Localization of valuations and Alesker's irreducibility theorem
Georg C. Hofstätter, Jonas Knoerr
TL;DR
This work delivers a self-contained, elementary proof of Alesker's Irreducibility Theorem by developing a localization framework for polynomial valuations and proving that smooth translation-invariant valuations arise from normal-cycle integration. The authors reduce the irreducibility problem to an explicit analysis of the action of $\mathfrak{sl}(n)$ on differential forms (via the SL(n) action on the sphere bundle) and establish a GL(n)-equivariant, SL(n)-invariant pairing to transfer algebraic irreducibility to topological irreducibility of valuation spaces. Key ingredients include Goodey–Weil distributions for polynomial valuations, Goodey–Weil distributions for polynomial valuations, the Rumin differential, and a detailed study of double forms under Lie algebra actions. The approach not only provides a new proof but also yields finer information about the SL(n) representation structure of valuation spaces, with potential applications to refined decomposition results and further representation-theoretic analyses in integral geometry.
Abstract
We provide a new proof of Alesker's Irreducibility Theorem. We first introduce a new localization technique for polynomial valuations on convex bodies, which we use to independently prove that smooth and translation invariant valuations are representable by integration with respect to the normal cycle. This allows us to reduce the statement to a corresponding result for the representation of $\mathfrak{sl}(n)$ on the space of these differential forms.