${\rm SL}(m,C)$-equivariant and translation covariant continuous tensor valuations

TL;DR

This paper classifies continuous tensor valuations on complex space that are equivariant under special linear group actions and covariant under translations, extending known classifications to complex settings.

Contribution

It provides a complete classification of ${ m SL}(m,C)$-equivariant, translation covariant tensor valuations, introducing the moment tensor valuation for higher ranks.

Findings

Classification involves the moment tensor valuation for rank r≥1.

Results are analogous to real space classifications but require different proof methods.

The space of such valuations is fully described, extending previous real-space results.

Abstract

The space of continuous, ${\rm SL}(m,C)$-equivariant, $m\geq 2$, and translation covariant valuations taking values in the space of real symmetric tensors on $C^m\cong R^{2m}$ of rank $r\geq 0$ is completely described. The classification involves the moment tensor valuation for $r\geq 1$ and is analogous to the classification of the corresponding tensor valuations that are ${\rm SL}(2m,R)$-equivariant, although the method of proof cannot be adapted.