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

Abstract

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