Shadows of Characteristic Cycles, Verma Modules, and Positivity of Chern-schwartz-macpherson Classes of Schubert Cells

Chern-Schwartz-MacPherson (CSM) classes generalize to singular and/or noncompact varieties the classical total homology Chern class of the tangent bundle of a smooth compact complex manifold. The theory of CSM classes has been extended to the equivariant setting by Ohmoto. We prove that for an arbitrary complex algebraic manifold X, the homogenized, torus equivariant CSM class of a constructible function φ is the restriction of the characteristic cycle of φ via the zero section of the cotangent bundle of X. This extends to the equivariant setting results of Ginzburg and Sabbah. We specialize X to be a (generalized) flag manifold G/B. In this case CSM classes are determined by a Demazure-Lusztig (DL) operator. We prove a ‘Hecke orthogonality’ of CSM classes, determined by the DL operator and its Poincaré adjoint. We further use the theory of holonomic DX -modules to show that the characteristic cycle of the Verma module, restricted to the zero section, gives the CSM class of a Schubert cell. Since the Verma characteristic cycles naturally identify with the Maulik and Okounkov’s stable envelopes, we establish an equivalence between CSM classes and stable envelopes; this reproves results of Rimányi and Varchenko. As an application, we obtain a Segre type formula for CSM classes. In the non-equivariant case this formula is manifestly positive, showing that the extension in the Schubert basis of the CSM class of a Schubert cell is effective. This proves a previous conjecture by Aluffi and Mihalcea, and it extends previous positivity results by J. Huh in the Grassmann manifold case. Finally, we generalize all of this to partial flag manifolds G/P .