DIFFERENTIAL-DIFFERENCE OPERATORS AND MONODROMY REPRESENTATIONS OF HECKE ALGEBRAS

Associated to any finite reflection group G on an Euclidean space ther e is a parametrized commutative algebra of differential-difference ope rators with as many parameters as there are conjugacy classes of refle ctions. The Hecke algebra of the group can be represented by monodromy action on the space of functions annihilated by each differential-dif ference operator in the algebra. For each irreducible representation o f G the differential-difference equations lead to a linear system of f irst-order meromorphic differential equations corresponding to an inte grable connection over the G-orbits of regular points in the complexif ication of the Euclidean space. The fundamental group is the generaliz ed Artin braid group belonging to G, and its monodromy representation factors over the Hecke algebra of G. Monodromy has long been of import ance in the study of special functions of several variables, for examp le, the hyperlogarithms of Lappo-Danilevsky are used to express the fl at sections and the work of Riemann on the monodromy of the hypergeome tric equation is applied to the case of dihedral groups.