Irreducibility of the monodromy representation of Lauricella's $F_C$

Abstract

Let $E_C$ be the hypergeometric system of differential equations satisfied by Lauricella's hypergeometric series $F_C$ of $m$ variables. This system is irreducible in the sense of $D$-modules if and only if $2^{m+1}$ non-integral conditions for parameters are satisfied. We find a linear transformation of the classically known $2^m$ solutions so that the transformed ones always form a fundamental system of solutions under the irreducibility conditions. By using this fundamental system, we give an elementary proof of the irreducibility of the monodromy representation of $E_C$. When one of the conditions is not satisfied, we specify a non-trivial invariant subspace, which implies that the monodromy representation is reducible in this case.