Sums of squares and moment problems in equivariant situations

We begin a systematic study of positivity and moment problems in an equivariant setting. Given a reductive group $G$ over $\mathbb{R}$ acting on an affine $\mathbb{R}$-variety $V$, we consider the induced dual action on the coordinate ring $\mathbb{R}[V]$ and on the linear dual space of $\mathbb{R}[V]$. In this setting, given an invariant closed semialgebraic subset $K$ of $V(\mathbb{R})$, we study the problem of representation of invariant nonnegative polynomials on $K$ by invariant sums of squares, and the closely related problem of representation of invariant linear functionals on $\mathbb{R}[V]$ by invariant measures supported on $K$. To this end, we analyse the relation between quadratic modules of $\mathbb{R}[V]$ and associated quadratic modules of the (finitely generated) subring $\mathbb{R}[V]^G$ of invariant polynomials. We apply our results to investigate the finite solvability of an equivariant version of the multidimensional $K$-moment problem. Most of our results are specific to the case where the group $G(\mathbb{R})$ is compact.