Generalized Verma modules, \(b\)-functions of semi-invariants and duality for twisted \(D\)-modules on generalized flag manifolds (Q699216)

From author's abstract: Let \(G\) be a connected semisimple algebraic group over \(\mathbb C\), \(P\) a parabolic subgroup, \(\mathfrak g\) and \(\mathfrak p\) their Lie algebras. We prove a microlocal version of Gyoja's conjectures [\textit{A. Gyoja}, Publ. Res. Inst. Math. Sci. 30, 353-400 (1994; Zbl 0821.22006)] about a relation between the irreducibility of generalized Verma modules on \(\mathfrak g\) induced from \(\mathfrak p \) and the zeroes of \(b\)-functions of \(P\)-semi-invariants on \(G\). Our method uses a duality for twisted \(\mathcal D\)-modules on generalized flag manifolds.