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Let $X=(X(n))_{n \in \mathbb{Z_+}}$ be a standard subproduct system of$C^*$-correspondences over a $C^*$-algebra $\mathcal M.$ Assume $T=(T_n)_{n \in\mathbb{Z_+}}$ to be a pure completely contractive, covariant representation of$X$ on a Hilbert space $\mathcal H,$ and $\mathcal S$ to be a non-trivialclosed subspace of $\mathcal H.$ Then $\mathcal{S}$ is invariant for $T$ if andonly if there exist a Hilbert space $\mathcal{D},$ a representation $\pi$ of$\mathcal M$ on $\mathcal D,$ and a partial isometry $\Pi:\mathcal{F}_X\bigotimes_{\pi}\mathcal{D}\to \mathcal{H} $ such that $$\Pi(S_n(\zeta)\otimes I_{\mathcal{D}})=T_n(\zeta)\Pi~\mbox{whenever}~\zeta\inX(n), ~n\in \mathbb{Z_+},~\mbox{and}$$ $\mathcal S$ is the range of $\Pi,$ orequivalently, $P_{\mathcal S}=\Pi\Pi^*.$ This result leads us to many importantconsequences including Beurling type theorem and other general observations onwandering subspaces. We extend the notion of curvature for completelycontractive, covariant representations and analyze it in terms of the aboveresults.

Covariant representations of subproduct systems: Invariant subspaces and curvature