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Characteristic cycles of constructible sheaves. (English) Zbl 0851.32011
Let $$X$$ be a real analytic manifold and $$\mathcal F$$ a constructible complex with respect to a subanalytic stratification on $$X$$. For this data Kashiwara defined $$CC({\mathcal F})$$, the characteristic cycle, of $$\mathcal F$$ as a Lagrangian cycle in $$T* X$$.
Let now $$j : U \hookrightarrow X$$ be the embedding of an open subanalytic subset $$U$$ of $$X$$. The main result of the paper describes the characteristic cycles $$CC(Rj_* {\mathcal F})$$ and $$CC(Rj_! {\mathcal F})$$. As a consequence descriptions of $$CC(Rf_* {\mathcal F})$$ and $$CC (f^* {\mathcal F})$$ analogous to that given by Kashiwara-Shapira are obtained for arbitrary morphisms $$f : X \to Y$$ in the semi-algebraic category.
A second application is the following: Let $$X$$ be the flag manifold of a complex semisimple Lie algebra $$g$$ and $$W$$ be the Weyl group of $$g$$. It is known that $$W$$ acts on the $$K$$-group of $$D^b(X)$$ (the category of semi-algebraically constructible sheaves) and this gives an operation on $$CC(D^b(X))$$. On the other hand Rossmann defined geometrically an action of $$W$$ on the group of all semi-algebraic Lagrangian cycles on $$T^* X$$. The authors prove that the two actions coincide on $$CC(D^b (X))$$.

##### MSC:
 32B20 Semi-analytic sets, subanalytic sets, and generalizations 58A07 Real-analytic and Nash manifolds 32C05 Real-analytic manifolds, real-analytic spaces 58A35 Stratified sets 35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
##### Keywords:
constructible sheaf; characteristic cycle
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