On the local monodromy of $A$-hypergeometric functions and some monodromy invariant subspaces

Abstract

We obtain an explicit formula for the characteristic polynomial of the local monodromy of $A$-hypergeometric functions with respect to small loops around a coordinate hyperplane $x_i=0$. This formula is similar to the one obtained by Ando, Esterov and Takeuchi for the local monodromy at infinity. Our proof is combinatorial and can be adapted to provide an alternative proof for the latter formula as well. On the other hand, we also prove that the solution space at a nonsingular point of certain irregular and irreducible $A$-hypergeometric $D$-modules has a nontrivial global monodromy invariant subspace.