Abstract view
Characteristic cycles in Hermitian symmetric spaces


Published:19970601
Printed: Jun 1997
Brian D. Boe
Joseph H. G. Fu
Abstract
We give explicit combinatorial expresssions for the characteristic
cycles associated to certain canonical sheaves on Schubert varieties
$X$ in the classical Hermitian symmetric spaces: namely the
intersection homology sheaves $IH_X$ and the constant sheaves $\Bbb
C_X$. 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
$CC(IH_X)$ 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 $CC(\Bbb C_X)$ by a direct geometric
method, then to use the combinatorics of the KazhdanLusztig
polynomials (simplified for Hermitian symmetric spaces) to compute
$CC(IH_X)$. The geometric method is based on the fundamental formula
$$CC(\Bbb C_X) = \lim_{r\downarrow 0} CC(\Bbb C_{X_r}),$$ where the
$X_r \downarrow X$ constitute a family of tubes around the variety
$X$. This formula leads at once to an expression for the coefficients
of $CC(\Bbb C_X)$ as the degrees of certain singular maps between
spheres.
MSC Classifications: 
14M15, 22E47, 53C65 show english descriptions
Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35] Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) [See also 17B10] Integral geometry [See also 52A22, 60D05]; differential forms, currents, etc. [See mainly 58Axx]
14M15  Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35] 22E47  Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) [See also 17B10] 53C65  Integral geometry [See also 52A22, 60D05]; differential forms, currents, etc. [See mainly 58Axx]
