Characteristic cycles of constructible sheaves (Q1911149)

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 \(CC(D^b (X))\).