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.