Title: Shadows of characteristic cycles, Verma modules, and positivity of Chern-Schwartz-MacPherson classes of Schubert cells,
Abstract: 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 projective manifold $X$, the homogenized, torus equivariant CSM class
of a constructible function $\varphi$ is the restriction of the characteristic
cycle of $\varphi$ 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
Poincar{é} adjoint. We further use the theory of holonomic
$\mathcal{D}_X$-modules to show that the characteristic cycle of a Verma
module, restricted to the zero section, gives the CSM class of the
corresponding 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
Rim{á}nyi 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 expansion 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
$G/P$.