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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 $\R$ acting on an affine $\R$-variety $V$, we consider the induced dual action on the coordinate ring $\R[V]$ and on the linear dual space of $\R[V]$. In this setting, given an invariant closed semialgebraic subset $K$ of $V(\R)$, 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 $\R[V]$ by invariant measures supported on $K$. To this end, we analyse the relation between quadratic modules of $\R[V]$ and associated quadratic modules of the (finitely generated) subring $\R[V]^G$ 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 $G(\R)$ is compact.