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Characteristic cycles in Hermitian symmetric spaces

  Published:1997-06-01
 Printed: Jun 1997
  • Brian D. Boe
  • Joseph H. G. Fu
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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 Kazhdan-Lusztig 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]
 

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