On the monodromy invariant Hermitian form for \(A\)-hypergeometric systems (Q2155676)

\(A\)-hypergeometric functions are a generalization (introduced by \textit{I. M. Gelfand} et al. [Adv. Math. 84, No. 2, 255--271 (1990; Zbl 0741.33011)]) of hypergeometric functions. An \(A\)-hypergeometric system comprises Euler operators and box operators, defined by means of a lattice in \(\mathbb{Z}^N\) satisfying certain conditions. \textit{F. Beukers} [J. Reine Angew. Math. 718, 183--206 (2016; Zbl 1355.33017)] has shown how to find (under very restrictive conditions) a subgroup of the full monodromy group using Mellin-Barnes integral solutions of the associated \(A\)-hypergeometric system. In the present paper an explicit construction of the invariant Hermitian form for the monodromy of an \(A\)-hypergeometric system is given provided that there is a Mellin-Barnes basis of solutions.