

• 5. Discrete local systems and the discrete Gauss-Manin connection

  • 5.1 Discrete flat connections and discrete local systems

  • 5.2 Discrete Gauss-Manin connectinon

  • 5.3 Discrete local system associated with the hypergeometric integrals

  • 5.4 Periodic sections of the homological bundle via the hypergeometric integral

• 6. Asymptotics of the hypergeometric maps

• 7. The quantum loop algebra ${U'_q}(\widetilde {\mathfrak{g}{\mathfrak{l}_2}})$ and the qKZ equation

  • 7.1 Highest weight $U_q(\mathfrak{sl}_2)$-modules

  • 7.2 The quantum loop algebra ${U'_q}(\widetilde {\mathfrak{g}{\mathfrak{l}_2}})$

  • 7.3 The trigonometric qKZ equation

  • 7.4 Tensor coordinates on the trigonometric hypergeometric spaces

• 8. The elliptic quantum group $E_{\rho, \gamma}(\mathfrak{sl}_2)$

  • 8.1 Modules over the elliptic quantum group $E_{\rho, \gamma}(\mathfrak{sl}_2)$

  • 8.2 Tensor coordinates on the elliptic hypergeometric spaces

  • 8.3 The hypergeometric maps

• 9. Asymptotic solutions of the qKZ equation

• A. Six determinant formulae

• B. The Jackson integrals via the hypergeometric integrals