We give an analogue of the Capelli identity. The analogue is constructed on certain scalar generalized Verma modules of the complex simple Lie algebras of Hermitian symmetric type, and has stronger compatibility with group actions than the original Capelli identity. As in the original case, the analogue expresses certain invariant operators as operators from the center of the universal enveloping algebra. We also give examples for all the possible cases except for $E_{7}$.

The main purpose of this paper is to give an explicit formula of a homology cobordism invariant of plumbed homology 3-spheres which was defined in a joint work with M. Furuta by using the Seiberg-Witten monopole equation on 4-dimensional V-manifolds [8]. This formula provides a sufficient condition for homology 3-spheres of plumbing type to have infinite order in the homology cobordism group.

McGibbon asked if for a connected finite complex $X$ there is a rational equivalence from the loop space of $X$ to a product of spheres and loop spaces of spheres. We will show that the answer is yes if it has only a finite number of nonzero rational homotopy groups or if spaces are localised at a prime. We will also give a clear picture of phantom maps out of the iterated loop space of a finite complex.