Associated to any finite
reflection group G on an Euclidean space there is a parametrized commutative
algebra of differential-difference operators with as many parameters as there
are conjugacy classes of reflections. The Hecke algebra of the group can be
represented by monodromy action on the space of functions annihilated
by each differential-difference operator in the algebra. For each irreducible
representation of G the differential-difference equations lead to a linear system of
first-order meromorphic differential equations corresponding to an integrable
connection over the G-orbits of regular points in the complexification of
the Euclidean space. The fundamental group is the generalized Artin braid
group belonging to G, and its monodromy representation factors over the
Hecke algebra of G. Monodromy has long been of importance in the study of
special functions of several variables, for example, the hyperlogarithms of
Lappo-Danilevsky are used to express the flat sections and the work of Riemann on
the monodromy of the hypergeometric equation is applied to the case of dihedral
groups.
