Invariant valuations on 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.