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Invariant valuations on Lie groups

Andreas Bernig, Dmitry Faifman, Jan Kotrbatý

TL;DR

This work develops a comprehensive framework for convolutions of invariant valuations on Lie groups by translating valuations into invariant differential forms and introducing a deformation of the wedge product via a formal convolution on forms. It provides an explicit convolution formula for left-invariant valuations on compact Lie groups, classifies connected groups with nontrivial bi-invariant valuations, and builds a unified convolution theory for unimodular groups that subsumes the Alesker–Bernig convolution on compact groups and the Bernig–Fu convolution on vector spaces. The results yield a complete description of bi-invariant valuations in key cases (notably S^3 and SO(3)) and establish the structural properties (unit, associativity, and duality) of the generalized convolution across unimodular Lie groups. The S^3 computations illustrate the nuanced differences between convolution and the product structure in the valuation algebras, highlighting both the reach and limitations of the developed framework.

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.

Invariant valuations on Lie groups

TL;DR

This work develops a comprehensive framework for convolutions of invariant valuations on Lie groups by translating valuations into invariant differential forms and introducing a deformation of the wedge product via a formal convolution on forms. It provides an explicit convolution formula for left-invariant valuations on compact Lie groups, classifies connected groups with nontrivial bi-invariant valuations, and builds a unified convolution theory for unimodular groups that subsumes the Alesker–Bernig convolution on compact groups and the Bernig–Fu convolution on vector spaces. The results yield a complete description of bi-invariant valuations in key cases (notably S^3 and SO(3)) and establish the structural properties (unit, associativity, and duality) of the generalized convolution across unimodular Lie groups. The S^3 computations illustrate the nuanced differences between convolution and the product structure in the valuation algebras, highlighting both the reach and limitations of the developed framework.

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.

Paper Structure

This paper contains 18 sections, 34 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Lie groups
  5. Smooth valuations
  6. Operations on smooth valuations
  7. Translation-invariant valuations
  8. Invariant differential forms
  9. Left-invariant and bi-invariant forms
  10. The differential on invariant forms
  11. Convolution of invariant differential forms
  12. Compact Lie groups
  13. Existence of bi-invariant valuations
  14. Convolution of smooth bi-invariant valuations on Lie groups
  15. Haar measures and forms
  16. ...and 3 more sections

Key Result

Theorem 1

Let $G$ be a compact Lie group. Let $\phi=\{c_\phi,\tau_\phi\} \in \mathcal{V}^\infty(G)^{G \times G}$ and $\psi=\{c_\psi,\tau_\psi\}=[[\mu_\psi,\omega_\psi]] \in \mathcal{V}^\infty(G)^{L_G}$ with $\omega_\psi$ left-invariant and $d\omega_\psi$ vertical. Denote $\mu(\psi)=\int_G \mu_\psi=\psi(G)\in

Theorems & Definitions (43)

  • Theorem 1
  • Theorem 2
  • Definition 1.1
  • Theorem 3
  • Proposition 1.2
  • Lemma 3.1
  • Lemma 3.2
  • Corollary 3.3
  • Definition 4.1
  • Definition 4.2
  • ...and 33 more