Invariant valuations on Lie groups

TL;DR

This paper studies invariant valuations on Lie groups, providing explicit convolution formulas for compact groups, characterizing when smooth bi-invariant valuations exist beyond basic measures, and unifying convolution operations across unimodular Lie groups.

Contribution

It offers an explicit convolution formula for invariant valuations on compact Lie groups and characterizes the existence of smooth bi-invariant valuations on connected Lie groups, unifying previous convolution definitions.

Findings

Explicit convolution formula for compact groups

Characterization of bi-invariant valuations beyond basic measures

Unified convolution framework for unimodular Lie groups

Abstract

Convolution of valuations was introduced by the first named author and Fu for linear spaces, and later by Alesker and the first named author for compact Lie groups. In this paper we study the convolution of invariant valuations on Lie groups. First, we obtain an explicit formula for the convolution of left-invariant valuations on compact groups in terms of differential forms. Independently, we show that a connected Lie group admits smooth bi-invariant valuations beyond the Euler characteristic and the Haar measure if and only if the group is the product of a compact group and a linear space. Finally, we use these two results to define the convolution of bi-invariant smooth valuations on an arbitrary unimodular Lie group, thus unifying both previously defined convolution operations.