The convolution algebra of constructible sheaves

TL;DR

This paper explores the structure of invertible constructible sheaves on a real vector space, revealing their duality, conditions for invertibility, and a microlocal transform that interacts with convolution.

Contribution

It characterizes invertible sheaves via duality and convex support, and introduces a microlocal transform compatible with convolution, advancing understanding of sheaf convolution algebra.

Findings

Inverse of an invertible sheaf is its antipodal dual.

A compactly supported constant sheaf is invertible iff its support is convex.

The microlocal transform B(F) is compatible with convolution, providing a necessary invertibility condition.

Abstract

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.