The convolution algebra of constructible sheaves
Let \(E\) be a finite-dimensional real vector space. We study invertible objects in the monoidal category of constructible sheaves on \(E\), endowed with the convolution product \(\star\). We show that the inverse of an invertible constructible sheaf \(F\) is the dual of its antipodal transform. We also prove that a compactly supported constant sheaf is invertible if and only if its support is convex. We also introduce a microlocal transform \(B(F)\), obtained by projecting the characteristic cycle of $F$ to \(E^*\), and prove that it is compatible with convolution. This yields a necessary condition for invertibility.
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