154 research outputs found
On the topology of components of some Springer fibers and their relation to Kazhdan-Lusztig theory
We describe the irreducible components of Springer fibers for hook and
two-row nilpotent elements of gl_n(C) as iterated bundles of flag manifolds and
Grassmannians. We then relate the topology (in particular, the intersection
homology Poincare' polynomials) of the intersections of these components with
the inner products of the Kazhdan-Lusztig basis elements of irreducible
representations of the rational Iwahori-Hecke algebra of type A corresponding
to the hook and two-row Young shapes.Comment: This work has been submitted to Advances in Mathematics (Academic
Press) for possible publication
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 projective manifold , the homogenized, torus equivariant CSM class
of a constructible function is the restriction of the characteristic
cycle of via the zero section of the cotangent bundle of . This
extends to the equivariant setting results of Ginzburg and Sabbah. We
specialize to be a (generalized) flag manifold . 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{\'e} adjoint. We further use the theory of holonomic
-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{\'a}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
.Comment: 40 pages; main changes in v2: removed some unnecessary compactness
hypotheses; added remarks 7.2 and 9.6 explaining how orthogonality of
characteristic cycles for transversal Schubert cell stratifications leads to
orthogonality of stable envelopes and that of CSM classe
Toda lattice, cohomology of compact Lie groups and finite Chevalley groups
In this paper, we describe a connection that exists among (a) the number of
singular points along the trajectory of Toda flow, (b) the cohomology of a
compact subgroup , and (c) the number of points of a Chevalley group
related to over a finite field . The Toda
lattice is defined for a real split semisimple Lie algebra , and
is a maximal compact Lie subgroup of associated to .
Relations are also obtained between the singularities of the Toda flow and the
integral cohomology of the real flag manifold with the Borel subgroup
of (here we have with a finite group ). We also compute the
maximal number of singularities of the Toda flow for any real split semisimple
algebra, and find that this number gives the multiplicity of the singularity at
the intersection of the varieties defined by the zero set of Schur polynomials.Comment: 28 pages, 5 figure
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