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Let $\Sigma_g$ be a closed oriented surface of genus g and let $H_\mathbb{Q}$denote $H_1(\Sigma_g;\mathbb{Q})$ which we understand to be the standardsymplectic vector space over $\mathbb{Q}$ of dimension $2g$. We introduce acanonical metric on the space $(H_\mathbb{Q}^{\otimes 2k})^{\mathrm{Sp}}$ ofsymplectic invariant tensors by analyzing the structure of the vector space$\mathbb{Q}\mathcal{D}^{\ell}(2k)$ generated by linear chord diagrams with $2k$vertices. This space, equipped with a certain inner product, serves as auniversal model for $(H^{\otimes 2k})^{\mathrm{Sp}}$ for any $g$. We decompose$\mathbb{Q}\mathcal{D}^\ell(2k)$ as an orthogonal direct sum of eigenspaces$E_\lambda$ where $\lambda$ is indexed by the set of all the Young diagramswith $k$ boxes. We give a formula for the eigenvalue $\mu_\lambda$ of$E_\lambda$ and thereby we obtain a complete description of how the spaces$(H_\mathbb{Q}^{\otimes 2k})^{\mathrm{Sp}}$ degenerate according as the genusdecreases from the stable range $g\geq k$ to the last case $g=1$ with thelargest eigenvalue $2g(2g+1) \cdots (2g+k-1)$.  As an application of our canonical metric, we obtain certain relations amongthe Mumford-Morita-Miller tautological classes, in a systematic way, which holdin the tautological algebra in cohomology of the moduli space of curves. Wealso indicate other possible applications such as characteristic classes oftransversely symplectic foliations and a project with T. Sakasai and M. Suzukiwhere we study the structure of the symplectic derivation Lie algebra.

Canonical metric on the space of symplectic invariant tensors and its applications