Characteristic cycles in Hermitian symmetric spaces

We give explicit combinatorial expressions for the characteristic cycles associated to certain canonical sheaves on Schubert varieties X in the classical Hermitian symmetric spaces: namely the intersection homology sheaves IHX and the constant sheaves C-X. The three main cases of interest are the Hermitian symmetric spaces for groups of type A(n), (the standard Grassmannian), C-n (the Lagrangian Grassmannian) and D-n. In particular we find that CC(IHX) is irreducible for all Schubert varieties X if and only if the associated Dynkin diagram is simply laced. The result for Schubert varieties in the standard Grassmannian had been established earlier by Bressler, Finkelberg and Lunts, while the computations in the C-n and D-n cases are new. Our approach is to compute CC(C-X) by a direct geometric method, then to use the combinatorics of the Kazhdan-Lusztig polynomials (simplified for Hermitian symmetric spaces) to compute CC(IHX). The geometric method is based on the fundamental formula CC(C-X) = (r down arrow 0)lim CC(C-Xr), where the X-r down arrow X constitute a family of tubes around the variety X. This formula leads at once to an expression for the coefficients of CC(C-X) as the degrees of certain singular maps between spheres.