Affine Extensions of Knizhnik-Zamolodchikov Equations and Lusztig’s Isomorphisms

Abstract

Some affine analogues of the Knizhnik-Zamolodchikov equations from the two-dimensional conformai field theory are discussed for arbitrary root systems. We demonstrate that they give some versions of Lusztig’s isomorphisms from affine Hecke algebras to their degenerate variants.

• Section 1. 𝕊 _n -invariant KZ equations.

• Section 2. Trigonometric KZ equations for root systems.

• Section 3. The monodromy and Lusztig’s isomorphisms.

In this paper we consider differential equations generalizing the affine KnizhnikZamolodchikov equation (KZ) from [1] for arbitrary root systems. These equations are also “trigonometric” extensions of the “rational” non-affine KZ equations from [2] and are particular cases of the differential-difference equations from [3]. Their monodromy representations result in Lusztig's type isomorphisms [4] from affine Hecke algebras to the corresponding degenerate algebras and are closely connected with a certain generalization of the Lusztig-Lascoux-Schiitzenberger operations (see [3]). We will introduce our equations as natural variants of the equation from [1] (in [3] they were obtained on the base of the LLS operations). The main points of these notes are the definitions, the proof of the consistency of arising equations and a calculation of the monodromy representations.

This paper was mainly prepared during my stay at RIMS (Kyoto University) and finished for my stay at MSRI (Berkeley). It is partially based on my talk at the Hayashibara Forum 90 on special functions. I am grateful for the kind invitation and for hospitality. I also express appreciation to G. Heckman, V. Jones, A. Matsuo, T. Miwa, to other my colleagues for useful discussions and to the secretaries in RIMS for the help in preparing the manuscript for publication.

On leave from A. N. Belozersky Lab., Bldg “A” Moscow State University, Moscow 119899, USSR, research at MSRI supported in part by NSF grant # DMS 8505550.