Microlocalization of ind-sheaves

Summary

Let X be a C^∞-manifold and T*X its cotangent bundle. We construct a microlocalization functor μ _X: D^b(I($\mathbb{K}_X$ )) → D^b(I($\mathbb{K}_{T*X}$ )), where D^b(I($\mathbb{K}_X$ )) denotes the bounded derived category of ind-sheaves of vector spaces on X over a field $\mathbb{K}$ . This functor satisfies Rℌom(μ _X(F), μ _X(G)), ⋍ μhom(F,G) for any F,F ∈ D^b($\mathbb{K}_X$ ), thus generalizing the classical theory of microlocalization. Then we discuss the functoriality of μ _X. The main result is the existence of a microlocal convolution morphism

$$ \mu _{X \times Y} \left( {\mathcal{K}_1 } \right)_ \circ ^a \mu _{Y \times Z} \left( {\mathcal{K}_2 } \right) \to \mu _{X \times Z} \left( {\mathcal{K}_1 \circ \mathcal{K}_2 } \right) $$

which is an isomorphism under suitable non-characteristic conditions on 171-10 and 171-11.

The author M.K. is partially supported by Grant-in-Aid for Scientific Research (B1) 13440006, Japan Society for the Promotion of Science and the 21st century COE program “Formation of an International Center of Excellence in the Frontier of Mathematics and Fostering of Researchers in Future Generations.”

The author I.W. is partially supported by the same 21st century COE program.