Special Functions, KZ Type Equations, and Representation Theory

Alexander Varchenko

Contents

1. Hypergeometric solutions of KZ`equations

1. Examples of hypergeometric integrals
2. Knizhnik-Zamolodchikov equations
3. Examples of solutions
4. Solutions of the KZ equation with values in Sing M^m [|m|-2k]
5. Solutions of the KZ equation with values in Sing L^m [|m|-2k]
6. The classical hypergeometric series
7. Identities for differential forms
8. Hyperplane arrangements

2. Cycles of integrals and the monodromy of the KZ equation

1. Cycles for solutions in Sing M^m[|m|-2]
2. The quantum group U_qIsl₂)
3. The Yang-Baxter equation and representations of braid groups
4. Quantum singular vectors
5. The monodromy of KZ equations
6. A remark on integration cycles

3. Selberg integral, determinant formulas, and dynamical equations

1. The Selberg integral
2. A connection with finite reflection groups
3. An example of a determinant formula
4. The determinant formula for solutions in a weight subspace
5. General determinant formulas
6. Resonances
7. Dynamical equations

4. Critical points of master functions and the Bethe ansatz

1. The Gaudin model of the Bethe ansatz
2. Asymptotic solutions and eigenvectors
3. Quasi-classical asymptotics of solutions to the KZ equation
4. The Shapovalov norm of Bethe vectors
5. The number of critical points of a product of powers of linear functions
6. Critical points of Phi_k,n(t,z,m) if m₁,..., m_n are natural numbers
7. Critical points and Fuchsian equations with polynomial solutions
8. Resonant local systems

5. Elliptic hypergeometric functions

1. Knizhnik-Zamolodchikov-Bernard equations
2. The case n = 1 and V = L_2p
3. Quasi-classical asymptotics of solutions to the KZB heat equation
4. Elliptic hypergeometric functions associated with one marked point
5. Integral representations for elliptic hypergeometric functions
6. Elliptic Selberg integrals
7. Transformations acting on the space of conformal blocks
8. Relations between theta functions
9. Basic relations between elliptic hypergeometric functions
10. Macdonald polynomials and the shift operator
11. Coefficients f^(k)_m,n and values of Macdonald polynomials
12. Modular transformations of elliptic hypergeometric functions
13. Trace functions for U_q(sl₂)

6. q-Hypergeometric solutions of qKZ equations

1. Quantum Knizhnik-Zamolodchikov equations
2. Quasi-calssical asymptotics of solutions and eigenvectors
3. An example of quantization of hypergeometric functions
4. q-Hypergeometric solutions, general case
5. The q-hypergeometric pairing
6. Quantization of the Kohno-Drinfeld theorem