Connections on the projective line whose differential Galois groups are as large as possible

It is a well-known classical result that a generic nth order polynomial equation with rational coefficients has Galois group as large as possible, namely equal to S_n . However, it is challenging to exhibit explicit examples of equations whose Galois group is S_n . In this paper, we discuss the analogue of this problem for differential Galois groups of linear differential equations with coefficients in the field of complex rational functions. The differential Galois group of an nth order linear differential equation is the symmetry group of its solutions; it is an algebraic subgroup of GL_n (C). More generally, if G is a complex connected reductive algebraic group, the differential Galois group of a G-connection is an algebraic subgroup of G. For G simple, we generalize earlier work of Katz in type A to give a criterion for the differential Galois group of a meromorphic G-connection to be “large”: usually all of G, but certain Dynkin diagrams allow one or two more possibilities depending on the symmetry properties of the connection. We will then apply this result to Coxeter G-connections on G_m, a class of connections that has appeared in recent work on the geometric Langlands program and includes both Airy and Kloosterman equations in type A. In particular, we will exhibit many explicit G-connections with differential Galois group G.