Sums of squares and moment problems in equivariant situations

Abstract

We begin a systematic study of positivity and moment problems in an equivariant setting. Given a reductive groupGGoverR\mathbb {R}acting on an affineR\mathbb {R}-varietyVV, we consider the induced dual action on the coordinate ringR[V]\mathbb {R}[V]and on the linear dual space ofR[V]\mathbb {R}[V]. In this setting, given an invariant closed semialgebraic subsetKKofV(R)V(\mathbb R), we study the problem of representation of invariant nonnegative polynomials onKKby invariant sums of squares, and the closely related problem of representation of invariant linear functionals onR[V]\mathbb {R}[V]by invariant measures supported onKK. To this end, we analyse the relation between quadratic modules ofR[V]\mathbb {R}[V]and associated quadratic modules of the (finitely generated) subringR[V]G\mathbb {R}[V]^Gof invariant polynomials. We apply our results to investigate the finite solvability of an equivariant version of the multidimensionalKK-moment problem. Most of our results are specific to the case where the groupG(R)G(\mathbb {R})is compact.